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Reliability Demonstration Test Design for Repairable Systems

2019-02-14T15:09:50Z

Richard House:

{{template:RGA BOOK|8|Reliability Demonstration Test Design for Repairable Systems}}
The RGA software provides a utility that allows you to design reliability demonstration tests for repairable systems. For example, you may want to design a test to demonstrate a specific cumulative MTBF for a specific operating period at a specific confidence level for a repairable system. The same applies for demonstrating an instantaneous MTBF or cumulative and instantaneous failure intensity at a given time <math>t.\,\!</math>

==Underlying Theory==
The failure process in a repairable system is considered to be a non-homogeneous Poisson process (NHPP) with power law failure intensity. The instantaneous failure intensity at time <math>t\,\!</math> is:

:<math>{{\lambda }_{i}}\left( t \right)=\lambda \beta {{t}^{\beta -1}}={{\lambda }_{c}}\left( t \right)\beta \,\!</math>

So, the cumulative failure intensity at time <math>t\,\!</math> is:

:<math>{{\lambda }_{c}}\left( t \right)=\lambda {{t}^{\beta -1}}=\frac{{{\lambda }_{i}}\left( t \right)}{\beta }\,\!</math>

The instantaneous MTBF is:

:<math>\begin{align}
MTB{{F}_{i}}\left( t \right)= & \frac{1}{{{\lambda }_{i\left( t \right)}}} \\
= & \frac{1}{\lambda \beta }{{t}^{1-\beta }} \\
= & \frac{MTB{{F}_{c}}\left( t \right)}{\beta }
\end{align}\,\!</math>

The cumulative MTBF at time <math>t\,\!</math> is:

:<math>MTB{{F}_{c}}\left( t \right)=\frac{1}{{{\lambda }_{c}}\left( t \right)}=\frac{1}{\lambda }{{t}^{1-\beta }}=MTB{{F}_{i}}\left( t \right)\beta \,\!</math>

The relation between the confidence level, required test time, number of systems under test and allowed total number of failures in the test is:

:<math>1-CL=\underset{i=0}{\overset{r}{\mathop \sum }}\,\frac{{{\left( m\lambda {{T}^{\beta }} \right)}^{i}}\exp (-m\lambda {{T}^{\beta }})}{i!}\,\!</math>

where:

*<math>T\,\!</math> is the total test time for each system.
*<math>m\,\!</math> is the number of systems under test.
*<math>r\,\!</math> is the number of allowed failures in the test.
*<math>CL\,\!</math> is the confidence level.

Given any three of the parameters, the equation above can be solved for the fourth unknown parameter. Note that when <math>\beta =1,\,\!</math> the number of failures is a homogeneous Poisson process, and the time between failures is given by the exponential distribution.

==Example: Solve for Time==
{{:Repairable_System_Test_Design_Example_-_Solve_for_Time}}

==Example: Solve for Number of Samples==
{{:Repairable_System_Test_Design_Example_-_Solve_for_Sample_Size}}

Arrhenius Relationship

2019-02-07T16:42:19Z

Richard House: /* Formulation */

{{template:ALTABOOK|4}}
The Arrhenius life-stress model (or relationship) is probably the most common life-stress relationship utilized in accelerated life testing. It has been widely used when the stimulus or acceleration variable (or stress) is thermal (i.e., temperature). It is derived from the Arrhenius reaction rate equation proposed by the Swedish physical chemist Svandte Arrhenius in 1887.

===Formulation===
The Arrhenius reaction rate equation is given by:

::<math>R(T)=A{{e}^{-\tfrac{{{E}_{a}}}{k\cdot T}}}\,\!</math>

where:

*<math>R\,\!</math> is the speed of reaction.

*<math>A\,\!</math> is an unknown nonthermal constant.

*<math>{{E}_{a}}\,\!</math> is the activation energy <math>(\text{eV})\,\!</math>.

*<math>k\,\!</math> is the Boltzmann's constant <math>(8.6173303\times {{10}^{-5}}\text{eV}{{\text{K}}^{-1}})\,\!</math>.

*<math>T\,\!</math> is the absolute temperature <math>(\text{K})\,\!</math>.

The activation energy is the energy that a molecule must have to participate in the reaction. In other words, the activation energy is a measure of the effect that temperature has on the reaction.

The Arrhenius life-stress model is formulated by assuming that life is proportional to the inverse reaction rate of the process, thus the Arrhenius life-stress relationship is given by:

::<math>L(V)=C{{e}^{\tfrac{B}{V}}}\,\!</math>

where:

*<math>L\,\!</math> represents a quantifiable life measure, such as mean life, characteristic life, median life, or <math>B(x)\,\!</math> life, etc.

*<math>V\,\!</math> represents the stress level (formulated for temperature and '''temperature values in absolute units, degrees Kelvin or degrees Rankine''').

*<math>C\,\!</math> is one of the model parameters to be determined, <math>(C>0)\,\!</math>.

*<math>B\,\!</math> is another model parameter to be determined.

[[Image:ALTA6.1.png|center|400px|Graphical look at the Arrhenius life-stress relationship (linear scale) for a different life characteristics, assuming a Weibull distribution.]]

Since the Arrhenius is a physics-based model derived for temperature dependence, it is used for temperature accelerated tests. For the same reason, temperature values must be in absolute units (Kelvin or Rankine), even though the Arrhenius equation is unitless.

===Life Stress Plots===
The Arrhenius relationship can be linearized and plotted on a Life vs. Stress plot, also called the Arrhenius plot. The relationship is linearized by taking the natural logarithm of both sides in the Arrhenius equation or:

::<math>ln(L(V))=ln(C)+\frac{B}{V}\,\!</math>

[[Image:ALTA6.2.png|center|400px|Arrhenius plot for Weibull life distribution.]]

In the linearized Arrhenius equation, <math>\ln (C)\,\!</math> is the intercept of the line and <math>B\,\!</math> is the slope of the line. Note that the inverse of the stress, and not the stress, is the variable. In the above figure, life is plotted versus stress and not versus the inverse stress. This is because the linearized Arrhenius equation was plotted on a reciprocal scale. On such a scale, the slope <math>B\,\!</math> appears to be negative even though it has a positive value. This is because <math>B\,\!</math> is actually the slope of the reciprocal of the stress and not the slope of the stress. The reciprocal of the stress is decreasing as stress is increasing ( <math>\tfrac{1}{V}\,\!</math> is decreasing as <math>V\,\!</math> is increasing). The two different axes are shown in the next figure.

[[Image:ALTA6.3.png|center|400px|An illustration of both reciprocal and non-reciprocal scales.]]

The Arrhenius relationship is plotted on a reciprocal scale for practical reasons. For example, in the above figure it is more convenient to locate the life corresponding to a stress level of 370K than to take the reciprocal of 370K (0.0027) first, and then locate the corresponding life.
The shaded areas shown in the above figure are the imposed at each test stress level. From such imposed ''pdfs'' one can see the range of the life at each test stress level, as well as the scatter in life. The next figure illustrates a case in which there is a significant scatter in life at each of the test stress levels.

[[Image:ALTA6.4.png|center|400px|An example of scatter in life at each test stress level.]]

===Activation Energy and the Parameter ''B'' ===
Depending on the application (and where the stress is exclusively thermal), the parameter <math>B\,\!</math> can be replaced by:

::<math>B=\frac{{{E}_{a}}}{k}=\frac{\text{activation energy}}{\text{Boltzman}{{\text{n}}^{\prime }}\text{s constant}}=\frac{\text{activation energy}}{8.617385\times {{10}^{-5}}\text{eV}{{\text{K}}^{-1}}}\,\!</math>

Note that in this formulation, the activation energy <math>{{E}_{a}}\,\!</math> must be known a priori. If the activation energy is known then there is only one model parameter remaining, <math>C.\,\!</math> Because in most real life situations this is rarely the case, all subsequent formulations will assume that this activation energy is unknown and treat <math>B\,\!</math> as one of the model parameters. <math>B\,\!</math> has the same properties as the activation energy. In other words, <math>B\,\!</math> is a measure of the effect that the stress (i.e. temperature) has on the life. The larger the value of <math>B,\,\!</math> the higher the dependency of the life on the specific stress. Parameter <math>B\,\!</math> may also take negative values. In that case, life is increasing with increasing stress. An example of this would be plasma filled bulbs, where low temperature is a higher stress on the bulbs than high temperature.

[[Image:ALTA6.5.png|center|400px|Behavior of the parameter ''B''.]]

===Acceleration Factor===
Most practitioners use the term acceleration factor to refer to the ratio of the life (or acceleration characteristic) between the use level and a higher test stress level or:

::<math>{{A}_{F}}=\frac{{{L}_{USE}}}{{{L}_{Accelerated}}}\,\!</math>

For the Arrhenius model this factor is:

::<math>{{A}_{F}}=\frac{{{L}_{USE}}}{{{L}_{Accelerated}}}=\frac{C\text{ }{{e}^{\tfrac{B}{{{V}_{u}}}}}}{C\text{ }{{e}^{\tfrac{B}{{{V}_{A}}}}}}=\frac{\text{ }{{e}^{\tfrac{B}{{{V}_{u}}}}}}{\text{ }{{e}^{\tfrac{B}{{{V}_{A}}}}}}={{e}^{\left( \tfrac{B}{{{V}_{u}}}-\tfrac{B}{{{V}_{A}}} \right)}}\,\!</math>

Thus, if <math>B\,\!</math> is assumed to be known a priori (using an activation energy), the assumed activation energy alone dictates this acceleration factor!

=Arrhenius-Exponential=
The ''pdf'' of the 1-parameter exponential distribution is given by:

::<math>\begin{align}
f(t)=\lambda {{e}^{-\lambda t}}
\end{align}\,\!</math>

It can be easily shown that the mean life for the 1-parameter exponential distribution (presented in detail [[Distributions used in Accelerated Testing#The Exponential Distribution|here]]) is given by:

::<math>\lambda =\frac{1}{m}\,\!</math>

thus:

::<math>f(t)=\frac{1}{m}{{e}^{-\tfrac{t}{m}}}\,\!</math>

The Arrhenius-exponential model ''pdf'' can then be obtained by setting <math>m=L(V)\,\!</math>:

Therefore:

::<math>m=L(V)=C{{e}^{\tfrac{B}{V}}}\,\!</math>

Substituting for <math>m\,\!</math> yields a ''pdf'' that is both a function of time and stress or:

::<math>f(t,V)=\frac{1}{C{{e}^{\tfrac{B}{V}}}}\cdot {{e}^{-\tfrac{1}{C{{e}^{\tfrac{B}{V}}}}\cdot t}}\,\!</math>

==Arrhenius-Exponential Statistical Properties Summary==
====Mean or MTTF====
The mean, <math>\overline{T},\,\!</math> or Mean Time To Failure (MTTF) of the Arrhenius-exponential is given by,

::<math>\begin{align}
\overline{T}=\int_{0}^{\infty }t\cdot f(t,V)dt=\int_{0}^{\infty }t\cdot \frac{1}{C{{e}^{\tfrac{B}{V}}}}{{e}^{-\tfrac{t}{C{{e}^{\tfrac{B}{V}}}}}}dt =\ C{{e}^{\tfrac{B}{V}}}
\end{align}\,\!</math>

====Median====
The median, <math>\breve{T}\,\!</math> of the Arrhenius-exponential model is given by:

::<math>\breve{T}=0.693\cdot C{{e}^{\tfrac{B}{V}}}\,\!</math>

====Mode====
The mode, <math>\tilde{T},\,\!</math> of the Arrhenius-exponential model is given by:

::<math>\tilde{T}=0\,\!</math>

====Standard Deviation====
The standard deviation, <math>{{\sigma }_{T}}\,\!</math>, of the Arrhenius-exponential model is given by:

::<math>{{\sigma }_{T}}=C{{e}^{\tfrac{B}{V}}}\,\!</math>

====Arrhenius-Exponential Reliability Function====
The Arrhenius-exponential reliability function is given by:

::<math>R(T,V)={{e}^{-\tfrac{T}{C{{e}^{\tfrac{B}{V}}}}}}\,\!</math>

This function is the complement of the Arrhenius-exponential cumulative distribution function or:

::<math>R(T,V)=1-Q(T,V)=1-\int_{0}^{T}f(T,V)dT\,\!</math>

and:

::<math>R(T,V)=1-\int_{0}^{T}\frac{1}{C{{e}^{\tfrac{B}{V}}}}{{e}^{-\tfrac{T}{C{{e}^{\tfrac{B}{V}}}}}}dT={{e}^{-\tfrac{T}{C{{e}^{\tfrac{B}{V}}}}}}\,\!</math>

====Conditional Reliability====
The Arrhenius-exponential conditional reliability function is given by:

::<math>R((t|T),V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\tfrac{t}{C{{e}^{\tfrac{B}{V}}}}}}\,\!</math>

====Reliable Life====
For the Arrhenius-exponential model, the reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}},\,\!</math> is given by:

::<math>R({{t}_{R}},V)={{e}^{-\tfrac{{{t}_{R}}}{C{{e}^{\tfrac{B}{V}}}}}}\,\!</math>

::<math>\ln [R({{t}_{R}},V)]=-\frac{{{t}_{R}}}{C{{e}^{\tfrac{B}{V}}}}\,\!</math>

or:

::<math>{{t}_{R}}=-C{{e}^{\tfrac{B}{V}}}\ln [R({{t}_{R}},V)]\,\!</math>

==Parameter Estimation==
====Maximum Likelihood Estimation Method====
The log-likelihood function for the exponential distribution is as shown next:

::<math>\begin{align}
& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \lambda {{e}^{-\lambda {{T}_{i}}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda T_{i}^{\prime } \
& \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })]
\end{align}\,\!</math>

where:

::<math>R_{Li}^{\prime \prime }={{e}^{-\lambda T_{Li}^{\prime \prime }}}\,\!</math>

::<math>R_{Ri}^{\prime \prime }={{e}^{-\lambda T_{Ri}^{\prime \prime }}}\,\!</math>

and:

*<math>{{F}_{e}}\,\!</math> is the number of groups of exact times-to-failure data points.

*<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group.

*<math>\lambda \,\!</math> is the failure rate parameter (unknown).

*<math>{{T}_{i}}\,\!</math> is the exact failure time of the <math>{{i}^{th}}\,\!</math> group.

*<math>S\,\!</math> is the number of groups of suspension data points.

*<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points.

*<math>T_{i}^{\prime }\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> suspension data group.

*<math>FI\,\!</math> is the number of interval data groups.

*<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals.

*<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval.

*<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval.

Substituting the Arrhenius-exponential model into the log-likelihood function yields:

::<math>\begin{align}
\Lambda = \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{e}^{-\tfrac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{T}_{i}}}} \right] -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}T_{i}^{\prime }+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
\end{align}\,\!</math>

where:

::<math>R_{Li}^{\prime \prime }={{e}^{-\tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}}}\,\!</math>

::<math>R_{Ri}^{\prime \prime }={{e}^{-\tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}}}\,\!</math>

The solution (parameter estimates) will be found by solving for the parameters <math>\widehat{B},\,\!</math> <math>\widehat{C}\,\!</math> so that <math>\tfrac{\partial \Lambda }{\partial B}=0\,\!</math> and <math>\tfrac{\partial \Lambda }{\partial C}=0\,\!</math>, where:

::<math>\begin{align}
& \frac{\partial \Lambda }{\partial B}= & \frac{1}{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{T}_{i}}}{{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}-\frac{C}{{{V}_{i}}} \right)+\frac{1}{C}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}} \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime }}{(R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime })C{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}
\end{align}\,\!</math>

::<math>\begin{align}
& \frac{\partial \Lambda }{\partial C}= & \frac{1}{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{T}_{i}}}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}-1 \right)+\frac{1}{{{C}^{2}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{{{e}^{\tfrac{B}{{{V}_{i}}}}}} \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime }}{(R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }){{C}^{2}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}
\end{align}\,\!</math>

=Arrhenius-Weibull=

The ''pdf'' for the 2-parameter Weibull distribution is given by:

::<math>f(t)=\frac{\beta }{\eta }{{\left( \frac{t}{\eta } \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t}{\eta } \right)}^{\beta }}}}\,\!</math>

The scale parameter (or characteristic life) of the Weibull distribution is <math>\eta \,\!</math>.

The Arrhenius-Weibull model ''pdf'' can then be obtained by setting <math>\eta =L(V)\,\!</math>:

::<math>\eta =L(V)=C\cdot {{e}^{\tfrac{B}{V}}}\,\!</math>

and substituting for <math>\eta \,\!</math> in the 2-parameter Weibull distribution equation:

::<math>f(t,V)=\frac{\beta }{C\cdot {{e}^{\tfrac{B}{V}}}}{{\left( \frac{t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}}}\,\!</math>

An illustration of the ''pdf'' for different stresses is shown in the next figure. As expected, the ''pdf'' at lower stress levels is more stretched to the right, with a higher scale parameter, while its shape remains the same (the shape parameter is approximately 3). This behavior is observed when the parameter <math>B\,\!</math> of the Arrhenius model is positive.

[[Image:ALTA6.6.png|center|400px|Behavior of the probability density function at different stresses and with the parameters held constant.]]

The advantage of using the Weibull distribution as the life distribution lies in its flexibility to assume different shapes. The Weibull distribution is presented in greater detail in [[The Weibull Distribution]].

==Arrhenius-Weibull Statistical Properties Summary==
====Mean or MTTF====
The mean, <math>\overline{T}\,\!</math> (also called <math>MTTF\,\!</math> by some authors), of the Arrhenius-Weibull relationship is given by:

::<math>\overline{T}=C\cdot {{e}^{\tfrac{B}{V}}}\cdot \Gamma \left( \frac{1}{\beta }+1 \right)\,\!</math>

where <math>\Gamma \left( \tfrac{1}{\beta }+1 \right)\,\!</math> is the gamma function evaluated at the value of <math>\left( \tfrac{1}{\beta }+1 \right)\,\!</math>.

====Median====
The median, <math>\breve{T},\,\!</math>
for the Arrhenius-Weibull model is given by:

::<math>\breve{T}=C\cdot {{e}^{\tfrac{B}{V}}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}}\,\!</math>

====Mode====
The mode, <math>\tilde{T},\,\!</math>
for the Arrhenius-Weibull model is given by:

::<math>\tilde{T}=C\cdot {{e}^{\tfrac{B}{V}}}{{\left( 1-\frac{1}{\beta } \right)}^{\tfrac{1}{\beta }}}\,\!</math>

====Standard Deviation====
The standard deviation, <math>{{\sigma }_{T}},\,\!</math> for the Arrhenius-Weibull model is given by:

::<math>{{\sigma }_{T}}=C\cdot {{e}^{\tfrac{B}{V}}}\cdot \sqrt{\Gamma \left( \frac{2}{\beta }+1 \right)-{{\left( \Gamma \left( \frac{1}{\beta }+1 \right) \right)}^{2}}}\,\!</math>

====Arrhenius-Weibull Reliability Function====
The Arrhenius-Weibull reliability function is given by:

::<math>R(T,V)={{e}^{-{{\left( \tfrac{T}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}}}\,\!</math>

If the parameter <math>B\,\!</math> is positive, then the reliability increases as stress decreases.

[[Image:ALTA6.7.png|center|500px|Behavior of the reliability function at different stress and constant parameter values.]]

The behavior of the reliability function of the Weibull distribution for different values of <math>\beta \,\!</math> was illustrated [[Distributions used in Accelerated Testing#The Weibull Distribution|here]]. In the case of the Arrhenius-Weibull model, however, the reliability is a function of stress also. A 3D plot such as the ones shown in the next figure is now needed to illustrate the effects of both the stress and <math>\beta .\,\!</math>

[[Image:ALTA6.8.png|center|800px|Reliability function for <math>\Beta<1 \,\!</math>, <math>\Beta=1 \,\!</math>, and <math>\Beta>1 \,\!</math>.]]

====Conditional Reliability Function====
The Arrhenius-Weibull conditional reliability function at a specified stress level is given by:

::<math>R((t|T),V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}}\,\!</math>

or:

::<math>R((t|T),V)={{e}^{-\left[ {{\left( \tfrac{T+t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}-{{\left( \tfrac{T}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }} \right]}}\,\!</math>

====Reliable Life====
For the Arrhenius-Weibull relationship, the reliable life, <math>{{t}_{R}}\,\!</math>, of a unit for a specified reliability and starting the mission at age zero is given by:

::<math>{{t}_{R}}=C\cdot {{e}^{\tfrac{B}{V}}}{{\left\{ -\ln \left[ R\left( {{t}_{R}},V \right) \right] \right\}}^{\tfrac{1}{\beta }}}\,\!</math>

This is the life for which the unit will function successfully with a reliability of <math>R({{t}_{R}})\,\!</math>. If <math>R({{t}_{R}})=0.50\,\!</math> then <math>{{t}_{R}}=\breve{T}\,\!</math>, the median life, or the life by which half of the units will survive.

====Arrhenius-Weibull Failure Rate Function====
The Arrhenius-Weibull failure rate function, <math>\lambda (T)\,\!</math>, is given by:

::<math>\lambda \left( T,V \right)=\frac{f\left( T,V \right)}{R\left( T,V \right)}=\frac{\beta }{C\cdot {{e}^{\tfrac{B}{V}}}}{{\left( \frac{T}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta -1}}\,\!</math>

[[Image:ALTA6.9.png|center|800px|Failure rate function for <math>\Beta<1 \,\!</math>, <math>\Beta=1 \,\!</math>, and <math>\Beta>1 \,\!</math>.]]

==Parameter Estimation==
====Maximum Likelihood Estimation Method====
The Arrhenius-Weibull log-likelihood function is as follows:

::<math>\begin{align}
& \Lambda = & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{\beta }{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}}} \right] \ -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
\end{align}\,\!</math>

where:

::<math>R_{Li}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}}}\,\!</math>

::<math>R_{Ri}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}}}\,\!</math>

and:

*<math>{{F}_{e}}\,\!</math> is the number of groups of exact times-to-failure data points.

*<math>{{N}_{i}}\,\!</math> is the number of times-to-failure data points in the <math>{{i}^{th}}\,\!</math> time-to-failure data group.

*<math>\beta \,\!</math> is the Weibull shape parameter (unknown, the first of three parameters to be estimated).

*<math>B\,\!</math> is the Arrhenius parameter (unknown, the second of three parameters to be estimated).

*<math>C\,\!</math> is the second Arrhenius parameter (unknown, the third of three parameters to be estimated).

*<math>{{V}_{i}}\,\!</math> is the stress level of the <math>{{i}^{th}}\,\!</math> group.

*<math>{{T}_{i}}\,\!</math> is the exact failure time of the <math>{{i}^{th}}\,\!</math> group.

*<math>S\,\!</math> is the number of groups of suspension data points.

*<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points.

*<math>T_{i}^{\prime }\,\!</math> is the running time of the <math>{{i}^{th}}\,\!</math> suspension data group.

*<math>FI\,\!</math> is the number of interval data groups.

*<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals.

*<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval.

*<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval.

The solution (parameter estimates) will be found by solving for <math>\widehat{\beta },\,\!</math> <math>\widehat{B},\,\!</math> <math>\widehat{C}\,\!</math> so that <math>\tfrac{\partial \Lambda }{\partial \beta }=0,\,\!</math> <math>\tfrac{\partial \Lambda }{\partial B}=0\,\!</math> and <math>\tfrac{\partial \Lambda }{\partial C}=0\,\!</math>, where:

::<math>\begin{align}
\frac{\partial \Lambda }{\partial \beta }=\ & \frac{1}{\beta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln \left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) -\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) -\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) \\
& \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{\left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)R_{Li}^{\prime \prime }-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)R_{Ri}^{\prime \prime }}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}
\end{align}\,\!</math>

::<math>\begin{align}
& \frac{\partial \Lambda }{\partial B}= -\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\frac{1}{{{V}_{i}}}+\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\frac{1}{{{V}_{i}}}{{\left( \frac{{{T}_{i}}}{\widehat{C}{{e}^{\tfrac{\widehat{B}}{{{V}_{i}}}}}} \right)}^{\beta }}+\beta \underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }\frac{1}{{{V}_{i}}}{{\left( \frac{T_{i}^{\prime }}{\widehat{C}{{e}^{\tfrac{\widehat{B}}{{{V}_{i}}}}}} \right)}^{\beta }} +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\beta }{{{V}_{i}}}\frac{{{(T_{Li}^{\prime \prime })}^{\beta }}R_{Li}^{\prime \prime }-{{(T_{Ri}^{\prime \prime })}^{\beta }}R_{Ri}^{\prime \prime }}{{{\left( C{{e}^{\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}\left( R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime } \right)}
\end{align}\,\!</math>

::<math>\begin{align}
& \frac{\partial \Lambda }{\partial C}= -\frac{\beta }{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\frac{\beta }{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}+\frac{\beta }{C}\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }} +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\beta }{C}\frac{{{(T_{Li}^{\prime \prime })}^{\beta }}R_{Li}^{\prime \prime }-{{(T_{Ri}^{\prime \prime })}^{\beta }}R_{Ri}^{\prime \prime }}{{{\left( C{{e}^{\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}\left( R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime } \right)}
\end{align}\,\!</math>

==Arrhenius-Weibull example==
{{:Arrhenius_Example}}

=Arrhenius-Lognormal=

The ''pdf'' of the lognormal distribution is given by:

::<math>f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\bar{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}\,\!</math>

where:

::<math>\begin{align}
{T}'=\ln(T)
\end{align}\,\!</math>

and:

*<math>T=\,\!</math> times-to-failure.

*<math>{T}'=\,\!</math> mean of the natural logarithms of the times-to-failure.

*<math>{{\sigma }_{{{T}'}}}=\,\!</math> standard deviation of the natural logarithms of the times-to-failure.

The median of the lognormal distribution is given by:

::<math>\breve{T}={{e}^{{{\overline{T}}^{\prime }}}}\,\!</math>

The Arrhenius-lognormal model ''pdf'' can be obtained first by setting <math>\breve{T}=L(V)\,\!</math>. Therefore:

::<math>\breve{T}=L(V)=C{{e}^{\tfrac{B}{V}}}\,\!</math>

or:

::<math>{{e}^{{{\overline{T}}^{\prime }}}}=C{{e}^{\tfrac{B}{V}}}\,\!</math>

Thus:

::<math>{{\overline{T}}^{\prime }}=\ln (C)+\frac{B}{V}\,\!</math>

Substituting the above equation into the lognormal ''pdf'' yields the Arrhenius-lognormal model ''pdf'' or:

::<math>f(T,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}\,\!</math>

Note that in the Arrhenius-lognormal ''pdf'', it was assumed that the standard deviation of the natural logarithms of the times-to-failure, <math>{{\sigma }_{{{T}'}}},\,\!</math> is independent of stress. This assumption implies that the shape of the distribution does not change with stress ( <math>{{\sigma }_{{{T}'}}}\,\!</math> is the shape parameter of the lognormal distribution).

==Arrhenius-Lognormal Statistical Properties Summary==
====The Mean====
*The mean life of the Arrhenius-lognormal model (mean of the times-to-failure), <math>\bar{T}\,\!</math>, is given by:

::<math>\begin{align}
& \bar{T}= & {{e}^{\bar{{T}'}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}} =\ & {{e}^{\ln (C)+\tfrac{B}{V}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}}
\end{align}\,\!</math>

*The mean of the natural logarithms of the times-to-failure, <math>{{\bar{T}}^{^{\prime }}}\,\!</math>, in terms of <math>\bar{T}\,\!</math> and <math>{{\sigma }_{T}}\,\!</math> is given by:

::<math>{{\bar{T}}^{\prime }}=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)\,\!</math>

====The Standard Deviation====
*The standard deviation of the Arrhenius-lognormal model (standard deviation of the times-to-failure), <math>{{\sigma }_{T}}\,\!</math>, is given by:

::<math>\begin{align}
& {{\sigma }_{T}}= & \sqrt{\left( {{e}^{2\bar{{T}'}+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)} =\ & \sqrt{\left( {{e}^{2\left( \ln (C)+\tfrac{B}{V} \right)+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)}
\end{align}\,\!</math>

*The standard deviation of the natural logarithms of the times-to-failure, <math>{{\sigma }_{{{T}'}}}\,\!</math>, in terms of <math>\bar{T}\,\!</math> and <math>{{\sigma }_{T}}\,\!</math> is given by:

::<math>{{\sigma }_{{{T}'}}}=\sqrt{\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)}\,\!</math>

====The Mode====
*The mode of the Arrhenius-lognormal model is given by:

::<math>\begin{align}
& \tilde{T}=\ {{e}^{{{\overline{T}}^{\prime }}-\sigma _{{{T}'}}^{2}}} =\ {{e}^{\ln (C)+\tfrac{B}{V}-\sigma _{{{T}'}}^{2}}}
\end{align}\,\!</math>

====Arrhenius-Lognormal Reliability Function====
The reliability for a mission of time <math>T\,\!</math>, starting at age 0, for the Arrhenius-lognormal model is determined by:

::<math>R(T,V)=\int_{T}^{\infty }f(t,V)dt\,\!</math>

or:

::<math>R(T,V)=\int_{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (C)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt\,\!</math>

There is no closed form solution for the lognormal reliability function. Solutions can be obtained via the use of standard normal tables. Since the application automatically solves for the reliability, we will not discuss manual solution methods.

====Reliable Life====
For the Arrhenius-lognormal model, the reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}},\,\!</math> is estimated by first solving the reliability equation with respect to time, as follows:

::<math>{{T}'_{R}}= ln(C)+\frac{B}{V}+z \cdot {{\sigma}_{{T}'}}\,\!</math>

where:

::<math>z={{\Phi }^{-1}}\left[ F\left( T_{R}^{\prime },V \right) \right]\,\!</math>

and:

::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({T}',V)}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt\,\!</math>

Since <math>{T}'=\ln (T)\,\!</math> the reliable life, <math>{{t}_{R}},\,\!</math> is given by:

::<math>{{t}_{R}}={{e}^{T_{R}^{\prime }}}\,\!</math>

====Arrhenius-Lognormal Failure Rate====
The Arrhenius-lognormal failure rate is given by:

::<math>\lambda (T,V)=\frac{f(T,V)}{R(T,V)}=\frac{\tfrac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}}{\int_{{{T}'}}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt}\,\!</math>

==Parameter Estimation==
====Maximum Likelihood Estimation Method====

The lognormal log-likelihood function for the Arrhenius-lognormal model is as follows:

::<math>\begin{align}
& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{{{\sigma }_{{{T}'}}}{{T}_{i}}}\phi \left( \frac{\ln \left( {{T}_{i}} \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \right] \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ 1-\Phi \left( \frac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \right] +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })]
\end{align}\,\!</math>

where:

::<math>z_{Li}^{\prime \prime }=\frac{\ln T_{Li}^{\prime \prime }-\ln C-\tfrac{B}{{{V}_{i}}}}{\sigma _{T}^{\prime }}\,\!</math>

::<math>z_{Ri}^{\prime \prime }=\frac{\ln T_{Ri}^{\prime \prime }-\ln C-\tfrac{B}{{{V}_{i}}}}{\sigma _{T}^{\prime }}\,\!</math>

and:

*<math>{{F}_{e}}\,\!</math> is the number of groups of exact times-to-failure data points.

*<math>{{N}_{i}}\,\!</math> is the number of times-to-failure data points in the <math>{{i}^{th}}\,\!</math> time-to-failure data group.

*<math>{{\sigma}_{{T}'}}\,\!</math> is the standard deviation of the natural logarithm of the times-to-failure (unknown, the first of three parameters to be estimated).

*<math>B\,\!</math> is the Arrhenius parameter (unknown, the second of three parameters to be estimated).

*<math>C\,\!</math> is the second Arrhenius parameter (unknown, the third of three parameters to be estimated).

*<math>{{V}_{i}}\,\!</math> is the stress level of the <math>{{i}^{th}}\,\!</math> group.

*<math>{{T}_{i}}\,\!</math> is the exact failure time of the <math>{{i}^{th}}\,\!</math> group.

*<math>S\,\!</math> is the number of groups of suspension data points.

*<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points.

*<math>T_{i}^{\prime }\,\!</math> is the running time of the <math>{{i}^{th}}\,\!</math> suspension data group.

*<math>FI\,\!</math> is the number of interval data groups.

*<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals.

*<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval.

*<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval.

The solution (parameter estimates) will be found by solving for <math>{{\widehat{\sigma }}_{{{T}'}}},\,\!</math> <math>\widehat{B},\,\!</math> <math>\widehat{C}\,\!</math> so that <math>\tfrac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}=0,\,\!</math> <math>\tfrac{\partial \Lambda }{\partial B}=0\,\!</math> and <math>\tfrac{\partial \Lambda }{\partial C}=0\,\!</math>, where:

::<math>\begin{align}
& \frac{\partial \Lambda }{\partial B}= \frac{1}{\sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\frac{1}{{{V}_{i}}}(\ln ({{T}_{i}})-\ln (C)-\frac{B}{{{V}_{i}}}) +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{1}{{{V}_{i}}}\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)} \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\varphi (z_{Ri}^{\prime \prime })-\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }{{V}_{i}}(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))}
\end{align}\,\!</math>

::<math>\begin{align}
& \frac{\partial \Lambda }{\partial C}= \frac{1}{C\cdot \sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}(\ln ({{T}_{i}})-\ln (C)-\frac{B}{{{V}_{i}}}) +\frac{1}{C\cdot {{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)} \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\varphi (z_{Ri}^{\prime \prime })-\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }C(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} \\
& & \\
& \frac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}= \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{\left( \ln ({{T}_{i}})-\ln (C)-\tfrac{B}{{{V}_{i}}} \right)}^{2}}}{\sigma _{{{T}'}}^{3}}-\frac{1}{{{\sigma }_{{{T}'}}}} \right) +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)} \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{z_{Ri}^{\prime \prime }\varphi (z_{Ri}^{\prime \prime })-z_{Li}^{\prime \prime }\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))}
\end{align}\,\!</math>

and:

::<math>\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( x \right)}^{2}}}}\,\!</math>

::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt\,\!</math>

=Arrhenius Confidence Bounds=
==Approximate Confidence Bounds for the Arrhenius-Exponential==
There are different methods for computing confidence bounds. ALTA utilizes confidence bounds that are based on the asymptotic theory for maximum likelihood estimates, most commonly referred to as the Fisher matrix bounds.
 
====Confidence Bounds on the Mean Life====

The Arrhenius-exponential distribution is given by setting <math>m=L(V)\,\!</math> in the exponential ''pdf'' equation. The upper <math>({{m}_{U}})\,\!</math> and lower <math>({{m}_{L}})\,\!</math> bounds on the mean life are then estimated by:

::<math>\begin{align}
& {{m}_{U}}= \widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \\
& {{m}_{L}}= \widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}
\end{align}\,\!</math>

where <math>{{K}_{\alpha }}\,\!</math> is defined by:

::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})\,\!</math>

If <math>\delta \,\!</math> is the confidence level (i.e., 95%=0.95), then <math>\alpha =\tfrac{1-\delta }{2}\,\!</math> for the two-sided bounds, and <math>\alpha =1-\delta \,\!</math> for the one-sided bounds. The variance of <math>\widehat{m}\,\!</math> is given by:

::<math>\begin{align}
& Var(\widehat{m})= & {{\left( \frac{\partial m}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial m}{\partial B} \right)}^{2}}Var(\widehat{B}) +2\left( \frac{\partial m}{\partial C} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{B},\widehat{C})
\end{align}\,\!</math>

or:

::<math>Var(\widehat{m})={{e}^{\tfrac{2\widehat{B}}{V}}}\left[ Var(\widehat{C})+\frac{{{\widehat{C}}^{2}}}{{{V}^{2}}}Var(\widehat{B})+\frac{2\widehat{C}}{V}Cov(\widehat{B},\widehat{C}) \right]\,\!</math>

The variances and covariance of <math>B\,\!</math> and <math>C\,\!</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{B}\,\!</math>, <math>\widehat{C})\,\!</math> as follows:

::<math>\left[ \begin{matrix}
Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\
Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\
\end{matrix} \right]={{\left[ \begin{matrix}
-\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\
\end{matrix} \right]}^{-1}}\,\!</math>

====Confidence Bounds on Reliability====
The bounds on reliability for any given time, <math>T\,\!</math>, are estimated by:

::<math>\begin{align}
& {{R}_{U}}(T)= & {{e}^{-\tfrac{T}{{{m}_{U}}}}} \\
& {{R}_{L}}(T)= & {{e}^{-\tfrac{T}{{{m}_{L}}}}}
\end{align}\,\!</math>

where <math>{{m}_{U}}\,\!</math> and <math>{{m}_{L}}\,\!</math> are estimated estimated by:

::<math>\begin{align}
& {{m}_{U}}= \widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \\
& {{m}_{L}}= \widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}
\end{align}\,\!</math>

====Confidence Bounds on Time====
The bounds on time (ML estimate of time) for a given reliability are estimated by first solving the reliability function with respect to time:

::<math>\widehat{T}=-\widehat{m}\cdot \ln (R)\,\!</math>

The corresponding confidence bounds are then estimated from:

::<math>\begin{align}
& {{T}_{U}}= -{{m}_{U}}\cdot \ln (R) \\
& {{T}_{L}}= -{{m}_{L}}\cdot \ln (R)
\end{align}\,\!</math>

where <math>{{m}_{U}}\,\!</math> and <math>{{m}_{L}}\,\!</math> are estimated estimated by:

::<math>\begin{align}
& {{m}_{U}}= \widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \\
& {{m}_{L}}= \widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}
\end{align}\,\!</math>

==Approximate Confidence Bounds for the Arrhenius-Weibull==
====Bounds on the Parameters====
From the asymptotically normal property of the maximum likelihood estimators, and since <math>\widehat{\beta },\,\!</math> and <math>\widehat{C}\,\!</math> are positive parameters, <math>\ln (\widehat{\beta }),\,\!</math> and <math>\ln (\widehat{C})\,\!</math> can then be treated as normally distributed. After performing this transformation, the bounds on the parameters can be estimated from:

::<math>\begin{align}
& {{\beta }_{U}}= \widehat{\beta }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \\
& {{\beta }_{L}}= \widehat{\beta }\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}}
\end{align}\,\!</math>

also:

::<math>\begin{align}
& {{B}_{U}}= \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \\
& {{B}_{L}}= \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}
\end{align}\,\!</math>

and:

::<math>\begin{align}
& {{C}_{U}}= \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \\
& {{C}_{L}}= \widehat{C}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}
\end{align}\,\!</math>

The variances and covariances of <math>\beta ,\,\!</math> <math>B,\,\!</math> and <math>C\,\!</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{\beta },\,\!</math> <math>\widehat{B},\,\!</math> <math>\widehat{C})\,\!</math>, as follows:

::<math>\left[ \begin{matrix}
Var(\widehat{\beta }) & Cov(\widehat{\beta },\widehat{B}) & Cov(\widehat{\beta },\widehat{C}) \\
Cov(\widehat{B},\widehat{\beta }) & Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\
Cov(\widehat{C},\widehat{\beta }) & Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\
\end{matrix} \right]={{\left[ \begin{matrix}
-\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial C} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\
\end{matrix} \right]}^{-1}}\,\!</math>

====Confidence Bounds on Reliability====
The reliability function for the Arrhenius-Weibull model (ML estimate) is given by:

::<math>\widehat{R}(T,V)={{e}^{-{{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}}}}\,\!</math>

or:

::<math>\widehat{R}(T)={{e}^{-{{e}^{\ln \left[ {{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \right]}}}}\,\!</math>

Setting:

::<math>\widehat{u}=\ln \left[ {{\left( \frac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \right]\,\!</math>

or:

::<math>\widehat{u}=\widehat{\beta }\left[ \ln (T)-\ln (\widehat{C})-\frac{\widehat{B}}{V} \right]\,\!</math>

The reliability function now becomes:

::<math>\widehat{R}(T,V)={{e}^{-{{e}^{\widehat{u}}}}}\,\!</math>

The next step is to find the upper and lower bounds on <math>\widehat{u}\ \ :\,\!</math>

::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}\,\!</math>

::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}\,\!</math>

where:

::<math>\begin{align}
& Var(\widehat{u})= {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C})
\end{align}\,\!</math>

or:

::<math>\begin{align}
& Var(\widehat{u})= {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\widehat{\beta }}{\widehat{C}} \right)}^{2}}Var(\widehat{C}) -\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B})-\frac{2\widehat{u}}{\widehat{C}}Cov(\widehat{\beta },\widehat{C})+\frac{2{{\widehat{\beta }}^{2}}}{V\widehat{C}}Cov(\widehat{B},\widehat{C})
\end{align}\,\!</math>

The upper and lower bounds on reliability are:

::<math>\begin{align}
& {{R}_{U}}(T,V)= {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\
& {{R}_{L}}(T,V)= {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}}
\end{align}\,\!</math>

====Confidence Bounds on Time====
The bounds on time for a given reliability are estimated by first solving the reliability function with respect to time:

::<math>\begin{align}
\ln (R)&= -{{\left( \frac{\widehat{T}}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \\
\ln (-\ln (R))&= \widehat{\beta }\left( \ln \widehat{T}-\ln \widehat{C}-\frac{\widehat{B}}{V} \right)
\end{align}\,\!</math>

or:

::<math>\widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))+\ln \widehat{C}+\frac{\widehat{B}}{V}\,\!</math>

where <math>\widehat{u}=\ln \widehat{T}\,\!</math>.

The upper and lower bounds on <math>u\,\!</math> are estimated from:

::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}\,\!</math>

::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}\,\!</math>

where:

::<math>\begin{align}
& Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C})
\end{align}\,\!</math>

or:

::<math>\begin{align}
& Var(\widehat{u})= & \frac{1}{{{\widehat{\beta }}^{4}}}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta })+\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{\widehat{C}}^{2}}}Var(\widehat{C})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}\widehat{C}}Cov(\widehat{\beta },\widehat{C}) +\frac{2}{V\widehat{C}}Cov(\widehat{B},\widehat{C})
\end{align}\,\!</math>

The upper and lower bounds on time can then found by:

::<math>\begin{align}
& {{T}_{U}}= & {{e}^{{{u}_{U}}}} \\
& {{T}_{L}}= & {{e}^{{{u}_{L}}}}
\end{align}\,\!</math>

==Approximate Confidence Bounds for the Arrhenius-Lognormal==
====Bounds on the Parameters====
The lower and upper bounds on <math>B\,\!</math> are estimated from:

::<math>\begin{align}
& {{B}_{U}}= \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Upper bound)} \\
& {{B}_{L}}= \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Lower bound)}
\end{align}\,\!</math>

Since the standard deviation, <math>{{\widehat{\sigma }}_{{{T}'}}}\,\!</math>, and the parameter <math>\widehat{C}\,\!</math> are positive parameters, <math>\ln ({{\widehat{\sigma }}_{{{T}'}}})\,\!</math> and <math>\ln (\widehat{C})\,\!</math> are treated as normally distributed. The bounds are estimated from:

::<math>\begin{align}
& {{C}_{U}}= \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}\text{ (Upper bound)} \\
& {{C}_{L}}= \frac{\widehat{C}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}}\text{ (Lower bound)}
\end{align}\,\!</math>

and:

::<math>\begin{align}
& {{\sigma }_{U}}= {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}\text{ (Upper bound)} \\
& {{\sigma }_{L}}= \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}\text{ (Lower bound)}
\end{align}\,\!</math>

The variances and covariances of <math>B,\,\!</math> <math>C,\,\!</math> and <math>{{\sigma }_{{{T}'}}}\,\!</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{B},\,\!</math> <math>\widehat{C}\,\!</math>, <math>{{\widehat{\sigma }}_{{{T}'}}}),\,\!</math> as follows:

::<math>\left[ \begin{matrix}
Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{B} \right) & Var\left( \widehat{B} \right) & Cov\left( \widehat{B},\widehat{C} \right) \\
Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{C} \right) & Cov\left( \widehat{C},\widehat{B} \right) & Var\left( \widehat{C} \right) \\
\end{matrix} \right]= {{\left[ \begin{matrix}
-\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _{{{T}'}}^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial C} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\
\end{matrix} \right]}^{-1}}\,\!</math>

====Bounds on Reliability====
The reliability of the lognormal distribution is:

::<math>R({T}',V;B,C,{{\sigma }_{{{T}'}}})=\int_{{{T}'}}^{\infty }\frac{1}{{{\widehat{\sigma }}_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}} \right)}^{2}}}}dt\,\!</math>

Let <math>\widehat{z}(t,V;B,C,{{\sigma }_{T}})=\tfrac{t-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}},\,\!</math> then <math>\frac{d \widehat{z}}{dt}=\frac{1}{{\widehat{\sigma }}_{{{T}'}}}\,\!</math>.

For <math>t={T}'\,\!</math>, <math>\widehat{z}=\tfrac{{T}'-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}}\,\!</math>, and for <math>t=\infty ,\,\!</math> <math>\widehat{z}=\infty .\,\!</math> The above equation then becomes:

::<math>R(\widehat{z})=\int_{\widehat{z}({T}')}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\,\!</math>

The bounds on <math>z\,\!</math> are estimated from:

::<math>\begin{align}
& {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\
& {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})}
\end{align}\,\!</math>

where:

::<math>\begin{align}
Var(\widehat{z})=& \left( \frac{\partial \widehat{z}}{\partial B} \right)_{\widehat{B}}^{2}Var(\widehat{B})+\left( \frac{\partial \widehat{z}}{\partial C} \right)_{\widehat{C}}^{2}Var(\widehat{C})+\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)_{{{\widehat{\sigma }}_{{{T}'}}}}^{2}Var({{\widehat{\sigma }}_{T}}) +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}Cov\left( \widehat{B},\widehat{C} \right) \\
& +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{B},{{\widehat{\sigma }}_{T}} \right) +2{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{C},{{\widehat{\sigma }}_{T}} \right)
\end{align}\,\!</math>

or:

::<math>\begin{align}
& Var(\widehat{z})= & \frac{1}{\widehat{\sigma }_{{{T}'}}^{2}}[\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) +\frac{2}{C\cdot V}Cov\left( \widehat{B},\widehat{C} \right)+\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)+\frac{2\widehat{z}}{C}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)]
\end{align}\,\!</math>

The upper and lower bounds on reliability are:

::<math>\begin{align}
& {{R}_{U}}= & \int_{{{z}_{L}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Upper bound)} \\
& {{R}_{L}}= & \int_{{{z}_{U}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Lower bound)}
\end{align}\,\!</math>

====Confidence Bounds on Time====
The bounds around time, for a given lognormal percentile (unreliability), are estimated by first solving the reliability equation with respect to time, as follows:

::<math>{T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})=\ln (\widehat{C})+\frac{\widehat{B}}{V}+z\cdot {{\widehat{\sigma }}_{{{T}'}}}\,\!</math>

where:

::<math>\begin{align}
{T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})=&\ \ln (T) \\
z= & \ {{\Phi }^{-1}}\left[ F({T}') \right]
\end{align}\,\!</math>

and:

::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\,\!</math>

The next step is to calculate the variance of <math>{T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}}):\,\!</math>

::<math>\begin{align}
Var({T}')= & {{\left( \frac{\partial {T}'}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial {T}'}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial C} \right)Cov\left( \widehat{B},\widehat{C} \right) \\
& +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) +2\left( \frac{\partial {T}'}{\partial C} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right)
\end{align}\,\!</math>

or:

::<math>\begin{align}
& Var({T}')= \frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) +\frac{2}{B\cdot C}Cov\left( \widehat{B},\widehat{C} \right) +\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) +\frac{2\widehat{z}}{C}Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right)
\end{align}\,\!</math>

The upper and lower bounds are then found by:

::<math>\begin{align}
& T_{U}^{\prime }= & \ln {{T}_{U}}={T}'+{{K}_{\alpha }}\sqrt{Var({T}')} \\
& T_{L}^{\prime }= & \ln {{T}_{L}}={T}'-{{K}_{\alpha }}\sqrt{Var({T}')}
\end{align}\,\!</math>

Solving for <math>{{T}_{U}}\,\!</math> and <math>{{T}_{L}}\,\!</math> yields:

::<math>\begin{align}
& {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (Upper bound)} \\
& {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)}
\end{align}\,\!</math>

XFMEA Examples

2018-06-07T17:30:32Z

Richard House:

[[Image:Xfmea_Examples_Banner.png|left|400px|link=Xfmea_Examples|alt=Xfmea Examples|Caption]] __NOTOC__

==Quick Start Guide==
The ''Xfmea & RCM++ Quick Start Guide'' has been designed to help you explore many of the software's key features by working through step-by-step instructions for some practical application examples. The guide is available as a free *.PDF download.
* [http://www.synthesisplatform.net/Xfmea/en/QS_Xfmea10.pdf Download the print-ready *.pdf file]

==Examples==
===Design for Reliability===
*Creating a Customized DFR Planner -- [http://www.synthesis8.com/Xfmea/en/QS_Xfmea10.pdf See Chapter 3]

===Design FMEA===
*DFMEA for a Single Light Pendant Chandelier -- [http://www.synthesis8.com/Xfmea/en/QS_Xfmea10.pdf See Chapter 5]

===Risk Discovery Analysis===
*[http://www.reliawiki.org/index.php/Xfmea_Risk_Discovery_Analysis_Example Risk Discovery Analysis for a Multi-Function Printer]
* Preliminary Risk Assessment for a Single Light Pendant Chandelier -- [http://www.synthesis8.com/Xfmea/en/QS_Xfmea10.pdf See Chapter 4]

===Failure Modes and Reliability Analysis===
*Estimating System Reliability for a Single Light Pendant Chandelier -- [http://www.synthesis8.com/Xfmea/en/QS_Xfmea10.pdf See Chapter 6]

ReliaSoft API Reference

2018-01-31T20:45:19Z

Richard House:

{{Template:API}}

<div style="border:1px solid #D0CDE8; background:#EEEDF7; border-radius:10px; width:auto; margin:0 0 1em 1em; padding:1em; float:right;">
*'''[[API Changelog|Changelog]]'''
*'''[[API Reference Documentation|API Documentation]]
*'''[[API Tutorials|Tutorials]]
</div>

The Synthesis Platform’s ''Application Programming Interface'' (API) enables you to programmatically read and write data to Synthesis repositories, as well as analyze and plot data sets from external sources. It is provided for Visual Basic for Applications (VBA) and .NET applications, allowing you to create automated solutions using popular Microsoft Office applications, such as Excel and PowerPoint, or create custom applications for your organization.

Implementing the Synthesis API allows you to connect data in Synthesis repositories with the systems and tools used within your organization. It provides the functionality to:

*'''Transfer and synchronize data via XML'''
:Use XML files to manage structured data (such as system hierarchies/BOMs, incidents and work orders) between ReliaSoft’s [http://xfracas.reliasoft.com/ XFRACAS] and a variety of different PLMs, ERPs, CMMS and other external systems. You can also use XML files to create reliability block diagrams (RBDs) in [http://www.reliasoft.com/products/reliability-analysis/blocksim BlockSim].
*'''Read/write specific data elements'''
:Read or write data to Synthesis resources (models, actions, URDs, etc.), system hierarchy items and FMEA records (causes, effects, failures, etc.). Initiate batch data processing and create custom solutions for importing/exporting data from Synthesis repositories, as well as load data into the Synthesis Data Warehouse (SDW).
*'''Obtain calculated results'''
:Read data from published Synthesis models and obtain calculated results such as reliability, probability of failure, BX% life, failure rate and mean time.
*'''Analyze data'''
:Use [http://www.reliasoft.com/products/reliability-analysis/weibull Weibull++'s] or [http://www.reliasoft.com/products/reliability-analysis/alta ALTA’s] analysis and plotting engines to analyze data from third-party sources, and incorporate the functionality into your own custom applications. (Requires a Synthesis API license.)

==Prerequisites==
*'''Getting the latest version'''
:The Synthesis API consists of a library file that contains the classes, methods and events that provide access to Synthesis applications. The file is located in the folder where the Synthesis application is installed. By default, this folder is at "C:\Program Files\ReliaSoft\Synthesis 11." For VBA, you will use the '''SynthesisAPI.tlb''' file; for .NET applications, you will use the '''SynthesisAPI.dll''' file.

:To obtain the latest version of the library file and the Synthesis application, choose '''File > Help > Check for Updates''' or download the latest version at http://www.ReliaSoft.com/synthesis/downloads.htm.

*'''Build environment'''
:For Visual Studio projects using Version 10 or 11 of the Synthesis Platform, at minimum you will need Visual Studio 2010 and the .NET Framework 4.0 runtime. Version 2018 and later require Visual Studio 2015 and the .NET Framework 4.6.1 developer pack and runtime.

*'''32- vs. 64-bit'''
:For VBA, the bit version of the Synthesis Platform that you've installed must match your installed version of Microsoft Office. If they do not match, you may see an error related to ActiveX controls. (See [http://www.reliasoft.com/support/rs20024.htm ReliaSoft KB article RS20024].)

:For Visual Studio projects targeting the .NET Framework 4.5 (or later) and compiled to run an any platform (Any CPU), turn off the '''Prefer 32-bit''' property of the build configuration if you want instances of your application to run as a 64-bit process; otherwise, the application will run as a 32-bit process on both 32- and 64-bit versions of Windows. Note that when using [http://www.synthesisplatform.net/licensing_locally_hosted.htm locally hosted licensing], it is necessary to ensure your application bitness matches the bitness of the Synthesis Platform installed wherever your application is run due to dependencies on platform-specific libraries.

*'''Running API applications on a server
:If you plan to run the API on a Windows server, you must use [http://www.synthesisplatform.net/licensing_locally_hosted.htm locally hosted licensing] and you’ll need to install the ReliaSoft API Ante Service to scale the implementation (the service is not required when running the API on a Windows desktop environment). For instructions on how to install and configure the service, see http://www.synthesisplatform.net/reliasoft_api_ante_service.htm.

*'''Access to calculations functionality
:Access to Weibull++'s and ALTA’s analysis and plotting engines require a Synthesis API license. To obtain a license, please [http://www.reliasoft.com/contact.htm contact ReliaSoft].

==Getting Started==

====Referencing the Synthesis API Library File====

<onlyinclude>To use the Synthesis API in your code, you must reference the Synthesis API library in your VBA or .NET projects. The file is located in the folder where the Synthesis application is installed. By default, this folder is at "C:\Program Files\ReliaSoft\Synthesis 11." For VBA, you will use the '''SynthesisAPI.tlb''' file; for .NET applications, you will use the '''SynthesisAPI.dll''' file.

'''To reference the library in VBA''':
:#Enter the VBA environment by clicking the '''Visual Basic''' command on the Developer tab (If the Developer tab is not visible, see [https://msdn.microsoft.com/en-us/library/bb608625.aspx this Microsoft article]).
:#In the VBA environment, choose '''Tools > References'''.
:#Click '''Browse''' and find the SynthesisAPI.tlb file on your computer. Click '''OK'''.

'''To reference the library in Visual Studio:'''
:#In the Solution Explorer, right-click the '''References''' node and choose '''Add References'''.
:#In the Reference Manager window, click the '''Browse''' tab and find the SynthesisAPI.dll file on your computer. Click '''OK'''.
:#Copy the following files from the Synthesis installation folder to your target folder (e.g., bin/Debug or bin/Release Folder).
:::*licncr.dll
:::*licncr.dll.nsign
:::*licshr.dll
:::*licshr.dll.nsign
:::*msvcp110.dll
:::*msvcr110.dll
:::*RSDB.dll
:::*RSLM8nCodeDotNetWrapper.dll
:::*utlibr.dll
</onlyinclude>
====Examples====
Sample applications that you can test and tweak yourself are included with the Synthesis Platform installation. The files are stored on your computer at C:\Users\Public\Public Documents\ReliaSoft\Examples11\API. (For Version 10, see C:\Users\Public\Public Documents\ReliaSoft\Examples10\API.)

The sample applications are available for Excel VBA and VB.NET projects. (To run the Excel examples, macros must be enabled. See [http://office.microsoft.com/en-us/excel-help/enable-or-disable-macros-in-office-documents-HA010031071.aspx this Microsoft article].)

====Tutorials====
[[API Tutorials|Tutorials]] are available to help you learn the core objects of the Synthesis API and start coding right away. The tutorials include code samples for both VBA and VB.NET.

==Object Library ==
Below is a summary of the core objects in the Synthesis API library. For an alphabetical listing of all objects in the latest version, see [[API Reference Documentation]]. (Click [[API Reference Documentation 10| here]] for the list of objects in Version 10.)

*The <code>'''[[Repository Class|Repository]]'''</code> object represents the Synthesis repository. All other objects use the functions in this class to read or write data to the repository. This class also contains the functions for importing/exporting XML files.

*The following objects represent the Synthesis resources. Instances of each class can be used to store data for an existing or new resource.
**<code>'''[[CAction Class|cAction]]'''</code>
**<code>'''[[CCorrectiveTask Class|cCorrectiveTask]]'''</code>
**<code>'''[[CScheduledTask Class|cScheduledTask]]'''</code>
**<code>'''[[CCrew Class|cCrew]]'''</code>
**<code>'''[[CModel Class|cModel]]'''</code>
**<code>'''[[CPool Class|cPool]]'''</code>
**<code>'''[[CProfile Class|cProfile]]'''</code>
**<code>'''[[CURD Class|cURD]]'''</code>
**<code>'''[[CVariable Class|cVariable]]'''</code>
**<code>'''[[CXfmeaControl Class|cXfmeaControl]]'''</code>

*The following objects build data sets for the Synthesis Data Warehouse (SDW).
**<code>'''[[RawData Class|RawData]]'''</code> represents a single data point.
**<code>'''[[RawDataSet Class|RawDataSet]]'''</code> serves as the data container for <code>RawData</code> objects.

*The <code>'''[[XfmeaItem Class|XfmeaItem]]'''</code> object represents a system hierarchy item. Instances of this class can be used to store data for a system or assembly, as well as build a multi-level system configuration.

*The following objects represent the FMEA records. Instances of each class can be used to store data for an existing or new FMEA analysis.
**<code>'''[[XfmeaFunction Class|XfmeaFunction]]'''</code>
**<code>'''[[XfmeaCause Class|XfmeaCause]]'''</code>
**<code>'''[[XfmeaEffect Class|XfmeaEffect]]'''</code>
**<code>'''[[XfmeaFailure Class|XfmeaFailure]]'''</code>

*The <code>'''[[WeibullDataSet Class|WeibullDataSet]]'''</code> object represents a Weibull++ data sheet for life data analysis. Instances of this class are used to define a data set and fit a statistical distribution to the data.

*The <code>'''[[ALTADataSet Class|ALTADataSet]]'''</code> object represents an ALTA data sheet for accelerated life testing data analysis. Instances of this class are used to define a data set, and fit a distribution and life-stress relationship to the data.

*The <code>'''[[WAPlots Class|WAPlots]]'''</code> object represents a plot based on a <code>cModel</code>, <code>WeibullDataSet</code> or <code>ALTADataSet</code> object. Instances of this class are used to display a plot of the analysis results.

ReliaSoft API Reference

2018-01-31T20:44:37Z

Richard House:

{{Template:API}}

<div style="border:1px solid #D0CDE8; background:#EEEDF7; border-radius:10px; width:auto; margin:0 0 1em 1em; padding:1em; float:right;">
*'''[[API Changelog|Changelog]]'''
*'''[[API Reference Documentation|API Documentation]]
*'''[[API Tutorials|Tutorials]]
</div>

The Synthesis Platform’s ''Application Programming Interface'' (API) enables you to programmatically read and write data to Synthesis repositories, as well as analyze and plot data sets from external sources. It is provided for Visual Basic for Applications (VBA) and .NET applications, allowing you to create automated solutions using popular Microsoft Office applications, such as Excel and PowerPoint, or create custom applications for your organization.

Implementing the Synthesis API allows you to connect data in Synthesis repositories with the systems and tools used within your organization. It provides the functionality to:

*'''Transfer and synchronize data via XML'''
:Use XML files to manage structured data (such as system hierarchies/BOMs, incidents and work orders) between ReliaSoft’s [http://xfracas.reliasoft.com/ XFRACAS] and a variety of different PLMs, ERPs, CMMS and other external systems. You can also use XML files to create reliability block diagrams (RBDs) in [http://www.reliasoft.com/products/reliability-analysis/blocksim BlockSim].
*'''Read/write specific data elements'''
:Read or write data to Synthesis resources (models, actions, URDs, etc.), system hierarchy items and FMEA records (causes, effects, failures, etc.). Initiate batch data processing and create custom solutions for importing/exporting data from Synthesis repositories, as well as load data into the Synthesis Data Warehouse (SDW).
*'''Obtain calculated results'''
:Read data from published Synthesis models and obtain calculated results such as reliability, probability of failure, BX% life, failure rate and mean time.
*'''Analyze data'''
:Use [http://weibull.reliasoft.com/ Weibull++'s] or [http://alta.reliasoft.com/ ALTA’s] analysis and plotting engines to analyze data from third-party sources, and incorporate the functionality into your own custom applications. (Requires a Synthesis API license.)

==Prerequisites==
*'''Getting the latest version'''
:The Synthesis API consists of a library file that contains the classes, methods and events that provide access to Synthesis applications. The file is located in the folder where the Synthesis application is installed. By default, this folder is at "C:\Program Files\ReliaSoft\Synthesis 11." For VBA, you will use the '''SynthesisAPI.tlb''' file; for .NET applications, you will use the '''SynthesisAPI.dll''' file.

:To obtain the latest version of the library file and the Synthesis application, choose '''File > Help > Check for Updates''' or download the latest version at http://www.ReliaSoft.com/synthesis/downloads.htm.

*'''Build environment'''
:For Visual Studio projects using Version 10 or 11 of the Synthesis Platform, at minimum you will need Visual Studio 2010 and the .NET Framework 4.0 runtime. Version 2018 and later require Visual Studio 2015 and the .NET Framework 4.6.1 developer pack and runtime.

*'''32- vs. 64-bit'''
:For VBA, the bit version of the Synthesis Platform that you've installed must match your installed version of Microsoft Office. If they do not match, you may see an error related to ActiveX controls. (See [http://www.reliasoft.com/support/rs20024.htm ReliaSoft KB article RS20024].)

:For Visual Studio projects targeting the .NET Framework 4.5 (or later) and compiled to run an any platform (Any CPU), turn off the '''Prefer 32-bit''' property of the build configuration if you want instances of your application to run as a 64-bit process; otherwise, the application will run as a 32-bit process on both 32- and 64-bit versions of Windows. Note that when using [http://www.synthesisplatform.net/licensing_locally_hosted.htm locally hosted licensing], it is necessary to ensure your application bitness matches the bitness of the Synthesis Platform installed wherever your application is run due to dependencies on platform-specific libraries.

*'''Running API applications on a server
:If you plan to run the API on a Windows server, you must use [http://www.synthesisplatform.net/licensing_locally_hosted.htm locally hosted licensing] and you’ll need to install the ReliaSoft API Ante Service to scale the implementation (the service is not required when running the API on a Windows desktop environment). For instructions on how to install and configure the service, see http://www.synthesisplatform.net/reliasoft_api_ante_service.htm.

*'''Access to calculations functionality
:Access to Weibull++'s and ALTA’s analysis and plotting engines require a Synthesis API license. To obtain a license, please [http://www.reliasoft.com/contact.htm contact ReliaSoft].

==Getting Started==

====Referencing the Synthesis API Library File====

<onlyinclude>To use the Synthesis API in your code, you must reference the Synthesis API library in your VBA or .NET projects. The file is located in the folder where the Synthesis application is installed. By default, this folder is at "C:\Program Files\ReliaSoft\Synthesis 11." For VBA, you will use the '''SynthesisAPI.tlb''' file; for .NET applications, you will use the '''SynthesisAPI.dll''' file.

'''To reference the library in VBA''':
:#Enter the VBA environment by clicking the '''Visual Basic''' command on the Developer tab (If the Developer tab is not visible, see [https://msdn.microsoft.com/en-us/library/bb608625.aspx this Microsoft article]).
:#In the VBA environment, choose '''Tools > References'''.
:#Click '''Browse''' and find the SynthesisAPI.tlb file on your computer. Click '''OK'''.

'''To reference the library in Visual Studio:'''
:#In the Solution Explorer, right-click the '''References''' node and choose '''Add References'''.
:#In the Reference Manager window, click the '''Browse''' tab and find the SynthesisAPI.dll file on your computer. Click '''OK'''.
:#Copy the following files from the Synthesis installation folder to your target folder (e.g., bin/Debug or bin/Release Folder).
:::*licncr.dll
:::*licncr.dll.nsign
:::*licshr.dll
:::*licshr.dll.nsign
:::*msvcp110.dll
:::*msvcr110.dll
:::*RSDB.dll
:::*RSLM8nCodeDotNetWrapper.dll
:::*utlibr.dll
</onlyinclude>
====Examples====
Sample applications that you can test and tweak yourself are included with the Synthesis Platform installation. The files are stored on your computer at C:\Users\Public\Public Documents\ReliaSoft\Examples11\API. (For Version 10, see C:\Users\Public\Public Documents\ReliaSoft\Examples10\API.)

The sample applications are available for Excel VBA and VB.NET projects. (To run the Excel examples, macros must be enabled. See [http://office.microsoft.com/en-us/excel-help/enable-or-disable-macros-in-office-documents-HA010031071.aspx this Microsoft article].)

====Tutorials====
[[API Tutorials|Tutorials]] are available to help you learn the core objects of the Synthesis API and start coding right away. The tutorials include code samples for both VBA and VB.NET.

==Object Library ==
Below is a summary of the core objects in the Synthesis API library. For an alphabetical listing of all objects in the latest version, see [[API Reference Documentation]]. (Click [[API Reference Documentation 10| here]] for the list of objects in Version 10.)

*The <code>'''[[Repository Class|Repository]]'''</code> object represents the Synthesis repository. All other objects use the functions in this class to read or write data to the repository. This class also contains the functions for importing/exporting XML files.

*The following objects represent the Synthesis resources. Instances of each class can be used to store data for an existing or new resource.
**<code>'''[[CAction Class|cAction]]'''</code>
**<code>'''[[CCorrectiveTask Class|cCorrectiveTask]]'''</code>
**<code>'''[[CScheduledTask Class|cScheduledTask]]'''</code>
**<code>'''[[CCrew Class|cCrew]]'''</code>
**<code>'''[[CModel Class|cModel]]'''</code>
**<code>'''[[CPool Class|cPool]]'''</code>
**<code>'''[[CProfile Class|cProfile]]'''</code>
**<code>'''[[CURD Class|cURD]]'''</code>
**<code>'''[[CVariable Class|cVariable]]'''</code>
**<code>'''[[CXfmeaControl Class|cXfmeaControl]]'''</code>

*The following objects build data sets for the Synthesis Data Warehouse (SDW).
**<code>'''[[RawData Class|RawData]]'''</code> represents a single data point.
**<code>'''[[RawDataSet Class|RawDataSet]]'''</code> serves as the data container for <code>RawData</code> objects.

*The <code>'''[[XfmeaItem Class|XfmeaItem]]'''</code> object represents a system hierarchy item. Instances of this class can be used to store data for a system or assembly, as well as build a multi-level system configuration.

*The following objects represent the FMEA records. Instances of each class can be used to store data for an existing or new FMEA analysis.
**<code>'''[[XfmeaFunction Class|XfmeaFunction]]'''</code>
**<code>'''[[XfmeaCause Class|XfmeaCause]]'''</code>
**<code>'''[[XfmeaEffect Class|XfmeaEffect]]'''</code>
**<code>'''[[XfmeaFailure Class|XfmeaFailure]]'''</code>

*The <code>'''[[WeibullDataSet Class|WeibullDataSet]]'''</code> object represents a Weibull++ data sheet for life data analysis. Instances of this class are used to define a data set and fit a statistical distribution to the data.

*The <code>'''[[ALTADataSet Class|ALTADataSet]]'''</code> object represents an ALTA data sheet for accelerated life testing data analysis. Instances of this class are used to define a data set, and fit a distribution and life-stress relationship to the data.

*The <code>'''[[WAPlots Class|WAPlots]]'''</code> object represents a plot based on a <code>cModel</code>, <code>WeibullDataSet</code> or <code>ALTADataSet</code> object. Instances of this class are used to display a plot of the analysis results.

ReliaSoft API Reference

2018-01-31T20:42:44Z

Richard House:

{{Template:API}}

<div style="border:1px solid #D0CDE8; background:#EEEDF7; border-radius:10px; width:auto; margin:0 0 1em 1em; padding:1em; float:right;">
*'''[[API Changelog|Changelog]]'''
*'''[[API Reference Documentation|API Documentation]]
*'''[[API Tutorials|Tutorials]]
</div>

The Synthesis Platform’s ''Application Programming Interface'' (API) enables you to programmatically read and write data to Synthesis repositories, as well as analyze and plot data sets from external sources. It is provided for Visual Basic for Applications (VBA) and .NET applications, allowing you to create automated solutions using popular Microsoft Office applications, such as Excel and PowerPoint, or create custom applications for your organization.

Implementing the Synthesis API allows you to connect data in Synthesis repositories with the systems and tools used within your organization. It provides the functionality to:

*'''Transfer and synchronize data via XML'''
:Use XML files to manage structured data (such as system hierarchies/BOMs, incidents and work orders) between ReliaSoft’s [http://xfracas.reliasoft.com/ XFRACAS] and a variety of different PLMs, ERPs, CMMS and other external systems. You can also use XML files to create reliability block diagrams (RBDs) in [http://blocksim.reliasoft.com/ BlockSim].
*'''Read/write specific data elements'''
:Read or write data to Synthesis resources (models, actions, URDs, etc.), system hierarchy items and FMEA records (causes, effects, failures, etc.). Initiate batch data processing and create custom solutions for importing/exporting data from Synthesis repositories, as well as load data into the Synthesis Data Warehouse (SDW).
*'''Obtain calculated results'''
:Read data from published Synthesis models and obtain calculated results such as reliability, probability of failure, BX% life, failure rate and mean time.
*'''Analyze data'''
:Use [http://weibull.reliasoft.com/ Weibull++'s] or [http://alta.reliasoft.com/ ALTA’s] analysis and plotting engines to analyze data from third-party sources, and incorporate the functionality into your own custom applications. (Requires a Synthesis API license.)

==Prerequisites==
*'''Getting the latest version'''
:The Synthesis API consists of a library file that contains the classes, methods and events that provide access to Synthesis applications. The file is located in the folder where the Synthesis application is installed. By default, this folder is at "C:\Program Files\ReliaSoft\Synthesis 11." For VBA, you will use the '''SynthesisAPI.tlb''' file; for .NET applications, you will use the '''SynthesisAPI.dll''' file.

:To obtain the latest version of the library file and the Synthesis application, choose '''File > Help > Check for Updates''' or download the latest version at http://www.ReliaSoft.com/synthesis/downloads.htm.

*'''Build environment'''
:For Visual Studio projects using Version 10 or 11 of the Synthesis Platform, at minimum you will need Visual Studio 2010 and the .NET Framework 4.0 runtime. Version 2018 and later require Visual Studio 2015 and the .NET Framework 4.6.1 developer pack and runtime.

*'''32- vs. 64-bit'''
:For VBA, the bit version of the Synthesis Platform that you've installed must match your installed version of Microsoft Office. If they do not match, you may see an error related to ActiveX controls. (See [http://www.reliasoft.com/support/rs20024.htm ReliaSoft KB article RS20024].)

:For Visual Studio projects targeting the .NET Framework 4.5 (or later) and compiled to run an any platform (Any CPU), turn off the '''Prefer 32-bit''' property of the build configuration if you want instances of your application to run as a 64-bit process; otherwise, the application will run as a 32-bit process on both 32- and 64-bit versions of Windows. Note that when using [http://www.synthesisplatform.net/licensing_locally_hosted.htm locally hosted licensing], it is necessary to ensure your application bitness matches the bitness of the Synthesis Platform installed wherever your application is run due to dependencies on platform-specific libraries.

*'''Running API applications on a server
:If you plan to run the API on a Windows server, you must use [http://www.synthesisplatform.net/licensing_locally_hosted.htm locally hosted licensing] and you’ll need to install the ReliaSoft API Ante Service to scale the implementation (the service is not required when running the API on a Windows desktop environment). For instructions on how to install and configure the service, see http://www.synthesisplatform.net/reliasoft_api_ante_service.htm.

*'''Access to calculations functionality
:Access to Weibull++'s and ALTA’s analysis and plotting engines require a Synthesis API license. To obtain a license, please [http://www.reliasoft.com/contact.htm contact ReliaSoft].

==Getting Started==

====Referencing the Synthesis API Library File====

<onlyinclude>To use the Synthesis API in your code, you must reference the Synthesis API library in your VBA or .NET projects. The file is located in the folder where the Synthesis application is installed. By default, this folder is at "C:\Program Files\ReliaSoft\Synthesis 11." For VBA, you will use the '''SynthesisAPI.tlb''' file; for .NET applications, you will use the '''SynthesisAPI.dll''' file.

'''To reference the library in VBA''':
:#Enter the VBA environment by clicking the '''Visual Basic''' command on the Developer tab (If the Developer tab is not visible, see [https://msdn.microsoft.com/en-us/library/bb608625.aspx this Microsoft article]).
:#In the VBA environment, choose '''Tools > References'''.
:#Click '''Browse''' and find the SynthesisAPI.tlb file on your computer. Click '''OK'''.

'''To reference the library in Visual Studio:'''
:#In the Solution Explorer, right-click the '''References''' node and choose '''Add References'''.
:#In the Reference Manager window, click the '''Browse''' tab and find the SynthesisAPI.dll file on your computer. Click '''OK'''.
:#Copy the following files from the Synthesis installation folder to your target folder (e.g., bin/Debug or bin/Release Folder).
:::*licncr.dll
:::*licncr.dll.nsign
:::*licshr.dll
:::*licshr.dll.nsign
:::*msvcp110.dll
:::*msvcr110.dll
:::*RSDB.dll
:::*RSLM8nCodeDotNetWrapper.dll
:::*utlibr.dll
</onlyinclude>
====Examples====
Sample applications that you can test and tweak yourself are included with the Synthesis Platform installation. The files are stored on your computer at C:\Users\Public\Public Documents\ReliaSoft\Examples11\API. (For Version 10, see C:\Users\Public\Public Documents\ReliaSoft\Examples10\API.)

The sample applications are available for Excel VBA and VB.NET projects. (To run the Excel examples, macros must be enabled. See [http://office.microsoft.com/en-us/excel-help/enable-or-disable-macros-in-office-documents-HA010031071.aspx this Microsoft article].)

====Tutorials====
[[API Tutorials|Tutorials]] are available to help you learn the core objects of the Synthesis API and start coding right away. The tutorials include code samples for both VBA and VB.NET.

==Object Library ==
Below is a summary of the core objects in the Synthesis API library. For an alphabetical listing of all objects in the latest version, see [[API Reference Documentation]]. (Click [[API Reference Documentation 10| here]] for the list of objects in Version 10.)

*The <code>'''[[Repository Class|Repository]]'''</code> object represents the Synthesis repository. All other objects use the functions in this class to read or write data to the repository. This class also contains the functions for importing/exporting XML files.

*The following objects represent the Synthesis resources. Instances of each class can be used to store data for an existing or new resource.
**<code>'''[[CAction Class|cAction]]'''</code>
**<code>'''[[CCorrectiveTask Class|cCorrectiveTask]]'''</code>
**<code>'''[[CScheduledTask Class|cScheduledTask]]'''</code>
**<code>'''[[CCrew Class|cCrew]]'''</code>
**<code>'''[[CModel Class|cModel]]'''</code>
**<code>'''[[CPool Class|cPool]]'''</code>
**<code>'''[[CProfile Class|cProfile]]'''</code>
**<code>'''[[CURD Class|cURD]]'''</code>
**<code>'''[[CVariable Class|cVariable]]'''</code>
**<code>'''[[CXfmeaControl Class|cXfmeaControl]]'''</code>

*The following objects build data sets for the Synthesis Data Warehouse (SDW).
**<code>'''[[RawData Class|RawData]]'''</code> represents a single data point.
**<code>'''[[RawDataSet Class|RawDataSet]]'''</code> serves as the data container for <code>RawData</code> objects.

*The <code>'''[[XfmeaItem Class|XfmeaItem]]'''</code> object represents a system hierarchy item. Instances of this class can be used to store data for a system or assembly, as well as build a multi-level system configuration.

*The following objects represent the FMEA records. Instances of each class can be used to store data for an existing or new FMEA analysis.
**<code>'''[[XfmeaFunction Class|XfmeaFunction]]'''</code>
**<code>'''[[XfmeaCause Class|XfmeaCause]]'''</code>
**<code>'''[[XfmeaEffect Class|XfmeaEffect]]'''</code>
**<code>'''[[XfmeaFailure Class|XfmeaFailure]]'''</code>

*The <code>'''[[WeibullDataSet Class|WeibullDataSet]]'''</code> object represents a Weibull++ data sheet for life data analysis. Instances of this class are used to define a data set and fit a statistical distribution to the data.

*The <code>'''[[ALTADataSet Class|ALTADataSet]]'''</code> object represents an ALTA data sheet for accelerated life testing data analysis. Instances of this class are used to define a data set, and fit a distribution and life-stress relationship to the data.

*The <code>'''[[WAPlots Class|WAPlots]]'''</code> object represents a plot based on a <code>cModel</code>, <code>WeibullDataSet</code> or <code>ALTADataSet</code> object. Instances of this class are used to display a plot of the analysis results.

Monte Carlo Simulation Example

2017-08-11T15:59:05Z

Richard House:

<noinclude>{{Banner Weibull Examples}}
''This example appears in the [[Risk_Analysis_and_Probabilistic_Design_with_Monte_Carlo_Simulation|Life Data Analysis Reference book]]''.

</noinclude>Monte Carlo simulation can be used to perform simple relationship-based simulations. This type of simulation has many applications in probabilistic design, risk analysis, quality control, etc. The Monte Carlo utility includes a ''User Defined'' distribution feature that allows you to specify an equation relating different random variables. The following example uses the Life Comparison tool to compare the ''pdf'' of two user-defined distributions. A variation of the example that demonstrates how to obtain the joint ''pdf'' of random variables is available in the [http://help.synthesis8.com/weibull_alta11/user-defined_equation_example.htm Weibull++/ALTA Help file]. A demonstration on how to perform the example using ReliaSoft's advanced stochastic event simulation software, [http://reno.reliasoft.com '''RENO'''], is also available ([http://www.reliasoft.com/reno/examples/renoexr3/index.htm view it in html]).

'''Monte Carlo Simulation: A Hinge Length Example'''

A hinge is made up of four components A, B, C, D, as shown next. Seven units of each component were taken from the assembly line and measurements (in cm) were recorded.

[[Image:WB.23 lda26.1.png|center|250px| ]]

The following table shows the measurements. Determine the probability that D will fall out of specifications.

::<center><math>\begin{matrix}
\text{Dimensions for A} & \text{Dimensions for B} & \text{Dimensions for C} & \text{Dimensions for D} \\
\text{2}\text{.0187} & \text{1}\text{.9795} & \text{30}\text{.4216} & \text{33}\text{.6573} \\
\text{1}\text{.9996} & \text{2}\text{.0288} & \text{29}\text{.9818} & \text{34}\text{.5432} \\
\text{2}\text{.0167} & \text{1}\text{.9883} & \text{29}\text{.9724} & \text{34}\text{.6218} \\
\text{2}\text{.0329} & \text{2}\text{.0327} & \text{30}\text{.192} & \text{34}\text{.7538} \\
\text{2}\text{.0233} & \text{2}\text{.0119} & \text{29}\text{.9421} & \text{35}\text{.1508} \\
\text{2}\text{.0273} & \text{2}\text{.0354} & \text{30}\text{.1343} & \text{35}\text{.2666} \\
\text{1}\text{.984} & \text{1}\text{.9908} & \text{30}\text{.0423} & \text{35}\text{.7111} \\
\end{matrix}\,\!</math></center>

'''Solution'''

In a Weibull++ standard folio, enter the parts dimensions measurements of each component into separate data sheets. Analyze each data sheet using the normal distribution and the RRX analysis method. The parameters are:

::<center><math>\begin{matrix}
\text{A} & \text{B} & \text{C} & \text{D} \\
\hat{\mu }=2.0146 & \hat{\mu }=2.0096 & \hat{\mu }=30.0981 & \hat{\mu }=34.8149 \\
\hat{\sigma }=0.0181 & \hat{\sigma }=0.0249 & \hat{\sigma }=0.1762 & \hat{\sigma }=0.7121 \\
\end{matrix}\,\!</math></center>

Next, perform a Monte Carlo simulation to estimate the probability that (A+B+C) will be greater than D. To do this, choose the '''User Defined''' distribution and enter its equation as follows. (Click the '''Insert Data Source''' button to insert the data sheets that contain the measurements for the components.)

[[Image:Rsik Analysis Example Monte Carlo Setting.png|center|550px| ]]

On the Settings tab, set the number of data points to '''100''', as shown next.

[[Image:Rsik Analysis Example Monte Carlo Number of Points.png|center|550px| ]]

Click '''Generate''' to create a data sheet that contains the generated data points. Rename the new data sheet to "Simulated A+B+C."

Follow the same procedure to generate 100 data points to represent the D measurements. Rename the new data sheet to "Simulated D."

[[Image:Rsik Analysis Example Monte Carlo D.png|center|550px| ]]

Analyze the two data sets, "Simulated A+B+C" and "Simulated D," using the normal distribution and the RRX analysis method.

Next, open the Life Comparison tool and choose to compare the two data sheets. The following picture shows the ''pdf'' curves of the two data sets.

[[Image:Rsik Analysis Example Selected Life Comparison Plot.png|center|550px| ]]

The following report shows that the probability that "Simulated A+B+C" will be greater than "Simulated D" is 16.033%. (Note that the results may vary because of the randomness in the simulation.)

[[Image:Rsik Analysis Example Selected Life Comparison Result.png|center|550px| ]]

Response Surface Methods for Optimization

2017-08-10T19:46:23Z

Richard House: /* Desirability Functions */

{{Template:Doebook|10}}
The experiment designs mentioned in [[Two_Level_Factorial_Experiments| Two Level Factorial Experiments]] and [[Highly_Fractional_Factorial_Designs| Highly Fractional Factorial Designs]] help the experimenter identify factors that affect the response. Once the important factors have been identified, the next step is to determine the settings for these factors that result in the optimum value of the response. The optimum value of the response may either be a maximum value or a minimum value, depending upon the product or process in question. For example, if the response in an experiment is the yield from a chemical process, then the objective might be to find the settings of the factors affecting the yield so that the yield is maximized. On the other hand, if the response in an experiment is the number of defects, then the goal would be to find the factor settings that minimize the number of defects. Methodologies that help the experimenter reach the goal of optimum response are referred to as ''response surface methods''. These methods are exclusively used to examine the "surface," or the relationship between the response and the factors affecting the response. Regression models are used for the analysis of the response, as the focus now is on the nature of the relationship between the response and the factors, rather than identification of the important factors.

Response surface methods usually involve the following steps:

#The experimenter needs to move from the present operating conditions to the vicinity of the operating conditions where the response is optimum. This is done using the ''method of steepest ascent'' in the case of maximizing the response. The same method can be used to minimize the response and is then referred to as the ''method of steepest descent''.
#Once in the vicinity of the optimum response the experimenter needs to fit a more elaborate model between the response and the factors. Special experiment designs, referred to as [[Response_Surface_Methods_for_Optimization#RSM_Designs|''RSM designs'']], are used to accomplish this. The fitted model is used to arrive at the best operating conditions that result in either a maximum or minimum response.
#It is possible that a number of responses may have to be optimized at the same time. For example, an experimenter may want to maximize strength, while keeping the number of defects to a minimum. The optimum settings for each of the responses in such cases may lead to conflicting settings for the factors. A balanced setting has to be found that gives the most appropriate values for all the responses. Desirability functions are useful in these cases.

==Method of Steepest Ascent==
The first step in obtaining the optimum response settings, after the important factors have been identified, is to explore the region around the current operating conditions to decide what direction needs to be taken to move towards the optimum region. Usually, a first order regression model (containing just the main effects and no interaction terms) is sufficient at the current operating conditions because the operating conditions are normally far from the optimum response settings. The experimenter needs to move from the current operating conditions to the optimum region in the most efficient way by using the minimum number of experiments. This is done using the method of steepest ascent. In this method, the contour plot of the first order model is used to decide the settings for the next experiment, in order to move towards the optimum conditions. Consider a process where the response has been found to be a function of two factors. To explore the region around the current operating conditions, the experimenter fits the following first order model between the response and the two factors:

::<math>y={{\beta }_{0}}+{{\beta }_{1}}{{x}_{1}}+{{\beta }_{2}}{{x}_{2}}+\epsilon \,\!</math>

The response surface plot for the model, along with the contours, is shown in the figure below. It can be seen in the figure that in order to maximize the response, the most efficient direction in which to move the experiment is along the line perpendicular to the contours. This line, also referred to as the path of steepest ascent, is the line along which the rate of increase of the response is maximum. The steps along this line to move towards the optimum region are proportional to the regression coefficients, <math>{{\beta }_{j}},\,\!</math> of the fitted first order model.

[[Image:doe9_1.png|center|504px|Path of steepest ascent for the model <math>y=\beta_0+\beta_1 x_1+\beta_2 x_2+\epsilon \,\!</math>.]]

Experiments are conducted along each step of the path of steepest ascent until an increase in the response is not seen. Then, a new first order model is fit at the region of the maximum response. If the first order model shows a lack of fit, then this indicates that the experimenter has reached the vicinity of the optimum. RSM designs are then used explore the region thoroughly and obtain the point of the maximum response. If the first order model does not show a lack of fit, then a new path of steepest ascent is determined and the process is repeated.

===Example===
The yield from a chemical process is found to be affected by two factors: reaction temperature and reaction time. The current reaction temperature is 230 <math>F\,\!</math> and the reaction time is 65 minutes. The experimenter wants to determine the settings of the two factors such that maximum yield can be obtained from the process. To explore the region around the current operating conditions, the experimenter decides to use a single replicate of the <math>2^{2}\,\!</math> design. The range of the factors for this design are chosen to be (225, 235) <math>F\,\!</math> for the reaction temperature and (55, 75) minutes for the reaction time. The unreplicated <math>2^{2}\,\!</math> design is also augmented with five runs at the center point to estimate the error sum of squares, <math>S{{S}_{E}}\,\!</math>, and check for model adequacy. The response values obtained for this design are shown next.

[[Image:doe9.2.png|center|256px|The <math>2^{2}\,\!</math> design augmented with five center points to explore the region around current operating conditions for a chemical process.|link=]]

In a DOE folio, this design can be set up using the properties shown next.

[[Image:doe9_3.png|center|809px|Design properties for the <math>2^{2}\,\!</math> design to explore the current operating conditions.|link=]]

The resulting design and the analysis results are shown next.

[[Image:doe9_4.png|center|778px|The <math>2^{2}\,\!</math> experiment design in to explore the current operating conditions.|link=]]

[[Image:doe9_5.png|center|774px|Results for the <math>2^{2}\,\!</math> experiment to explore the current operating conditions.|link=]]

Note that the results shown are in terms of the coded values of the factors (taking -1 as the value of the lower settings for reaction temperature and reaction time and +1 as the value for the higher settings for these two factors). The results show that the factors, <math>A\,\!</math> (temperature) and <math>B\,\!</math> (time), affect the response significantly but their interaction does not affect the response. Therefore the interaction term can be dropped from the model for this experiment. The results also show that Curvature is not a significant factor. This indicates that the first order model is adequate for the experiment at the current operating conditions. Using these two conclusions, the model for the current operating conditions, in terms the coded variables is:

:<math>\hat{y}=36.6375+1.1635x_{1}+0.4875x_{2}\,\!</math>

where <math>\hat{y}\,\!</math> represents the yield and <math>x_{1}\,\!</math> and <math>x_{2}\,\!</math> are the predictor variables for the two factors, <math>A\,\!</math> and <math>B\,\!</math>, respectively. To further confirm the adequacy of the model of the equation given above, the experiment can be analyzed again after dropping the interaction term, <math>AB\,\!</math>. The results are shown next.

[[Image:doe9_6.png|center|774px|Results for the <math>2^{2}\,\!</math> experiment after the interaction term is dropped from the model.|link=]]

The results show that the lack-of-fit for this model (because of the deficiency created in the model by the absence of the interaction term) is not significant, confirming that the model is adequate.

==Path of Steepest Ascent==
The contour plot for the model used in the above example is shown next.

[[Image:doe9_7.png|center|650px|Results for the <math>2^{2}\,\!</math> experiment after the interaction term is dropped from the model.|link=]]

The regression coefficients for the model are <math>\hat{\beta}_{1}= 1.1625\,\!</math> and <math>\hat{\beta}_{2}= 0.4875\,\!</math>. To move towards the optimum, the experimenter needs to move along the path of steepest ascent, which lies perpendicular to the contours. This path is the line through the center point of the current operating conditions (<math>x_{1}=0\,\!</math>, <math>x_{2}=0\,\!</math>) with a slope of <math>\hat{\beta}_{2}/\hat{\beta}_{1}\,\!</math>. Therefore, in terms of the coded variables, the experiment should be moved 1.1625 units in the <math>x_{1}\,\!</math> direction for every 0.4875 units in the <math>x_{2}\,\!</math> direction. To move along this path, the experimenter decides to use a step-size of 10 minutes for the reaction time, <math>x_{2}\,\!</math>. The coded value for this step size can be obtained as follows. Recall from [[Multiple_Linear_Regression_Analysis|Multiple Linear Regression Analysis]] that the relationship between coded and actual values is:

::<math>coded\ value=\frac{actual\ value - mean}{half\ of\ range}</math>

or

::<math>coded\ value=\frac{step\ size\ in\ actual\ value}{half\ of\ range}</math>

Thus, for a step-size of 10 minutes, the equivalent step size in coded value for <math>x_{2}\,\!</math> is:

::<math>=\frac{10}{half\ of\ range\ of\ x_{2}}\,\!</math>
::<math>=\frac{10}{(75-55)/2}\,\!</math>
::<math>=1\,\!</math>

In terms of the coded variables, the path of steepest ascent requires a move of 1.1625 units in the <math>x_{1}\,\!</math> direction for every 0.4875 units in the <math>x_{2}\,\!</math> direction. The step-size for <math>x_{1}\,\!</math>, in terms of the coded value corresponding to any step-size in <math>x_{2}\,\!</math>, is:

::<math>=\frac{\hat{\beta_{1}}}{\hat{\beta_{2}}}\cdot(step\ size\ in\ x_{2})\,\!</math>

Therefore, the step-size for the reaction temperature, <math>x_{1}\,\!</math>, in terms of the coded variables is:

::<math>=\frac{\hat{\beta_{1}}}{\hat{\beta_{2}}}\cdot\Delta x_{2}\,\!</math>
::<math>=\frac{1.1625}{0.4875}\cdot1\,\!</math>
::<math>=\frac{\hat{\beta_{1}}}{\hat{\beta_{2}}}\cdot\Delta x_{2}\,\!</math>

This corresponds to a step of approximately 12 <math>F\,\!</math> for temperature in terms of the actual value as shown next:

::<math>\begin{align}
step\ size\ in\ actual\ value = & (coded\ value)\cdot(half\ of\ the\ range\ of\ x_{1})\\
= & 2.39\cdot10/2\\
= & 11.95
\end{align}\,\!</math>

Using a step of 12 <math>F\,\!</math> and 10 minutes, the experimenter conducts experiments until no further increase is observed in the yield. The yield values at each step are shown in the table given next.

[[Image:doet9.1.png|center|341px|Response values at each step of the path of steepest ascent for the experiment to investigate the yield of a chemical process. Units for factor levels and the response have been omitted.]]

The yield starts decreasing after the reaction temperature of 350 <math>F\,\!</math> and the reaction time of 165 minutes, indicating that this point may lie close to the optimum region. To analyze the vicinity of this point, a <math>2^{2}\,\!</math> design augmented by five center points is selected. The range of exploration is chosen to be 345 to 355 <math>F\,\!</math> for reaction temperature and 155 to 175 minutes for reaction time. The response values recorded are shown next.

[[Image:doe9.8.png|center|348px|The <math>2^{2}\,\!</math> design augmented with five center points to explore the region of maximum response obtained from the path of steepest ascent. Note that the center point of this design is the new origin.]]

The results for this design are shown next.

[[Image:doe9_9.png|center|760px|Results for the <math>2^{2}\,\!</math> experiment to explore the region of maximum response.|link=]]

In the results, Curvature is displayed as a significant factor. This indicates that the first order model is not adequate for this region of the experiment and a higher order model is required. As a result, the methodology of steepest ascent can no longer be used. The presence of curvature indicates that the experiment region may be close to the optimum. Special designs that allow the use of second order models are needed at this point.

==RSM Designs==
A second order model is generally used to approximate the response once it is realized that the experiment is close to the optimum response region where a first order model is no longer adequate. The second order model is usually sufficient for the optimum region, as third order and higher effects are seldom important. The second order regression model takes the following form for <math>k\,\!</math> factors:

::<math>\begin{align}
y= & {{\beta }_{0}}+{{\beta }_{1}}{{x}_{1}}+{{\beta }_{2}}{{x}_{2}}+...+{{\beta }_{k}}{{x}_{k}}+ \\
& {{\beta }_{11}}x_{1}^{2}+{{\beta }_{22}}x_{2}^{2}+...+{{\beta }_{kk}}x_{k}^{2}+{{\beta }_{12}}{{x}_{1}}{{x}_{2}}+...+{{\beta }_{k-1,k}}{{x}_{k-1}}{{x}_{k}}+\epsilon
\end{align}\,\!</math>

The model contains <math>p=(k+1)(k+2)/2\,\!</math> regression parameters that include coefficients for main effects (<math>{{\beta }_{1}},{{\beta }_{2}}...{{\beta }_{k}}\,\!</math>), coefficients for quadratic main effects (<math>{{\beta }_{11}},{{\beta }_{22}}...{{\beta }_{kk}}\,\!</math>) and coefficients for two factor interaction effects (<math>{{\beta }_{12}},{{\beta }_{13}}\,\!</math>... <math>{{\beta }_{k-1,k}}\,\!</math>). A full factorial design with all factors at three levels would provide estimation of all the required regression parameters. However, full factorial three level designs are expensive to use as the number of runs increases rapidly with the number of factors. For example, a three factor full factorial design with each factor at three levels would require <math>{{3}^{3}}=27\,\!</math> runs while a design with four factors would require <math>{{3}^{4}}=81\,\!</math> runs. Additionally, these designs will estimate a number of higher order effects which are usually not of much importance to the experimenter. Therefore, for the purpose of analysis of response surfaces, special designs are used that help the experimenter fit the second order model to the response with the use of a minimum number of runs. Examples of these designs are the central composite and Box-Behnken designs.

===Central Composite Designs===
Central composite designs are two level full factorial (<math>2^{k}\,\!</math>) or fractional factorial (<math>2^{k-f}\,\!</math>) designs augmented by a number of center points and other chosen runs. These designs are such that they allow the estimation of all the regression parameters required to fit a second order model to a given response.

The simplest of the central composite designs can be used to fit a second order model to a response with two factors. The design consists of a <math>2^{2}\,\!</math> full factorial design augmented by a few runs at the center point (such a design is shown in figure (a) given below). A central composite design is obtained when runs at points (<math>-1,0\,\!</math>), (<math>1,0\,\!</math>), (<math>0,-1\,\!</math>) and (<math>0,1\,\!</math>) are added to this design. These points are referred to as ''axial points'' or ''star points'' and represent runs where all but one of the factors are set at their mid-levels. The number of axial points in a central composite design having <math>k\,\!</math> factors is <math>2k\,\!</math>. The distance of the axial points from the center point is denoted by <math>\alpha \,\!</math> and is always specified in terms of coded values. For example, the central composite design in figure (b) given below has <math>\alpha =1\,\!</math>, while for the design of figure (c) <math>\alpha =1.414\,\!</math>.

[[Image:doe9.10.png|center|227px|Central composite designs: (a) shows the <math>2^2\,\!</math> design with center point runs, (b) shows the two factor central composite design with <math>\alpha=1\,\!</math> and (c) shows the two factor central composite design with <math>\alpha=\sqrt{2}\,\!</math>.]]

It can be noted that when <math>\alpha >1\,\!</math>, each factor is run at five levels (<math>-\alpha \,\!</math>, <math>-1\,\!</math>, <math>0\,\!</math>, <math>1\,\!</math> and <math>\alpha \,\!</math>) instead of the three levels of <math>-1\,\!</math>, <math>0\,\!</math> and <math>1\,\!</math>. The reason for running central composite designs with <math>\alpha >1\,\!</math> is to have a rotatable design, which is explained next.

====Rotatability====
A central composite design is said to be ''rotatable'' if the variance of any predicted value of the response, <math>{{\hat{y}}_{p}}\,\!</math>, for any level of the factors depends only on the distance of the point from the center of the design, regardless of the direction. In other words, a rotatable central composite design provides constant variance of the estimated response corresponding to all new observation points that are at the same distance from the center point of the design (in terms of the coded variables).

The variance of the predicted response at any point, <math>{{x}_{p}}\,\!</math>, is given as follows:

::<math>V[{{\hat{y}}_{p}}]={{\sigma }^{2}}x_{p}^{\prime }{{({{X}^{\prime }}X)}^{-1}}{{x}_{p}}\,\!</math>

The contours of <math>V[{{\hat{y}}_{p}}]\,\!</math> for the central composite design in figure (c) above are shown in the figure below. The contours are concentric circles indicating that the central composite design of figure (c) is rotatable. Rotatability is a desirable property because the experimenter does not have any prior information about the location of the optimum. Therefore, a design that provides equal precision of estimation in all directions would be preferred. Such a design will assure the experimenter that no matter what direction is taken to search for the optimum, he/she will be able to estimate the response value with equal precision. A central composite design is rotatable if the value of <math>\alpha \,\!</math> for the design satisfies the following equation:

::<math>\alpha ={{\left[ \frac{{{2}^{k-f}}({{n}_{f}})}{{{n}_{s}}} \right]}^{1/4}}\,\!</math>

where <math>{{n}_{f}}\,\!</math> is the number of replicates of the runs in the original factorial design and <math>{{n}_{s}}\,\!</math> is the number of replicates of the runs at the axial points. For example, a central composite design with two factors, having a single replicate of the original factorial design, and a single replicate of all the axial points, would be rotatable for the following <math>\alpha \,\!</math> value:

::<math>\begin{align}
\alpha = & {{\left[ \frac{{{2}^{k-f}}({{n}_{f}})}{{{n}_{s}}} \right]}^{1/4}} \\
= & {{\left[ \frac{{{2}^{2}}(1)}{1} \right]}^{1/4}} \\
= & \sqrt{2} \\
= & 1.414
\end{align}\,\!</math>

Thus, a central composite design in two factors, having a single replicate of the original <math>2^{2}\,\!</math> design and axial points, and with <math>\alpha =1.414\,\!</math>, is a rotatable design. This design is shown in figure (c) above.

[[Image:doe9.11.png|center|266px|The countours of <math>V[\hat{y}_p] \,\!</math> for the rotatable two factor central composite design.]]

====Spherical Design====
A central composite design is said to be ''spherical'' if all factorial and axial points are at same distance from the center of the design. Spherical central composite designs are obtained by setting <math>\alpha =\sqrt{k}\,\!</math>. For example, the rotatable design in the figure above (c) is also a spherical design because for this design <math>\alpha =\sqrt{k}=\sqrt{2}=1.414\,\!</math>.

====Face-centered Design====
Central composite designs in which the axial points represent the mid levels for all but one of the factors are also referred to as ''face-centered'' central composite designs. For these designs, <math>\alpha =1\,\!</math> and all factors are run at three levels, which are <math>-1\,\!</math>, <math>0\,\!</math> and <math>1,\,\!</math> in terms of the coded values (see the figure below).

[[Image:doe9.12.png|center|300px|Face-centered central composite design for three factors.]]

===Box-Behnken Designs===
In [[Highly_Fractional_Factorial_Designs| Highly Fractional Factorial Designs]], highly fractional designs introduced by Plackett and Burman were discussed. Plackett-Burman designs are used to estimate main effects in the case of two level fractional factorial experiments using very few runs. [[DOE_References|[G. E. P. Box and D. W. Behnken (1960)]]] introduced similar designs for three level factors that are widely used in response surface methods to fit second-order models to the response. The designs are referred to as Box-Behnken designs. The designs were developed by the combination of two level factorial designs with incomplete block designs. For example, the figure below shows the Box-Behnken design for three factors. The design is obtained by the combination of <math>2^{2}\,\!</math> design with a balanced incomplete block design having three treatments and three blocks (for details see [[DOE_References|[Box 1960, Montgomery 2001]]]).

[[Image:doe9.13.png|center|400px|Box-Behnken design for three factors: (a) shows the geometric representation and (b) shows the design.]]

The advantages of Box-Behnken designs include the fact that they are all spherical designs and require factors to be run at only three levels. The designs are also rotatable or nearly rotatable. Some of these designs also provide orthogonal blocking. Thus, if there is a need to separate runs into blocks for the Box-Behnken design, then designs are available that allow blocks to be used in such a way that the estimation of the regression parameters for the factor effects are not affected by the blocks. In other words, in these designs the block effects are orthogonal to the other factor effects. Yet another advantage of these designs is that there are no runs where all factors are at either the <math>+1\,\!</math> or <math>-1\,\!</math> levels. For example, in the figure below the representation of the Box-Behnken design for three factors clearly shows that there are no runs at the corner points. This could be advantageous when the corner points represent runs that are expensive or inconvenient because they lie at the end of the range of the factor levels. A few of the Box-Behnken designs available in DOE++ are presented in [[Box-Behnken_Designs|Appendix F]].

====Example====
Continuing with the example in [[Response_Surface_Methods_for_Optimization#Method_of_Steepest_Ascent| Method of Steepest Ascent]], the first order model was found to be inadequate for the region near the optimum. Once the experimenter realized that the first order model was not adequate (for the region with a reaction temperature of 350 <math>F\,\!</math> and reaction time of 165 minutes), it was decided to augment the experiment with axial runs to be able to complete a central composite design and fit a second order model to the response. Notice the advantage of using a central composite design, as the experimenter only had to add the axial runs to the <math>2^{2}\,\!</math> design with center point runs, and did not have to begin a new experiment. The experimenter decided to use <math>\alpha =1.4142\,\!</math> to get a rotatable design. The obtained response values are shown in the figure below.

[[Image:doe9.14.png|center|280px|Response values for the two factor central composite design in the [[Response_Surface_Methods_for_Optimization#Example_2| example]].]]

Such a design can be set up in a Weibull++ DOE folio using the properties shown in the figure below.

[[Image:doe9_15.png|center|744px|Properties for the central composite design in the [[Response_Surface_Methods_for_Optimization#Example_2| example]].|link=]]

The resulting design is shown in the figure shown next.

[[Image:doe9_16.png|center|800px|Central composite design for the experiment in the [[Response_Surface_Methods_for_Optimization#Example_2| example]].|link=]]

Results from the analysis of the design are shown in the next figure.

[[Image:doe9_17.png|center|744px|Results for the central composite design in the [[Response_Surface_Methods_for_Optimization#Example_2| example]].|link=]]

The results in the figure above show that the main effects, <math>A\,\!</math> and <math>B\,\!</math>, the interaction, <math>AB\,\!</math>, and the quadratic main effects, <math>{{A}^{2}}\,\!</math> and <math>{{B}^{2}}\,\!</math>, (represented as AA and BB in the figure) are significant. The lack-of-fit test also shows that the second order model with these terms is adequate and a higher order model is not needed. Using these results, the model for the experiment in terms of the coded values is:

::<math>\hat{y}=94.91+0.74{{x}_{1}}+1.53{{x}_{2}}+0.45{{x}_{1}}{{x}_{2}}-1.52x_{1}^{2}-2.08x_{2}^{2}\,\!</math>

The response surface and the contour plot for this model, in terms of the actual variables, are shown in the below figures (a) and (b), respectively.

[[Image:doe9_18_a.png|center|650px|link=]]

[[Image:doe9_18_b.png|center|650px|Response surface and countour plot for the experiment in the [[Response_Surface_Methods_for_Optimization#Example_2| example]].|link=]]

==Analysis of the Second Order Model==
Once a second order model is fit to the response, the next step is to locate the point of maximum or minimum response. The second order model for <math>k\,\!</math> factors can be written as:

::<math>\hat{y}={{\hat{\beta }}_{0}}+\underset{i=1}{\overset{k}{\mathop{\sum }}}\,{{\hat{\beta }}_{i}}{{x}_{i}}+\underset{i=1}{\overset{k}{\mathop{\sum }}}\,{{\hat{\beta }}_{ii}}x_{i}^{2}+\underset{i=1}{\overset{k}{\mathop{\sum }}}\,\underset{j=1}{\overset{i-1}{\mathop{\sum }}}\,{{\hat{\beta }}_{ij}}{{x}_{i}}{{x}_{j}}\,\!</math>

The point for which the response, <math>\hat{y}\,\!</math>, is optimized is the point at which the partial derivatives, <math>\partial \hat{y}/\partial {{x}_{1}}\,\!</math>, <math>\partial \hat{y}/\partial {{x}_{2}}\,\!</math>, <math>...\partial \hat{y}/\partial {{x}_{k}},\,\!</math> are all equal to zero. This point is called the ''stationary point''. The stationary point may be a point of maximum response, minimum response or a saddle point. These three conditions are shown in the following figures (a), (b) and (c) respectively.

[[Image:doe9.19.png|center|300px|Types of second order response surfaces and their contour plots. (a) shows the surface with a maximum point, (b) shows the surface with a minimum point and (c) shows the surface with a saddle point.]]

Notice that these conditions are easy to identify, in the case of two factor experiments, by the inspection of the contour plots. However, when more than two factors exist in an experiment, then the general mathematical solution for the location of the stationary point has to be used. The equation given above can be written in matrix notation as:

::<math>\hat{y}={{\hat{\beta }}_{0}}+{{x}^{\prime }}b+{{x}^{\prime }}Bx\,\!</math>

where:

::<math>x=\left[ \begin{matrix}
{{x}_{1}} \\
{{x}_{2}} \\
. \\
. \\
. \\
{{x}_{k}} \\
\end{matrix} \right]\text{ }b=\left[ \begin{matrix}
{{{\hat{\beta }}}_{1}} \\
{{{\hat{\beta }}}_{2}} \\
. \\
. \\
. \\
{{{\hat{\beta }}}_{k}} \\
\end{matrix} \right]\text{ and }B=\left[ \begin{matrix}
{{{\hat{\beta }}}_{11}} & {{{\hat{\beta }}}_{12}}/2 & . & . & . & {{{\hat{\beta }}}_{1k}}/2 \\
{} & {{{\hat{\beta }}}_{22}} & . & . & . & {{{\hat{\beta }}}_{2k}}/2 \\
{} & {} & . & . & . & . \\
{} & {} & {} & . & . & . \\
{} & {} & {} & {} & . & . \\
\text{sym}\text{.} & {} & {} & {} & {} & {{{\hat{\beta }}}_{kk}} \\
\end{matrix} \right]\,\!</math>

Then the stationary point can be determined as follows:

::<math>\frac{\partial \hat{y}}{\partial x}=b+2Bx=0\,\!</math>

Thus, the stationary point is:

::<math>{{x}_{s}}=-\frac{1}{2}{{B}^{-1}}b\,\!</math>

The optimum response is the response corresponding to <math>{{x}_{s}}\,\!</math>. The optimum response can be obtained to get:

::<math>{{\hat{y}}_{s}}={{\hat{\beta }}_{0}}+\frac{1}{2}x_{s}^{\prime }b\,\!</math>

Once the stationary point is known, it is necessary to determine if it is a maximum or minimum or saddle point. To do this, the second order model has to be transformed to the canonical form. This is done by transforming the model to a new coordinate system such that the origin lies at the stationary point and the axes are parallel to the principal axes of the fitted response surface, shown next.

[[Image:doe9.20.png|center|247px|The second order model in canonical form.]]

The resulting model equation then takes the following form:

::<math>\hat{y}={{\hat{y}}_{s}}+{{\lambda }_{1}}w_{1}^{2}+{{\lambda }_{2}}w_{2}^{2}+...+{{\lambda }_{k}}w_{k}^{2}\,\!</math>

where the <math>{{w}_{i}}\,\!</math> s are the transformed independent variables, and <math>{{\lambda }_{i}}\,\!</math>s are constants that are also the eigenvalues of the matrix <math>B\,\!</math>. The nature of the stationary point is known by looking at the signs of the <math>{{\lambda }_{i}}\,\!</math>s. If the <math>{{\lambda }_{i}}\,\!</math> s are all negative, then <math>{{x}_{s}}\,\!</math> is a point of maximum response. If the <math>{{\lambda }_{i}}\,\!</math>s are all positive then <math>{{x}_{s}}\,\!</math> is a point of minimum response. If the <math>{{\lambda }_{i}}\,\!</math>s have different signs, then <math>{{x}_{s}}\,\!</math> is a saddle point.

====Example====
Continuing with the [[Response_Surface_Methods_for_Optimization#Example_2| example]] in [[Response_Surface_Methods_for_Optimization#Method_of_Steepest_Ascent| Method of Steepest Ascent]], the second order model fitted to the response, in terms of the coded variables, was obtained as:

::<math>\hat{y}=94.91+0.74{{x}_{1}}+1.53{{x}_{2}}+0.45{{x}_{1}}{{x}_{2}}-1.52x_{1}^{2}-2.08x_{2}^{2}\,\!</math>

Then the <math>b\,\!</math> and <math>B\,\!</math> matrices for this model are:

::<math>b=\left[ \begin{matrix}
0.74 \\
1.53 \\
\end{matrix} \right]\text{ }B=\left[ \begin{matrix}
-1.52 & 0.45 \\
0.45 & -2.08 \\
\end{matrix} \right]\,\!</math>
 

The stationary point is:

<math>\begin{align}
{{x}_{s}}= & -\frac{1}{2}{{B}^{-1}}b \\
= & -\frac{1}{2}{{\left[ \begin{matrix}
-1.52 & 0.45 \\
0.45 & -2.08 \\
\end{matrix} \right]}^{-1}}\left[ \begin{matrix}
0.74 \\
1.53 \\
\end{matrix} \right] \\
= & \left[ \begin{matrix}
0.3 \\
0.4 \\
\end{matrix} \right]
\end{align}\,\!</math>

Then, in terms of the actual values, the stationary point can be found as:

::<math>\begin{align}
\text{Reaction Temperature:} & 0.3=\frac{T-350}{(355-345)/2} \\
T= & 351.5\text{ }F
\end{align}\,\!</math>

::<math>\begin{align}
\text{Reaction Time:} & 0.4=\frac{t-165}{(175-155)/2} \\
t= & 169.0\text{ minutes}
\end{align}\,\!</math>

To find the nature of the stationary point the eigenvalues of the <math>B\,\!</math> matrix can be obtained as follows using the determinant of the matrix <math>B-\lambda I\,\!</math>:

::<math>\begin{align}
\left| B-\lambda I \right|= & 0 \\
\left| \begin{matrix}
-1.52-\lambda & 0.45 \\
0.45 & -2.08-\lambda \\
\end{matrix} \right|= & 0
\end{align}\,\!</math>

This gives us:

::<math>{{\lambda }^{2}}+3.61\lambda +2.9743=0\,\!</math>

Solving the quadratic equation in <math>\lambda \,\!</math> returns the eigenvalues <math>{{\lambda }_{1}}=-1.2723\,\!</math> and <math>{{\lambda }_{2}}=-2.3377\,\!</math>. Since both the eigenvalues are negative, it can be concluded that the stationary point is a point of maximum response. The predicted value of the maximum response can be obtained as:

::<math>\begin{align}
{{{\hat{y}}}_{s}}= & {{{\hat{\beta }}}_{0}}+\frac{1}{2}x_{s}^{\prime }b \\
= & 94.91+\frac{1}{2}{{\left[ \begin{matrix}
0.3 \\
0.4 \\
\end{matrix} \right]}^{\prime }}\left[ \begin{matrix}
0.74 \\
1.53 \\
\end{matrix} \right] \\
= & 95.3
\end{align}\,\!</math>

In a DOE folio, the maximum response can be obtained by entering the required values as shown in the figure below. In the figure, the goal is to maximize the response and the limits of the search range for maximizing the response are entered as 90 and 100. The value of the maximum response and the corresponding values of the factors obtained are shown in the second figure following. These values match the values calculated in this example.

[[Image:doe9_21.png|center|543px|Settings to obtain the maximum value of the response in the [[Response_Surface_Methods_for_Optimization#Example_3| example]].|link=]]

[[Image:doe9_22.png|center|650px|Plot of the maximum response in the [[Response_Surface_Methods_for_Optimization#Example_3| example]] against the factors, temperature and time.|link=]]

==Multiple Responses==
In many cases, the experimenter has to optimize a number of responses at the same time. For the example in [[Response_Surface_Methods_for_Optimization#Method_of_Steepest_Ascent| Method of Steepest Ascent]], assume that the experimenter has to also consider two other responses: cost of the product (which should be minimized) and the pH of the product (which should be close to 7 so that the product is neither acidic nor basic). The data is presented in the figure below.

[[Image:doe9.23.png|center|400px|Data for the additional responses of cost and pH for the example to investigate the yield of a chemical process.]]

The problem in dealing with multiple responses is that now there might be conflicting objectives because of the different requirements of each of the responses. The experimenter needs to come up with a solution that satisfies each of the requirements as much as possible without compromising too much on any of the requirements. The approach used in a Weibull++ DOE folio to deal with optimization of multiple responses involves the use of desirability functions that are discussed next (for details see [[DOE_References|[Derringer and Suich, 1980]]]).

===Desirability Functions===
Under this approach, each <math>i\,\!</math>th response is assigned a desirability function, <math>{{d}_{i}}\,\!</math>, where the value of <math>{{d}_{i}}\,\!</math> varies between 0 and 1. The function, <math>{{d}_{i}},\,\!</math> is defined differently based on the objective of the response. If the response is to be maximized, as in the case of the previous example where the yield had to be maximized, then <math>{{d}_{i}}\,\!</math> is defined as follows:

::<math>{{d}_{i}}=\left\{ \begin{matrix}
0 & {{y}_{i}}<L \\
{{\left( \frac{{{y}_{i}}-L}{T-L} \right)}^{\omega }} & L\le {{y}_{i}}\le T \\
1 & {{y}_{i}}>T \\
\end{matrix} \right.</math>

where <math>T\,\!</math> represents the target value of the <math>i\,\!</math>th response, <math>{{y}_{i}}\,\!</math>, <math>L\,\!</math> represents the acceptable lower limit value for this response and <math>\omega \,\!</math> represents the weight. When <math>\omega =1\,\!</math> the function <math>{{d}_{i}}\,\!</math> is linear. If <math>\omega >1\,\!</math> then more importance is placed on achieving the target for the response, <math>{{y}_{i}}\,\!</math>. When <math>\omega <1\,\!</math>, less weight is assigned to achieving the target for the response, <math>{{y}_{i}}\,\!</math>. A graphical representation is shown in figure (a) below.

[[Image:doe9.24.png|center|400px|Desirability function plots for different response optimizations: (a) the goal is to maximize the response, (b) the goal is to minimize the response and (c) the goal is to get the response to a target value.]]

If the response is to be minimized, as in the case when the response is cost, <math>{{d}_{i}}\,\!</math> is defined as follows:

::<math>{{d}_{i}}=\left\{ \begin{matrix}
1 & {{y}_{i}}<T \\
{{\left( \frac{U-{{y}_{i}}}{U-T} \right)}^{\omega }} & T\le {{y}_{i}}\le U \\
0 & {{y}_{i}}>U \\
\end{matrix} \right.</math>

Here <math>U\,\!</math> represents the acceptable upper limit for the response (see figure (b) above).

There may be times when the experimenter wants the response to be neither maximized nor minimized, but instead stay as close to a specified target as possible. For example, in the case where the experimenter wants the product to be neither acidic nor basic, there is a requirement to keep the pH close to the neutral value of 7. In such cases, the desirability function is defined as follows (see figure (c) above):

::<math>{{d}_{i}}=\left\{ \begin{matrix}
0 & {{y}_{i}}<L \\
{{\left( \frac{{{y}_{i}}-L}{T-L} \right)}^{{{\omega }_{1}}}} & L\le {{y}_{i}}\le T \\
{{\left( \frac{U-{{y}_{i}}}{U-T} \right)}^{{{\omega }_{2}}}} & T\le {{y}_{i}}\le U \\
0 & {{y}_{i}}>U \\
\end{matrix} \right.</math>

Once a desirability function is defined for each of the responses, assuming that there are <math>m\,\!</math> responses, an overall desirability function is obtained as follows:

::<math>D={{(d_{1}^{{{r}_{1}}}\cdot d_{2}^{{{r}_{2}}}\cdot ...\cdot d_{m}^{{{r}_{m}}})}^{1/({{r}_{1}}+{{r}_{2}}+...+{{r}_{m}})}}\,\!</math>

where the <math>{{r}_{i}}\,\!</math> s represent the importance of each response. The greater the value of <math>{{r}_{i}}\,\!</math>, the more important the response with respect to the other responses. The objective is to now find the settings that return the maximum value of <math>D\,\!</math>.

To illustrate the use of desirability functions, consider the previous example with the three responses of yield, cost and pH. The response surfaces for the two additional responses of cost and pH are shown next in the figures (a) and (b), respectively.

[[Image:doe9_25.png|center|500px|Response surfaces for (a) cost and (b) pH.]]

In terms of actual variables, the models obtained for all three responses are as shown next:

::<math>\begin{align}
\text{Yield: } & {{{\hat{y}}}_{1}}=-7480.24+41.24{{x}_{1}}+3.88{{x}_{2}}+0.009{{x}_{1}}{{x}_{2}} \\
& -0.060825x_{1}^{2}-0.020831x_{2}^{2} \\
\text{Cost: } & {{{\hat{y}}}_{2}}=15560.49-82.48{{x}_{1}}-7.76{{x}_{2}}-0.02{{x}_{1}}{{x}_{2}} \\
& +0.12165x_{1}^{2}+0.041663x_{2}^{2} \\
\text{pH: } & {{{\hat{y}}}_{3}}=38.87-0.0709{{x}_{1}}-0.041{{x}_{2}}
\end{align}\,\!</math>

Assume that the experimenter wants to have a target yield value of 95, although any value of yield greater than 94 is acceptable. Then the desirability function for yield is:

::<math>{{d}_{1}}=\left\{ \begin{matrix}
0 & {{y}_{1}}<94 \\
{{\left( \frac{{{y}_{1}}-94}{95-94} \right)}^{\omega }} & 94\le {{y}_{1}}\le 95 \\
1 & {{y}_{1}}>95 \\
\end{matrix} \right.</math>

For the cost, assume that the experimenter wants to lower the cost to 400, although any cost value below 415 is acceptable. Then the desirability function for cost is:

::<math>{{d}_{2}}=\left\{ \begin{matrix}
1 & {{y}_{2}}<400 \\
{{\left( \frac{415-{{y}_{2}}}{415-400} \right)}^{\omega }} & 400\le {{y}_{2}}\le 415 \\
0 & {{y}_{2}}>415 \\
\end{matrix} \right.</math>

For the pH, a target of 7 is desired but values between 6.9 and 7.1 are also acceptable. Thus, the desirability function here is:

::<math>{{d}_{3}}=\left\{ \begin{matrix}
0 & {{y}_{3}}<L \\
{{\left( \frac{{{y}_{3}}-6.9}{7-6.9} \right)}^{{{\omega }_{1}}}} & 6.9\le {{y}_{3}}\le 7 \\
{{\left( \frac{7.1-{{y}_{3}}}{7.1-7} \right)}^{{{\omega }_{2}}}} & 7\le {{y}_{3}}\le 7.1 \\
0 & {{y}_{3}}>7.1 \\
\end{matrix} \right.</math>

Notice that in the previous equations all weights used (<math>{{\omega }_{i}}\,\!</math> s) are 1. Thus, all three desirability functions are linear. The overall desirability function, assuming equal importance (<math>{{r}_{1}}={{r}_{2}}={{r}_{3}}=1\,\!</math>) for all the responses, is:

::<math>\begin{align}
D= & {{({{d}_{1}}\cdot {{d}_{2}}\cdot {{d}_{3}})}^{1/3}} \\
= & f({{{\hat{y}}}_{1}},{{{\hat{y}}}_{2}},{{{\hat{y}}}_{3}})
\end{align}\,\!</math>

The objective of the experimenter is to find the settings of <math>{{x}_{1}}\,\!</math> and <math>{{x}_{2}}\,\!</math> such that the overall desirability, <math>D\,\!</math>, is maximum. In a DOE folio, the settings for the desirability functions for each of the three responses can be entered as shown in the next figure.

[[Image:doe9_26.png|center|750px|Optimization settings for the three responses of yield, cost, and pH.]]

Based on these settings, the Weibull++ DOE folio solves this optimization problem to obtain the following solution:

[[Image:doe9_27.png|center|650px|Optimum solution from Weibull++ for the three responses of yield, cost, and pH.]]

::<math>\begin{align}
Temperature= & 351.5\text{ }F\text{ } \\
Time= & 169\text{ }minutes
\end{align}\,\!</math>

The overall desirability achieved with this solution can be calculated easily. The values of each of the response for these settings are:

::<math>\begin{align}
\text{ }{{{\hat{y}}}_{1}}= & -7480.24+41.24{{x}_{1}}+3.88{{x}_{2}}+0.009{{x}_{1}}{{x}_{2}} \\
& -0.060825x_{1}^{2}-0.020831x_{2}^{2} \\
= & -7480.24+41.24(351.5)+3.88(169)+0.009(351.3)(169) \\
& -0.060825{{(351.5)}^{2}}-0.020831{{(169)}^{2}} \\
= & 95.3 \\
\text{ }{{{\hat{y}}}_{2}}= & 15560.49-82.48{{x}_{1}}-7.76{{x}_{2}}-0.02{{x}_{1}}{{x}_{2}} \\
& +0.12165x_{1}^{2}+0.041663x_{2}^{2} \\
= & 15560.49-82.48(351.5)-7.76(169)-0.02(351.5)(169) \\
& +0.12165{{(351.5)}^{2}}+0.041663{{(169)}^{2}} \\
= & 409.35 \\
\text{ }{{{\hat{y}}}_{3}}= & 38.87-0.0709{{x}_{1}}-0.041{{x}_{2}} \\
= & 38.87-0.0709(351.5)-0.041(169) \\
= & 7.00
\end{align}\,\!</math>

Based on the response values, the individual desirability functions are:

::<math>\begin{align}
{{d}_{1}}= & 1\text{ (since }{{{\hat{y}}}_{1}}>95\text{)} \\
{{d}_{2}}= & \left( \frac{415-{{{\hat{y}}}_{2}}}{415-400} \right)\text{ (since }400\le {{y}_{2}}\le 415\text{)} \\
= & \left( \frac{415-409.35}{415-400} \right) \\
= & 0.3767 \\
{{d}_{3}}= & 1\text{ (since }{{{\hat{y}}}_{3}}\approx 7\text{)}
\end{align}\,\!</math>

Then the overall desirability is:

::<math>\begin{align}
D= & {{({{d}_{1}}\cdot {{d}_{2}}\cdot {{d}_{3}})}^{1/3}} \\
= & {{(1\cdot 0.3767\cdot 1)}^{1/3}} \\
= & 0.72
\end{align}\,\!</math>

This is the same as the Global Desirability displayed by the Weibull++ DOE folio in the figure above. At times, a number of solutions may be obtained from the DOE folio, and it is up to the experimenter to choose the most feasible one.

Competing Failure Modes Analysis

2017-08-10T18:40:16Z

Richard House: /* Complex Failure Modes Analysis */

{{template:LDABOOK|18|Competing Failure Modes Models}}
Often, a group of products will fail due to more than one failure mode. One can take the view that the products could have failed due to any one of the possible failure modes, but since an item cannot fail more than one time, there can only be one failure mode for each failed product. In this view, the failure modes compete as to which causes the failure for each particular item. This can be viewed as a series system reliability model, with each failure mode composing a block of the series system. Competing failure modes (CFM) analysis segregates the analyses of failure modes and then combines the results to provide an overall model for the product in question.
==CFM Analysis Approach==
In order to begin analyzing data sets with more than one competing failure mode, one must perform a separate analysis for each failure mode. During each of these analyses, the failure times for all other failure modes not being analyzed are considered to be suspensions. This is because the units under test would have failed at some time in the future due to the failure mode being analyzed, had the unrelated (not analyzed) mode not occurred. Thus, in this case, the information available is that the mode under consideration did not occur and the unit under consideration accumulated test time without a failure due to the mode under consideration (or a suspension due to that mode).

Once the analysis for each separate failure mode has been completed (using the same principles as before), the resulting reliability equation for all modes is the product of the reliability equation for each mode, or:

::<math>R(t)={{R}_{1}}(t)\cdot {{R}_{2}}(t)\cdot ...\cdot {{R}_{n}}(t)\,\!</math>

where <math>n\,\!</math> is the total number of failure modes considered. This is the product rule for the reliability of series systems with statistically independent components, which states that the reliability for a series system is equal to the product of the reliability values of the components comprising the system. Do note that the above equation is the reliability function based on any assumed life distribution. In Weibull++ this life distribution can be either the 2-parameter Weibull, lognormal, normal or the 1-parameter exponential.

===CFM Example===

The following example demonstrates how you can use the reliability equation to determine the overall reliability of a component. (This example has been abstracted from Example 15.6 from the Meeker and Escobar textbook Statistical Methods for Reliability Data [[Appendix:_Life_Data_Analysis_References|[27]]].)

An electronic component has two competing failure modes. One failure mode is due to random voltage spikes, which cause failure by overloading the system. The other failure mode is due to wearout failures, which usually happen only after the system has run for many cycles. The objective is to determine the overall reliability for the component at 100,000 cycles.

30 units are tested, and the failure times are recorded in the following table. The failures that are due to the random voltage spikes are denoted by a V. The failures that are due to wearout failures are denoted by a W.

<center><math>\begin{matrix}
Number & Failure & Failure & Number & Failure & Failure \\
in State & Time* & Mode & in State & Time* & Mode \\
\text{1} & \text{2} & \text{V} & \text{1} & \text{147} & \text{W} \\
\text{1} & \text{10} & \text{V} & \text{1} & \text{173} & \text{V} \\
\text{1} & \text{13} & \text{V} & \text{1} & \text{181} & \text{W} \\
\text{2} & \text{23} & \text{V} & \text{1} & \text{212} & \text{W} \\
\text{1} & \text{28} & \text{V} & \text{1} & \text{245} & \text{W} \\
\text{1} & \text{30} & \text{V} & \text{1} & \text{247} & \text{V} \\
\text{1} & \text{65} & \text{V} & \text{1} & \text{261} & \text{V} \\
\text{1} & \text{80} & \text{V} & \text{1} & \text{266} & \text{W} \\
\text{1} & \text{88} & \text{V} & \text{1} & \text{275} & \text{W} \\
\text{1} & \text{106} & \text{V} & \text{1} & \text{293} & \text{W} \\
\text{1} & \text{143} & \text{V} & \text{8} & \text{300} & \text{suspended} \\
\end{matrix}\,\!</math></center>

''*Failure times given are in thousands of cycles.''

'''Solution'''

To obtain the overall reliability of the component, we will first need to analyze the data set due to each failure mode. For example, to obtain the reliability of the component due to voltage spikes, we must consider all of the failures for the wear-out mode to be suspensions. We do the same for analyzing the wear-out failure mode, counting only the wear-out data as failures and assuming that the voltage spike failures are suspensions. Once we have obtained the reliability of the component due to each mode, we can use the system Reliability Equation to determine the overall component reliability.

The following analysis shows the data set for the voltage spikes. Using the Weibull distribution and the MLE analysis method (recommended due to the number of suspensions in the data), the parameters are <math>{{\beta }_{V}}=0.671072\,\!</math> and <math>{{\eta }_{V}}=449.427230\,\!</math>. The reliability for this failure mode at <math>t=100\,\!</math> is <math>{{R}_{V}}(100)=0.694357\,\!</math>.

[[Image:Competing Failiure Mode V Mode Data.png|center|550px| ]]

The following analysis shows the data set for the wearout failure mode. Using the same analysis settings (i.e., Weibull distribution and MLE analysis method), the parameters are <math>{{\beta }_{W}}=4.337278\,\!</math> and <math>{{\eta }_{W}}=340.384242\,\!</math>. The reliability for this failure mode at <math>t=100\,\!</math> is <math>{{R}_{W}}(100)=0.995084\,\!</math>.

[[Image:Competing Failiure Mode W Mode Data.png|center|550px| ]]

Using the Reliability Equation to obtain the overall component reliability at 100,000 cycles, we get:
::<math>\begin{align}
& {{R}_{sys}}(100)= {{R}_{V}}(100)\cdot {{R}_{W}}(100) \\
& = 0.694357\cdot 0.995084 \\
& = 0.690943
\end{align}\,\!</math>

Or the reliability of the unit (or system) under both modes is <math>{{R}_{sys}}(100)=69.094%\,\!</math>.

You can also perform this analysis using Weibull++'s built-in CFM analysis options, which allow you to generate a probability plot that contains the combined mode line as well as the individual mode lines.

<div class="noprint">
{{Examples Box|Weibull++ Examples|More CFM examples are available! See also:
{{Examples Both|http://www.reliasoft.com/Weibull/examples/rc10/index.htm|Competing Failure Modes Analysis|http://www.reliasoft.tv/weibull/appexamples/weibull_app_ex_10.html|Watch the video...}}<nowiki/>
}}
</div>

==Confidence Bounds for CFM Analysis==

The method available in Weibull++ for estimating the different types of confidence bounds, for competing failure modes analysis, is the Fisher matrix method, and is presented in this section.

===Variance/Covariance Matrix===

The variances and covariances of the parameters are estimated from the inverse local Fisher matrix, as follows:

::<math>\begin{align}
& \left( \begin{matrix}
Var({{{\hat{a}}}_{1}}) & Cov({{{\hat{a}}}_{1}},{{{\hat{b}}}_{1}}) & 0 & 0 & 0 & 0 & 0 \\
Cov({{{\hat{a}}}_{1}},{{{\hat{b}}}_{1}}) & Var({{{\hat{b}}}_{1}}) & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \cdot & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \cdot & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & \cdot & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & Var({{{\hat{a}}}_{n}}) & Cov({{{\hat{a}}}_{n}},{{{\hat{b}}}_{n}}) \\
0 & 0 & 0 & 0 & 0 & Cov({{{\hat{a}}}_{n}},{{{\hat{b}}}_{n}}) & Var({{{\hat{b}}}_{n}}) \\
\end{matrix} \right) \\
& ={\left( \begin{matrix}
-\frac{{{\partial }^{2}}\Lambda }{\partial a_{1}^{2}} & -\frac{{{\partial }^{2}}\Lambda }{\partial a_{1}^{{}}\partial {{b}_{1}}} & 0 & 0 & 0 & 0 & 0 \\
-\frac{{{\partial }^{2}}\Lambda }{\partial a_{1}^{{}}\partial {{b}_{1}}} & -\frac{{{\partial }^{2}}\Lambda }{\partial b_{1}^{2}} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \cdot & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \cdot & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & \cdot & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -\frac{{{\partial }^{2}}\Lambda }{\partial a_{n}^{2}} & -\frac{{{\partial }^{2}}\Lambda }{\partial a_{n}^{{}}\partial {{b}_{n}}} \\
0 & 0 & 0 & 0 & 0 & -\frac{{{\partial }^{2}}\Lambda }{\partial a_{n}^{{}}\partial {{b}_{n}}} & -\frac{{{\partial }^{2}}\Lambda }{\partial b_{n}^{2}} \\
\end{matrix} \right)}^{-1} \\
\end{align}\,\!</math>

where <math>\Lambda \,\!</math> is the log-likelihood function of the failure distribution, described in [[Parameter Estimation]].

===Bounds on Reliability===

The competing failure modes reliability function is given by:

::<math>\widehat{R}=\underset{i=1}{\overset{n}{\mathop \prod }}\,{{\hat{R}}_{i}}\,\!</math>

where:
* <math>{{R}_{i}}\,\!</math> is the reliability of the <math>{{i}^{th}}\,\!</math> mode.
* <math>n\,\!</math> is the number of failure modes.

The upper and lower bounds on reliability are estimated using the logit transformation:

::<math>\begin{align}
& {{R}_{U}}= & \frac{\widehat{R}}{\widehat{R}+(1-\widehat{R}){{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})}}{\widehat{R}(1-\widehat{R})}}}} \\
& {{R}_{L}}= & \frac{\widehat{R}}{\widehat{R}+(1-\widehat{R}){{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{R})}}{\widehat{R}(1-\widehat{R})}}}}
\end{align}\,\!</math>

where <math>\widehat{R}\,\!</math> is calculated using the reliability equation for competing failure modes.
<math>{{K}_{\alpha }}\,\!</math> is defined by:

::<math>\alpha =\frac{1}{\sqrt{2\pi }}\underset{{{K}_{\alpha }}}{\overset{\infty }{\mathop \int }}\,{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})\,\!</math>

(If <math>\delta \,\!</math> is the confidence level, then <math>\alpha =\tfrac{1-\delta }{2}\,\!</math> for the two-sided bounds, and <math>\alpha =1-\delta \,\!</math> for the one-sided bounds.)

The variance of <math>\widehat{R}\,\!</math> is estimated by:

::<math>Var(\widehat{R})=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{\left( \frac{\partial R}{\partial {{R}_{i}}} \right)}^{2}}Var({{\hat{R}}_{i}})\,\!</math>

::<math>\frac{\partial R}{\partial {{R}_{i}}}=\underset{j=1,j\ne i}{\overset{n}{\mathop \prod }}\,\widehat{{{R}_{j}}}\,\!</math>

Thus:

::<math>Var(\widehat{R})=\underset{i=1}{\overset{n}{\mathop \sum }}\,\left( \underset{j=1,j\ne i}{\overset{n}{\mathop \prod }}\,\widehat{R}_{j}^{2} \right)Var({{\hat{R}}_{i}})\,\!</math>

::<math>Var({{\hat{R}}_{i}})=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{\left( \frac{\partial {{R}_{i}}}{\partial {{a}_{i}}} \right)}^{2}}Var({{\hat{a}}_{i}})\,\!</math>

where <math>\widehat{{{a}_{i}}}\,\!</math> is an element of the model parameter vector.

Therefore, the value of <math>Var({{\hat{R}}_{i}})\,\!</math> is dependent on the underlying distribution.

For the Weibull distribution:

::<math>Var({{\hat{R}}_{i}})={{\left( {{{\hat{R}}}_{i}}{{e}^{{{{\hat{u}}}_{i}}}} \right)}^{2}}Var({{\hat{u}}_{i}})\,\!</math>

where:

::<math>{{\hat{u}}_{i}}={{\hat{\beta }}_{i}}(\ln (t-{{\hat{\gamma }}_{i}})-\ln {{\hat{\eta }}_{i}})\,\!</math>

and <math>Var(\widehat{{{u}_{i}}})\,\!</math> is given in [[The Weibull Distribution]].

For the exponential distribution:

::<math>Var({{\hat{R}}_{i}})={{\left( {{{\hat{R}}}_{i}}(t-{{{\hat{\gamma }}}_{i}}) \right)}^{2}}Var({{\hat{\lambda }}_{i}})\,\!</math>

where <math>Var(\widehat{{{\lambda }_{i}}})\,\!</math> is given in [[The Exponential Distribution]].

For the normal distribution:

::<math>Var({{\hat{R}}_{i}})={{\left( f({{{\hat{z}}}_{i}})\hat{\sigma } \right)}^{2}}Var({{\hat{z}}_{i}})\,\!</math>

::<math>{{\hat{z}}_{i}}=\frac{t-{{{\hat{\mu }}}_{i}}}{{{{\hat{\sigma }}}_{i}}}\,\!</math>

where <math>Var(\widehat{{{z}_{i}}})\,\!</math> is given in [[The Normal Distribution]].

For the lognormal distribution:

::<math>Var({{\hat{R}}_{i}})={{\left( f({{{\hat{z}}}_{i}})\cdot {{{\hat{\sigma }}}^{\prime }} \right)}^{2}}Var({{\hat{z}}_{i}})\,\!</math>

::<math>{{\hat{z}}_{i}}=\frac{\ln \text{(}t)-\hat{\mu }_{i}^{\prime }}{\hat{\sigma }_{i}^{\prime }}\,\!</math>

where <math>Var(\widehat{{{z}_{i}}})\,\!</math> is given in [[The Lognormal Distribution]].

===Bounds on Time===
The bounds on time are estimate by solving the reliability equation with respect to time. From the reliabilty equation for competing faiure modes, we have that:

::<math>\hat{t}=\varphi (R,{{\hat{a}}_{i}},{{\hat{b}}_{i}})\,\!</math>

::<math>i=1,...,n\,\!</math>

where:
:• <math>\varphi \,\!</math> is inverse function for the reliabilty equation for competing faiure modes.
:• for the Weibull distribution <math>{{\hat{a}}_{i}}\,\!</math> is <math>{{\hat{\beta }}_{i}}\,\!</math>, and <math>{{\hat{b}}_{i}}\,\!</math> is <math>{{\hat{\eta }}_{i}}\,\!</math>
:• for the exponential distribution <math>{{\hat{a}}_{i}}\,\!</math> is <math>{{\hat{\lambda }}_{i}}\,\!</math>, and <math>{{\hat{b}}_{i}}\,\!</math> =0
:• for the normal distribution <math>{{\hat{a}}_{i}}\,\!</math> is <math>{{\hat{\mu }}_{i}}\,\!</math>, and <math>{{\hat{b}}_{i}}\,\!</math> is <math>{{\hat{\sigma }}_{i}}\,\!</math>, and
:• for the lognormal distribution <math>{{\hat{a}}_{i}}\,\!</math> is <math>\hat{\mu }_{i}^{\prime }\,\!</math>, and <math>{{\hat{b}}_{i}}\,\!</math> is <math>\hat{\sigma }_{i}^{\prime }\,\!</math>

Set:

::<math>\begin{align}
u=\ln (t)
\end{align}\,\!</math>

The bounds on <math>u\,\!</math> are estimated from:

::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}\,\!</math>

and:

::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}\,\!</math>

Then the upper and lower bounds on time are found by using the equations:

::<math>{{t}_{U}}={{e}^{{{u}_{U}}}}\,\!</math>

and:

::<math>{{t}_{L}}={{e}^{{{u}_{L}}}}\,\!</math>

<math>{{K}_{\alpha }}\,\!</math> is calculated using the inverse standard normal distribution and <math>Var(\widehat{u})\,\!</math> is computed as:

::<math>Var(\widehat{u})=\underset{i=1}{\overset{n}{\mathop \sum }}\,\left( {{\left( \frac{\partial u}{\partial {{a}_{i}}} \right)}^{2}}Var(\widehat{{{a}_{i}}})+{{\left( \frac{\partial u}{\partial {{b}_{i}}} \right)}^{2}}Var(\widehat{{{b}_{i}}})+2\frac{\partial u}{\partial {{a}_{i}}}\frac{\partial u}{\partial {{b}_{i}}}Cov(\widehat{{{a}_{i}}},\widehat{{{b}_{i}}}) \right)\,\!</math>

==Complex Failure Modes Analysis==
In addition to being viewed as a series system, the relationship between the different competing failures modes can be more complex. After performing separate analysis for each failure mode, a diagram that describes how each failure mode can result in a product failure can be used to perform analysis for the item in question. Such diagrams are usually referred to as Reliability Block Diagrams (RBD) (for more on RBDs see [[System_Analysis_Reference|ReliaSoft's System Analysis Reference]] and [http://www.reliasoft.com/BlockSim/ ReliaSoft's BlockSim software]).

A reliability block diagram is made of blocks that represent the failure modes and arrows and connects the blocks in different configurations. Note that the blocks can also be used to represent different components or subsystems that make up the product. Weibull++ provides the capability to use a diagram to model, series, parallel, k-out-of-n configurations in addition to any complex combinations of these configurations.

In this analysis, the failure modes are assumed to be statistically independent. (Note: In the context of this reference, statistically independent implies that failure information for one failure mode provides no information about, i.e. does not affect, other failure mode). Analysis of dependent modes is more complex. Advanced RBD software such as ReliaSoft's BlockSim can handle and analyze such dependencies, as well as provide more advanced constructs and analyses (see http://www.ReliaSoft.com/BlockSim).

===Failure Modes Configurations===

'''Series Configuration'''

The basic competing failure modes configuration, which has already been discussed, is a series configuration. In a series configuration, the occurrence of any failure mode results in failure for the product.

[[Image:WB.18 series configuration.png|center|400px| ]]

The equation that describes series configuration is:

::<math>R(t)={{R}_{1}}(t)\cdot {{R}_{2}}(t)\cdot ...\cdot {{R}_{n}}(t)\,\!</math>

where <math>n\,\!</math> is the total number of failure modes considered.

'''Parallel'''

In a simple parallel configuration, at least one of the failure modes must not occur for the product to continue operation.

[[Image:WB.18 parallel.png|center|200px| ]]

The equation that describes the parallel configuration is:

::<math>R(t)=1-\underset{i=1}{\overset{n}{\mathop \prod }}\,(1-{{R}_{i}}(t))\,\!</math>

where <math>n\,\!</math> is the total number of failure modes considered.

'''Combination of Series and Parallel'''

While many smaller products can be accurately represented by either a simple series or parallel configuration, there may be larger products that involve both series and parallel configurations in the overall model of the product. Such products can be analyzed by calculating the reliabilities for the individual series and parallel sections and then combining them in the appropriate manner.

[[Image:WB.18 series parallel.png|center|250px| ]]

'''k-out-of-n Parallel Configuration='''

The k-out-of-n configuration is a special case of parallel redundancy. This type of configuration requires that at least <math>k\,\!</math> failure modes do not happen out of the total <math>n\,\!</math> parallel failure modes for the product to succeed.
The simplest case of a k-out-of-n configuration is when the failure modes are independent and identical and have the same failure distribution and uncertainties about the parameters (in other words they are derived from the same test data). In this case, the reliability of the product with such a configuration can be evaluated using the binomial distribution, or:

::<math>R(t)=\overset{n}{\mathop{\underset{r=k}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,\left( \underset{k}{\mathop{\overset{n}{\mathop{{}}}\,}}\, \right){{R}^{r}}(t){{(1-R(t))}^{n-r}}\,\!</math>

In the case where the k-out-of-n failure modes are not identical, other approaches for calculating the reliability must be used (e.g. the event space method). Discussion of these is beyond the scope of this reference. Interested readers can consult the [[RBDs_and_Analytical_System_Reliability#k-out-of-n_Parallel_Configuration| System Analysis Reference book]].

'''Complex Systems'''

In many cases, it is not easy to recognize which components are in series and which are in parallel in a complex system.

[[Image:WB.18 complex systems.png|center|400px| ]]

The previous configuration cannot be broken down into a group of series and parallel configurations. This is primarily due to the fact that failure mode C has two paths leading away from it, whereas B and D have only one. Several methods exist for obtaining the reliability of a complex configuration including the decomposition method, the event space method and the path-tracing method. Discussion of these is beyond the scope of this reference. Interested readers can consult the [[RBDs_and_Analytical_System_Reliability#Complex_Systems|System Analysis Reference book]].

===Complex Failure Modes Example===
{{:Complex Failure Modes Example}}

Resolution III Design Example

2017-08-10T17:45:09Z

Richard House:

<noinclude>{{Banner DOE Examples}}
''This example appears in the [[Two_Level_Factorial_Experiments#Resolution_III_Designs|Experiment Design and Analysis Reference book]]''. </noinclude>

A baker wants to investigate the factors that most affect the taste of the cakes made in his bakery. He chooses to investigate seven factors, each at two levels: flour type (factor <math>A\,\!</math>), conditioner type (factor <math>B\,\!</math>), sugar quantity (factor <math>C\,\!</math>), egg quantity (factor <math>D\,\!</math>), preservative type (factor <math>E\,\!</math>), bake time (factor <math>F\,\!</math>) and bake temperature (factor <math>G\,\!</math>). The baker expects most of these factors and all higher order interactions to be inactive. On the basis of this, he decides to run a screening experiment using a 2 <math>_{\text{III}}^{7-4}\,\!</math> design that requires just 8 runs. The cakes are rated on a scale of 1 to 10. The design properties for the 2 <math>_{\text{III}}^{7-4}\,\!</math> design (with generators <math>D=AB\,\!</math>, <math>E=AC\,\!</math>, <math>F=BC\,\!</math> and <math>G=ABC\,\!</math>) are shown in the following figure.

[[Image:doe7_44.png|center|800px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_8| example]].|link=]]

The resulting design along with the rating of the cakes corresponding to each run is shown in the following figure.

[[Image:doe7_45.png|center|700px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_8| example]].|link=]]

The normal probability plot of effects for the unreplicated design shows main effects <math>B\,\!</math>, <math>C\,\!</math>, <math>D\,\!</math> and <math>G\,\!</math> to be significant, as shown in the next figure.

[[Image:doe7_46.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_8| example]].|link=]]

However, for this design, the following alias relations exist for the main effects:

<center><math>\begin{align}
& A= & A+BD+CE+FG \\
& B= & B+AD+CF+EG \\
& C= & C+AE+BF+DG \\
& D= & D+AB+CG+EF \\
& E= & E+AC+BG+DF \\
& F= & F+BC+AG+DE \\
& G= & G+CD+BE+AF
\end{align}\,\!</math></center>

Based on the alias structure, three separate possible conclusions can be drawn. It can be concluded that effect <math>CD\,\!</math> is active instead of <math>G\,\!</math> so that effects <math>C\,\!</math>, <math>D\,\!</math> and their interaction, <math>CD\,\!</math>, are the significant effects. Another conclusion can be that effect <math>DG\,\!</math> is active instead of <math>C\,\!</math> so that effects <math>D\,\!</math>, <math>G\,\!</math> and their interaction, <math>DG\,\!</math>, are significant. Yet another conclusion can be that effects <math>C\,\!</math>, <math>G\,\!</math> and their interaction, <math>CG\,\!</math>, are significant. To accurately discover the active effects, the baker decides to a run a fold-over of the present design and base his conclusions on the effect values calculated once results from both the designs are available.

The present design is shown next.

[[Image:doe7_47.png|center|612px|Effect values for the experiment in the [[Two_Level_Factorial_Experiments#Example_8| example]].|link=]]

Using the alias relations, the effects obtained from the DOE folio for the present design can be expressed as:

<center><math>\begin{align}
& Effec{{t}_{A}}= & 0.025=A+BD+CE+FG \\
& Effec{{t}_{B}}= & -0.225=B+AD+CF+EG \\
& Effec{{t}_{C}}= & 2.075=C+AE+BF+DG \\
& Effec{{t}_{D}}= & 2.875=D+AB+CG+EF \\
& Effec{{t}_{E}}= & -0.025=E+AC+BG+DF \\
& Effec{{t}_{F}}= & -0.075=F+BC+AG+DE \\
& Effec{{t}_{G}}= & 3.825=G+CD+BE+AF
\end{align}\,\!</math></center>

The fold-over design for the experiment is obtained by reversing the signs of the columns <math>D\,\!</math>, <math>E\,\!</math>, and <math>F\,\!</math>. In a DOE folio, you can fold over a design using the following window.

[[Image:doe7_48_1.png|center|311px|Fold-over design window|link=]]

The resulting design and the corresponding response values obtained are shown in the following figures.

[[Image:doe7_48.png|center|722px|Fold-over design for the experiment in the [[Two_Level_Factorial_Experiments#Example_8| example]].|link=]]

[[Image:doe7_49_1.png|center|700px|Effect values for the fold-over design in the [[Two_Level_Factorial_Experiments#Example_8| example]].|link=]]

Comparing the absolute values of the effects, the active effects are <math>B\,\!</math>, <math>C\,\!</math>, <math>D\,\!</math> and the interaction <math>CD\,\!</math>. Therefore, the most important factors affecting the taste of the cakes in the present case are sugar quantity, egg quantity and their interaction.

Two Level Factorial Experiments

2017-08-10T17:42:11Z

Richard House: /* Example */

{{Template:Doebook|8}}
Two level factorial experiments are factorial experiments in which each factor is investigated at only two levels. The early stages of experimentation usually involve the investigation of a large number of potential factors to discover the "vital few" factors. Two level factorial experiments are used during these stages to quickly filter out unwanted effects so that attention can then be focused on the important ones.

==2''k'' Designs==
The factorial experiments, where all combination of the levels of the factors are run, are usually referred to as ''full factorial experiments''. Full factorial two level experiments are also referred to as <math>{2}^{k}\,\!</math> designs where <math>k\,\!</math> denotes the number of factors being investigated in the experiment. In Weibull++ DOE folios, these designs are referred to as 2 Level Factorial Designs as shown in the figure below.

[[Image:doe7_1.png|center|487px|Selection of full factorial experiments with two levels in Weibull++.|link=]]

A full factorial two level design with <math>k\,\!</math> factors requires <math>{{2}^{k}}\,\!</math> runs for a single replicate. For example, a two level experiment with three factors will require <math>2\times 2\times 2={{2}^{3}}=8\,\!</math> runs. The choice of the two levels of factors used in two level experiments depends on the factor; some factors naturally have two levels. For example, if gender is a factor, then male and female are the two levels. For other factors, the limits of the range of interest are usually used. For example, if temperature is a factor that varies from <math>{45}^{o}C\,\!</math> to <math>{90}^{o}C\,\!</math>, then the two levels used in the <math>{2}^{k}\,\!</math> design for this factor would be <math>{45}^{o}\,\!C\,\!</math> and <math>{90}^{o}\,\!C\,\!</math>.

The two levels of the factor in the <math>{2}^{k}\,\!</math> design are usually represented as <math>-1\,\!</math> (for the first level) and <math>1\,\!</math> (for the second level). Note that this representation is reversed from the coding used in [[General Full Factorial Designs]] for the indicator variables that represent two level factors in ANOVA models. For ANOVA models, the first level of the factor was represented using a value of <math>1\,\!</math> for the indicator variable, while the second level was represented using a value of <math>-1\,\!</math>. For details on the notation used for two level experiments refer to [[Two_Level_Factorial_Experiments#Notation| Notation]].

===The 22 Design===

The simplest of the two level factorial experiments is the <math>{2}^{2}\,\!</math> design where two factors (say factor <math>A\,\!</math> and factor <math>B\,\!</math>) are investigated at two levels. A single replicate of this design will require four runs (<math>{{2}^{2}}=2\times 2=4\,\!</math>) The effects investigated by this design are the two main effects, <math>A\,\!</math> and <math>B,\,\!</math> and the interaction effect <math>AB\,\!</math>. The treatments for this design are shown in figure (a) below. In figure (a), letters are used to represent the treatments. The presence of a letter indicates the high level of the corresponding factor and the absence indicates the low level. For example, (1) represents the treatment combination where all factors involved are at the low level or the level represented by <math>-1\,\!</math> ; <math>a\,\!</math> represents the treatment combination where factor <math>A\,\!</math> is at the high level or the level of <math>1\,\!</math>, while the remaining factors (in this case, factor <math>B\,\!</math>) are at the low level or the level of <math>-1\,\!</math>. Similarly, <math>b\,\!</math> represents the treatment combination where factor <math>B\,\!</math> is at the high level or the level of <math>1\,\!</math>, while factor <math>A\,\!</math> is at the low level and <math>ab\,\!</math> represents the treatment combination where factors <math>A\,\!</math> and <math>B\,\!</math> are at the high level or the level of the 1. Figure (b) below shows the design matrix for the <math>{2}^{2}\,\!</math> design. It can be noted that the sum of the terms resulting from the product of any two columns of the design matrix is zero. As a result the <math>{2}^{2}\,\!</math> design is an ''orthogonal design''. In fact, all <math>{2}^{k}\,\!</math> designs are orthogonal designs. This property of the <math>{2}^{k}\,\!</math> designs offers a great advantage in the analysis because of the simplifications that result from orthogonality. These simplifications are explained later on in this chapter.
The <math>{2}^{2}\,\!</math> design can also be represented geometrically using a square with the four treatment combinations lying at the four corners, as shown in figure (c) below.

[[Image:doe7.2.png|center|400px|The <math>2^2\,\!</math> design. Figure (a) displays the experiment design, (b) displays the design matrix and (c) displays the geometric representation for the design. In Figure (b), the column names I, A, B and AB are used. Column I represents the intercept term. Columns A and B represent the respective factor settings. Column AB represents the interaction and is the product of columns A and B.]]
 

===The 23 Design===

The <math>{2}^{3}\,\!</math> design is a two level factorial experiment design with three factors (say factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>). This design tests three (<math>k=3\,\!</math>) main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math> ; three (<math>(_{2}^{k})=\,\!</math> <math>(_{2}^{3})=3\,\!</math>) two factor interaction effects, <math>AB\,\!</math>, <math>BC\,\!</math>, <math>AC\,\!</math> ; and one (<math>(_{3}^{k})=\,\!</math> <math>(_{3}^{3})=1\,\!</math>) three factor interaction effect, <math>ABC\,\!</math>. The design requires eight runs per replicate. The eight treatment combinations corresponding to these runs are <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math>, <math>ab\,\!</math>, <math>c\,\!</math>, <math>ac\,\!</math>, <math>bc\,\!</math> and <math>abc\,\!</math>. Note that the treatment combinations are written in such an order that factors are introduced one by one with each new factor being combined with the preceding terms. This order of writing the treatments is called the ''standard order'' or ''Yates' order''. The <math>{2}^{3}\,\!</math> design is shown in figure (a) below. The design matrix for the <math>{2}^{3}\,\!</math> design is shown in figure (b). The design matrix can be constructed by following the standard order for the treatment combinations to obtain the columns for the main effects and then multiplying the main effects columns to obtain the interaction columns.

[[Image:doe7.3.png|center|324px|The <math>2^3\,\!</math> design. Figure (a) shows the experiment design and (b) shows the design matrix.]]

[[Image:doe7.4.png|center|290px|Geometric representation of the <math>2^3\,\!</math> design.]]

The <math>{2}^{3}\,\!</math> design can also be represented geometrically using a cube with the eight treatment combinations lying at the eight corners as shown in the figure above.

==Analysis of 2''k'' Designs==

The <math>{2}^{k}\,\!</math> designs are a special category of the factorial experiments where all the factors are at two levels. The fact that these designs contain factors at only two levels and are orthogonal greatly simplifies their analysis even when the number of factors is large. The use of <math>{2}^{k}\,\!</math> designs in investigating a large number of factors calls for a revision of the notation used previously for the ANOVA models. The case for revised notation is made stronger by the fact that the ANOVA and multiple linear regression models are identical for <math>{2}^{k}\,\!</math> designs because all factors are only at two levels. Therefore, the notation of the regression models is applied to the ANOVA models for these designs, as explained next.

===Notation===

Based on the notation used in [[General Full Factorial Designs]], the ANOVA model for a two level factorial experiment with three factors would be as follows:

::<math>\begin{align}
& Y= & \mu +{{\tau }_{1}}\cdot {{x}_{1}}+{{\delta }_{1}}\cdot {{x}_{2}}+{{(\tau \delta )}_{11}}\cdot {{x}_{1}}{{x}_{2}}+{{\gamma }_{1}}\cdot {{x}_{3}} \\
& & +{{(\tau \gamma )}_{11}}\cdot {{x}_{1}}{{x}_{3}}+{{(\delta \gamma )}_{11}}\cdot {{x}_{2}}{{x}_{3}}+{{(\tau \delta \gamma )}_{111}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where:
 
:• <math>\mu \,\!</math> represents the overall mean
:• <math>{{\tau }_{1}}\,\!</math> represents the independent effect of the first factor (factor <math>A\,\!</math>) out of the two effects <math>{{\tau }_{1}}\,\!</math> and <math>{{\tau }_{2}}\,\!</math>
:• <math>{{\delta }_{1}}\,\!</math> represents the independent effect of the second factor (factor <math>B\,\!</math>) out of the two effects <math>{{\delta }_{1}}\,\!</math> and <math>{{\delta }_{2}}\,\!</math>
:• <math>{{(\tau \delta )}_{11}}\,\!</math> represents the independent effect of the interaction <math>AB\,\!</math> out of the other interaction effects
:• <math>{{\gamma }_{1}}\,\!</math> represents the effect of the third factor (factor <math>C\,\!</math>) out of the two effects <math>{{\gamma }_{1}}\,\!</math> and <math>{{\gamma }_{2}}\,\!</math>
:• <math>{{(\tau \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>AC\,\!</math> out of the other interaction effects
:• <math>{{(\delta \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>BC\,\!</math> out of the other interaction effects
:• <math>{{(\tau \delta \gamma )}_{111}}\,\!</math> represents the effect of the interaction <math>ABC\,\!</math> out of the other interaction effects
and <math>\epsilon \,\!</math> is the random error term.

 
The notation for a linear regression model having three predictor variables with interactions is:

::<math>\begin{align}
& Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& & +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

The notation for the regression model is much more convenient, especially for the case when a large number of higher order interactions are present. In two level experiments, the ANOVA model requires only one indicator variable to represent each factor for both qualitative and quantitative factors. Therefore, the notation for the multiple linear regression model can be applied to the ANOVA model of the experiment that has all the factors at two levels. For example, for the experiment of the ANOVA model given above, <math>{{\beta }_{0}}\,\!</math> can represent the overall mean instead of <math>\mu \,\!</math>, and <math>{{\beta }_{1}}\,\!</math> can represent the independent effect, <math>{{\tau }_{1}}\,\!</math>, of factor <math>A\,\!</math>. Other main effects can be represented in a similar manner. The notation for the interaction effects is much more simplified (e.g., <math>{{\beta }_{123}}\,\!</math> can be used to represent the three factor interaction effect, <math>{{(\tau \beta \gamma )}_{111}}\,\!</math>).

As mentioned earlier, it is important to note that the coding for the indicator variables for the ANOVA models of two level factorial experiments is reversed from the coding followed in [[General Full Factorial Designs]]. Here <math>-1\,\!</math> represents the first level of the factor while <math>1\,\!</math> represents the second level. This is because for a two level factor a single variable is needed to represent the factor for both qualitative and quantitative factors. For quantitative factors, using <math>-1\,\!</math> for the first level (which is the low level) and 1 for the second level (which is the high level) keeps the coding consistent with the numerical value of the factors. The change in coding between the two coding schemes does not affect the analysis except that signs of the estimated effect coefficients will be reversed (i.e., numerical values of <math>{{\hat{\tau }}_{1}}\,\!</math>, obtained based on the coding of [[General Full Factorial Designs]], and <math>{{\hat{\beta }}_{1}}\,\!</math>, obtained based on the new coding, will be the same but their signs would be opposite).

::<math>\begin{align}
& & \text{Factor }A\text{ Coding (two level factor)} \\
& &
\end{align}\,\!</math>

::<math>\begin{matrix}
\text{Previous Coding} & {} & {} & {} & \text{Coding for }{{\text{2}}^{k}}\text{ Designs} \\
{} & {} & {} & {} & {} \\
Effect\text{ }{{\tau }_{1}}\ \ :\ \ {{x}_{1}}=1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{1}}\text{ (or }-{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=-1\text{ } \\
Effect\text{ }{{\tau }_{2}}\ \ :\ \ {{x}_{1}}=-1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{2}}\text{ (or }{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=1\text{ } \\
\end{matrix}\,\!</math>

In summary, the ANOVA model for the experiments with all factors at two levels is different from the ANOVA models for other experiments in terms of the notation in the following two ways:
 

:• The notation of the regression models is used for the effect coefficients.
:• The coding of the indicator variables is reversed.

===Special Features===

Consider the design matrix, <math>X\,\!</math>, for the <math>{2}^{3}\,\!</math> design discussed above. The (<math>{{X}^{\prime }}X\,\!</math>) <math>^{-1}\,\!</math> matrix is:

<center><math>{{({{X}^{\prime }}X)}^{-1}}=\left[ \begin{matrix}
0.125 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0.125 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0.125 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0.125 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0.125 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0.125 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0.125 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.125 \\
\end{matrix} \right]\,\!</math></center>

Notice that, due to the orthogonal design of the <math>X\,\!</math> matrix, the <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> has been simplified to a diagonal matrix which can be written as:

::<math>\begin{align}
{{({{X}^{\prime }}X)}^{-1}}= & 0.125\cdot I = & \frac{1}{8}\cdot I = & \frac{1}{{{2}^{3}}}\cdot I
\end{align}\,\!</math>

where <math>I\,\!</math> represents the identity matrix of the same order as the design matrix, <math>X\,\!</math>. Since there are eight observations per replicate of the <math>{2}^{3}\,\!</math> design, the <math>(X\,\!</math> ' <math>X{{)}^{-1}}\,\!</math> matrix for <math>m\,\!</math> replicates of this design can be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{3}}\cdot m)}\cdot I\,\!</math>

The <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> matrix for any <math>{2}^{k}\,\!</math> design can now be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{k}}\cdot m)}\cdot I\,\!</math>

Then the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is:

::<math>\begin{align}
C= & {{{\hat{\sigma }}}^{2}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & M{{S}_{E}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & \frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}\cdot I
\end{align}\,\!</math>

Note that the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is also a diagonal matrix. Therefore, the estimated effect coefficients (<math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math>, <math>{{\beta }_{12}},\,\!</math> etc.) for these designs are uncorrelated. This implies that the terms in the <math>{2}^{k}\,\!</math> design (main effects, interactions) are independent of each other. Consequently, the extra sum of squares for each of the terms in these designs is independent of the sequence of terms in the model, and also independent of the presence of other terms in the model. As a result the sequential and partial sum of squares for the terms are identical for these designs and will always add up to the model sum of squares. Multicollinearity is also not an issue for these designs.

It can also be noted from the equation given above, that in addition to the <math>C\,\!</math> matrix being diagonal, all diagonal elements of the <math>C\,\!</math> matrix are identical. This means that the variance (or its square root, the standard error) of all estimated effect coefficients are the same. The standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>, for all the coefficients is:

::<math>\begin{align}
se({{{\hat{\beta }}}_{j}})= & \sqrt{{{C}_{jj}}} = & \sqrt{\frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}}\text{ }for\text{ }all\text{ }j
\end{align}\,\!</math>

This property is used to construct the normal probability plot of effects in <math>{2}^{k}\,\!</math> designs and identify significant effects using graphical techniques. For details on the normal probability plot of effects in a Weibull++ DOE folio, refer to [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]].

====Example====
To illustrate the analysis of a full factorial <math>{2}^{k}\,\!</math> design, consider a three factor experiment to investigate the effect of honing pressure, number of strokes and cycle time on the surface finish of automobile brake drums. Each of these factors is investigated at two levels. The honing pressure is investigated at levels of 200 <math>psi\,\!</math> and 400 <math>psi\,\!</math>, the number of strokes used is 3 and 5 and the two levels of the cycle time are 3 and 5 seconds. The design for this experiment is set up in a Weibull++ DOE folio as shown in the first two following figures. It is decided to run two replicates for this experiment. The surface finish data collected from each run (using randomization) and the complete design is shown in the third following figure. The analysis of the experiment data is explained next.

[[Image:doe7_5.png|center|877px|Design properties for the experiment in the example.|link=]]

[[Image:doe7_6.png|center|859px|Design summary for the experiment in the example.|link=]]

[[Image:doe7_7.png|center|778px|Experiment design for the example to investigate the surface finish of automobile brake drums.|link=]]

The applicable model using the notation for <math>{2}^{k}\,\!</math> designs is:

::<math>\begin{align}
Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where the indicator variable, <math>{{x}_{1,}}\,\!</math> represents factor <math>A\,\!</math> (honing pressure), <math>{{x}_{1}}=-1\,\!</math> represents the low level of 200 <math>psi\,\!</math> and <math>{{x}_{1}}=1\,\!</math> represents the high level of 400 <math>psi\,\!</math>. Similarly, <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> represent factors <math>B\,\!</math> (number of strokes) and <math>C\,\!</math> (cycle time), respectively. <math>{{\beta }_{0}}\,\!</math> is the overall mean, while <math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math> and <math>{{\beta }_{3}}\,\!</math> are the effect coefficients for the main effects of factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. <math>{{\beta }_{12}}\,\!</math>, <math>{{\beta }_{13}}\,\!</math> and <math>{{\beta }_{23}}\,\!</math> are the effect coefficients for the <math>AB\,\!</math>, <math>AC\,\!</math> and <math>BC\,\!</math> interactions, while <math>{{\beta }_{123}}\,\!</math> represents the <math>ABC\,\!</math> interaction.

 
If the subscripts for the run (<math>i\,\!</math> ; <math>i=\,\!</math> 1 to 8) and replicates (<math>j\,\!</math> ; <math>j=\,\!</math> 1,2) are included, then the model can be written as:

::<math>\begin{align}
{{Y}_{ij}}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{ij1}}+{{\beta }_{2}}\cdot {{x}_{ij2}}+{{\beta }_{12}}\cdot {{x}_{ij1}}{{x}_{ij2}}+{{\beta }_{3}}\cdot {{x}_{ij3}} \\
& +{{\beta }_{13}}\cdot {{x}_{ij1}}{{x}_{ij3}}+{{\beta }_{23}}\cdot {{x}_{ij2}}{{x}_{ij3}}+{{\beta }_{123}}\cdot {{x}_{ij1}}{{x}_{ij2}}{{x}_{ij3}}+{{\epsilon }_{ij}}
\end{align}\,\!</math>

To investigate how the given factors affect the response, the following hypothesis tests need to be carried:

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{1}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{1}}\ne 0\,\!</math>

This test investigates the main effect of factor <math>A\,\!</math> (honing pressure). The statistic for this test is:

::<math>{{({{F}_{0}})}_{A}}=\frac{M{{S}_{A}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{A}}\,\!</math> is the mean square for factor <math>A\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other main effects, <math>B\,\!</math> and <math>C\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{12}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{12}}\ne 0\,\!</math>

This test investigates the two factor interaction <math>AB\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{AB}}=\frac{M{{S}_{AB}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the interaction <math>AB\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other two factor interactions, <math>AC\,\!</math> and <math>BC\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{123}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{123}}\ne 0\,\!</math>

This test investigates the three factor interaction <math>ABC\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{ABC}}=\frac{M{{S}_{ABC}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{ABC}}\,\!</math> is the mean square for the interaction <math>ABC\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square.
To calculate the test statistics, it is convenient to express the ANOVA model in the form <math>y=X\beta +\epsilon \,\!</math>.

====Expression of the ANOVA Model as <math>y=X\beta +\epsilon \,\!</math>====

In matrix notation, the ANOVA model can be expressed as:

::<math>y=X\beta +\epsilon \,\!</math>

where:

<center><math>y=\left[ \begin{matrix}
{{Y}_{11}} \\
{{Y}_{21}} \\
. \\
{{Y}_{81}} \\
{{Y}_{12}} \\
. \\
{{Y}_{82}} \\
\end{matrix} \right]=\left[ \begin{matrix}
90 \\
90 \\
. \\
90 \\
86 \\
. \\
80 \\
\end{matrix} \right]\text{ }X=\left[ \begin{matrix}
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\end{matrix} \right]\,\!</math></center>

<center><math>\beta =\left[ \begin{matrix}
{{\beta }_{0}} \\
{{\beta }_{1}} \\
{{\beta }_{2}} \\
{{\beta }_{12}} \\
{{\beta }_{3}} \\
{{\beta }_{13}} \\
{{\beta }_{23}} \\
{{\beta }_{123}} \\
\end{matrix} \right]\text{ }\epsilon =\left[ \begin{matrix}
{{\epsilon }_{11}} \\
{{\epsilon }_{21}} \\
. \\
{{\epsilon }_{81}} \\
{{\epsilon }_{12}} \\
. \\
. \\
{{\epsilon }_{82}} \\
\end{matrix} \right]\,\!</math></center>

====Calculation of the Extra Sum of Squares for the Factors====

Knowing the matrices <math>y\,\!</math>, <math>X\,\!</math> and <math>\beta \,\!</math>, the extra sum of squares for the factors can be calculated. These are used to calculate the mean squares that are used to obtain the test statistics. Since the experiment design is orthogonal, the partial and sequential extra sum of squares are identical. The extra sum of squares for each effect can be calculated as shown next. As an example, the extra sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & Model\text{ }Sum\text{ }of\text{ }Squares - Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }main\text{ }effect\text{ }of\text{ }A \\
= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }A}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }A}}={{X}_{\tilde{\ }A}}{{(X_{\tilde{\ }A}^{\prime }{{X}_{\tilde{\ }A}})}^{-1}}X_{\tilde{\ }A}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }A}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the main effect of factor <math>A\,\!</math>. Thus, the sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y \\
= & 654.4375-549.375 \\
= & 105.0625
\end{align}\,\!</math>

Similarly, the extra sum of squares for the interaction effect <math>AB\,\!</math> is:

::<math>\begin{align}
S{{S}_{AB}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }AB}}-(1/16)J]y \\
= & 654.4375-636.375 \\
= & 18.0625
\end{align}\,\!</math>

The extra sum of squares for other effects can be obtained in a similar manner.

====Calculation of the Test Statistics====

Knowing the extra sum of squares, the test statistic for the effects can be calculated. For example, the test statistic for the interaction <math>AB\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{AB}}= & \frac{M{{S}_{AB}}}{M{{S}_{E}}} \\
= & \frac{S{{S}_{AB}}/dof(S{{S}_{AB}})}{S{{S}_{E}}/dof(S{{S}_{E}})} \\
= & \frac{18.0625/1}{147.5/8} \\
= & 0.9797
\end{align}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the <math>AB\,\!</math> interaction and <math>M{{S}_{E}}\,\!</math> is the error mean square. The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{AB}}=0.9797\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{AB}}) \\
= & 1-0.6487 \\
= & 0.3513
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that the interaction between honing pressure and number of strokes does not affect the surface finish of the brake drums. Tests for other effects can be carried out in a similar manner. The results are shown in the ANOVA Table in the following figure. The values S, R-sq and R-sq(adj) in the figure indicate how well the model fits the data. The value of S represents the standard error of the model, R-sq represents the coefficient of multiple determination and R-sq(adj) represents the adjusted coefficient of multiple determination. For details on these values refer to [[Multiple_Linear_Regression_Analysis|Multiple Linear Regression Analysis]].

[[Image:doe7_8.png|center|799px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

====Calculation of Effect Coefficients====

The estimate of effect coefficients can also be obtained:

<center><math>\begin{align}
\hat{\beta }= & {{({{X}^{\prime }}X)}^{-1}}{{X}^{\prime }}y \\
= & \left[ \begin{matrix}
86.4375 \\
2.5625 \\
-4.9375 \\
1.0625 \\
-1.0625 \\
2.4375 \\
-1.3125 \\
-0.1875 \\
\end{matrix} \right]
\end{align}\,\!</math></center>

[[Image:doe7_9.png|center|800px|Regression Information table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

The coefficients and related results are shown in the Regression Information table above. In the table, the Effect column displays the effects, which are simply twice the coefficients. The Standard Error column displays the standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>. The Low CI and High CI columns display the confidence interval on the coefficients. The interval shown is the 90% interval as the significance is chosen as 0.1. The T Value column displays the <math>t\,\!</math> statistic, <math>{{t}_{0}}\,\!</math>, corresponding to the coefficients. The P Value column displays the <math>p\,\!</math> value corresponding to the <math>t\,\!</math> statistic. (For details on how these results are calculated, refer to [[General Full Factorial Designs]]). Plots of residuals can also be obtained from the DOE folio to ensure that the assumptions related to the ANOVA model are not violated.

====Model Equation====

From the analysis results in the above figure within [[Two_Level_Factorial_Experiments#Calculation_of_Effect_Coefficients|calculation of effect coefficients]] section, it is seen that effects <math>A\,\!</math>, <math>B\,\!</math> and <math>AC\,\!</math> are significant. In a DOE folio, the <math>p\,\!</math> values for the significant effects are displayed in red in the ANOVA Table for easy identification. Using the values of the estimated effect coefficients, the model for the present <math>{2}^{3}\,\!</math> design in terms of the coded values can be written as:

::<math>\begin{align}
\hat{y}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}} \\
= & 86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+2.4375{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

To make the model hierarchical, the main effect, <math>C\,\!</math>, needs to be included in the model (because the interaction <math>AC\,\!</math> is included in the model). The resulting model is:

::<math>\hat{y}=86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+1.0625{{x}_{3}}+2.4375{{x}_{1}}{{x}_{3}}\,\!</math>

This equation can be viewed in a DOE folio, as shown in the following figure, using the Show Analysis Summary icon in the Control Panel. The equation shown in the figure will match the hierarchical model once the required terms are selected using the Select Effects icon.

[[Image:doe7_10.png|center|557px|The model equation for the experiment of the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

==Replicated and Repeated Runs==

In the case of replicated experiments, it is important to note the difference between replicated runs and repeated runs. Both repeated and replicated runs are multiple response readings taken at the same factor levels. However, repeated runs are response observations taken at the same time or in succession. Replicated runs are response observations recorded in a random order. Therefore, replicated runs include more variation than repeated runs. For example, a baker, who wants to investigate the effect of two factors on the quality of cakes, will have to bake four cakes to complete one replicate of a <math>{2}^{2}\,\!</math> design. Assume that the baker bakes eight cakes in all. If, for each of the four treatments of the <math>{2}^{2}\,\!</math> design, the baker selects one treatment at random and then bakes two cakes for this treatment at the same time then this is a case of two repeated runs. If, however, the baker bakes all the eight cakes randomly, then the eight cakes represent two sets of replicated runs.
For repeated measurements, the average values of the response for each treatment should be entered into a DOE folio as shown in the following figure (a) when the two cakes for a particular treatment are baked together. For replicated measurements, when all the cakes are baked randomly, the data is entered as shown in the following figure (b).

[[Image:doe7.11.png|center|300px|Data entry for repeated and replicated runs. Figure (a) shows repeated runs and (b) shows replicated runs.]]

==Unreplicated 2''k'' Designs==

If a factorial experiment is run only for a single replicate then it is not possible to test hypotheses about the main effects and interactions as the error sum of squares cannot be obtained. This is because the number of observations in a single replicate equals the number of terms in the ANOVA model. Hence the model fits the data perfectly and no degrees of freedom are available to obtain the error sum of squares.

However, sometimes it is only possible to run a single replicate of the <math>{2}^{k}\,\!</math> design because of constraints on resources and time. In the absence of the error sum of squares, hypothesis tests to identify significant factors cannot be conducted. A number of methods of analyzing information obtained from unreplicated <math>{2}^{k}\,\!</math> designs are available. These include pooling higher order interactions, using the normal probability plot of effects or including center point replicates in the design.

===Pooling Higher Order Interactions===

One of the ways to deal with unreplicated <math>{2}^{k}\,\!</math> designs is to use the sum of squares of some of the higher order interactions as the error sum of squares provided these higher order interactions can be assumed to be insignificant. By dropping some of the higher order interactions from the model, the degrees of freedom corresponding to these interactions can be used to estimate the error mean square. Once the error mean square is known, the test statistics to conduct hypothesis tests on the factors can be calculated.

===Normal Probability Plot of Effects===
Another way to use unreplicated <math>{2}^{k}\,\!</math> designs to identify significant effects is to construct the normal probability plot of the effects. As mentioned in [[Two_Level_Factorial_Experiments#Special_Features| Special Features]], the standard error for all effect coefficients in the <math>{2}^{k}\,\!</math> designs is the same. Therefore, on a normal probability plot of effect coefficients, all non-significant effect coefficients (with <math>\beta =0\,\!</math>) will fall along the straight line representative of the normal distribution, N(<math>0,{{\sigma }^{2}}/({{2}^{k}}\cdot m)\,\!</math>). Effect coefficients that show large deviations from this line will be significant since they do not come from this normal distribution. Similarly, since effects <math>=2\times \,\!</math> effect coefficients, all non-significant effects will also follow a straight line on the normal probability plot of effects. For replicated designs, the Effects Probability plot of a DOE folio plots the normalized effect values (or the T Values) on the standard normal probability line, N(0,1). However, in the case of unreplicated <math>{2}^{k}\,\!</math> designs, <math>{{\sigma }^{2}}\,\!</math> remains unknown since <math>M{{S}_{E}}\,\!</math> cannot be obtained. Lenth's method is used in this case to estimate the variance of the effects. For details on Lenth's method, please refer to [[DOE References| Montgomery (2001)]]. The DOE folio then uses this variance value to plot effects along the N(0, Lenth's effect variance) line. The
method is illustrated in the following example.

====Example====

Vinyl panels, used as instrument panels in a certain automobile, are seen to develop defects after a certain amount of time. To investigate the issue, it is decided to carry out a two level factorial experiment. Potential factors to be investigated in the experiment are vacuum rate (factor <math>A\,\!</math>), material temperature (factor <math>B\,\!</math>), element intensity (factor <math>C\,\!</math>) and pre-stretch (factor <math>D\,\!</math>). The two levels of the factors used in the experiment are as shown in below.

[[Image:doet7.1.png|center|300px|Factors to investigate defects in vinyl panels.]]

With a <math>{2}^{4}\,\!</math> design requiring 16 runs per replicate it is only feasible for the manufacturer to run a single replicate.

The experiment design and data, collected as percent defects, are shown in the following figure. Since the present experiment design contains only a single replicate, it is not possible to obtain an estimate of the error sum of squares, <math>S{{S}_{E}}\,\!</math>. It is decided to use the normal probability plot of effects to identify the significant effects. The effect values for each term are obtained as shown in the following figure.

[[Image:doe7_13.png|center|650px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

Lenth's method uses these values to estimate the variance. As described in [[DOE_References|[Lenth, 1989]]], if all effects are arranged in ascending order, using their absolute values, then <math>{{s}_{0}}\,\!</math> is defined as 1.5 times the median value:

::<math>\begin{align}
{{s}_{0}}= & 1.5\cdot median(\left| effect \right|) \\
= & 1.5\cdot 2 \\
= & 3
\end{align}\,\!</math>

Using <math>{{s}_{0}}\,\!</math>, the "pseudo standard error" (<math>PSE\,\!</math>) is calculated as 1.5 times the median value of all effects that are less than 2.5 <math>{{s}_{0}}\,\!</math> :

::<math>\begin{align}
PSE= & 1.5\cdot median(\left| effect \right|\ \ :\ \ \left| effect \right|<2.5{{s}_{0}}) \\
= & 1.5\cdot 1.5 \\
= & 2.25
\end{align}\,\!</math>

Using <math>PSE\,\!</math> as an estimate of the effect variance, the effect variance is 2.25. Knowing the effect variance, the normal probability plot of effects for the present unreplicated experiment can be constructed as shown in the following figure. The line on this plot is the line N(0, 2.25). The plot shows that the effects <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math> do not follow the distribution represented by this line. Therefore, these effects are significant.

The significant effects can also be identified by comparing individual effect values to the margin of error or the threshold value using the pareto chart (see the third following figure). If the required significance is 0.1, then:

::<math>margin\text{ }of\text{ }error={{t}_{\alpha /2,d}}\cdot PSE\,\!</math>

The <math>t\,\!</math> statistic, <math>{{t}_{\alpha /2,d}}\,\!</math>, is calculated at a significance of <math>\alpha /2\,\!</math> (for the two-sided hypothesis) and degrees of freedom <math>d=(\,\!</math> number of effects <math>)/3\,\!</math>. Thus:

::<math>\begin{align}
margin\text{ }of\text{ }error= & {{t}_{0.05,5}}\cdot PSE \\
= & 2.015\cdot 2.25 \\
= & 4.534
\end{align}\,\!</math>

The value of 4.534 is shown as the critical value line in the third following figure. All effects with absolute values greater than the margin of error can be considered to be significant. These effects are <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math>. Therefore, the vacuum rate, the pre-stretch and their interaction have a significant effect on the defects of the vinyl panels.

[[Image:doe7_14.png|center|800px|Effect values for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_15.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_16.png|center|650px|Pareto chart for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

===Center Point Replicates===

Another method of dealing with unreplicated <math>{2}^{k}\,\!</math> designs that only have quantitative factors is to use replicated runs at the center point. The center point is the response corresponding to the treatment exactly midway between the two levels of all factors. Running multiple replicates at this point provides an estimate of pure error. Although running multiple replicates at any treatment level can provide an estimate of pure error, the other advantage of running center point replicates in the <math>{2}^{k}\,\!</math> design is in checking for the presence of curvature. The test for curvature investigates whether the model between the response and the factors is linear and is discussed in [[Two_Level_Factorial_Experiments#Using_Center_Point_Replicates_to_Test_Curvature| Center Pt. Replicates to Test Curvature]].

====Example: Use Center Point to Get Pure Error====

Consider a <math>{2}^{2}\,\!</math> experiment design to investigate the effect of two factors, <math>A\,\!</math> and <math>B\,\!</math>, on a certain response. The energy consumed when the treatments of the <math>{2}^{2}\,\!</math> design are run is considerably larger than the energy consumed for the center point run (because at the center point the factors are at their middle levels). Therefore, the analyst decides to run only a single replicate of the design and augment the design by five replicated runs at the center point as shown in the following figure. The design properties for this experiment are shown in the second following figure. The complete experiment design is shown in the third following figure. The center points can be used in the identification of significant effects as shown next.

[[Image:doe7.17.png||center|300px|<math>2^2\,\!</math> design augmented by five center point runs.]]
[[Image:doe7_18.png|center|537px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

[[Image:doe7_19.png|center|630px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

Since the present <math>{2}^{2}\,\!</math> design is unreplicated, there are no degrees of freedom available to calculate the error sum of squares. By augmenting this design with five center points, the response values at the center points, <math>y_{i}^{c}\,\!</math>, can be used to obtain an estimate of pure error, <math>S{{S}_{PE}}\,\!</math>. Let <math>{{\bar{y}}^{c}}\,\!</math> represent the average response for the five replicates at the center. Then:

::<math>S{{S}_{PE}}=Sum\text{ }of\text{ }Squares\text{ }for\text{ }center\text{ }points\,\!</math>

::<math>\begin{align}
S{{S}_{PE}}= & \underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}} \\
= & {{(25.2-25.26)}^{2}}+...+{{(25.3-25.26)}^{2}} \\
= & 0.052
\end{align}\,\!</math>

Then the corresponding mean square is:

::<math>\begin{align}
M{{S}_{PE}}= & \frac{S{{S}_{PE}}}{degrees\text{ }of\text{ }freedom} \\
= & \frac{0.052}{5-1} \\
= & 0.013
\end{align}\,\!</math>

Alternatively, <math>M{{S}_{PE}}\,\!</math> can be directly obtained by calculating the variance of the response values at the center points:

::<math>\begin{align}
M{{S}_{PE}}= & {{s}^{2}} \\
= & \frac{\underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}}}{5-1}
\end{align}\,\!</math>

Once <math>M{{S}_{PE}}\,\!</math> is known, it can be used as the error mean square, <math>M{{S}_{E}}\,\!</math>, to carry out the test of significance for each effect. For example, to test the significance of the main effect of factor <math>A,\,\!</math> the sum of squares corresponding to this effect is obtained in the usual manner by considering only the four runs of the original <math>{2}^{2}\,\!</math> design.

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/4)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/4)J]y \\
= & 0.5625
\end{align}\,\!</math>

Then, the test statistic to test the significance of the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{A}}= & \frac{M{{S}_{A}}}{M{{S}_{E}}} \\
= & \frac{0.5625/1}{0.052/4} \\
= & 43.2692
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{A}}=43.2692\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{A}}) \\
= & 1-0.9972 \\
= & 0.0028
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value < 0.1, it can be concluded that the main effect of factor <math>A\,\!</math> significantly affects the response. This result is displayed in the ANOVA table as shown in the following figure. Test for the significance of other factors can be carried out in a similar manner.

[[Image:doe7_20.png|center|649px|Results for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

===Using Center Point Replicates to Test Curvature===

Center point replicates can also be used to check for curvature in replicated or unreplicated <math>{2}^{k}\,\!</math> designs. The test for curvature investigates whether the model between the response and the factors is linear. The way DOE++ handles center point replicates is similar to its handling of blocks. The center point replicates are treated as an additional factor in the model. The factor is labeled as Curvature in the results of the DOE folio. If Curvature turns out to be a significant factor in the results, then this indicates the presence of curvature in the model.

====Example: Use Center Point to Test Curvature====

To illustrate the use of center point replicates in testing for curvature, consider again the data of the single replicate <math>{2}^{2}\,\!</math> experiment from a preceding figure(labeled "<math>2^2</math> design augmented by five center point runs"). Let <math>{{x}_{1}}\,\!</math> be the indicator variable to indicate if the run is a center point:

::<math>\begin{matrix}
{{x}_{1}}=0 & {} & \text{Center point run} \\
{{x}_{1}}=1 & {} & \text{Other run} \\
\end{matrix}\,\!</math>

If <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> are the indicator variables representing factors <math>A\,\!</math> and <math>B\,\!</math>, respectively, then the model for this experiment is:

::<math>Y={{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}\,\!</math>

To investigate the presence of curvature, the following hypotheses need to be tested:

::<math>\begin{align}
& {{H}_{0}}: & {{\beta }_{1}}=0\text{ (Curvature is absent)} \\
& {{H}_{1}}: & {{\beta }_{1}}\ne 0
\end{align}\,\!</math>

The test statistic to be used for this test is:

::<math>{{({{F}_{0}})}_{curvature}}=\frac{M{{S}_{curvature}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{curvature}}\,\!</math> is the mean square for Curvature and <math>M{{S}_{E}}\,\!</math> is the error mean square.

'''Calculation of the Sum of Squares'''

The <math>X\,\!</math> matrix and <math>y\,\!</math> vector for this experiment are:

<center><math>X=\left[ \begin{matrix}
1 & 1 & -1 & -1 & 1 \\
1 & 1 & 1 & -1 & -1 \\
1 & 1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
\end{matrix} \right]\text{ }y=\left[ \begin{matrix}
24.6 \\
25.4 \\
25.0 \\
25.7 \\
25.2 \\
25.3 \\
25.4 \\
25.1 \\
25.3 \\
\end{matrix} \right]\,\!</math></center>

The sum of squares can now be calculated. For example, the error sum of squares is:

::<math>\begin{align}
& S{{S}_{E}}= & {{y}^{\prime }}[I-H]y \\
& = & 0.052
\end{align}\,\!</math>

where <math>I\,\!</math> is the identity matrix and <math>H\,\!</math> is the hat matrix. It can be seen that this is equal to <math>S{{S}_{PE\text{ }}}\,\!</math> (the sum of squares due to pure error) because of the replicates at the center point, as obtained in the [[Two_Level_Factorial_Experiments#Example_3| example]]. The number of degrees of freedom associated with <math>S{{S}_{E}}\,\!</math>, <math>dof(S{{S}_{E}})\,\!</math> is four. The extra sum of squares corresponding to the center point replicates (or Curvature) is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & Model\text{ }Sum\text{ }of\text{ }Squares- \\
& & Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }center\text{ }point \\
& = & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Curvature}}-(1/9)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }Curvature}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }Curvature}}={{X}_{\tilde{\ }Curv}}{{(X_{\tilde{\ }Curv}^{\prime }{{X}_{\tilde{\ }Curv}})}^{-1}}X_{\tilde{\ }Curv}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }Curv}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the center point. Thus, the extra sum of squares corresponding to Curvature is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Center}}-(1/9)J]y \\
& = & 0.7036-0.6875 \\
& = & 0.0161
\end{align}\,\!</math>

This extra sum of squares can be used to test for the significance of curvature. The corresponding mean square is:

::<math>\begin{align}
& M{{S}_{Curvature}}= & \frac{Sum\text{ }of\text{ }squares\text{ }corresponding\text{ }to\text{ }Curvature}{degrees\text{ }of\text{ }freedom} \\
& = & \frac{0.0161}{1} \\
& = & 0.0161
\end{align}\,\!</math>

'''Calculation of the Test Statistic'''

Knowing the mean squares, the statistic to check the significance of curvature can be calculated.

::<math>\begin{align}
& {{({{f}_{0}})}_{Curvature}}= & \frac{M{{S}_{Curvature}}}{M{{S}_{E}}} \\
& = & \frac{0.0161/1}{0.052/4} \\
& = & 1.24
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{Curvature}}=1.24\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and four degrees of freedom in the denominator is:

::<math>\begin{align}
& p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{Curvature}}) \\
& = & 1-0.6713 \\
& = & 0.3287
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that curvature does not exist for this design. This results is shown in the ANOVA table in the figure above. The surface of the fitted model based on these results, along with the observed response values, is shown in the figure below.

[[Image:doe7.21.png|center|400px|Model surface and observed response values for the design in the [[Two_Level_Factorial_Experiments#Example_4| example]].]]

 

==Blocking in 2k Designs==

Blocking can be used in the <math>{2}^{k}\,\!</math> designs to deal with cases when replicates cannot be run under identical conditions. Randomized complete block designs that were discussed in [[Randomization and Blocking in DOE]] for factorial experiments are also applicable here. At times, even with just two levels per factor, it is not possible to run all treatment combinations for one replicate of the experiment under homogeneous conditions. For example, each replicate of the <math>{2}^{2}\,\!</math> design requires four runs. If each run requires two hours and testing facilities are available for only four hours per day, two days of testing would be required to run one complete replicate. Blocking can be used to separate the treatment runs on the two different days. Blocks that do not contain all treatments of a replicate are called incomplete blocks. In incomplete block designs, the block effect is confounded with certain effect(s) under investigation. For the <math>{2}^{2}\,\!</math> design assume that treatments <math>(1)\,\!</math> and <math>ab\,\!</math> were run on the first day and treatments <math>a\,\!</math> and <math>b\,\!</math> were run on the second day. Then, the incomplete block design for this experiment is:

::<math>\begin{matrix}
\text{Block 1} & {} & \text{Block 2} \\
\left[ \begin{matrix}
(1) \\
ab \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
b \\
\end{matrix} \right] \\
\end{matrix}\,\!</math>

For this design the block effect may be calculated as:

::<math>\begin{align}
& Block\text{ }Effect= & Average\text{ }response\text{ }for\text{ }Block\text{ }1- \\
& & Average\text{ }response\text{ }for\text{ }Block\text{ }2 \\
& = & \frac{(1)+ab}{2}-\frac{a+b}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The <math>AB\,\!</math> interaction effect is:

::<math>\begin{align}
& AB= & Average\text{ }response\text{ }at\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{low}}}- \\
& & Average\text{ }response\text{ }at\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{low}}} \\
& = & \frac{ab+(1)}{2}-\frac{b+a}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The two equations given above show that, in this design, the <math>AB\,\!</math> interaction effect cannot be distinguished from the block effect because the formulas to calculate these effects are the same. In other words, the <math>AB\,\!</math> interaction is said to be confounded with the block effect and it is not possible to say if the effect calculated based on these equations is due to the <math>AB\,\!</math> interaction effect, the block effect or both. In incomplete block designs some effects are always confounded with the blocks. Therefore, it is important to design these experiments in such a way that the important effects are not confounded with the blocks. In most cases, the experimenter can assume that higher order interactions are unimportant. In this case, it would better to use incomplete block designs that confound these effects with the blocks.
One way to design incomplete block designs is to use defining contrasts as shown next:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}+...+{{\alpha }_{k}}{{q}_{k}}\,\!</math>

where the <math>{{\alpha }_{i}}\,\!</math> s are the exponents for the factors in the effect that is to be confounded with the block effect and the <math>{{q}_{i}}\,\!</math> s are values based on the level of the <math>i\,\!</math> the factor (in a treatment that is to be allocated to a block). For <math>{2}^{k}\,\!</math> designs the <math>{{\alpha }_{i}}\,\!</math> s are either 0 or 1 and the <math>{{q}_{i}}\,\!</math> s have a value of 0 for the low level of the <math>i\,\!</math> th factor and a value of 1 for the high level of the factor in the treatment under consideration. As an example, consider the <math>{2}^{2}\,\!</math> design where the interaction effect <math>AB\,\!</math> is confounded with the block. Since there are two factors, <math>k=2\,\!</math>, with <math>i=1\,\!</math> representing factor <math>A\,\!</math> and <math>i=2\,\!</math> representing factor <math>B\,\!</math>. Therefore:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}\,\!</math>

The value of <math>{{\alpha }_{1}}\,\!</math> is one because the exponent of factor <math>A\,\!</math> in the confounded interaction <math>AB\,\!</math> is one. Similarly, the value of <math>{{\alpha }_{2}}\,\!</math> is one because the exponent of factor <math>B\,\!</math> in the confounded interaction <math>AB\,\!</math> is also one. Therefore, the defining contrast for this design can be written as:

::<math>\begin{align}
& L= & {{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}} \\
& = & 1\cdot {{q}_{1}}+1\cdot {{q}_{2}} \\
& = & {{q}_{1}}+{{q}_{2}}
\end{align}\,\!</math>

Once the defining contrast is known, it can be used to allocate treatments to the blocks. For the <math>{2}^{2}\,\!</math> design, there are four treatments <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math> and <math>ab\,\!</math>. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. In order to decide which block the treatment <math>(1)\,\!</math> belongs to, the levels of factors <math>A\,\!</math> and <math>B\,\!</math> for this run are used. Since factor <math>A\,\!</math> is at the low level in this treatment, <math>{{q}_{1}}=0\,\!</math>. Similarly, since factor <math>B\,\!</math> is also at the low level in this treatment, <math>{{q}_{2}}=0\,\!</math>. Therefore:

::<math>\begin{align}
& L= & {{q}_{1}}+{{q}_{2}} \\
& = & 0+0=0\text{ (mod 2)}
\end{align}\,\!</math>

Note that the value of <math>L\,\!</math> used to decide the block allocation is "mod 2" of the original value. This value is obtained by taking the value of 1 for odd numbers and 0 otherwise. Based on the value of <math>L\,\!</math>, treatment <math>(1)\,\!</math> is assigned to block 1. Other treatments can be assigned using the following calculations:

::<math>\begin{align}
& (1): & \text{ }L=0+0=0=0\text{ (mod 2)} \\
& a: & \text{ }L=1+0=1=1\text{ (mod 2)} \\
& b: & \text{ }L=0+1=1=1\text{ (mod 2)} \\
& ab: & \text{ }L=1+1=2=0\text{ (mod 2)}
\end{align}\,\!</math>

Therefore, to confound the interaction <math>AB\,\!</math> with the block effect in the <math>{2}^{2}\,\!</math> incomplete block design, treatments <math>(1)\,\!</math> and <math>ab\,\!</math> (with <math>L=0\,\!</math>) should be assigned to block 2 and treatment combinations <math>a\,\!</math> and <math>b\,\!</math> (with <math>L=1\,\!</math>) should be assigned to block 1.

====Example: Two Level Factorial Design with Two Blocks====

This example illustrates how treatments can be allocated to two blocks for an unreplicated <math>{2}^{k}\,\!</math> design. Consider the unreplicated <math>{2}^{4}\,\!</math> design to investigate the four factors affecting the defects in automobile vinyl panels discussed in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments required for this experiment were run by two different operators with each operator conducting 8 runs. This experiment is an example of an incomplete block design. The analyst in charge of this experiment assumed that the interaction <math>ABCD\,\!</math> was not significant and decided to allocate treatments to the two operators so that the <math>ABCD\,\!</math> interaction was confounded with the block effect (the two operators are the blocks). The allocation scheme to assign treatments to the two operators can be obtained as follows.
 
The defining contrast for the <math>{2}^{4}\,\!</math> design where the <math>ABCD\,\!</math> interaction is confounded with the blocks is:

::<math>L={{q}_{1}}+{{q}_{2}}+{{q}_{3}}+{{q}_{4}}\,\!</math>

The treatments can be allocated to the two operators using the values of the defining contrast. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. Then the value of the defining contrast for treatment <math>a\,\!</math> is:

::<math>a\ \ :\ \ \text{ }L=1+0+0+0=1=1\text{ (mod 2)}\,\!</math>

Therefore, treatment <math>a\,\!</math> should be assigned to Block 1 or the first operator. Similarly, for treatment <math>ab\,\!</math> we have:

::<math>ab\ \ :\ \ \text{ }L=1+1+0+0=2=0\text{ (mod 2)}\,\!</math>

[[Image:doe7.22.png|center|200px| Allocation of treatments to two blocks for the <math>2^4</math> design in the example by confounding interaction of <math>ABCD</math> with the blocks.]]

Therefore, <math>ab\,\!</math> should be assigned to Block 2 or the second operator. Other treatments can be allocated to the two operators in a similar manner to arrive at the allocation scheme shown in the figure below.
In DOE++, to confound the <math>ABCD\,\!</math> interaction for the <math>{2}^{4}\,\!</math> design into two blocks, the number of blocks are specified as shown in the figure below. Then the interaction <math>ABCD\,\!</math> is entered in the Block Generator window (second following figure) which is available using the Block Generator button in the following figure. The design generated by the Weibull++ DOE folio is shown in the third of the following figures. This design matches the allocation scheme of the preceding figure.

[[Image:doe7_23.png|center|700px| Adding block properties for the experiment in the example.|link=]]

[[Image:doe7_24.png|center|700px|Specifying the interaction ABCD as the interaction to be confounded with the blocks for the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

[[Image:doe7_25.png|center|800px|Two block design for the experiment in the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

For the analysis of this design, the sum of squares for all effects are calculated assuming no blocking. Then, to account for blocking, the sum of squares corresponding to the <math>ABCD\,\!</math> interaction is considered as the sum of squares due to blocks and <math>ABCD\,\!</math>. In DOE++ this is done by displaying this sum of squares as the sum of squares due to the blocks. This is shown in the following figure where the sum of squares in question is obtained as 72.25 and is displayed against Block. The interaction ABCD, which is confounded with the blocks, is not displayed. Since the design is unreplicated, any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_26.png|center|793px|ANOVA table for the experiment of the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

===Unreplicated 2''k'' Designs in 2''p'' Blocks===

A single replicate of the <math>{2}^{k}\,\!</math> design can be run in up to <math>{2}^{p}\,\!</math> blocks where <math>p<k\,\!</math>. The number of effects confounded with the blocks equals the degrees of freedom associated with the block effect.

If two blocks are used (the block effect has two levels), then one (<math>2-1=1)\,\!</math> effect is confounded with the blocks. If four blocks are used, then three (<math>4-1=3\,\!</math>) effects are confounded with the blocks and so on. For example an unreplicated <math>{2}^{4}\,\!</math> design may be confounded in <math>{2}^{2}\,\!</math> (four) blocks using two contrasts, <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}}\,\!</math>. Let <math>AC\,\!</math> and <math>BD\,\!</math> be the effects to be confounded with the blocks. Corresponding to these two effects, the contrasts are respectively:

::<math>\begin{align}
& {{L}_{1}}= & {{q}_{1}}+{{q}_{3}} \\
& {{L}_{2}}= & {{q}_{2}}+{{q}_{4}}
\end{align}\,\!</math>

Based on the values of <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}},\,\!</math> the treatments can be assigned to the four blocks as follows:

<center><math>\begin{matrix}
\text{Block 4} & {} & \text{Block 3} & {} & \text{Block 2} & {} & \text{Block 1} \\
{{L}_{1}}=0,{{L}_{2}}=0 & {} & {{L}_{1}}=1,{{L}_{2}}=0 & {} & {{L}_{1}}=0,{{L}_{2}}=1 & {} & {{L}_{1}}=1,{{L}_{2}}=1 \\
{} & {} & {} & {} & {} & {} & {} \\
\left[ \begin{matrix}
(1) \\
ac \\
bd \\
abcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
c \\
abd \\
bcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
b \\
abc \\
d \\
acd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
ab \\
bc \\
ad \\
cd \\
\end{matrix} \right] \\
\end{matrix}\,\!</math></center>

Since the block effect has three degrees of freedom, three effects are confounded with the block effect. In addition to <math>AC\,\!</math> and <math>BD\,\!</math>, the third effect confounded with the block effect is their generalized interaction, <math>(AC)(BD)=ABCD\,\!</math>.
In general, when an unreplicated <math>{2}^{k}\,\!</math> design is confounded in <math>{2}^{p}\,\!</math> blocks, <math>p\,\!</math> contrasts are needed (<math>{{L}_{1}},{{L}_{2}}...{{L}_{p}}\,\!</math>). <math>p\,\!</math> effects are selected to define these contrasts such that none of these effects are the generalized interaction of the others. The <math>{2}^{p}\,\!</math> blocks can then be assigned the treatments using the <math>p\,\!</math> contrasts. <math>{{2}^{p}}-(p+1)\,\!</math> effects, that are also confounded with the blocks, are then obtained as the generalized interaction of the <math>p\,\!</math> effects. In the statistical analysis of these designs, the sum of squares are computed as if no blocking were used. Then the block sum of squares is obtained by adding the sum of squares for all the effects confounded with the blocks.

====Example: 2 Level Factorial Design with Four Blocks====

This example illustrates how a DOE folio obtains the sum of squares when treatments for an unreplicated <math>{2}^{k}\,\!</math> design are allocated among four blocks. Consider again the unreplicated <math>{2}^{4}\,\!</math> design used to investigate the defects in automobile vinyl panels presented in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments needed to complete the experiment were run by four operators. Therefore, there are four blocks. Assume that the treatments were allocated to the blocks using the generators mentioned in the previous section, i.e., treatments were allocated among the four operators by confounding the effects, <math>AC\,\!</math> and <math>BD,\,\!</math> with the blocks. These effects can be specified as Block Generators as shown in the following figure. (The generalized interaction of these two effects, interaction <math>ABCD\,\!</math>, will also get confounded with the blocks.) The resulting design is shown in the second following figure and matches the allocation scheme obtained in the previous section.

[[Image:doe7_27.png|center|700px|Specifying the interactions AC and BD as block generators for the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

The sum of squares in this case can be obtained by calculating the sum of squares for each of the effects assuming there is no blocking. Once the individual sum of squares have been obtained, the block sum of squares can be calculated. The block sum of squares is the sum of the sum of squares of effects, <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, since these effects are confounded with the block effect. As shown in the second following figure, this sum of squares is 92.25 and is displayed against Block. The interactions <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, which are confounded with the blocks, are not displayed. Since the present design is unreplicated any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_28.png|center|800px|Design for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

[[Image:doe7_29.png|center|793px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

==Variability Analysis==

For replicated two level factorial experiments, DOE++ provides the option of conducting variability analysis (using the Variability Analysis icon under the Data menu). The analysis is used to identify the treatment that results in the least amount of variation in the product or process being investigated. Variability analysis is conducted by treating the standard deviation of the response for each treatment of the experiment as an additional response. The standard deviation for a treatment is obtained by using the replicated response values at that treatment run. As an example, consider the <math>{2}^{3}\,\!</math> design shown in the following figure where each run is replicated four times. A variability analysis can be conducted for this design. DOE++ calculates eight standard deviation values corresponding to each treatment of the design (see second following figure). Then, the design is analyzed as an unreplicated <math>{2}^{3}\,\!</math> design with the standard deviations (displayed as Y Standard Deviation. in second following figure) as the response. The normal probability plot of effects identifies <math>AC\,\!</math> as the effect that influences variability (see third figure following). Based on the effect coefficients obtained in the fourth figure following, the model for Y Std. is:

::<math>\begin{align}
& \text{Y Std}\text{.}= & 0.6779+0.2491\cdot AC \\
& = & 0.6779+0.2491{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

Based on the model, the experimenter has two choices to minimize variability (by minimizing Y Std.). The first choice is that <math>{{x}_{1}}\,\!</math> should be <math>1\,\!</math> (i.e., <math>A\,\!</math> should be set at the high level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the low level). The second choice is that <math>{{x}_{1}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>A\,\!</math> should be set at the low level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the high level). The experimenter can select the most feasible choice.

[[Image:doe7.30.png|center|391px|A <math>2^3\,\!</math> design with four replicated response values that can be used to conduct a variability analysis.]]
 

[[Image:doe7_31.png|center|727px|Variability analysis in DOE++.|link=]]
 

[[Image:doe7_32.png|center|650px|Normal probability plot of effects for the variability analysis example.|link=]]
 

[[Image:doe7_33.png|center|727px|Effect coefficients for the variability analysis example.|link=]]

 

==Two Level Fractional Factorial Designs==

As the number of factors in a two level factorial design increases, the number of runs for even a single replicate of the <math>{2}^{k}\,\!</math> design becomes very large. For example, a single replicate of an eight factor two level experiment would require 256 runs. Fractional factorial designs can be used in these cases to draw out valuable conclusions from fewer runs. The basis of fractional factorial designs is the ''sparsity of effects'' principle.[[DOE_References|[Wu, 2000]]] The principle states that, most of the time, responses are affected by a small number of main effects and lower order interactions, while higher order interactions are relatively unimportant. Fractional factorial designs are used as screening experiments during the initial stages of experimentation. At these stages, a large number of factors have to be investigated and the focus is on the main effects and two factor interactions. These designs obtain information about main effects and lower order interactions with fewer experiment runs by confounding these effects with unimportant higher order interactions. As an example, consider a <math>{2}^{8}\,\!</math> design that requires 256 runs. This design allows for the investigation of 8 main effects and 28 two factor interactions. However, 219 degrees of freedom are devoted to three factor or higher order interactions. This full factorial design can prove to be very inefficient when these higher order interactions can be assumed to be unimportant. Instead, a fractional design can be used here to identify the important factors that can then be investigated more thoroughly in subsequent experiments. In unreplicated fractional factorial designs, no degrees of freedom are available to calculate the error sum of squares and the techniques mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] should be employed for the analysis of these designs.

==Half-fraction Designs==

A half-fraction of the <math>{2}^{k}\,\!</math> design involves running only half of the treatments of the full factorial design. For example, consider a <math>{2}^{3}\,\!</math> design that requires eight runs in all. The design matrix for this design is shown in the figure (a) below. A half-fraction of this design is the design in which only four of the eight treatments are run. The fraction is denoted as <math>{2}^{3-1}\,\!</math> with the "<math>-1\,\!</math>" in the index denoting a half-fraction. Assume that the treatments chosen for the half-fraction design are the ones where the interaction <math>ABC\,\!</math> is at the high level (i.e., only those rows are chosen from the following figure (a) where the column for <math>ABC\,\!</math> has entries of 1). The resulting <math>{2}^{3-1}\,\!</math> design has a design matrix as shown in figure (b) below.

[[Image:doe7.34.png|center|330px|Half-fractions of the <math>2^3\,\!</math> design. (a) shows the full factorial <math>2^3\,\!</math> design, (b) shows the <math>2^{3-1}\,\!</math> design with the defining relation <math>I=ABC\,\!</math> and (c) shows the <math>{2}^{3-1}\,\!</math> design with the defining relation <math>I=-ABC\,\!</math>.]]

In the <math>{2}^{3-1}\,\!</math> design of figure (b), since the interaction <math>ABC\,\!</math> is always included at the same level (the high level represented by 1), it is not possible to measure this interaction effect. The effect, <math>ABC\,\!</math>, is called the ''generator'' or ''word'' for this design. It can be noted that, in the design matrix of the following figure (b), the column corresponding to the intercept, <math>I\,\!</math>, and column corresponding to the interaction <math>ABC\,\!</math>, are identical. The identical columns are written as <math>I=ABC\,\!</math> and this equation is called the ''defining relation'' for the design. In a DOE folio, the present <math>{2}^{3-1}\,\!</math> design can be obtained by specifying the design properties as shown in the following figure.

[[Image:doe7_35.png|center|700px|Design properties for the <math>2^{3-1}\,\!</math> design.|link=]]

The defining relation, <math>I=ABC\,\!</math>, is entered in the Fraction Generator window as shown next.

[[Image:doe7_36.png|center|700px|Specifying the defining relation for the <math>2^{3-1}\,\!</math> design.|link=]]

Note that in the figure following that, the defining relation is specified as <math>C=AB\,\!</math>. This relation is obtained by multiplying the defining relation, <math>I=ABC\,\!</math>, by the last factor, <math>C\,\!</math>, of the design.

===Calculation of Effects===

Using the four runs of the <math>{2}^{3-1}\,\!</math> design in figure (b) discussed above, the main effects can be calculated as follows:

::<math>\begin{align}
& A= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& B= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& C= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

where <math>a\,\!</math>, <math>b\,\!</math>, <math>c\,\!</math> and <math>abc\,\!</math> are the treatments included in the <math>{2}^{3-1}\,\!</math> design.

Similarly, the two factor interactions can also be obtained as:

::<math>\begin{align}
& BC= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& AC= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& AB= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

The equations for <math>A\,\!</math> and <math>BC\,\!</math> above result in the same effect values showing that effects <math>A\,\!</math> and <math>BC\,\!</math> are confounded in the present <math>{2}^{3-1}\,\!</math> design. Thus, the quantity, <math>\tfrac{1}{2}(a-b-c+abc),\,\!</math> estimates <math>A+BC\,\!</math> (i.e., both the main effect <math>A\,\!</math> and the two-factor interaction <math>BC\,\!</math>). The effects, <math>A\,\!</math> and <math>BC,\,\!</math> are called ''aliases''. From the remaining equations given above, it can be seen that the other aliases for this design are <math>B\,\!</math> and <math>AC\,\!</math>, and <math>C\,\!</math> and <math>AB\,\!</math>. Therefore, the equations to calculate the effects in the present <math>{2}^{3-1}\,\!</math> design can be written as follows:

::<math>\begin{align}
& A+BC= & \frac{1}{2}(a-b-c+abc) \\
& B+AC= & \frac{1}{2}(-a+b-c+abc) \\
& C+AB= & \frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

===Calculation of Aliases===

Aliases for a fractional factorial design can be obtained using the defining relation for the design. The defining relation for the present <math>{2}^{3-1}\,\!</math> design is:

::<math>I=ABC\,\!</math>

Multiplying both sides of the previous equation by the main effect, <math>A,\,\!</math> gives the alias effect of <math>A\,\!</math> :

::<math>\begin{align}
& A\cdot I= & A\cdot ABC \\
& A= & {{A}^{2}}BC \\
& A= & BC
\end{align}\,\!</math>

Note that in calculating the alias effects, any effect multiplied by <math>I\,\!</math> remains the same (<math>A\cdot I=A\,\!</math>), while an effect multiplied by itself results in <math>I\,\!</math> (<math>{{A}^{2}}=I\,\!</math>). Other aliases can also be obtained:

::<math>\begin{align}
& B\cdot I= & B\cdot ABC \\
& B= & A{{B}^{2}}C \\
& B= & AC
\end{align}\,\!</math>

:and:

::<math>\begin{align}
& C\cdot I= & C\cdot ABC \\
& C= & AB{{C}^{2}} \\
& C= & AB
\end{align}\,\!</math>

===Fold-over Design===

If it can be assumed for this design that the two-factor interactions are unimportant, then in the absence of <math>BC\,\!</math>, <math>AC\,\!</math> and <math>AB\,\!</math>, the equations for (A+BC), (B+AC) and (C+AB) can be used to estimate the main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. However, if such an assumption is not applicable, then to uncouple the main effects from their two factor aliases, the alternate fraction that contains runs having <math>ABC\,\!</math> at the lower level should be run. The design matrix for this design is shown in the preceding figure (c). The defining relation for this design is <math>I=-ABC\,\!</math> because the four runs for this design are obtained by selecting the rows of the preceding figure (a) for which the value of the <math>ABC\,\!</math> column is <math>-1\,\!</math>. The aliases for this fraction can be obtained as explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]] as <math>A=-BC\,\!</math>, <math>B=-AC\,\!</math> and <math>C=-AB\,\!</math>. The effects for this design can be calculated as:

::<math>\begin{align}
& A-BC= & \frac{1}{2}(ab+ac-(1)-bc) \\
& B-AC= & \frac{1}{2}(ab-ac+(1)-bc) \\
& C-AB= & \frac{1}{2}(-ab+ac-(1)+bc)
\end{align}\,\!</math>

These equations can be combined with the equations for (A+BC), (B+AC) and (C+AB) to obtain the de-aliased main effects and two factor interactions. For example, adding equations (A+BC) and (A-BC) returns the main effect <math>A\,\!</math>.

::<math>\begin{align}
& 2A= & \frac{1}{2}(a-b-c+abc)+ \\
& & \frac{1}{2}(ab+ac-(1)-bc)
\end{align}\,\!</math>

The process of augmenting a fractional factorial design by a second fraction of the same size by simply reversing the signs (of all effect columns except <math>I\,\!</math>) is called ''folding over''. The combined design is referred to as a ''fold-over design''.

==Quarter and Smaller Fraction Designs==

At times, the number of runs even for a half-fraction design are very large. In these cases, smaller fractions are used. A quarter-fraction design, denoted as <math>{2}^{k-2}\,\!</math>, consists of a fourth of the runs of the full factorial design. Quarter-fraction designs require two defining relations. The first defining relation returns the half-fraction or the <math>{2}^{k-1}\,\!</math> design. The second defining relation selects half of the runs of the <math>{2}^{k-1}\,\!</math> design to give the quarter-fraction. For example, consider the <math>{2}^{4}\,\!</math> design. To obtain a <math>{2}^{4-2}\,\!</math> design from this design, first a half-fraction of this design is obtained by using a defining relation. Assume that the defining relation used is <math>I=ABCD\,\!</math>. The design matrix for the resulting <math>{2}^{4-1}\,\!</math> design is shown in figure (a) below. Now, a quarter-fraction can be obtained from the <math>{2}^{4-1}\,\!</math> design shown in figure (a) below using a second defining relation <math>I=AD\,\!</math>. The resulting <math>{2}^{4-2}\,\!</math> design obtained is shown in figure (b) below.

[[Image:doe7.37.png|center|465px|Fractions of the <math>2^4\,\!</math> design - Figure (a) shows the <math>2^{4-1}</math> design with the defining relation <math>I=ABCD\,\!</math> and (b) shows the <math>2^{4-2}\,\!</math> design with the defining relation <math>I=ABCD=AD=BC\,\!</math>.]]

The complete defining relation for this <math>{2}^{4-2}\,\!</math> design is:

::<math>I=ABCD=AD=BC\,\!</math>

Note that the effect, <math>BC,\,\!</math> in the defining relation is the generalized interaction of <math>ABCD\,\!</math> and <math>AD\,\!</math> and is obtained using <math>(ABCD)(AD)={{A}^{2}}BC{{D}^{2}}=BC\,\!</math>. In general, a <math>{2}^{k-p}\,\!</math> fractional factorial design requires <math>p\,\!</math> independent generators. The defining relation for the design consists of the <math>p\,\!</math> independent generators and their <math>{2}^{p}\,\!</math> - (<math>p\,\!</math> +1) generalized interactions.

===Calculation of Aliases===

The alias structure for the present <math>{2}^{4-2}\,\!</math> design can be obtained using the defining relation of equation (I=ABCD=AD=BC) following the procedure explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]]. For example, multiplying the defining relation by <math>A\,\!</math> returns the effects aliased with the main effect, <math>A\,\!</math>, as follows:

::<math>\begin{align}
& A\cdot I= & A\cdot ABCD=A\cdot AD=A\cdot BC \\
& A= & {{A}^{2}}BCD={{A}^{2}}D=ABC \\
& A= & BCD=D=ABC
\end{align}\,\!</math>

Therefore, in the present <math>{2}^{4-2}\,\!</math> design, it is not possible to distinguish between effects <math>A\,\!</math>, <math>D\,\!</math>, <math>BCD\,\!</math> and <math>ABC\,\!</math>. Similarly, multiplying the defining relation by <math>B\,\!</math> and <math>AB\,\!</math> returns the effects that are aliased with these effects:

::<math>\begin{align}
& B= & ACD=ABD=C \\
& AB= & CD=AD=AC
\end{align}\,\!</math>

Other aliases can be obtained in a similar way. It can be seen that each effect in this design has three aliases. In general, each effect in a <math>{2}^{k-p}\,\!</math> design has <math>{2}^{p-1}\,\!</math> aliases.
The aliases for the <math>{2}^{4-2}\,\!</math> design show that in this design the main effects are aliased with each other (<math>A\,\!</math> is aliased with <math>D\,\!</math> and <math>B\,\!</math> is aliased with <math>C\,\!</math>). Therefore, this design is not a useful design and is not available in a Weibull++ DOE folio. It is important to ensure that main effects and lower order interactions of interest are not aliased in a fractional factorial design. This is known by looking at the resolution of the fractional factorial design.

==Design Resolution==

The resolution of a fractional factorial design is defined as the number of factors in the lowest order effect in the defining relation. For example, in the defining relation <math>I=ABCD=AD=BC\,\!</math> of the previous <math>{2}^{4-2}\,\!</math> design, the lowest-order effect is either <math>AD\,\!</math> or <math>BC,\,\!</math> containing two factors. Therefore, the resolution of this design is equal to two. The resolution of a fractional factorial design is represented using Roman numerals. For example, the previously mentioned <math>{2}^{4-2}\,\!</math> design with a resolution of two can be represented as 2 <math>_{\text{II}}^{4-2}\,\!</math>. The resolution provides information about the confounding in the design as explained next:
 
 
#'''Resolution III Designs''' In these designs, the lowest order effect in the defining relation has three factors (e.g., a <math>{2}^{5-2}\,\!</math> design with the defining relation <math>I=ABDE=ABC=CDE\,\!</math>). In resolution III designs, no main effects are aliased with any other main effects, but main effects are aliased with two factor interactions. In addition, some two factor interactions are aliased with each other.
#'''Resolution IV Designs''' In these designs, the lowest order effect in the defining relation has four factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABDE\,\!</math>). In resolution IV designs, no main effects are aliased with any other main effects or two factor interactions. However, some main effects are aliased with three factor interactions and the two factor interactions are aliased with each other.
#'''Resolution V Designs''' In these designs the lowest order effect in the defining relation has five factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABCDE\,\!</math>). In resolution V designs, no main effects or two factor interactions are aliased with any other main effects or two factor interactions. However, some main effects are aliased with four factor interactions and the two factor interactions are aliased with three factor interactions.

Fractional factorial designs with the highest resolution possible should be selected because the higher the resolution of the design, the less severe the degree of confounding. In general, designs with a resolution less than III are never used because in these designs some of the main effects are aliased with each other. The table below shows fractional factorial designs with the highest available resolution for three to ten factor designs along with their defining relations.

[[Image:doet7.3.png|center|474px|Highest resolution designs available for fractional factorial designs with 3 to 10 factors.]]

All of the two level fractional factorial designs available in a DOE folio are shown next.

[[Image:doe7.38.png|center|471px|Two level fractional factorial designs available in Weibull++ and their resolutions.]]

===Minimum Aberration Designs===
At times, different designs with the same resolution but different aliasing may be available. The best design to select in such a case is the minimum aberration design. For example, all <math>{2}^{7-2}\,\!</math> designs in the fourth table have a resolution of four (since the generator with the minimum number of factors in each design has four factors). Design <math>1\,\!</math> has three generators of length four (<math>ABCF,\,\!</math> <math>BCDG,\,\!</math> <math>ADFG\,\!</math>). Design <math>2\,\!</math> has two generators of length four (<math>ABCF,\,\!</math> <math>ADEG\,\!</math>). Design <math>3\,\!</math> has one generator of length four (<math>CEFG\,\!</math>). Therefore, design <math>3\,\!</math> has the least number of generators with the minimum length of four. Design <math>3\,\!</math> is called the minimum aberration design. It can be seen that the alias structure for design <math>3\,\!</math> is less involved compared to the other designs. For details refer to [[DOE_References|[Wu, 2000]]].

[[Image:doet7.4.png|center|432px|Three <math>2_{IV}^{7-2}\,\!</math> designs with different defining relations.]]

====Example====

The design of an automobile fuel cone is thought to be affected by six factors in the manufacturing process: cavity temperature (factor <math>A\,\!</math>), core temperature (factor <math>B\,\!</math>), melt temperature (factor <math>C\,\!</math>), hold pressure (factor <math>D\,\!</math>), injection speed (factor <math>E\,\!</math>) and cool time (factor <math>F\,\!</math>). The manufacturer of the fuel cone is unable to run the <math>{2}^{6}=64\,\!</math> runs required to complete one replicate for a two level full factorial experiment with six factors. Instead, they decide to run a fractional factorial design. Considering that three factor and higher order interactions are likely to be inactive, the manufacturer selects a <math>{2}^{6-2}\,\!</math> design that will require only 16 runs. The manufacturer chooses the resolution IV design which will ensure that all main effects are free from aliasing (assuming three factor and higher order interactions are absent). However, in this design the two factor interactions may be aliased with each other. It is decided that, if important two factor interactions are found to be present, additional experiment trials may be conducted to separate the aliased effects. The performance of the fuel cone is measured on a scale of 1 to 15. In a Weibull++ DOE folio, the design for this experiment is set up using the properties shown in the following figure. The Fraction Generators for the design, <math>E=ABC\,\!</math> and <math>F=BCD\,\!</math>, are the same as the defaults used in a DOE folio. The resulting <math>{2}^{6-2}\,\!</math> design and the corresponding response values are shown in the following two figures.

[[Image:doe7_39.png|center|644px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

[[Image:doe7_40.png|center|534px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The complete alias structure for the 2 <math>_{\text{IV}}^{6-2}\,\!</math> design is shown next.

<center><math>I=ABCE=ADEF=BCDF\,\!</math></center>

 
<center><math>\begin{align}
& A= & BCE=DEF=ABCDF \\
& B= & ACE=CDF=ABDEF \\
& C= & ABE=BDF=ACDEF \\
& D= & AEF=BCF=ABCDE \\
& E= & ABC=ADF=BCDEF \\
& F= & ADE=BCD=ABCEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& AB= & CE=ACDF=BDEF \\
& AC= & BE=ABDF=CDEF \\
& AD= & EF=ABCF=BCDE \\
& AE= & BC=DF=ABCDEF \\
& AF= & DE=ABCD=BCEF \\
& BD= & CF=ABEF=ACDE \\
& BF= & CD=ABDE=ACEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& ABD= & ACF=BEF=CDE \\
& ABF= & ACD=BDE=CEF
\end{align}\,\!</math></center>

In a DOE folio, the alias structure is displayed in the Design Summary and as part of the Design Evaluation result, as shown next:

[[Image:doe7_41.png|center|700px|Alias structure for the experiment design in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The normal probability plot of effects for this unreplicated design shows the main effects of factors <math>C\,\!</math> and <math>D\,\!</math> and the interaction effect, <math>BF\,\!</math>, to be significant (see the following figure).

[[Image:doe7_42.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

From the alias structure, it can be seen that for the present design interaction effect, <math>BF,\,\!</math> is confounded with <math>CD\,\!</math>. Therefore, the actual source of this effect cannot be known on the basis of the present experiment. However because neither factor <math>B\,\!</math> nor <math>F\,\!</math> is found to be significant there is an indication the observed effect is likely due to interaction, <math>CD\,\!</math>. To confirm this, a follow-up <math>{2}^{2}\,\!</math> experiment is run involving only factors <math>B\,\!</math> and <math>F\,\!</math>. The interaction, <math>BF\,\!</math>, is found to be inactive, leading to the conclusion that the interaction effect in the original experiment is effect, <math>CD\,\!</math>. Given these results, the fitted regression model for the fuel cone design as per the coefficients obtained from the DOE folio is shown next.

::<math>\hat{y}=7.6875+C+2\cdot D+2.1875\cdot CD\,\!</math>

[[Image:doe7_43.png|center|700px|Effect coefficients for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

==Projection==

Projection refers to the reduction of a fractional factorial design to a full factorial design by dropping out some of the factors of the design. Any fractional factorial design of resolution, <math>R,\,\!</math> can be reduced to complete factorial designs in any subset of <math>R-1\,\!</math> factors. For example, consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. The resolution of this design is four. Therefore, this design can be reduced to full factorial designs in any three (<math>4-1=3\,\!</math>) of the original seven factors (by dropping the remaining four of factors). Further, a fractional factorial design can also be reduced to a full factorial design in any <math>R\,\!</math> of the original factors, as long as these <math>R\,\!</math> factors are not part of the generator in the defining relation. Again consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. This design can be reduced to a full factorial design in four factors provided these four factors do not appear together as a generator in the defining relation. The complete defining relation for this design is:

::<math>\begin{align}
& I= & ABCE=BCDF=ACDG=ADEF=ABFG=BDEG=CEFG
\end{align}\,\!</math>

Therefore, there are seven four factor combinations out of the 35 (<math>(_{7}^{4})=35\,\!</math>) possible four-factor combinations that are used as generators in the defining relation. The designs with the remaining 28 four factor combinations would be full factorial 16-run designs. For example, factors <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math> do not occur as a generator in the defining relation of the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. If the remaining factors, <math>E\,\!</math>, <math>F\,\!</math> and <math>G\,\!</math>, are dropped, the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design will reduce to a full factorial design in <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math>.

==Resolution III Designs==

At times, the factors to be investigated in screening experiments are so large that even running a fractional factorial design is impractical. This can be partially solved by using resolution III fractional factorial designs in the cases where three factor and higher order interactions can be assumed to be unimportant. Resolution III designs, such as the 2 <math>_{\text{III}}^{3-1}\,\!</math> design, can be used to estimate <math>k\,\!</math> main effects using just <math>k+1\,\!</math> runs. In these designs, the main effects are aliased with two factor interactions. Once the results from these designs are obtained, and knowing that three factor and higher order interactions are unimportant, the experimenter can decide if there is a need to run a fold-over design to de-alias the main effects from the two factor interactions. Thus, the 2 <math>_{\text{III}}^{3-1}\,\!</math> design can be used to investigate three factors in four runs, the 2 <math>_{\text{III}}^{7-4}\,\!</math> design can be used to investigate seven factors in eight runs, the 2 <math>_{\text{III}}^{15-11}\,\!</math> design can be used to investigate fifteen factors in sixteen runs and so on.

====Example====

{{:Resolution_III_Design_Example}}

==Alias Matrix==
In [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction designs]] and [[Two_Level_Factorial_Experiments#Quarter_and_Smaller_Fraction_Designs| Quarter and Smaller Fraction Designs]], the alias structure for fractional factorial designs was obtained using the defining relation. However, this method of obtaining the alias structure is not very efficient when the alias structure is very complex or when partial aliasing is involved. One of the ways to obtain the alias structure for any design, regardless of its complexity, is to use the alias matrix. The alias matrix for a design is calculated using <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> where <math>{{X}_{1}}\,\!</math> is the portion of the design matrix, <math>X,\,\!</math> that contains the effects for which the aliases need to be calculated, and <math>{{X}_{2}}\,\!</math> contains the remaining columns of the design matrix, other than those included in <math>{{X}_{1}}\,\!</math>.

To illustrate the use of the alias matrix, consider the design matrix for the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design (using the defining relation <math>I=ABCD\,\!</math>) shown next:

[[Image:Chapter7__879.png|center|link=]]

The alias structure for this design can be obtained by defining <math>{{X}_{1}}\,\!</math> using eight columns since the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design estimates eight effects. If the first eight columns of <math>X\,\!</math> are used then <math>{{X}_{1}}\,\!</math> is:

[[Image:Chapter7__884.png|center|link=]]

<math>{{X}_{2}}\,\!</math> is obtained using the remaining columns as:

[[Image:Chapter7__886.png|center|link=]]

Then the alias matrix <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> is:

[[Image:Chapter7__888.png|center|link=]]

The alias relations can be easily obtained by observing the alias matrix as:

<center><math>\begin{align}
& I= & ABCD \\
& A= & BCD \\
& B= & ACD \\
& AB= & CD \\
& C= & ABD \\
& AC= & BD \\
& BC= & AD \\
& D= & ABC
\end{align}\,\!</math></center>

Two Level Factorial Experiments

2017-08-10T17:39:20Z

Richard House: /* Example */

{{Template:Doebook|8}}
Two level factorial experiments are factorial experiments in which each factor is investigated at only two levels. The early stages of experimentation usually involve the investigation of a large number of potential factors to discover the "vital few" factors. Two level factorial experiments are used during these stages to quickly filter out unwanted effects so that attention can then be focused on the important ones.

==2''k'' Designs==
The factorial experiments, where all combination of the levels of the factors are run, are usually referred to as ''full factorial experiments''. Full factorial two level experiments are also referred to as <math>{2}^{k}\,\!</math> designs where <math>k\,\!</math> denotes the number of factors being investigated in the experiment. In Weibull++ DOE folios, these designs are referred to as 2 Level Factorial Designs as shown in the figure below.

[[Image:doe7_1.png|center|487px|Selection of full factorial experiments with two levels in Weibull++.|link=]]

A full factorial two level design with <math>k\,\!</math> factors requires <math>{{2}^{k}}\,\!</math> runs for a single replicate. For example, a two level experiment with three factors will require <math>2\times 2\times 2={{2}^{3}}=8\,\!</math> runs. The choice of the two levels of factors used in two level experiments depends on the factor; some factors naturally have two levels. For example, if gender is a factor, then male and female are the two levels. For other factors, the limits of the range of interest are usually used. For example, if temperature is a factor that varies from <math>{45}^{o}C\,\!</math> to <math>{90}^{o}C\,\!</math>, then the two levels used in the <math>{2}^{k}\,\!</math> design for this factor would be <math>{45}^{o}\,\!C\,\!</math> and <math>{90}^{o}\,\!C\,\!</math>.

The two levels of the factor in the <math>{2}^{k}\,\!</math> design are usually represented as <math>-1\,\!</math> (for the first level) and <math>1\,\!</math> (for the second level). Note that this representation is reversed from the coding used in [[General Full Factorial Designs]] for the indicator variables that represent two level factors in ANOVA models. For ANOVA models, the first level of the factor was represented using a value of <math>1\,\!</math> for the indicator variable, while the second level was represented using a value of <math>-1\,\!</math>. For details on the notation used for two level experiments refer to [[Two_Level_Factorial_Experiments#Notation| Notation]].

===The 22 Design===

The simplest of the two level factorial experiments is the <math>{2}^{2}\,\!</math> design where two factors (say factor <math>A\,\!</math> and factor <math>B\,\!</math>) are investigated at two levels. A single replicate of this design will require four runs (<math>{{2}^{2}}=2\times 2=4\,\!</math>) The effects investigated by this design are the two main effects, <math>A\,\!</math> and <math>B,\,\!</math> and the interaction effect <math>AB\,\!</math>. The treatments for this design are shown in figure (a) below. In figure (a), letters are used to represent the treatments. The presence of a letter indicates the high level of the corresponding factor and the absence indicates the low level. For example, (1) represents the treatment combination where all factors involved are at the low level or the level represented by <math>-1\,\!</math> ; <math>a\,\!</math> represents the treatment combination where factor <math>A\,\!</math> is at the high level or the level of <math>1\,\!</math>, while the remaining factors (in this case, factor <math>B\,\!</math>) are at the low level or the level of <math>-1\,\!</math>. Similarly, <math>b\,\!</math> represents the treatment combination where factor <math>B\,\!</math> is at the high level or the level of <math>1\,\!</math>, while factor <math>A\,\!</math> is at the low level and <math>ab\,\!</math> represents the treatment combination where factors <math>A\,\!</math> and <math>B\,\!</math> are at the high level or the level of the 1. Figure (b) below shows the design matrix for the <math>{2}^{2}\,\!</math> design. It can be noted that the sum of the terms resulting from the product of any two columns of the design matrix is zero. As a result the <math>{2}^{2}\,\!</math> design is an ''orthogonal design''. In fact, all <math>{2}^{k}\,\!</math> designs are orthogonal designs. This property of the <math>{2}^{k}\,\!</math> designs offers a great advantage in the analysis because of the simplifications that result from orthogonality. These simplifications are explained later on in this chapter.
The <math>{2}^{2}\,\!</math> design can also be represented geometrically using a square with the four treatment combinations lying at the four corners, as shown in figure (c) below.

[[Image:doe7.2.png|center|400px|The <math>2^2\,\!</math> design. Figure (a) displays the experiment design, (b) displays the design matrix and (c) displays the geometric representation for the design. In Figure (b), the column names I, A, B and AB are used. Column I represents the intercept term. Columns A and B represent the respective factor settings. Column AB represents the interaction and is the product of columns A and B.]]
 

===The 23 Design===

The <math>{2}^{3}\,\!</math> design is a two level factorial experiment design with three factors (say factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>). This design tests three (<math>k=3\,\!</math>) main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math> ; three (<math>(_{2}^{k})=\,\!</math> <math>(_{2}^{3})=3\,\!</math>) two factor interaction effects, <math>AB\,\!</math>, <math>BC\,\!</math>, <math>AC\,\!</math> ; and one (<math>(_{3}^{k})=\,\!</math> <math>(_{3}^{3})=1\,\!</math>) three factor interaction effect, <math>ABC\,\!</math>. The design requires eight runs per replicate. The eight treatment combinations corresponding to these runs are <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math>, <math>ab\,\!</math>, <math>c\,\!</math>, <math>ac\,\!</math>, <math>bc\,\!</math> and <math>abc\,\!</math>. Note that the treatment combinations are written in such an order that factors are introduced one by one with each new factor being combined with the preceding terms. This order of writing the treatments is called the ''standard order'' or ''Yates' order''. The <math>{2}^{3}\,\!</math> design is shown in figure (a) below. The design matrix for the <math>{2}^{3}\,\!</math> design is shown in figure (b). The design matrix can be constructed by following the standard order for the treatment combinations to obtain the columns for the main effects and then multiplying the main effects columns to obtain the interaction columns.

[[Image:doe7.3.png|center|324px|The <math>2^3\,\!</math> design. Figure (a) shows the experiment design and (b) shows the design matrix.]]

[[Image:doe7.4.png|center|290px|Geometric representation of the <math>2^3\,\!</math> design.]]

The <math>{2}^{3}\,\!</math> design can also be represented geometrically using a cube with the eight treatment combinations lying at the eight corners as shown in the figure above.

==Analysis of 2''k'' Designs==

The <math>{2}^{k}\,\!</math> designs are a special category of the factorial experiments where all the factors are at two levels. The fact that these designs contain factors at only two levels and are orthogonal greatly simplifies their analysis even when the number of factors is large. The use of <math>{2}^{k}\,\!</math> designs in investigating a large number of factors calls for a revision of the notation used previously for the ANOVA models. The case for revised notation is made stronger by the fact that the ANOVA and multiple linear regression models are identical for <math>{2}^{k}\,\!</math> designs because all factors are only at two levels. Therefore, the notation of the regression models is applied to the ANOVA models for these designs, as explained next.

===Notation===

Based on the notation used in [[General Full Factorial Designs]], the ANOVA model for a two level factorial experiment with three factors would be as follows:

::<math>\begin{align}
& Y= & \mu +{{\tau }_{1}}\cdot {{x}_{1}}+{{\delta }_{1}}\cdot {{x}_{2}}+{{(\tau \delta )}_{11}}\cdot {{x}_{1}}{{x}_{2}}+{{\gamma }_{1}}\cdot {{x}_{3}} \\
& & +{{(\tau \gamma )}_{11}}\cdot {{x}_{1}}{{x}_{3}}+{{(\delta \gamma )}_{11}}\cdot {{x}_{2}}{{x}_{3}}+{{(\tau \delta \gamma )}_{111}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where:
 
:• <math>\mu \,\!</math> represents the overall mean
:• <math>{{\tau }_{1}}\,\!</math> represents the independent effect of the first factor (factor <math>A\,\!</math>) out of the two effects <math>{{\tau }_{1}}\,\!</math> and <math>{{\tau }_{2}}\,\!</math>
:• <math>{{\delta }_{1}}\,\!</math> represents the independent effect of the second factor (factor <math>B\,\!</math>) out of the two effects <math>{{\delta }_{1}}\,\!</math> and <math>{{\delta }_{2}}\,\!</math>
:• <math>{{(\tau \delta )}_{11}}\,\!</math> represents the independent effect of the interaction <math>AB\,\!</math> out of the other interaction effects
:• <math>{{\gamma }_{1}}\,\!</math> represents the effect of the third factor (factor <math>C\,\!</math>) out of the two effects <math>{{\gamma }_{1}}\,\!</math> and <math>{{\gamma }_{2}}\,\!</math>
:• <math>{{(\tau \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>AC\,\!</math> out of the other interaction effects
:• <math>{{(\delta \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>BC\,\!</math> out of the other interaction effects
:• <math>{{(\tau \delta \gamma )}_{111}}\,\!</math> represents the effect of the interaction <math>ABC\,\!</math> out of the other interaction effects
and <math>\epsilon \,\!</math> is the random error term.

 
The notation for a linear regression model having three predictor variables with interactions is:

::<math>\begin{align}
& Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& & +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

The notation for the regression model is much more convenient, especially for the case when a large number of higher order interactions are present. In two level experiments, the ANOVA model requires only one indicator variable to represent each factor for both qualitative and quantitative factors. Therefore, the notation for the multiple linear regression model can be applied to the ANOVA model of the experiment that has all the factors at two levels. For example, for the experiment of the ANOVA model given above, <math>{{\beta }_{0}}\,\!</math> can represent the overall mean instead of <math>\mu \,\!</math>, and <math>{{\beta }_{1}}\,\!</math> can represent the independent effect, <math>{{\tau }_{1}}\,\!</math>, of factor <math>A\,\!</math>. Other main effects can be represented in a similar manner. The notation for the interaction effects is much more simplified (e.g., <math>{{\beta }_{123}}\,\!</math> can be used to represent the three factor interaction effect, <math>{{(\tau \beta \gamma )}_{111}}\,\!</math>).

As mentioned earlier, it is important to note that the coding for the indicator variables for the ANOVA models of two level factorial experiments is reversed from the coding followed in [[General Full Factorial Designs]]. Here <math>-1\,\!</math> represents the first level of the factor while <math>1\,\!</math> represents the second level. This is because for a two level factor a single variable is needed to represent the factor for both qualitative and quantitative factors. For quantitative factors, using <math>-1\,\!</math> for the first level (which is the low level) and 1 for the second level (which is the high level) keeps the coding consistent with the numerical value of the factors. The change in coding between the two coding schemes does not affect the analysis except that signs of the estimated effect coefficients will be reversed (i.e., numerical values of <math>{{\hat{\tau }}_{1}}\,\!</math>, obtained based on the coding of [[General Full Factorial Designs]], and <math>{{\hat{\beta }}_{1}}\,\!</math>, obtained based on the new coding, will be the same but their signs would be opposite).

::<math>\begin{align}
& & \text{Factor }A\text{ Coding (two level factor)} \\
& &
\end{align}\,\!</math>

::<math>\begin{matrix}
\text{Previous Coding} & {} & {} & {} & \text{Coding for }{{\text{2}}^{k}}\text{ Designs} \\
{} & {} & {} & {} & {} \\
Effect\text{ }{{\tau }_{1}}\ \ :\ \ {{x}_{1}}=1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{1}}\text{ (or }-{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=-1\text{ } \\
Effect\text{ }{{\tau }_{2}}\ \ :\ \ {{x}_{1}}=-1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{2}}\text{ (or }{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=1\text{ } \\
\end{matrix}\,\!</math>

In summary, the ANOVA model for the experiments with all factors at two levels is different from the ANOVA models for other experiments in terms of the notation in the following two ways:
 

:• The notation of the regression models is used for the effect coefficients.
:• The coding of the indicator variables is reversed.

===Special Features===

Consider the design matrix, <math>X\,\!</math>, for the <math>{2}^{3}\,\!</math> design discussed above. The (<math>{{X}^{\prime }}X\,\!</math>) <math>^{-1}\,\!</math> matrix is:

<center><math>{{({{X}^{\prime }}X)}^{-1}}=\left[ \begin{matrix}
0.125 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0.125 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0.125 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0.125 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0.125 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0.125 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0.125 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.125 \\
\end{matrix} \right]\,\!</math></center>

Notice that, due to the orthogonal design of the <math>X\,\!</math> matrix, the <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> has been simplified to a diagonal matrix which can be written as:

::<math>\begin{align}
{{({{X}^{\prime }}X)}^{-1}}= & 0.125\cdot I = & \frac{1}{8}\cdot I = & \frac{1}{{{2}^{3}}}\cdot I
\end{align}\,\!</math>

where <math>I\,\!</math> represents the identity matrix of the same order as the design matrix, <math>X\,\!</math>. Since there are eight observations per replicate of the <math>{2}^{3}\,\!</math> design, the <math>(X\,\!</math> ' <math>X{{)}^{-1}}\,\!</math> matrix for <math>m\,\!</math> replicates of this design can be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{3}}\cdot m)}\cdot I\,\!</math>

The <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> matrix for any <math>{2}^{k}\,\!</math> design can now be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{k}}\cdot m)}\cdot I\,\!</math>

Then the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is:

::<math>\begin{align}
C= & {{{\hat{\sigma }}}^{2}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & M{{S}_{E}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & \frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}\cdot I
\end{align}\,\!</math>

Note that the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is also a diagonal matrix. Therefore, the estimated effect coefficients (<math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math>, <math>{{\beta }_{12}},\,\!</math> etc.) for these designs are uncorrelated. This implies that the terms in the <math>{2}^{k}\,\!</math> design (main effects, interactions) are independent of each other. Consequently, the extra sum of squares for each of the terms in these designs is independent of the sequence of terms in the model, and also independent of the presence of other terms in the model. As a result the sequential and partial sum of squares for the terms are identical for these designs and will always add up to the model sum of squares. Multicollinearity is also not an issue for these designs.

It can also be noted from the equation given above, that in addition to the <math>C\,\!</math> matrix being diagonal, all diagonal elements of the <math>C\,\!</math> matrix are identical. This means that the variance (or its square root, the standard error) of all estimated effect coefficients are the same. The standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>, for all the coefficients is:

::<math>\begin{align}
se({{{\hat{\beta }}}_{j}})= & \sqrt{{{C}_{jj}}} = & \sqrt{\frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}}\text{ }for\text{ }all\text{ }j
\end{align}\,\!</math>

This property is used to construct the normal probability plot of effects in <math>{2}^{k}\,\!</math> designs and identify significant effects using graphical techniques. For details on the normal probability plot of effects in a Weibull++ DOE folio, refer to [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]].

====Example====
To illustrate the analysis of a full factorial <math>{2}^{k}\,\!</math> design, consider a three factor experiment to investigate the effect of honing pressure, number of strokes and cycle time on the surface finish of automobile brake drums. Each of these factors is investigated at two levels. The honing pressure is investigated at levels of 200 <math>psi\,\!</math> and 400 <math>psi\,\!</math>, the number of strokes used is 3 and 5 and the two levels of the cycle time are 3 and 5 seconds. The design for this experiment is set up in a Weibull++ DOE folio as shown in the first two following figures. It is decided to run two replicates for this experiment. The surface finish data collected from each run (using randomization) and the complete design is shown in the third following figure. The analysis of the experiment data is explained next.

[[Image:doe7_5.png|center|877px|Design properties for the experiment in the example.|link=]]

[[Image:doe7_6.png|center|859px|Design summary for the experiment in the example.|link=]]

[[Image:doe7_7.png|center|778px|Experiment design for the example to investigate the surface finish of automobile brake drums.|link=]]

The applicable model using the notation for <math>{2}^{k}\,\!</math> designs is:

::<math>\begin{align}
Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where the indicator variable, <math>{{x}_{1,}}\,\!</math> represents factor <math>A\,\!</math> (honing pressure), <math>{{x}_{1}}=-1\,\!</math> represents the low level of 200 <math>psi\,\!</math> and <math>{{x}_{1}}=1\,\!</math> represents the high level of 400 <math>psi\,\!</math>. Similarly, <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> represent factors <math>B\,\!</math> (number of strokes) and <math>C\,\!</math> (cycle time), respectively. <math>{{\beta }_{0}}\,\!</math> is the overall mean, while <math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math> and <math>{{\beta }_{3}}\,\!</math> are the effect coefficients for the main effects of factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. <math>{{\beta }_{12}}\,\!</math>, <math>{{\beta }_{13}}\,\!</math> and <math>{{\beta }_{23}}\,\!</math> are the effect coefficients for the <math>AB\,\!</math>, <math>AC\,\!</math> and <math>BC\,\!</math> interactions, while <math>{{\beta }_{123}}\,\!</math> represents the <math>ABC\,\!</math> interaction.

 
If the subscripts for the run (<math>i\,\!</math> ; <math>i=\,\!</math> 1 to 8) and replicates (<math>j\,\!</math> ; <math>j=\,\!</math> 1,2) are included, then the model can be written as:

::<math>\begin{align}
{{Y}_{ij}}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{ij1}}+{{\beta }_{2}}\cdot {{x}_{ij2}}+{{\beta }_{12}}\cdot {{x}_{ij1}}{{x}_{ij2}}+{{\beta }_{3}}\cdot {{x}_{ij3}} \\
& +{{\beta }_{13}}\cdot {{x}_{ij1}}{{x}_{ij3}}+{{\beta }_{23}}\cdot {{x}_{ij2}}{{x}_{ij3}}+{{\beta }_{123}}\cdot {{x}_{ij1}}{{x}_{ij2}}{{x}_{ij3}}+{{\epsilon }_{ij}}
\end{align}\,\!</math>

To investigate how the given factors affect the response, the following hypothesis tests need to be carried:

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{1}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{1}}\ne 0\,\!</math>

This test investigates the main effect of factor <math>A\,\!</math> (honing pressure). The statistic for this test is:

::<math>{{({{F}_{0}})}_{A}}=\frac{M{{S}_{A}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{A}}\,\!</math> is the mean square for factor <math>A\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other main effects, <math>B\,\!</math> and <math>C\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{12}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{12}}\ne 0\,\!</math>

This test investigates the two factor interaction <math>AB\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{AB}}=\frac{M{{S}_{AB}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the interaction <math>AB\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other two factor interactions, <math>AC\,\!</math> and <math>BC\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{123}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{123}}\ne 0\,\!</math>

This test investigates the three factor interaction <math>ABC\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{ABC}}=\frac{M{{S}_{ABC}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{ABC}}\,\!</math> is the mean square for the interaction <math>ABC\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square.
To calculate the test statistics, it is convenient to express the ANOVA model in the form <math>y=X\beta +\epsilon \,\!</math>.

====Expression of the ANOVA Model as <math>y=X\beta +\epsilon \,\!</math>====

In matrix notation, the ANOVA model can be expressed as:

::<math>y=X\beta +\epsilon \,\!</math>

where:

<center><math>y=\left[ \begin{matrix}
{{Y}_{11}} \\
{{Y}_{21}} \\
. \\
{{Y}_{81}} \\
{{Y}_{12}} \\
. \\
{{Y}_{82}} \\
\end{matrix} \right]=\left[ \begin{matrix}
90 \\
90 \\
. \\
90 \\
86 \\
. \\
80 \\
\end{matrix} \right]\text{ }X=\left[ \begin{matrix}
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\end{matrix} \right]\,\!</math></center>

<center><math>\beta =\left[ \begin{matrix}
{{\beta }_{0}} \\
{{\beta }_{1}} \\
{{\beta }_{2}} \\
{{\beta }_{12}} \\
{{\beta }_{3}} \\
{{\beta }_{13}} \\
{{\beta }_{23}} \\
{{\beta }_{123}} \\
\end{matrix} \right]\text{ }\epsilon =\left[ \begin{matrix}
{{\epsilon }_{11}} \\
{{\epsilon }_{21}} \\
. \\
{{\epsilon }_{81}} \\
{{\epsilon }_{12}} \\
. \\
. \\
{{\epsilon }_{82}} \\
\end{matrix} \right]\,\!</math></center>

====Calculation of the Extra Sum of Squares for the Factors====

Knowing the matrices <math>y\,\!</math>, <math>X\,\!</math> and <math>\beta \,\!</math>, the extra sum of squares for the factors can be calculated. These are used to calculate the mean squares that are used to obtain the test statistics. Since the experiment design is orthogonal, the partial and sequential extra sum of squares are identical. The extra sum of squares for each effect can be calculated as shown next. As an example, the extra sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & Model\text{ }Sum\text{ }of\text{ }Squares - Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }main\text{ }effect\text{ }of\text{ }A \\
= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }A}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }A}}={{X}_{\tilde{\ }A}}{{(X_{\tilde{\ }A}^{\prime }{{X}_{\tilde{\ }A}})}^{-1}}X_{\tilde{\ }A}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }A}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the main effect of factor <math>A\,\!</math>. Thus, the sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y \\
= & 654.4375-549.375 \\
= & 105.0625
\end{align}\,\!</math>

Similarly, the extra sum of squares for the interaction effect <math>AB\,\!</math> is:

::<math>\begin{align}
S{{S}_{AB}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }AB}}-(1/16)J]y \\
= & 654.4375-636.375 \\
= & 18.0625
\end{align}\,\!</math>

The extra sum of squares for other effects can be obtained in a similar manner.

====Calculation of the Test Statistics====

Knowing the extra sum of squares, the test statistic for the effects can be calculated. For example, the test statistic for the interaction <math>AB\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{AB}}= & \frac{M{{S}_{AB}}}{M{{S}_{E}}} \\
= & \frac{S{{S}_{AB}}/dof(S{{S}_{AB}})}{S{{S}_{E}}/dof(S{{S}_{E}})} \\
= & \frac{18.0625/1}{147.5/8} \\
= & 0.9797
\end{align}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the <math>AB\,\!</math> interaction and <math>M{{S}_{E}}\,\!</math> is the error mean square. The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{AB}}=0.9797\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{AB}}) \\
= & 1-0.6487 \\
= & 0.3513
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that the interaction between honing pressure and number of strokes does not affect the surface finish of the brake drums. Tests for other effects can be carried out in a similar manner. The results are shown in the ANOVA Table in the following figure. The values S, R-sq and R-sq(adj) in the figure indicate how well the model fits the data. The value of S represents the standard error of the model, R-sq represents the coefficient of multiple determination and R-sq(adj) represents the adjusted coefficient of multiple determination. For details on these values refer to [[Multiple_Linear_Regression_Analysis|Multiple Linear Regression Analysis]].

[[Image:doe7_8.png|center|799px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

====Calculation of Effect Coefficients====

The estimate of effect coefficients can also be obtained:

<center><math>\begin{align}
\hat{\beta }= & {{({{X}^{\prime }}X)}^{-1}}{{X}^{\prime }}y \\
= & \left[ \begin{matrix}
86.4375 \\
2.5625 \\
-4.9375 \\
1.0625 \\
-1.0625 \\
2.4375 \\
-1.3125 \\
-0.1875 \\
\end{matrix} \right]
\end{align}\,\!</math></center>

[[Image:doe7_9.png|center|800px|Regression Information table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

The coefficients and related results are shown in the Regression Information table above. In the table, the Effect column displays the effects, which are simply twice the coefficients. The Standard Error column displays the standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>. The Low CI and High CI columns display the confidence interval on the coefficients. The interval shown is the 90% interval as the significance is chosen as 0.1. The T Value column displays the <math>t\,\!</math> statistic, <math>{{t}_{0}}\,\!</math>, corresponding to the coefficients. The P Value column displays the <math>p\,\!</math> value corresponding to the <math>t\,\!</math> statistic. (For details on how these results are calculated, refer to [[General Full Factorial Designs]]). Plots of residuals can also be obtained from the DOE folio to ensure that the assumptions related to the ANOVA model are not violated.

====Model Equation====

From the analysis results in the above figure within [[Two_Level_Factorial_Experiments#Calculation_of_Effect_Coefficients|calculation of effect coefficients]] section, it is seen that effects <math>A\,\!</math>, <math>B\,\!</math> and <math>AC\,\!</math> are significant. In a DOE folio, the <math>p\,\!</math> values for the significant effects are displayed in red in the ANOVA Table for easy identification. Using the values of the estimated effect coefficients, the model for the present <math>{2}^{3}\,\!</math> design in terms of the coded values can be written as:

::<math>\begin{align}
\hat{y}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}} \\
= & 86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+2.4375{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

To make the model hierarchical, the main effect, <math>C\,\!</math>, needs to be included in the model (because the interaction <math>AC\,\!</math> is included in the model). The resulting model is:

::<math>\hat{y}=86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+1.0625{{x}_{3}}+2.4375{{x}_{1}}{{x}_{3}}\,\!</math>

This equation can be viewed in a DOE folio, as shown in the following figure, using the Show Analysis Summary icon in the Control Panel. The equation shown in the figure will match the hierarchical model once the required terms are selected using the Select Effects icon.

[[Image:doe7_10.png|center|557px|The model equation for the experiment of the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

==Replicated and Repeated Runs==

In the case of replicated experiments, it is important to note the difference between replicated runs and repeated runs. Both repeated and replicated runs are multiple response readings taken at the same factor levels. However, repeated runs are response observations taken at the same time or in succession. Replicated runs are response observations recorded in a random order. Therefore, replicated runs include more variation than repeated runs. For example, a baker, who wants to investigate the effect of two factors on the quality of cakes, will have to bake four cakes to complete one replicate of a <math>{2}^{2}\,\!</math> design. Assume that the baker bakes eight cakes in all. If, for each of the four treatments of the <math>{2}^{2}\,\!</math> design, the baker selects one treatment at random and then bakes two cakes for this treatment at the same time then this is a case of two repeated runs. If, however, the baker bakes all the eight cakes randomly, then the eight cakes represent two sets of replicated runs.
For repeated measurements, the average values of the response for each treatment should be entered into a DOE folio as shown in the following figure (a) when the two cakes for a particular treatment are baked together. For replicated measurements, when all the cakes are baked randomly, the data is entered as shown in the following figure (b).

[[Image:doe7.11.png|center|300px|Data entry for repeated and replicated runs. Figure (a) shows repeated runs and (b) shows replicated runs.]]

==Unreplicated 2''k'' Designs==

If a factorial experiment is run only for a single replicate then it is not possible to test hypotheses about the main effects and interactions as the error sum of squares cannot be obtained. This is because the number of observations in a single replicate equals the number of terms in the ANOVA model. Hence the model fits the data perfectly and no degrees of freedom are available to obtain the error sum of squares.

However, sometimes it is only possible to run a single replicate of the <math>{2}^{k}\,\!</math> design because of constraints on resources and time. In the absence of the error sum of squares, hypothesis tests to identify significant factors cannot be conducted. A number of methods of analyzing information obtained from unreplicated <math>{2}^{k}\,\!</math> designs are available. These include pooling higher order interactions, using the normal probability plot of effects or including center point replicates in the design.

===Pooling Higher Order Interactions===

One of the ways to deal with unreplicated <math>{2}^{k}\,\!</math> designs is to use the sum of squares of some of the higher order interactions as the error sum of squares provided these higher order interactions can be assumed to be insignificant. By dropping some of the higher order interactions from the model, the degrees of freedom corresponding to these interactions can be used to estimate the error mean square. Once the error mean square is known, the test statistics to conduct hypothesis tests on the factors can be calculated.

===Normal Probability Plot of Effects===
Another way to use unreplicated <math>{2}^{k}\,\!</math> designs to identify significant effects is to construct the normal probability plot of the effects. As mentioned in [[Two_Level_Factorial_Experiments#Special_Features| Special Features]], the standard error for all effect coefficients in the <math>{2}^{k}\,\!</math> designs is the same. Therefore, on a normal probability plot of effect coefficients, all non-significant effect coefficients (with <math>\beta =0\,\!</math>) will fall along the straight line representative of the normal distribution, N(<math>0,{{\sigma }^{2}}/({{2}^{k}}\cdot m)\,\!</math>). Effect coefficients that show large deviations from this line will be significant since they do not come from this normal distribution. Similarly, since effects <math>=2\times \,\!</math> effect coefficients, all non-significant effects will also follow a straight line on the normal probability plot of effects. For replicated designs, the Effects Probability plot of a DOE folio plots the normalized effect values (or the T Values) on the standard normal probability line, N(0,1). However, in the case of unreplicated <math>{2}^{k}\,\!</math> designs, <math>{{\sigma }^{2}}\,\!</math> remains unknown since <math>M{{S}_{E}}\,\!</math> cannot be obtained. Lenth's method is used in this case to estimate the variance of the effects. For details on Lenth's method, please refer to [[DOE References| Montgomery (2001)]]. The DOE folio then uses this variance value to plot effects along the N(0, Lenth's effect variance) line. The
method is illustrated in the following example.

====Example====

Vinyl panels, used as instrument panels in a certain automobile, are seen to develop defects after a certain amount of time. To investigate the issue, it is decided to carry out a two level factorial experiment. Potential factors to be investigated in the experiment are vacuum rate (factor <math>A\,\!</math>), material temperature (factor <math>B\,\!</math>), element intensity (factor <math>C\,\!</math>) and pre-stretch (factor <math>D\,\!</math>). The two levels of the factors used in the experiment are as shown in below.

[[Image:doet7.1.png|center|300px|Factors to investigate defects in vinyl panels.]]

With a <math>{2}^{4}\,\!</math> design requiring 16 runs per replicate it is only feasible for the manufacturer to run a single replicate.

The experiment design and data, collected as percent defects, are shown in the following figure. Since the present experiment design contains only a single replicate, it is not possible to obtain an estimate of the error sum of squares, <math>S{{S}_{E}}\,\!</math>. It is decided to use the normal probability plot of effects to identify the significant effects. The effect values for each term are obtained as shown in the following figure.

[[Image:doe7_13.png|center|650px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

Lenth's method uses these values to estimate the variance. As described in [[DOE_References|[Lenth, 1989]]], if all effects are arranged in ascending order, using their absolute values, then <math>{{s}_{0}}\,\!</math> is defined as 1.5 times the median value:

::<math>\begin{align}
{{s}_{0}}= & 1.5\cdot median(\left| effect \right|) \\
= & 1.5\cdot 2 \\
= & 3
\end{align}\,\!</math>

Using <math>{{s}_{0}}\,\!</math>, the "pseudo standard error" (<math>PSE\,\!</math>) is calculated as 1.5 times the median value of all effects that are less than 2.5 <math>{{s}_{0}}\,\!</math> :

::<math>\begin{align}
PSE= & 1.5\cdot median(\left| effect \right|\ \ :\ \ \left| effect \right|<2.5{{s}_{0}}) \\
= & 1.5\cdot 1.5 \\
= & 2.25
\end{align}\,\!</math>

Using <math>PSE\,\!</math> as an estimate of the effect variance, the effect variance is 2.25. Knowing the effect variance, the normal probability plot of effects for the present unreplicated experiment can be constructed as shown in the following figure. The line on this plot is the line N(0, 2.25). The plot shows that the effects <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math> do not follow the distribution represented by this line. Therefore, these effects are significant.

The significant effects can also be identified by comparing individual effect values to the margin of error or the threshold value using the pareto chart (see the third following figure). If the required significance is 0.1, then:

::<math>margin\text{ }of\text{ }error={{t}_{\alpha /2,d}}\cdot PSE\,\!</math>

The <math>t\,\!</math> statistic, <math>{{t}_{\alpha /2,d}}\,\!</math>, is calculated at a significance of <math>\alpha /2\,\!</math> (for the two-sided hypothesis) and degrees of freedom <math>d=(\,\!</math> number of effects <math>)/3\,\!</math>. Thus:

::<math>\begin{align}
margin\text{ }of\text{ }error= & {{t}_{0.05,5}}\cdot PSE \\
= & 2.015\cdot 2.25 \\
= & 4.534
\end{align}\,\!</math>

The value of 4.534 is shown as the critical value line in the third following figure. All effects with absolute values greater than the margin of error can be considered to be significant. These effects are <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math>. Therefore, the vacuum rate, the pre-stretch and their interaction have a significant effect on the defects of the vinyl panels.

[[Image:doe7_14.png|center|800px|Effect values for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_15.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_16.png|center|650px|Pareto chart for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

===Center Point Replicates===

Another method of dealing with unreplicated <math>{2}^{k}\,\!</math> designs that only have quantitative factors is to use replicated runs at the center point. The center point is the response corresponding to the treatment exactly midway between the two levels of all factors. Running multiple replicates at this point provides an estimate of pure error. Although running multiple replicates at any treatment level can provide an estimate of pure error, the other advantage of running center point replicates in the <math>{2}^{k}\,\!</math> design is in checking for the presence of curvature. The test for curvature investigates whether the model between the response and the factors is linear and is discussed in [[Two_Level_Factorial_Experiments#Using_Center_Point_Replicates_to_Test_Curvature| Center Pt. Replicates to Test Curvature]].

====Example: Use Center Point to Get Pure Error====

Consider a <math>{2}^{2}\,\!</math> experiment design to investigate the effect of two factors, <math>A\,\!</math> and <math>B\,\!</math>, on a certain response. The energy consumed when the treatments of the <math>{2}^{2}\,\!</math> design are run is considerably larger than the energy consumed for the center point run (because at the center point the factors are at their middle levels). Therefore, the analyst decides to run only a single replicate of the design and augment the design by five replicated runs at the center point as shown in the following figure. The design properties for this experiment are shown in the second following figure. The complete experiment design is shown in the third following figure. The center points can be used in the identification of significant effects as shown next.

[[Image:doe7.17.png||center|300px|<math>2^2\,\!</math> design augmented by five center point runs.]]
[[Image:doe7_18.png|center|537px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

[[Image:doe7_19.png|center|630px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

Since the present <math>{2}^{2}\,\!</math> design is unreplicated, there are no degrees of freedom available to calculate the error sum of squares. By augmenting this design with five center points, the response values at the center points, <math>y_{i}^{c}\,\!</math>, can be used to obtain an estimate of pure error, <math>S{{S}_{PE}}\,\!</math>. Let <math>{{\bar{y}}^{c}}\,\!</math> represent the average response for the five replicates at the center. Then:

::<math>S{{S}_{PE}}=Sum\text{ }of\text{ }Squares\text{ }for\text{ }center\text{ }points\,\!</math>

::<math>\begin{align}
S{{S}_{PE}}= & \underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}} \\
= & {{(25.2-25.26)}^{2}}+...+{{(25.3-25.26)}^{2}} \\
= & 0.052
\end{align}\,\!</math>

Then the corresponding mean square is:

::<math>\begin{align}
M{{S}_{PE}}= & \frac{S{{S}_{PE}}}{degrees\text{ }of\text{ }freedom} \\
= & \frac{0.052}{5-1} \\
= & 0.013
\end{align}\,\!</math>

Alternatively, <math>M{{S}_{PE}}\,\!</math> can be directly obtained by calculating the variance of the response values at the center points:

::<math>\begin{align}
M{{S}_{PE}}= & {{s}^{2}} \\
= & \frac{\underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}}}{5-1}
\end{align}\,\!</math>

Once <math>M{{S}_{PE}}\,\!</math> is known, it can be used as the error mean square, <math>M{{S}_{E}}\,\!</math>, to carry out the test of significance for each effect. For example, to test the significance of the main effect of factor <math>A,\,\!</math> the sum of squares corresponding to this effect is obtained in the usual manner by considering only the four runs of the original <math>{2}^{2}\,\!</math> design.

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/4)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/4)J]y \\
= & 0.5625
\end{align}\,\!</math>

Then, the test statistic to test the significance of the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{A}}= & \frac{M{{S}_{A}}}{M{{S}_{E}}} \\
= & \frac{0.5625/1}{0.052/4} \\
= & 43.2692
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{A}}=43.2692\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{A}}) \\
= & 1-0.9972 \\
= & 0.0028
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value < 0.1, it can be concluded that the main effect of factor <math>A\,\!</math> significantly affects the response. This result is displayed in the ANOVA table as shown in the following figure. Test for the significance of other factors can be carried out in a similar manner.

[[Image:doe7_20.png|center|649px|Results for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

===Using Center Point Replicates to Test Curvature===

Center point replicates can also be used to check for curvature in replicated or unreplicated <math>{2}^{k}\,\!</math> designs. The test for curvature investigates whether the model between the response and the factors is linear. The way DOE++ handles center point replicates is similar to its handling of blocks. The center point replicates are treated as an additional factor in the model. The factor is labeled as Curvature in the results of the DOE folio. If Curvature turns out to be a significant factor in the results, then this indicates the presence of curvature in the model.

====Example: Use Center Point to Test Curvature====

To illustrate the use of center point replicates in testing for curvature, consider again the data of the single replicate <math>{2}^{2}\,\!</math> experiment from a preceding figure(labeled "<math>2^2</math> design augmented by five center point runs"). Let <math>{{x}_{1}}\,\!</math> be the indicator variable to indicate if the run is a center point:

::<math>\begin{matrix}
{{x}_{1}}=0 & {} & \text{Center point run} \\
{{x}_{1}}=1 & {} & \text{Other run} \\
\end{matrix}\,\!</math>

If <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> are the indicator variables representing factors <math>A\,\!</math> and <math>B\,\!</math>, respectively, then the model for this experiment is:

::<math>Y={{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}\,\!</math>

To investigate the presence of curvature, the following hypotheses need to be tested:

::<math>\begin{align}
& {{H}_{0}}: & {{\beta }_{1}}=0\text{ (Curvature is absent)} \\
& {{H}_{1}}: & {{\beta }_{1}}\ne 0
\end{align}\,\!</math>

The test statistic to be used for this test is:

::<math>{{({{F}_{0}})}_{curvature}}=\frac{M{{S}_{curvature}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{curvature}}\,\!</math> is the mean square for Curvature and <math>M{{S}_{E}}\,\!</math> is the error mean square.

'''Calculation of the Sum of Squares'''

The <math>X\,\!</math> matrix and <math>y\,\!</math> vector for this experiment are:

<center><math>X=\left[ \begin{matrix}
1 & 1 & -1 & -1 & 1 \\
1 & 1 & 1 & -1 & -1 \\
1 & 1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
\end{matrix} \right]\text{ }y=\left[ \begin{matrix}
24.6 \\
25.4 \\
25.0 \\
25.7 \\
25.2 \\
25.3 \\
25.4 \\
25.1 \\
25.3 \\
\end{matrix} \right]\,\!</math></center>

The sum of squares can now be calculated. For example, the error sum of squares is:

::<math>\begin{align}
& S{{S}_{E}}= & {{y}^{\prime }}[I-H]y \\
& = & 0.052
\end{align}\,\!</math>

where <math>I\,\!</math> is the identity matrix and <math>H\,\!</math> is the hat matrix. It can be seen that this is equal to <math>S{{S}_{PE\text{ }}}\,\!</math> (the sum of squares due to pure error) because of the replicates at the center point, as obtained in the [[Two_Level_Factorial_Experiments#Example_3| example]]. The number of degrees of freedom associated with <math>S{{S}_{E}}\,\!</math>, <math>dof(S{{S}_{E}})\,\!</math> is four. The extra sum of squares corresponding to the center point replicates (or Curvature) is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & Model\text{ }Sum\text{ }of\text{ }Squares- \\
& & Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }center\text{ }point \\
& = & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Curvature}}-(1/9)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }Curvature}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }Curvature}}={{X}_{\tilde{\ }Curv}}{{(X_{\tilde{\ }Curv}^{\prime }{{X}_{\tilde{\ }Curv}})}^{-1}}X_{\tilde{\ }Curv}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }Curv}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the center point. Thus, the extra sum of squares corresponding to Curvature is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Center}}-(1/9)J]y \\
& = & 0.7036-0.6875 \\
& = & 0.0161
\end{align}\,\!</math>

This extra sum of squares can be used to test for the significance of curvature. The corresponding mean square is:

::<math>\begin{align}
& M{{S}_{Curvature}}= & \frac{Sum\text{ }of\text{ }squares\text{ }corresponding\text{ }to\text{ }Curvature}{degrees\text{ }of\text{ }freedom} \\
& = & \frac{0.0161}{1} \\
& = & 0.0161
\end{align}\,\!</math>

'''Calculation of the Test Statistic'''

Knowing the mean squares, the statistic to check the significance of curvature can be calculated.

::<math>\begin{align}
& {{({{f}_{0}})}_{Curvature}}= & \frac{M{{S}_{Curvature}}}{M{{S}_{E}}} \\
& = & \frac{0.0161/1}{0.052/4} \\
& = & 1.24
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{Curvature}}=1.24\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and four degrees of freedom in the denominator is:

::<math>\begin{align}
& p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{Curvature}}) \\
& = & 1-0.6713 \\
& = & 0.3287
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that curvature does not exist for this design. This results is shown in the ANOVA table in the figure above. The surface of the fitted model based on these results, along with the observed response values, is shown in the figure below.

[[Image:doe7.21.png|center|400px|Model surface and observed response values for the design in the [[Two_Level_Factorial_Experiments#Example_4| example]].]]

 

==Blocking in 2k Designs==

Blocking can be used in the <math>{2}^{k}\,\!</math> designs to deal with cases when replicates cannot be run under identical conditions. Randomized complete block designs that were discussed in [[Randomization and Blocking in DOE]] for factorial experiments are also applicable here. At times, even with just two levels per factor, it is not possible to run all treatment combinations for one replicate of the experiment under homogeneous conditions. For example, each replicate of the <math>{2}^{2}\,\!</math> design requires four runs. If each run requires two hours and testing facilities are available for only four hours per day, two days of testing would be required to run one complete replicate. Blocking can be used to separate the treatment runs on the two different days. Blocks that do not contain all treatments of a replicate are called incomplete blocks. In incomplete block designs, the block effect is confounded with certain effect(s) under investigation. For the <math>{2}^{2}\,\!</math> design assume that treatments <math>(1)\,\!</math> and <math>ab\,\!</math> were run on the first day and treatments <math>a\,\!</math> and <math>b\,\!</math> were run on the second day. Then, the incomplete block design for this experiment is:

::<math>\begin{matrix}
\text{Block 1} & {} & \text{Block 2} \\
\left[ \begin{matrix}
(1) \\
ab \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
b \\
\end{matrix} \right] \\
\end{matrix}\,\!</math>

For this design the block effect may be calculated as:

::<math>\begin{align}
& Block\text{ }Effect= & Average\text{ }response\text{ }for\text{ }Block\text{ }1- \\
& & Average\text{ }response\text{ }for\text{ }Block\text{ }2 \\
& = & \frac{(1)+ab}{2}-\frac{a+b}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The <math>AB\,\!</math> interaction effect is:

::<math>\begin{align}
& AB= & Average\text{ }response\text{ }at\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{low}}}- \\
& & Average\text{ }response\text{ }at\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{low}}} \\
& = & \frac{ab+(1)}{2}-\frac{b+a}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The two equations given above show that, in this design, the <math>AB\,\!</math> interaction effect cannot be distinguished from the block effect because the formulas to calculate these effects are the same. In other words, the <math>AB\,\!</math> interaction is said to be confounded with the block effect and it is not possible to say if the effect calculated based on these equations is due to the <math>AB\,\!</math> interaction effect, the block effect or both. In incomplete block designs some effects are always confounded with the blocks. Therefore, it is important to design these experiments in such a way that the important effects are not confounded with the blocks. In most cases, the experimenter can assume that higher order interactions are unimportant. In this case, it would better to use incomplete block designs that confound these effects with the blocks.
One way to design incomplete block designs is to use defining contrasts as shown next:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}+...+{{\alpha }_{k}}{{q}_{k}}\,\!</math>

where the <math>{{\alpha }_{i}}\,\!</math> s are the exponents for the factors in the effect that is to be confounded with the block effect and the <math>{{q}_{i}}\,\!</math> s are values based on the level of the <math>i\,\!</math> the factor (in a treatment that is to be allocated to a block). For <math>{2}^{k}\,\!</math> designs the <math>{{\alpha }_{i}}\,\!</math> s are either 0 or 1 and the <math>{{q}_{i}}\,\!</math> s have a value of 0 for the low level of the <math>i\,\!</math> th factor and a value of 1 for the high level of the factor in the treatment under consideration. As an example, consider the <math>{2}^{2}\,\!</math> design where the interaction effect <math>AB\,\!</math> is confounded with the block. Since there are two factors, <math>k=2\,\!</math>, with <math>i=1\,\!</math> representing factor <math>A\,\!</math> and <math>i=2\,\!</math> representing factor <math>B\,\!</math>. Therefore:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}\,\!</math>

The value of <math>{{\alpha }_{1}}\,\!</math> is one because the exponent of factor <math>A\,\!</math> in the confounded interaction <math>AB\,\!</math> is one. Similarly, the value of <math>{{\alpha }_{2}}\,\!</math> is one because the exponent of factor <math>B\,\!</math> in the confounded interaction <math>AB\,\!</math> is also one. Therefore, the defining contrast for this design can be written as:

::<math>\begin{align}
& L= & {{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}} \\
& = & 1\cdot {{q}_{1}}+1\cdot {{q}_{2}} \\
& = & {{q}_{1}}+{{q}_{2}}
\end{align}\,\!</math>

Once the defining contrast is known, it can be used to allocate treatments to the blocks. For the <math>{2}^{2}\,\!</math> design, there are four treatments <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math> and <math>ab\,\!</math>. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. In order to decide which block the treatment <math>(1)\,\!</math> belongs to, the levels of factors <math>A\,\!</math> and <math>B\,\!</math> for this run are used. Since factor <math>A\,\!</math> is at the low level in this treatment, <math>{{q}_{1}}=0\,\!</math>. Similarly, since factor <math>B\,\!</math> is also at the low level in this treatment, <math>{{q}_{2}}=0\,\!</math>. Therefore:

::<math>\begin{align}
& L= & {{q}_{1}}+{{q}_{2}} \\
& = & 0+0=0\text{ (mod 2)}
\end{align}\,\!</math>

Note that the value of <math>L\,\!</math> used to decide the block allocation is "mod 2" of the original value. This value is obtained by taking the value of 1 for odd numbers and 0 otherwise. Based on the value of <math>L\,\!</math>, treatment <math>(1)\,\!</math> is assigned to block 1. Other treatments can be assigned using the following calculations:

::<math>\begin{align}
& (1): & \text{ }L=0+0=0=0\text{ (mod 2)} \\
& a: & \text{ }L=1+0=1=1\text{ (mod 2)} \\
& b: & \text{ }L=0+1=1=1\text{ (mod 2)} \\
& ab: & \text{ }L=1+1=2=0\text{ (mod 2)}
\end{align}\,\!</math>

Therefore, to confound the interaction <math>AB\,\!</math> with the block effect in the <math>{2}^{2}\,\!</math> incomplete block design, treatments <math>(1)\,\!</math> and <math>ab\,\!</math> (with <math>L=0\,\!</math>) should be assigned to block 2 and treatment combinations <math>a\,\!</math> and <math>b\,\!</math> (with <math>L=1\,\!</math>) should be assigned to block 1.

====Example: Two Level Factorial Design with Two Blocks====

This example illustrates how treatments can be allocated to two blocks for an unreplicated <math>{2}^{k}\,\!</math> design. Consider the unreplicated <math>{2}^{4}\,\!</math> design to investigate the four factors affecting the defects in automobile vinyl panels discussed in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments required for this experiment were run by two different operators with each operator conducting 8 runs. This experiment is an example of an incomplete block design. The analyst in charge of this experiment assumed that the interaction <math>ABCD\,\!</math> was not significant and decided to allocate treatments to the two operators so that the <math>ABCD\,\!</math> interaction was confounded with the block effect (the two operators are the blocks). The allocation scheme to assign treatments to the two operators can be obtained as follows.
 
The defining contrast for the <math>{2}^{4}\,\!</math> design where the <math>ABCD\,\!</math> interaction is confounded with the blocks is:

::<math>L={{q}_{1}}+{{q}_{2}}+{{q}_{3}}+{{q}_{4}}\,\!</math>

The treatments can be allocated to the two operators using the values of the defining contrast. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. Then the value of the defining contrast for treatment <math>a\,\!</math> is:

::<math>a\ \ :\ \ \text{ }L=1+0+0+0=1=1\text{ (mod 2)}\,\!</math>

Therefore, treatment <math>a\,\!</math> should be assigned to Block 1 or the first operator. Similarly, for treatment <math>ab\,\!</math> we have:

::<math>ab\ \ :\ \ \text{ }L=1+1+0+0=2=0\text{ (mod 2)}\,\!</math>

[[Image:doe7.22.png|center|200px| Allocation of treatments to two blocks for the <math>2^4</math> design in the example by confounding interaction of <math>ABCD</math> with the blocks.]]

Therefore, <math>ab\,\!</math> should be assigned to Block 2 or the second operator. Other treatments can be allocated to the two operators in a similar manner to arrive at the allocation scheme shown in the figure below.
In DOE++, to confound the <math>ABCD\,\!</math> interaction for the <math>{2}^{4}\,\!</math> design into two blocks, the number of blocks are specified as shown in the figure below. Then the interaction <math>ABCD\,\!</math> is entered in the Block Generator window (second following figure) which is available using the Block Generator button in the following figure. The design generated by the Weibull++ DOE folio is shown in the third of the following figures. This design matches the allocation scheme of the preceding figure.

[[Image:doe7_23.png|center|700px| Adding block properties for the experiment in the example.|link=]]

[[Image:doe7_24.png|center|700px|Specifying the interaction ABCD as the interaction to be confounded with the blocks for the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

[[Image:doe7_25.png|center|800px|Two block design for the experiment in the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

For the analysis of this design, the sum of squares for all effects are calculated assuming no blocking. Then, to account for blocking, the sum of squares corresponding to the <math>ABCD\,\!</math> interaction is considered as the sum of squares due to blocks and <math>ABCD\,\!</math>. In DOE++ this is done by displaying this sum of squares as the sum of squares due to the blocks. This is shown in the following figure where the sum of squares in question is obtained as 72.25 and is displayed against Block. The interaction ABCD, which is confounded with the blocks, is not displayed. Since the design is unreplicated, any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_26.png|center|793px|ANOVA table for the experiment of the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

===Unreplicated 2''k'' Designs in 2''p'' Blocks===

A single replicate of the <math>{2}^{k}\,\!</math> design can be run in up to <math>{2}^{p}\,\!</math> blocks where <math>p<k\,\!</math>. The number of effects confounded with the blocks equals the degrees of freedom associated with the block effect.

If two blocks are used (the block effect has two levels), then one (<math>2-1=1)\,\!</math> effect is confounded with the blocks. If four blocks are used, then three (<math>4-1=3\,\!</math>) effects are confounded with the blocks and so on. For example an unreplicated <math>{2}^{4}\,\!</math> design may be confounded in <math>{2}^{2}\,\!</math> (four) blocks using two contrasts, <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}}\,\!</math>. Let <math>AC\,\!</math> and <math>BD\,\!</math> be the effects to be confounded with the blocks. Corresponding to these two effects, the contrasts are respectively:

::<math>\begin{align}
& {{L}_{1}}= & {{q}_{1}}+{{q}_{3}} \\
& {{L}_{2}}= & {{q}_{2}}+{{q}_{4}}
\end{align}\,\!</math>

Based on the values of <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}},\,\!</math> the treatments can be assigned to the four blocks as follows:

<center><math>\begin{matrix}
\text{Block 4} & {} & \text{Block 3} & {} & \text{Block 2} & {} & \text{Block 1} \\
{{L}_{1}}=0,{{L}_{2}}=0 & {} & {{L}_{1}}=1,{{L}_{2}}=0 & {} & {{L}_{1}}=0,{{L}_{2}}=1 & {} & {{L}_{1}}=1,{{L}_{2}}=1 \\
{} & {} & {} & {} & {} & {} & {} \\
\left[ \begin{matrix}
(1) \\
ac \\
bd \\
abcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
c \\
abd \\
bcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
b \\
abc \\
d \\
acd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
ab \\
bc \\
ad \\
cd \\
\end{matrix} \right] \\
\end{matrix}\,\!</math></center>

Since the block effect has three degrees of freedom, three effects are confounded with the block effect. In addition to <math>AC\,\!</math> and <math>BD\,\!</math>, the third effect confounded with the block effect is their generalized interaction, <math>(AC)(BD)=ABCD\,\!</math>.
In general, when an unreplicated <math>{2}^{k}\,\!</math> design is confounded in <math>{2}^{p}\,\!</math> blocks, <math>p\,\!</math> contrasts are needed (<math>{{L}_{1}},{{L}_{2}}...{{L}_{p}}\,\!</math>). <math>p\,\!</math> effects are selected to define these contrasts such that none of these effects are the generalized interaction of the others. The <math>{2}^{p}\,\!</math> blocks can then be assigned the treatments using the <math>p\,\!</math> contrasts. <math>{{2}^{p}}-(p+1)\,\!</math> effects, that are also confounded with the blocks, are then obtained as the generalized interaction of the <math>p\,\!</math> effects. In the statistical analysis of these designs, the sum of squares are computed as if no blocking were used. Then the block sum of squares is obtained by adding the sum of squares for all the effects confounded with the blocks.

====Example: 2 Level Factorial Design with Four Blocks====

This example illustrates how a DOE folio obtains the sum of squares when treatments for an unreplicated <math>{2}^{k}\,\!</math> design are allocated among four blocks. Consider again the unreplicated <math>{2}^{4}\,\!</math> design used to investigate the defects in automobile vinyl panels presented in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments needed to complete the experiment were run by four operators. Therefore, there are four blocks. Assume that the treatments were allocated to the blocks using the generators mentioned in the previous section, i.e., treatments were allocated among the four operators by confounding the effects, <math>AC\,\!</math> and <math>BD,\,\!</math> with the blocks. These effects can be specified as Block Generators as shown in the following figure. (The generalized interaction of these two effects, interaction <math>ABCD\,\!</math>, will also get confounded with the blocks.) The resulting design is shown in the second following figure and matches the allocation scheme obtained in the previous section.

[[Image:doe7_27.png|center|700px|Specifying the interactions AC and BD as block generators for the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

The sum of squares in this case can be obtained by calculating the sum of squares for each of the effects assuming there is no blocking. Once the individual sum of squares have been obtained, the block sum of squares can be calculated. The block sum of squares is the sum of the sum of squares of effects, <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, since these effects are confounded with the block effect. As shown in the second following figure, this sum of squares is 92.25 and is displayed against Block. The interactions <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, which are confounded with the blocks, are not displayed. Since the present design is unreplicated any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_28.png|center|800px|Design for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

[[Image:doe7_29.png|center|793px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

==Variability Analysis==

For replicated two level factorial experiments, DOE++ provides the option of conducting variability analysis (using the Variability Analysis icon under the Data menu). The analysis is used to identify the treatment that results in the least amount of variation in the product or process being investigated. Variability analysis is conducted by treating the standard deviation of the response for each treatment of the experiment as an additional response. The standard deviation for a treatment is obtained by using the replicated response values at that treatment run. As an example, consider the <math>{2}^{3}\,\!</math> design shown in the following figure where each run is replicated four times. A variability analysis can be conducted for this design. DOE++ calculates eight standard deviation values corresponding to each treatment of the design (see second following figure). Then, the design is analyzed as an unreplicated <math>{2}^{3}\,\!</math> design with the standard deviations (displayed as Y Standard Deviation. in second following figure) as the response. The normal probability plot of effects identifies <math>AC\,\!</math> as the effect that influences variability (see third figure following). Based on the effect coefficients obtained in the fourth figure following, the model for Y Std. is:

::<math>\begin{align}
& \text{Y Std}\text{.}= & 0.6779+0.2491\cdot AC \\
& = & 0.6779+0.2491{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

Based on the model, the experimenter has two choices to minimize variability (by minimizing Y Std.). The first choice is that <math>{{x}_{1}}\,\!</math> should be <math>1\,\!</math> (i.e., <math>A\,\!</math> should be set at the high level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the low level). The second choice is that <math>{{x}_{1}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>A\,\!</math> should be set at the low level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the high level). The experimenter can select the most feasible choice.

[[Image:doe7.30.png|center|391px|A <math>2^3\,\!</math> design with four replicated response values that can be used to conduct a variability analysis.]]
 

[[Image:doe7_31.png|center|727px|Variability analysis in DOE++.|link=]]
 

[[Image:doe7_32.png|center|650px|Normal probability plot of effects for the variability analysis example.|link=]]
 

[[Image:doe7_33.png|center|727px|Effect coefficients for the variability analysis example.|link=]]

 

==Two Level Fractional Factorial Designs==

As the number of factors in a two level factorial design increases, the number of runs for even a single replicate of the <math>{2}^{k}\,\!</math> design becomes very large. For example, a single replicate of an eight factor two level experiment would require 256 runs. Fractional factorial designs can be used in these cases to draw out valuable conclusions from fewer runs. The basis of fractional factorial designs is the ''sparsity of effects'' principle.[[DOE_References|[Wu, 2000]]] The principle states that, most of the time, responses are affected by a small number of main effects and lower order interactions, while higher order interactions are relatively unimportant. Fractional factorial designs are used as screening experiments during the initial stages of experimentation. At these stages, a large number of factors have to be investigated and the focus is on the main effects and two factor interactions. These designs obtain information about main effects and lower order interactions with fewer experiment runs by confounding these effects with unimportant higher order interactions. As an example, consider a <math>{2}^{8}\,\!</math> design that requires 256 runs. This design allows for the investigation of 8 main effects and 28 two factor interactions. However, 219 degrees of freedom are devoted to three factor or higher order interactions. This full factorial design can prove to be very inefficient when these higher order interactions can be assumed to be unimportant. Instead, a fractional design can be used here to identify the important factors that can then be investigated more thoroughly in subsequent experiments. In unreplicated fractional factorial designs, no degrees of freedom are available to calculate the error sum of squares and the techniques mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] should be employed for the analysis of these designs.

==Half-fraction Designs==

A half-fraction of the <math>{2}^{k}\,\!</math> design involves running only half of the treatments of the full factorial design. For example, consider a <math>{2}^{3}\,\!</math> design that requires eight runs in all. The design matrix for this design is shown in the figure (a) below. A half-fraction of this design is the design in which only four of the eight treatments are run. The fraction is denoted as <math>{2}^{3-1}\,\!</math> with the "<math>-1\,\!</math>" in the index denoting a half-fraction. Assume that the treatments chosen for the half-fraction design are the ones where the interaction <math>ABC\,\!</math> is at the high level (i.e., only those rows are chosen from the following figure (a) where the column for <math>ABC\,\!</math> has entries of 1). The resulting <math>{2}^{3-1}\,\!</math> design has a design matrix as shown in figure (b) below.

[[Image:doe7.34.png|center|330px|Half-fractions of the <math>2^3\,\!</math> design. (a) shows the full factorial <math>2^3\,\!</math> design, (b) shows the <math>2^{3-1}\,\!</math> design with the defining relation <math>I=ABC\,\!</math> and (c) shows the <math>{2}^{3-1}\,\!</math> design with the defining relation <math>I=-ABC\,\!</math>.]]

In the <math>{2}^{3-1}\,\!</math> design of figure (b), since the interaction <math>ABC\,\!</math> is always included at the same level (the high level represented by 1), it is not possible to measure this interaction effect. The effect, <math>ABC\,\!</math>, is called the ''generator'' or ''word'' for this design. It can be noted that, in the design matrix of the following figure (b), the column corresponding to the intercept, <math>I\,\!</math>, and column corresponding to the interaction <math>ABC\,\!</math>, are identical. The identical columns are written as <math>I=ABC\,\!</math> and this equation is called the ''defining relation'' for the design. In a DOE folio, the present <math>{2}^{3-1}\,\!</math> design can be obtained by specifying the design properties as shown in the following figure.

[[Image:doe7_35.png|center|700px|Design properties for the <math>2^{3-1}\,\!</math> design.|link=]]

The defining relation, <math>I=ABC\,\!</math>, is entered in the Fraction Generator window as shown next.

[[Image:doe7_36.png|center|700px|Specifying the defining relation for the <math>2^{3-1}\,\!</math> design.|link=]]

Note that in the figure following that, the defining relation is specified as <math>C=AB\,\!</math>. This relation is obtained by multiplying the defining relation, <math>I=ABC\,\!</math>, by the last factor, <math>C\,\!</math>, of the design.

===Calculation of Effects===

Using the four runs of the <math>{2}^{3-1}\,\!</math> design in figure (b) discussed above, the main effects can be calculated as follows:

::<math>\begin{align}
& A= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& B= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& C= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

where <math>a\,\!</math>, <math>b\,\!</math>, <math>c\,\!</math> and <math>abc\,\!</math> are the treatments included in the <math>{2}^{3-1}\,\!</math> design.

Similarly, the two factor interactions can also be obtained as:

::<math>\begin{align}
& BC= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& AC= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& AB= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

The equations for <math>A\,\!</math> and <math>BC\,\!</math> above result in the same effect values showing that effects <math>A\,\!</math> and <math>BC\,\!</math> are confounded in the present <math>{2}^{3-1}\,\!</math> design. Thus, the quantity, <math>\tfrac{1}{2}(a-b-c+abc),\,\!</math> estimates <math>A+BC\,\!</math> (i.e., both the main effect <math>A\,\!</math> and the two-factor interaction <math>BC\,\!</math>). The effects, <math>A\,\!</math> and <math>BC,\,\!</math> are called ''aliases''. From the remaining equations given above, it can be seen that the other aliases for this design are <math>B\,\!</math> and <math>AC\,\!</math>, and <math>C\,\!</math> and <math>AB\,\!</math>. Therefore, the equations to calculate the effects in the present <math>{2}^{3-1}\,\!</math> design can be written as follows:

::<math>\begin{align}
& A+BC= & \frac{1}{2}(a-b-c+abc) \\
& B+AC= & \frac{1}{2}(-a+b-c+abc) \\
& C+AB= & \frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

===Calculation of Aliases===

Aliases for a fractional factorial design can be obtained using the defining relation for the design. The defining relation for the present <math>{2}^{3-1}\,\!</math> design is:

::<math>I=ABC\,\!</math>

Multiplying both sides of the previous equation by the main effect, <math>A,\,\!</math> gives the alias effect of <math>A\,\!</math> :

::<math>\begin{align}
& A\cdot I= & A\cdot ABC \\
& A= & {{A}^{2}}BC \\
& A= & BC
\end{align}\,\!</math>

Note that in calculating the alias effects, any effect multiplied by <math>I\,\!</math> remains the same (<math>A\cdot I=A\,\!</math>), while an effect multiplied by itself results in <math>I\,\!</math> (<math>{{A}^{2}}=I\,\!</math>). Other aliases can also be obtained:

::<math>\begin{align}
& B\cdot I= & B\cdot ABC \\
& B= & A{{B}^{2}}C \\
& B= & AC
\end{align}\,\!</math>

:and:

::<math>\begin{align}
& C\cdot I= & C\cdot ABC \\
& C= & AB{{C}^{2}} \\
& C= & AB
\end{align}\,\!</math>

===Fold-over Design===

If it can be assumed for this design that the two-factor interactions are unimportant, then in the absence of <math>BC\,\!</math>, <math>AC\,\!</math> and <math>AB\,\!</math>, the equations for (A+BC), (B+AC) and (C+AB) can be used to estimate the main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. However, if such an assumption is not applicable, then to uncouple the main effects from their two factor aliases, the alternate fraction that contains runs having <math>ABC\,\!</math> at the lower level should be run. The design matrix for this design is shown in the preceding figure (c). The defining relation for this design is <math>I=-ABC\,\!</math> because the four runs for this design are obtained by selecting the rows of the preceding figure (a) for which the value of the <math>ABC\,\!</math> column is <math>-1\,\!</math>. The aliases for this fraction can be obtained as explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]] as <math>A=-BC\,\!</math>, <math>B=-AC\,\!</math> and <math>C=-AB\,\!</math>. The effects for this design can be calculated as:

::<math>\begin{align}
& A-BC= & \frac{1}{2}(ab+ac-(1)-bc) \\
& B-AC= & \frac{1}{2}(ab-ac+(1)-bc) \\
& C-AB= & \frac{1}{2}(-ab+ac-(1)+bc)
\end{align}\,\!</math>

These equations can be combined with the equations for (A+BC), (B+AC) and (C+AB) to obtain the de-aliased main effects and two factor interactions. For example, adding equations (A+BC) and (A-BC) returns the main effect <math>A\,\!</math>.

::<math>\begin{align}
& 2A= & \frac{1}{2}(a-b-c+abc)+ \\
& & \frac{1}{2}(ab+ac-(1)-bc)
\end{align}\,\!</math>

The process of augmenting a fractional factorial design by a second fraction of the same size by simply reversing the signs (of all effect columns except <math>I\,\!</math>) is called ''folding over''. The combined design is referred to as a ''fold-over design''.

==Quarter and Smaller Fraction Designs==

At times, the number of runs even for a half-fraction design are very large. In these cases, smaller fractions are used. A quarter-fraction design, denoted as <math>{2}^{k-2}\,\!</math>, consists of a fourth of the runs of the full factorial design. Quarter-fraction designs require two defining relations. The first defining relation returns the half-fraction or the <math>{2}^{k-1}\,\!</math> design. The second defining relation selects half of the runs of the <math>{2}^{k-1}\,\!</math> design to give the quarter-fraction. For example, consider the <math>{2}^{4}\,\!</math> design. To obtain a <math>{2}^{4-2}\,\!</math> design from this design, first a half-fraction of this design is obtained by using a defining relation. Assume that the defining relation used is <math>I=ABCD\,\!</math>. The design matrix for the resulting <math>{2}^{4-1}\,\!</math> design is shown in figure (a) below. Now, a quarter-fraction can be obtained from the <math>{2}^{4-1}\,\!</math> design shown in figure (a) below using a second defining relation <math>I=AD\,\!</math>. The resulting <math>{2}^{4-2}\,\!</math> design obtained is shown in figure (b) below.

[[Image:doe7.37.png|center|465px|Fractions of the <math>2^4\,\!</math> design - Figure (a) shows the <math>2^{4-1}</math> design with the defining relation <math>I=ABCD\,\!</math> and (b) shows the <math>2^{4-2}\,\!</math> design with the defining relation <math>I=ABCD=AD=BC\,\!</math>.]]

The complete defining relation for this <math>{2}^{4-2}\,\!</math> design is:

::<math>I=ABCD=AD=BC\,\!</math>

Note that the effect, <math>BC,\,\!</math> in the defining relation is the generalized interaction of <math>ABCD\,\!</math> and <math>AD\,\!</math> and is obtained using <math>(ABCD)(AD)={{A}^{2}}BC{{D}^{2}}=BC\,\!</math>. In general, a <math>{2}^{k-p}\,\!</math> fractional factorial design requires <math>p\,\!</math> independent generators. The defining relation for the design consists of the <math>p\,\!</math> independent generators and their <math>{2}^{p}\,\!</math> - (<math>p\,\!</math> +1) generalized interactions.

===Calculation of Aliases===

The alias structure for the present <math>{2}^{4-2}\,\!</math> design can be obtained using the defining relation of equation (I=ABCD=AD=BC) following the procedure explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]]. For example, multiplying the defining relation by <math>A\,\!</math> returns the effects aliased with the main effect, <math>A\,\!</math>, as follows:

::<math>\begin{align}
& A\cdot I= & A\cdot ABCD=A\cdot AD=A\cdot BC \\
& A= & {{A}^{2}}BCD={{A}^{2}}D=ABC \\
& A= & BCD=D=ABC
\end{align}\,\!</math>

Therefore, in the present <math>{2}^{4-2}\,\!</math> design, it is not possible to distinguish between effects <math>A\,\!</math>, <math>D\,\!</math>, <math>BCD\,\!</math> and <math>ABC\,\!</math>. Similarly, multiplying the defining relation by <math>B\,\!</math> and <math>AB\,\!</math> returns the effects that are aliased with these effects:

::<math>\begin{align}
& B= & ACD=ABD=C \\
& AB= & CD=AD=AC
\end{align}\,\!</math>

Other aliases can be obtained in a similar way. It can be seen that each effect in this design has three aliases. In general, each effect in a <math>{2}^{k-p}\,\!</math> design has <math>{2}^{p-1}\,\!</math> aliases.
The aliases for the <math>{2}^{4-2}\,\!</math> design show that in this design the main effects are aliased with each other (<math>A\,\!</math> is aliased with <math>D\,\!</math> and <math>B\,\!</math> is aliased with <math>C\,\!</math>). Therefore, this design is not a useful design and is not available in a Weibull++ DOE folio. It is important to ensure that main effects and lower order interactions of interest are not aliased in a fractional factorial design. This is known by looking at the resolution of the fractional factorial design.

==Design Resolution==

The resolution of a fractional factorial design is defined as the number of factors in the lowest order effect in the defining relation. For example, in the defining relation <math>I=ABCD=AD=BC\,\!</math> of the previous <math>{2}^{4-2}\,\!</math> design, the lowest-order effect is either <math>AD\,\!</math> or <math>BC,\,\!</math> containing two factors. Therefore, the resolution of this design is equal to two. The resolution of a fractional factorial design is represented using Roman numerals. For example, the previously mentioned <math>{2}^{4-2}\,\!</math> design with a resolution of two can be represented as 2 <math>_{\text{II}}^{4-2}\,\!</math>. The resolution provides information about the confounding in the design as explained next:
 
 
#'''Resolution III Designs''' In these designs, the lowest order effect in the defining relation has three factors (e.g., a <math>{2}^{5-2}\,\!</math> design with the defining relation <math>I=ABDE=ABC=CDE\,\!</math>). In resolution III designs, no main effects are aliased with any other main effects, but main effects are aliased with two factor interactions. In addition, some two factor interactions are aliased with each other.
#'''Resolution IV Designs''' In these designs, the lowest order effect in the defining relation has four factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABDE\,\!</math>). In resolution IV designs, no main effects are aliased with any other main effects or two factor interactions. However, some main effects are aliased with three factor interactions and the two factor interactions are aliased with each other.
#'''Resolution V Designs''' In these designs the lowest order effect in the defining relation has five factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABCDE\,\!</math>). In resolution V designs, no main effects or two factor interactions are aliased with any other main effects or two factor interactions. However, some main effects are aliased with four factor interactions and the two factor interactions are aliased with three factor interactions.

Fractional factorial designs with the highest resolution possible should be selected because the higher the resolution of the design, the less severe the degree of confounding. In general, designs with a resolution less than III are never used because in these designs some of the main effects are aliased with each other. The table below shows fractional factorial designs with the highest available resolution for three to ten factor designs along with their defining relations.

[[Image:doet7.3.png|center|474px|Highest resolution designs available for fractional factorial designs with 3 to 10 factors.]]

All of the two level fractional factorial designs available in a DOE folio are shown next.

[[Image:doe7.38.png|center|471px|Two level fractional factorial designs available in Weibull++ and their resolutions.]]

===Minimum Aberration Designs===
At times, different designs with the same resolution but different aliasing may be available. The best design to select in such a case is the minimum aberration design. For example, all <math>{2}^{7-2}\,\!</math> designs in the fourth table have a resolution of four (since the generator with the minimum number of factors in each design has four factors). Design <math>1\,\!</math> has three generators of length four (<math>ABCF,\,\!</math> <math>BCDG,\,\!</math> <math>ADFG\,\!</math>). Design <math>2\,\!</math> has two generators of length four (<math>ABCF,\,\!</math> <math>ADEG\,\!</math>). Design <math>3\,\!</math> has one generator of length four (<math>CEFG\,\!</math>). Therefore, design <math>3\,\!</math> has the least number of generators with the minimum length of four. Design <math>3\,\!</math> is called the minimum aberration design. It can be seen that the alias structure for design <math>3\,\!</math> is less involved compared to the other designs. For details refer to [[DOE_References|[Wu, 2000]]].

[[Image:doet7.4.png|center|432px|Three <math>2_{IV}^{7-2}\,\!</math> designs with different defining relations.]]

====Example====

The design of an automobile fuel cone is thought to be affected by six factors in the manufacturing process: cavity temperature (factor <math>A\,\!</math>), core temperature (factor <math>B\,\!</math>), melt temperature (factor <math>C\,\!</math>), hold pressure (factor <math>D\,\!</math>), injection speed (factor <math>E\,\!</math>) and cool time (factor <math>F\,\!</math>). The manufacturer of the fuel cone is unable to run the <math>{2}^{6}=64\,\!</math> runs required to complete one replicate for a two level full factorial experiment with six factors. Instead, they decide to run a fractional factorial design. Considering that three factor and higher order interactions are likely to be inactive, the manufacturer selects a <math>{2}^{6-2}\,\!</math> design that will require only 16 runs. The manufacturer chooses the resolution IV design which will ensure that all main effects are free from aliasing (assuming three factor and higher order interactions are absent). However, in this design the two factor interactions may be aliased with each other. It is decided that, if important two factor interactions are found to be present, additional experiment trials may be conducted to separate the aliased effects. The performance of the fuel cone is measured on a scale of 1 to 15. In DOE++, the design for this experiment is set up using the properties shown in the following figure. The Fraction Generators for the design, <math>E=ABC\,\!</math> and <math>F=BCD\,\!</math>, are the same as the defaults used in DOE++. The resulting <math>{2}^{6-2}\,\!</math> design and the corresponding response values are shown in the following two figures.

[[Image:doe7_39.png|center|644px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

[[Image:doe7_40.png|center|534px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The complete alias structure for the 2 <math>_{\text{IV}}^{6-2}\,\!</math> design is shown next.

<center><math>I=ABCE=ADEF=BCDF\,\!</math></center>

 
<center><math>\begin{align}
& A= & BCE=DEF=ABCDF \\
& B= & ACE=CDF=ABDEF \\
& C= & ABE=BDF=ACDEF \\
& D= & AEF=BCF=ABCDE \\
& E= & ABC=ADF=BCDEF \\
& F= & ADE=BCD=ABCEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& AB= & CE=ACDF=BDEF \\
& AC= & BE=ABDF=CDEF \\
& AD= & EF=ABCF=BCDE \\
& AE= & BC=DF=ABCDEF \\
& AF= & DE=ABCD=BCEF \\
& BD= & CF=ABEF=ACDE \\
& BF= & CD=ABDE=ACEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& ABD= & ACF=BEF=CDE \\
& ABF= & ACD=BDE=CEF
\end{align}\,\!</math></center>

In a DOE folio, the alias structure is displayed in the Design Summary and as part of the Design Evaluation result, as shown next:

[[Image:doe7_41.png|center|700px|Alias structure for the experiment design in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The normal probability plot of effects for this unreplicated design shows the main effects of factors <math>C\,\!</math> and <math>D\,\!</math> and the interaction effect, <math>BF\,\!</math>, to be significant (see the following figure).

[[Image:doe7_42.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

From the alias structure, it can be seen that for the present design interaction effect, <math>BF,\,\!</math> is confounded with <math>CD\,\!</math>. Therefore, the actual source of this effect cannot be known on the basis of the present experiment. However because neither factor <math>B\,\!</math> nor <math>F\,\!</math> is found to be significant there is an indication the observed effect is likely due to interaction, <math>CD\,\!</math>. To confirm this, a follow-up <math>{2}^{2}\,\!</math> experiment is run involving only factors <math>B\,\!</math> and <math>F\,\!</math>. The interaction, <math>BF\,\!</math>, is found to be inactive, leading to the conclusion that the interaction effect in the original experiment is effect, <math>CD\,\!</math>. Given these results, the fitted regression model for the fuel cone design as per the coefficients obtained from DOE++ is shown next.

::<math>\hat{y}=7.6875+C+2\cdot D+2.1875\cdot CD\,\!</math>

[[Image:doe7_43.png|center|700px|Effect coefficients for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

==Projection==

Projection refers to the reduction of a fractional factorial design to a full factorial design by dropping out some of the factors of the design. Any fractional factorial design of resolution, <math>R,\,\!</math> can be reduced to complete factorial designs in any subset of <math>R-1\,\!</math> factors. For example, consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. The resolution of this design is four. Therefore, this design can be reduced to full factorial designs in any three (<math>4-1=3\,\!</math>) of the original seven factors (by dropping the remaining four of factors). Further, a fractional factorial design can also be reduced to a full factorial design in any <math>R\,\!</math> of the original factors, as long as these <math>R\,\!</math> factors are not part of the generator in the defining relation. Again consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. This design can be reduced to a full factorial design in four factors provided these four factors do not appear together as a generator in the defining relation. The complete defining relation for this design is:

::<math>\begin{align}
& I= & ABCE=BCDF=ACDG=ADEF=ABFG=BDEG=CEFG
\end{align}\,\!</math>

Therefore, there are seven four factor combinations out of the 35 (<math>(_{7}^{4})=35\,\!</math>) possible four-factor combinations that are used as generators in the defining relation. The designs with the remaining 28 four factor combinations would be full factorial 16-run designs. For example, factors <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math> do not occur as a generator in the defining relation of the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. If the remaining factors, <math>E\,\!</math>, <math>F\,\!</math> and <math>G\,\!</math>, are dropped, the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design will reduce to a full factorial design in <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math>.

==Resolution III Designs==

At times, the factors to be investigated in screening experiments are so large that even running a fractional factorial design is impractical. This can be partially solved by using resolution III fractional factorial designs in the cases where three factor and higher order interactions can be assumed to be unimportant. Resolution III designs, such as the 2 <math>_{\text{III}}^{3-1}\,\!</math> design, can be used to estimate <math>k\,\!</math> main effects using just <math>k+1\,\!</math> runs. In these designs, the main effects are aliased with two factor interactions. Once the results from these designs are obtained, and knowing that three factor and higher order interactions are unimportant, the experimenter can decide if there is a need to run a fold-over design to de-alias the main effects from the two factor interactions. Thus, the 2 <math>_{\text{III}}^{3-1}\,\!</math> design can be used to investigate three factors in four runs, the 2 <math>_{\text{III}}^{7-4}\,\!</math> design can be used to investigate seven factors in eight runs, the 2 <math>_{\text{III}}^{15-11}\,\!</math> design can be used to investigate fifteen factors in sixteen runs and so on.

====Example====

{{:Resolution_III_Design_Example}}

==Alias Matrix==
In [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction designs]] and [[Two_Level_Factorial_Experiments#Quarter_and_Smaller_Fraction_Designs| Quarter and Smaller Fraction Designs]], the alias structure for fractional factorial designs was obtained using the defining relation. However, this method of obtaining the alias structure is not very efficient when the alias structure is very complex or when partial aliasing is involved. One of the ways to obtain the alias structure for any design, regardless of its complexity, is to use the alias matrix. The alias matrix for a design is calculated using <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> where <math>{{X}_{1}}\,\!</math> is the portion of the design matrix, <math>X,\,\!</math> that contains the effects for which the aliases need to be calculated, and <math>{{X}_{2}}\,\!</math> contains the remaining columns of the design matrix, other than those included in <math>{{X}_{1}}\,\!</math>.

To illustrate the use of the alias matrix, consider the design matrix for the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design (using the defining relation <math>I=ABCD\,\!</math>) shown next:

[[Image:Chapter7__879.png|center|link=]]

The alias structure for this design can be obtained by defining <math>{{X}_{1}}\,\!</math> using eight columns since the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design estimates eight effects. If the first eight columns of <math>X\,\!</math> are used then <math>{{X}_{1}}\,\!</math> is:

[[Image:Chapter7__884.png|center|link=]]

<math>{{X}_{2}}\,\!</math> is obtained using the remaining columns as:

[[Image:Chapter7__886.png|center|link=]]

Then the alias matrix <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> is:

[[Image:Chapter7__888.png|center|link=]]

The alias relations can be easily obtained by observing the alias matrix as:

<center><math>\begin{align}
& I= & ABCD \\
& A= & BCD \\
& B= & ACD \\
& AB= & CD \\
& C= & ABD \\
& AC= & BD \\
& BC= & AD \\
& D= & ABC
\end{align}\,\!</math></center>

Two Level Factorial Experiments

2017-08-10T17:38:25Z

Richard House: /* Design Resolution */

{{Template:Doebook|8}}
Two level factorial experiments are factorial experiments in which each factor is investigated at only two levels. The early stages of experimentation usually involve the investigation of a large number of potential factors to discover the "vital few" factors. Two level factorial experiments are used during these stages to quickly filter out unwanted effects so that attention can then be focused on the important ones.

==2''k'' Designs==
The factorial experiments, where all combination of the levels of the factors are run, are usually referred to as ''full factorial experiments''. Full factorial two level experiments are also referred to as <math>{2}^{k}\,\!</math> designs where <math>k\,\!</math> denotes the number of factors being investigated in the experiment. In Weibull++ DOE folios, these designs are referred to as 2 Level Factorial Designs as shown in the figure below.

[[Image:doe7_1.png|center|487px|Selection of full factorial experiments with two levels in Weibull++.|link=]]

A full factorial two level design with <math>k\,\!</math> factors requires <math>{{2}^{k}}\,\!</math> runs for a single replicate. For example, a two level experiment with three factors will require <math>2\times 2\times 2={{2}^{3}}=8\,\!</math> runs. The choice of the two levels of factors used in two level experiments depends on the factor; some factors naturally have two levels. For example, if gender is a factor, then male and female are the two levels. For other factors, the limits of the range of interest are usually used. For example, if temperature is a factor that varies from <math>{45}^{o}C\,\!</math> to <math>{90}^{o}C\,\!</math>, then the two levels used in the <math>{2}^{k}\,\!</math> design for this factor would be <math>{45}^{o}\,\!C\,\!</math> and <math>{90}^{o}\,\!C\,\!</math>.

The two levels of the factor in the <math>{2}^{k}\,\!</math> design are usually represented as <math>-1\,\!</math> (for the first level) and <math>1\,\!</math> (for the second level). Note that this representation is reversed from the coding used in [[General Full Factorial Designs]] for the indicator variables that represent two level factors in ANOVA models. For ANOVA models, the first level of the factor was represented using a value of <math>1\,\!</math> for the indicator variable, while the second level was represented using a value of <math>-1\,\!</math>. For details on the notation used for two level experiments refer to [[Two_Level_Factorial_Experiments#Notation| Notation]].

===The 22 Design===

The simplest of the two level factorial experiments is the <math>{2}^{2}\,\!</math> design where two factors (say factor <math>A\,\!</math> and factor <math>B\,\!</math>) are investigated at two levels. A single replicate of this design will require four runs (<math>{{2}^{2}}=2\times 2=4\,\!</math>) The effects investigated by this design are the two main effects, <math>A\,\!</math> and <math>B,\,\!</math> and the interaction effect <math>AB\,\!</math>. The treatments for this design are shown in figure (a) below. In figure (a), letters are used to represent the treatments. The presence of a letter indicates the high level of the corresponding factor and the absence indicates the low level. For example, (1) represents the treatment combination where all factors involved are at the low level or the level represented by <math>-1\,\!</math> ; <math>a\,\!</math> represents the treatment combination where factor <math>A\,\!</math> is at the high level or the level of <math>1\,\!</math>, while the remaining factors (in this case, factor <math>B\,\!</math>) are at the low level or the level of <math>-1\,\!</math>. Similarly, <math>b\,\!</math> represents the treatment combination where factor <math>B\,\!</math> is at the high level or the level of <math>1\,\!</math>, while factor <math>A\,\!</math> is at the low level and <math>ab\,\!</math> represents the treatment combination where factors <math>A\,\!</math> and <math>B\,\!</math> are at the high level or the level of the 1. Figure (b) below shows the design matrix for the <math>{2}^{2}\,\!</math> design. It can be noted that the sum of the terms resulting from the product of any two columns of the design matrix is zero. As a result the <math>{2}^{2}\,\!</math> design is an ''orthogonal design''. In fact, all <math>{2}^{k}\,\!</math> designs are orthogonal designs. This property of the <math>{2}^{k}\,\!</math> designs offers a great advantage in the analysis because of the simplifications that result from orthogonality. These simplifications are explained later on in this chapter.
The <math>{2}^{2}\,\!</math> design can also be represented geometrically using a square with the four treatment combinations lying at the four corners, as shown in figure (c) below.

[[Image:doe7.2.png|center|400px|The <math>2^2\,\!</math> design. Figure (a) displays the experiment design, (b) displays the design matrix and (c) displays the geometric representation for the design. In Figure (b), the column names I, A, B and AB are used. Column I represents the intercept term. Columns A and B represent the respective factor settings. Column AB represents the interaction and is the product of columns A and B.]]
 

===The 23 Design===

The <math>{2}^{3}\,\!</math> design is a two level factorial experiment design with three factors (say factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>). This design tests three (<math>k=3\,\!</math>) main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math> ; three (<math>(_{2}^{k})=\,\!</math> <math>(_{2}^{3})=3\,\!</math>) two factor interaction effects, <math>AB\,\!</math>, <math>BC\,\!</math>, <math>AC\,\!</math> ; and one (<math>(_{3}^{k})=\,\!</math> <math>(_{3}^{3})=1\,\!</math>) three factor interaction effect, <math>ABC\,\!</math>. The design requires eight runs per replicate. The eight treatment combinations corresponding to these runs are <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math>, <math>ab\,\!</math>, <math>c\,\!</math>, <math>ac\,\!</math>, <math>bc\,\!</math> and <math>abc\,\!</math>. Note that the treatment combinations are written in such an order that factors are introduced one by one with each new factor being combined with the preceding terms. This order of writing the treatments is called the ''standard order'' or ''Yates' order''. The <math>{2}^{3}\,\!</math> design is shown in figure (a) below. The design matrix for the <math>{2}^{3}\,\!</math> design is shown in figure (b). The design matrix can be constructed by following the standard order for the treatment combinations to obtain the columns for the main effects and then multiplying the main effects columns to obtain the interaction columns.

[[Image:doe7.3.png|center|324px|The <math>2^3\,\!</math> design. Figure (a) shows the experiment design and (b) shows the design matrix.]]

[[Image:doe7.4.png|center|290px|Geometric representation of the <math>2^3\,\!</math> design.]]

The <math>{2}^{3}\,\!</math> design can also be represented geometrically using a cube with the eight treatment combinations lying at the eight corners as shown in the figure above.

==Analysis of 2''k'' Designs==

The <math>{2}^{k}\,\!</math> designs are a special category of the factorial experiments where all the factors are at two levels. The fact that these designs contain factors at only two levels and are orthogonal greatly simplifies their analysis even when the number of factors is large. The use of <math>{2}^{k}\,\!</math> designs in investigating a large number of factors calls for a revision of the notation used previously for the ANOVA models. The case for revised notation is made stronger by the fact that the ANOVA and multiple linear regression models are identical for <math>{2}^{k}\,\!</math> designs because all factors are only at two levels. Therefore, the notation of the regression models is applied to the ANOVA models for these designs, as explained next.

===Notation===

Based on the notation used in [[General Full Factorial Designs]], the ANOVA model for a two level factorial experiment with three factors would be as follows:

::<math>\begin{align}
& Y= & \mu +{{\tau }_{1}}\cdot {{x}_{1}}+{{\delta }_{1}}\cdot {{x}_{2}}+{{(\tau \delta )}_{11}}\cdot {{x}_{1}}{{x}_{2}}+{{\gamma }_{1}}\cdot {{x}_{3}} \\
& & +{{(\tau \gamma )}_{11}}\cdot {{x}_{1}}{{x}_{3}}+{{(\delta \gamma )}_{11}}\cdot {{x}_{2}}{{x}_{3}}+{{(\tau \delta \gamma )}_{111}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where:
 
:• <math>\mu \,\!</math> represents the overall mean
:• <math>{{\tau }_{1}}\,\!</math> represents the independent effect of the first factor (factor <math>A\,\!</math>) out of the two effects <math>{{\tau }_{1}}\,\!</math> and <math>{{\tau }_{2}}\,\!</math>
:• <math>{{\delta }_{1}}\,\!</math> represents the independent effect of the second factor (factor <math>B\,\!</math>) out of the two effects <math>{{\delta }_{1}}\,\!</math> and <math>{{\delta }_{2}}\,\!</math>
:• <math>{{(\tau \delta )}_{11}}\,\!</math> represents the independent effect of the interaction <math>AB\,\!</math> out of the other interaction effects
:• <math>{{\gamma }_{1}}\,\!</math> represents the effect of the third factor (factor <math>C\,\!</math>) out of the two effects <math>{{\gamma }_{1}}\,\!</math> and <math>{{\gamma }_{2}}\,\!</math>
:• <math>{{(\tau \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>AC\,\!</math> out of the other interaction effects
:• <math>{{(\delta \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>BC\,\!</math> out of the other interaction effects
:• <math>{{(\tau \delta \gamma )}_{111}}\,\!</math> represents the effect of the interaction <math>ABC\,\!</math> out of the other interaction effects
and <math>\epsilon \,\!</math> is the random error term.

 
The notation for a linear regression model having three predictor variables with interactions is:

::<math>\begin{align}
& Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& & +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

The notation for the regression model is much more convenient, especially for the case when a large number of higher order interactions are present. In two level experiments, the ANOVA model requires only one indicator variable to represent each factor for both qualitative and quantitative factors. Therefore, the notation for the multiple linear regression model can be applied to the ANOVA model of the experiment that has all the factors at two levels. For example, for the experiment of the ANOVA model given above, <math>{{\beta }_{0}}\,\!</math> can represent the overall mean instead of <math>\mu \,\!</math>, and <math>{{\beta }_{1}}\,\!</math> can represent the independent effect, <math>{{\tau }_{1}}\,\!</math>, of factor <math>A\,\!</math>. Other main effects can be represented in a similar manner. The notation for the interaction effects is much more simplified (e.g., <math>{{\beta }_{123}}\,\!</math> can be used to represent the three factor interaction effect, <math>{{(\tau \beta \gamma )}_{111}}\,\!</math>).

As mentioned earlier, it is important to note that the coding for the indicator variables for the ANOVA models of two level factorial experiments is reversed from the coding followed in [[General Full Factorial Designs]]. Here <math>-1\,\!</math> represents the first level of the factor while <math>1\,\!</math> represents the second level. This is because for a two level factor a single variable is needed to represent the factor for both qualitative and quantitative factors. For quantitative factors, using <math>-1\,\!</math> for the first level (which is the low level) and 1 for the second level (which is the high level) keeps the coding consistent with the numerical value of the factors. The change in coding between the two coding schemes does not affect the analysis except that signs of the estimated effect coefficients will be reversed (i.e., numerical values of <math>{{\hat{\tau }}_{1}}\,\!</math>, obtained based on the coding of [[General Full Factorial Designs]], and <math>{{\hat{\beta }}_{1}}\,\!</math>, obtained based on the new coding, will be the same but their signs would be opposite).

::<math>\begin{align}
& & \text{Factor }A\text{ Coding (two level factor)} \\
& &
\end{align}\,\!</math>

::<math>\begin{matrix}
\text{Previous Coding} & {} & {} & {} & \text{Coding for }{{\text{2}}^{k}}\text{ Designs} \\
{} & {} & {} & {} & {} \\
Effect\text{ }{{\tau }_{1}}\ \ :\ \ {{x}_{1}}=1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{1}}\text{ (or }-{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=-1\text{ } \\
Effect\text{ }{{\tau }_{2}}\ \ :\ \ {{x}_{1}}=-1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{2}}\text{ (or }{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=1\text{ } \\
\end{matrix}\,\!</math>

In summary, the ANOVA model for the experiments with all factors at two levels is different from the ANOVA models for other experiments in terms of the notation in the following two ways:
 

:• The notation of the regression models is used for the effect coefficients.
:• The coding of the indicator variables is reversed.

===Special Features===

Consider the design matrix, <math>X\,\!</math>, for the <math>{2}^{3}\,\!</math> design discussed above. The (<math>{{X}^{\prime }}X\,\!</math>) <math>^{-1}\,\!</math> matrix is:

<center><math>{{({{X}^{\prime }}X)}^{-1}}=\left[ \begin{matrix}
0.125 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0.125 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0.125 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0.125 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0.125 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0.125 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0.125 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.125 \\
\end{matrix} \right]\,\!</math></center>

Notice that, due to the orthogonal design of the <math>X\,\!</math> matrix, the <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> has been simplified to a diagonal matrix which can be written as:

::<math>\begin{align}
{{({{X}^{\prime }}X)}^{-1}}= & 0.125\cdot I = & \frac{1}{8}\cdot I = & \frac{1}{{{2}^{3}}}\cdot I
\end{align}\,\!</math>

where <math>I\,\!</math> represents the identity matrix of the same order as the design matrix, <math>X\,\!</math>. Since there are eight observations per replicate of the <math>{2}^{3}\,\!</math> design, the <math>(X\,\!</math> ' <math>X{{)}^{-1}}\,\!</math> matrix for <math>m\,\!</math> replicates of this design can be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{3}}\cdot m)}\cdot I\,\!</math>

The <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> matrix for any <math>{2}^{k}\,\!</math> design can now be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{k}}\cdot m)}\cdot I\,\!</math>

Then the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is:

::<math>\begin{align}
C= & {{{\hat{\sigma }}}^{2}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & M{{S}_{E}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & \frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}\cdot I
\end{align}\,\!</math>

Note that the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is also a diagonal matrix. Therefore, the estimated effect coefficients (<math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math>, <math>{{\beta }_{12}},\,\!</math> etc.) for these designs are uncorrelated. This implies that the terms in the <math>{2}^{k}\,\!</math> design (main effects, interactions) are independent of each other. Consequently, the extra sum of squares for each of the terms in these designs is independent of the sequence of terms in the model, and also independent of the presence of other terms in the model. As a result the sequential and partial sum of squares for the terms are identical for these designs and will always add up to the model sum of squares. Multicollinearity is also not an issue for these designs.

It can also be noted from the equation given above, that in addition to the <math>C\,\!</math> matrix being diagonal, all diagonal elements of the <math>C\,\!</math> matrix are identical. This means that the variance (or its square root, the standard error) of all estimated effect coefficients are the same. The standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>, for all the coefficients is:

::<math>\begin{align}
se({{{\hat{\beta }}}_{j}})= & \sqrt{{{C}_{jj}}} = & \sqrt{\frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}}\text{ }for\text{ }all\text{ }j
\end{align}\,\!</math>

This property is used to construct the normal probability plot of effects in <math>{2}^{k}\,\!</math> designs and identify significant effects using graphical techniques. For details on the normal probability plot of effects in a Weibull++ DOE folio, refer to [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]].

====Example====
To illustrate the analysis of a full factorial <math>{2}^{k}\,\!</math> design, consider a three factor experiment to investigate the effect of honing pressure, number of strokes and cycle time on the surface finish of automobile brake drums. Each of these factors is investigated at two levels. The honing pressure is investigated at levels of 200 <math>psi\,\!</math> and 400 <math>psi\,\!</math>, the number of strokes used is 3 and 5 and the two levels of the cycle time are 3 and 5 seconds. The design for this experiment is set up in a Weibull++ DOE folio as shown in the first two following figures. It is decided to run two replicates for this experiment. The surface finish data collected from each run (using randomization) and the complete design is shown in the third following figure. The analysis of the experiment data is explained next.

[[Image:doe7_5.png|center|877px|Design properties for the experiment in the example.|link=]]

[[Image:doe7_6.png|center|859px|Design summary for the experiment in the example.|link=]]

[[Image:doe7_7.png|center|778px|Experiment design for the example to investigate the surface finish of automobile brake drums.|link=]]

The applicable model using the notation for <math>{2}^{k}\,\!</math> designs is:

::<math>\begin{align}
Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where the indicator variable, <math>{{x}_{1,}}\,\!</math> represents factor <math>A\,\!</math> (honing pressure), <math>{{x}_{1}}=-1\,\!</math> represents the low level of 200 <math>psi\,\!</math> and <math>{{x}_{1}}=1\,\!</math> represents the high level of 400 <math>psi\,\!</math>. Similarly, <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> represent factors <math>B\,\!</math> (number of strokes) and <math>C\,\!</math> (cycle time), respectively. <math>{{\beta }_{0}}\,\!</math> is the overall mean, while <math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math> and <math>{{\beta }_{3}}\,\!</math> are the effect coefficients for the main effects of factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. <math>{{\beta }_{12}}\,\!</math>, <math>{{\beta }_{13}}\,\!</math> and <math>{{\beta }_{23}}\,\!</math> are the effect coefficients for the <math>AB\,\!</math>, <math>AC\,\!</math> and <math>BC\,\!</math> interactions, while <math>{{\beta }_{123}}\,\!</math> represents the <math>ABC\,\!</math> interaction.

 
If the subscripts for the run (<math>i\,\!</math> ; <math>i=\,\!</math> 1 to 8) and replicates (<math>j\,\!</math> ; <math>j=\,\!</math> 1,2) are included, then the model can be written as:

::<math>\begin{align}
{{Y}_{ij}}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{ij1}}+{{\beta }_{2}}\cdot {{x}_{ij2}}+{{\beta }_{12}}\cdot {{x}_{ij1}}{{x}_{ij2}}+{{\beta }_{3}}\cdot {{x}_{ij3}} \\
& +{{\beta }_{13}}\cdot {{x}_{ij1}}{{x}_{ij3}}+{{\beta }_{23}}\cdot {{x}_{ij2}}{{x}_{ij3}}+{{\beta }_{123}}\cdot {{x}_{ij1}}{{x}_{ij2}}{{x}_{ij3}}+{{\epsilon }_{ij}}
\end{align}\,\!</math>

To investigate how the given factors affect the response, the following hypothesis tests need to be carried:

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{1}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{1}}\ne 0\,\!</math>

This test investigates the main effect of factor <math>A\,\!</math> (honing pressure). The statistic for this test is:

::<math>{{({{F}_{0}})}_{A}}=\frac{M{{S}_{A}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{A}}\,\!</math> is the mean square for factor <math>A\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other main effects, <math>B\,\!</math> and <math>C\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{12}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{12}}\ne 0\,\!</math>

This test investigates the two factor interaction <math>AB\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{AB}}=\frac{M{{S}_{AB}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the interaction <math>AB\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other two factor interactions, <math>AC\,\!</math> and <math>BC\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{123}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{123}}\ne 0\,\!</math>

This test investigates the three factor interaction <math>ABC\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{ABC}}=\frac{M{{S}_{ABC}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{ABC}}\,\!</math> is the mean square for the interaction <math>ABC\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square.
To calculate the test statistics, it is convenient to express the ANOVA model in the form <math>y=X\beta +\epsilon \,\!</math>.

====Expression of the ANOVA Model as <math>y=X\beta +\epsilon \,\!</math>====

In matrix notation, the ANOVA model can be expressed as:

::<math>y=X\beta +\epsilon \,\!</math>

where:

<center><math>y=\left[ \begin{matrix}
{{Y}_{11}} \\
{{Y}_{21}} \\
. \\
{{Y}_{81}} \\
{{Y}_{12}} \\
. \\
{{Y}_{82}} \\
\end{matrix} \right]=\left[ \begin{matrix}
90 \\
90 \\
. \\
90 \\
86 \\
. \\
80 \\
\end{matrix} \right]\text{ }X=\left[ \begin{matrix}
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\end{matrix} \right]\,\!</math></center>

<center><math>\beta =\left[ \begin{matrix}
{{\beta }_{0}} \\
{{\beta }_{1}} \\
{{\beta }_{2}} \\
{{\beta }_{12}} \\
{{\beta }_{3}} \\
{{\beta }_{13}} \\
{{\beta }_{23}} \\
{{\beta }_{123}} \\
\end{matrix} \right]\text{ }\epsilon =\left[ \begin{matrix}
{{\epsilon }_{11}} \\
{{\epsilon }_{21}} \\
. \\
{{\epsilon }_{81}} \\
{{\epsilon }_{12}} \\
. \\
. \\
{{\epsilon }_{82}} \\
\end{matrix} \right]\,\!</math></center>

====Calculation of the Extra Sum of Squares for the Factors====

Knowing the matrices <math>y\,\!</math>, <math>X\,\!</math> and <math>\beta \,\!</math>, the extra sum of squares for the factors can be calculated. These are used to calculate the mean squares that are used to obtain the test statistics. Since the experiment design is orthogonal, the partial and sequential extra sum of squares are identical. The extra sum of squares for each effect can be calculated as shown next. As an example, the extra sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & Model\text{ }Sum\text{ }of\text{ }Squares - Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }main\text{ }effect\text{ }of\text{ }A \\
= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }A}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }A}}={{X}_{\tilde{\ }A}}{{(X_{\tilde{\ }A}^{\prime }{{X}_{\tilde{\ }A}})}^{-1}}X_{\tilde{\ }A}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }A}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the main effect of factor <math>A\,\!</math>. Thus, the sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y \\
= & 654.4375-549.375 \\
= & 105.0625
\end{align}\,\!</math>

Similarly, the extra sum of squares for the interaction effect <math>AB\,\!</math> is:

::<math>\begin{align}
S{{S}_{AB}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }AB}}-(1/16)J]y \\
= & 654.4375-636.375 \\
= & 18.0625
\end{align}\,\!</math>

The extra sum of squares for other effects can be obtained in a similar manner.

====Calculation of the Test Statistics====

Knowing the extra sum of squares, the test statistic for the effects can be calculated. For example, the test statistic for the interaction <math>AB\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{AB}}= & \frac{M{{S}_{AB}}}{M{{S}_{E}}} \\
= & \frac{S{{S}_{AB}}/dof(S{{S}_{AB}})}{S{{S}_{E}}/dof(S{{S}_{E}})} \\
= & \frac{18.0625/1}{147.5/8} \\
= & 0.9797
\end{align}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the <math>AB\,\!</math> interaction and <math>M{{S}_{E}}\,\!</math> is the error mean square. The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{AB}}=0.9797\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{AB}}) \\
= & 1-0.6487 \\
= & 0.3513
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that the interaction between honing pressure and number of strokes does not affect the surface finish of the brake drums. Tests for other effects can be carried out in a similar manner. The results are shown in the ANOVA Table in the following figure. The values S, R-sq and R-sq(adj) in the figure indicate how well the model fits the data. The value of S represents the standard error of the model, R-sq represents the coefficient of multiple determination and R-sq(adj) represents the adjusted coefficient of multiple determination. For details on these values refer to [[Multiple_Linear_Regression_Analysis|Multiple Linear Regression Analysis]].

[[Image:doe7_8.png|center|799px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

====Calculation of Effect Coefficients====

The estimate of effect coefficients can also be obtained:

<center><math>\begin{align}
\hat{\beta }= & {{({{X}^{\prime }}X)}^{-1}}{{X}^{\prime }}y \\
= & \left[ \begin{matrix}
86.4375 \\
2.5625 \\
-4.9375 \\
1.0625 \\
-1.0625 \\
2.4375 \\
-1.3125 \\
-0.1875 \\
\end{matrix} \right]
\end{align}\,\!</math></center>

[[Image:doe7_9.png|center|800px|Regression Information table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

The coefficients and related results are shown in the Regression Information table above. In the table, the Effect column displays the effects, which are simply twice the coefficients. The Standard Error column displays the standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>. The Low CI and High CI columns display the confidence interval on the coefficients. The interval shown is the 90% interval as the significance is chosen as 0.1. The T Value column displays the <math>t\,\!</math> statistic, <math>{{t}_{0}}\,\!</math>, corresponding to the coefficients. The P Value column displays the <math>p\,\!</math> value corresponding to the <math>t\,\!</math> statistic. (For details on how these results are calculated, refer to [[General Full Factorial Designs]]). Plots of residuals can also be obtained from the DOE folio to ensure that the assumptions related to the ANOVA model are not violated.

====Model Equation====

From the analysis results in the above figure within [[Two_Level_Factorial_Experiments#Calculation_of_Effect_Coefficients|calculation of effect coefficients]] section, it is seen that effects <math>A\,\!</math>, <math>B\,\!</math> and <math>AC\,\!</math> are significant. In a DOE folio, the <math>p\,\!</math> values for the significant effects are displayed in red in the ANOVA Table for easy identification. Using the values of the estimated effect coefficients, the model for the present <math>{2}^{3}\,\!</math> design in terms of the coded values can be written as:

::<math>\begin{align}
\hat{y}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}} \\
= & 86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+2.4375{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

To make the model hierarchical, the main effect, <math>C\,\!</math>, needs to be included in the model (because the interaction <math>AC\,\!</math> is included in the model). The resulting model is:

::<math>\hat{y}=86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+1.0625{{x}_{3}}+2.4375{{x}_{1}}{{x}_{3}}\,\!</math>

This equation can be viewed in a DOE folio, as shown in the following figure, using the Show Analysis Summary icon in the Control Panel. The equation shown in the figure will match the hierarchical model once the required terms are selected using the Select Effects icon.

[[Image:doe7_10.png|center|557px|The model equation for the experiment of the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

==Replicated and Repeated Runs==

In the case of replicated experiments, it is important to note the difference between replicated runs and repeated runs. Both repeated and replicated runs are multiple response readings taken at the same factor levels. However, repeated runs are response observations taken at the same time or in succession. Replicated runs are response observations recorded in a random order. Therefore, replicated runs include more variation than repeated runs. For example, a baker, who wants to investigate the effect of two factors on the quality of cakes, will have to bake four cakes to complete one replicate of a <math>{2}^{2}\,\!</math> design. Assume that the baker bakes eight cakes in all. If, for each of the four treatments of the <math>{2}^{2}\,\!</math> design, the baker selects one treatment at random and then bakes two cakes for this treatment at the same time then this is a case of two repeated runs. If, however, the baker bakes all the eight cakes randomly, then the eight cakes represent two sets of replicated runs.
For repeated measurements, the average values of the response for each treatment should be entered into a DOE folio as shown in the following figure (a) when the two cakes for a particular treatment are baked together. For replicated measurements, when all the cakes are baked randomly, the data is entered as shown in the following figure (b).

[[Image:doe7.11.png|center|300px|Data entry for repeated and replicated runs. Figure (a) shows repeated runs and (b) shows replicated runs.]]

==Unreplicated 2''k'' Designs==

If a factorial experiment is run only for a single replicate then it is not possible to test hypotheses about the main effects and interactions as the error sum of squares cannot be obtained. This is because the number of observations in a single replicate equals the number of terms in the ANOVA model. Hence the model fits the data perfectly and no degrees of freedom are available to obtain the error sum of squares.

However, sometimes it is only possible to run a single replicate of the <math>{2}^{k}\,\!</math> design because of constraints on resources and time. In the absence of the error sum of squares, hypothesis tests to identify significant factors cannot be conducted. A number of methods of analyzing information obtained from unreplicated <math>{2}^{k}\,\!</math> designs are available. These include pooling higher order interactions, using the normal probability plot of effects or including center point replicates in the design.

===Pooling Higher Order Interactions===

One of the ways to deal with unreplicated <math>{2}^{k}\,\!</math> designs is to use the sum of squares of some of the higher order interactions as the error sum of squares provided these higher order interactions can be assumed to be insignificant. By dropping some of the higher order interactions from the model, the degrees of freedom corresponding to these interactions can be used to estimate the error mean square. Once the error mean square is known, the test statistics to conduct hypothesis tests on the factors can be calculated.

===Normal Probability Plot of Effects===
Another way to use unreplicated <math>{2}^{k}\,\!</math> designs to identify significant effects is to construct the normal probability plot of the effects. As mentioned in [[Two_Level_Factorial_Experiments#Special_Features| Special Features]], the standard error for all effect coefficients in the <math>{2}^{k}\,\!</math> designs is the same. Therefore, on a normal probability plot of effect coefficients, all non-significant effect coefficients (with <math>\beta =0\,\!</math>) will fall along the straight line representative of the normal distribution, N(<math>0,{{\sigma }^{2}}/({{2}^{k}}\cdot m)\,\!</math>). Effect coefficients that show large deviations from this line will be significant since they do not come from this normal distribution. Similarly, since effects <math>=2\times \,\!</math> effect coefficients, all non-significant effects will also follow a straight line on the normal probability plot of effects. For replicated designs, the Effects Probability plot of a DOE folio plots the normalized effect values (or the T Values) on the standard normal probability line, N(0,1). However, in the case of unreplicated <math>{2}^{k}\,\!</math> designs, <math>{{\sigma }^{2}}\,\!</math> remains unknown since <math>M{{S}_{E}}\,\!</math> cannot be obtained. Lenth's method is used in this case to estimate the variance of the effects. For details on Lenth's method, please refer to [[DOE References| Montgomery (2001)]]. The DOE folio then uses this variance value to plot effects along the N(0, Lenth's effect variance) line. The
method is illustrated in the following example.

====Example====

Vinyl panels, used as instrument panels in a certain automobile, are seen to develop defects after a certain amount of time. To investigate the issue, it is decided to carry out a two level factorial experiment. Potential factors to be investigated in the experiment are vacuum rate (factor <math>A\,\!</math>), material temperature (factor <math>B\,\!</math>), element intensity (factor <math>C\,\!</math>) and pre-stretch (factor <math>D\,\!</math>). The two levels of the factors used in the experiment are as shown in below.

[[Image:doet7.1.png|center|300px|Factors to investigate defects in vinyl panels.]]

With a <math>{2}^{4}\,\!</math> design requiring 16 runs per replicate it is only feasible for the manufacturer to run a single replicate.

The experiment design and data, collected as percent defects, are shown in the following figure. Since the present experiment design contains only a single replicate, it is not possible to obtain an estimate of the error sum of squares, <math>S{{S}_{E}}\,\!</math>. It is decided to use the normal probability plot of effects to identify the significant effects. The effect values for each term are obtained as shown in the following figure.

[[Image:doe7_13.png|center|650px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

Lenth's method uses these values to estimate the variance. As described in [[DOE_References|[Lenth, 1989]]], if all effects are arranged in ascending order, using their absolute values, then <math>{{s}_{0}}\,\!</math> is defined as 1.5 times the median value:

::<math>\begin{align}
{{s}_{0}}= & 1.5\cdot median(\left| effect \right|) \\
= & 1.5\cdot 2 \\
= & 3
\end{align}\,\!</math>

Using <math>{{s}_{0}}\,\!</math>, the "pseudo standard error" (<math>PSE\,\!</math>) is calculated as 1.5 times the median value of all effects that are less than 2.5 <math>{{s}_{0}}\,\!</math> :

::<math>\begin{align}
PSE= & 1.5\cdot median(\left| effect \right|\ \ :\ \ \left| effect \right|<2.5{{s}_{0}}) \\
= & 1.5\cdot 1.5 \\
= & 2.25
\end{align}\,\!</math>

Using <math>PSE\,\!</math> as an estimate of the effect variance, the effect variance is 2.25. Knowing the effect variance, the normal probability plot of effects for the present unreplicated experiment can be constructed as shown in the following figure. The line on this plot is the line N(0, 2.25). The plot shows that the effects <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math> do not follow the distribution represented by this line. Therefore, these effects are significant.

The significant effects can also be identified by comparing individual effect values to the margin of error or the threshold value using the pareto chart (see the third following figure). If the required significance is 0.1, then:

::<math>margin\text{ }of\text{ }error={{t}_{\alpha /2,d}}\cdot PSE\,\!</math>

The <math>t\,\!</math> statistic, <math>{{t}_{\alpha /2,d}}\,\!</math>, is calculated at a significance of <math>\alpha /2\,\!</math> (for the two-sided hypothesis) and degrees of freedom <math>d=(\,\!</math> number of effects <math>)/3\,\!</math>. Thus:

::<math>\begin{align}
margin\text{ }of\text{ }error= & {{t}_{0.05,5}}\cdot PSE \\
= & 2.015\cdot 2.25 \\
= & 4.534
\end{align}\,\!</math>

The value of 4.534 is shown as the critical value line in the third following figure. All effects with absolute values greater than the margin of error can be considered to be significant. These effects are <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math>. Therefore, the vacuum rate, the pre-stretch and their interaction have a significant effect on the defects of the vinyl panels.

[[Image:doe7_14.png|center|800px|Effect values for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_15.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_16.png|center|650px|Pareto chart for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

===Center Point Replicates===

Another method of dealing with unreplicated <math>{2}^{k}\,\!</math> designs that only have quantitative factors is to use replicated runs at the center point. The center point is the response corresponding to the treatment exactly midway between the two levels of all factors. Running multiple replicates at this point provides an estimate of pure error. Although running multiple replicates at any treatment level can provide an estimate of pure error, the other advantage of running center point replicates in the <math>{2}^{k}\,\!</math> design is in checking for the presence of curvature. The test for curvature investigates whether the model between the response and the factors is linear and is discussed in [[Two_Level_Factorial_Experiments#Using_Center_Point_Replicates_to_Test_Curvature| Center Pt. Replicates to Test Curvature]].

====Example: Use Center Point to Get Pure Error====

Consider a <math>{2}^{2}\,\!</math> experiment design to investigate the effect of two factors, <math>A\,\!</math> and <math>B\,\!</math>, on a certain response. The energy consumed when the treatments of the <math>{2}^{2}\,\!</math> design are run is considerably larger than the energy consumed for the center point run (because at the center point the factors are at their middle levels). Therefore, the analyst decides to run only a single replicate of the design and augment the design by five replicated runs at the center point as shown in the following figure. The design properties for this experiment are shown in the second following figure. The complete experiment design is shown in the third following figure. The center points can be used in the identification of significant effects as shown next.

[[Image:doe7.17.png||center|300px|<math>2^2\,\!</math> design augmented by five center point runs.]]
[[Image:doe7_18.png|center|537px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

[[Image:doe7_19.png|center|630px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

Since the present <math>{2}^{2}\,\!</math> design is unreplicated, there are no degrees of freedom available to calculate the error sum of squares. By augmenting this design with five center points, the response values at the center points, <math>y_{i}^{c}\,\!</math>, can be used to obtain an estimate of pure error, <math>S{{S}_{PE}}\,\!</math>. Let <math>{{\bar{y}}^{c}}\,\!</math> represent the average response for the five replicates at the center. Then:

::<math>S{{S}_{PE}}=Sum\text{ }of\text{ }Squares\text{ }for\text{ }center\text{ }points\,\!</math>

::<math>\begin{align}
S{{S}_{PE}}= & \underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}} \\
= & {{(25.2-25.26)}^{2}}+...+{{(25.3-25.26)}^{2}} \\
= & 0.052
\end{align}\,\!</math>

Then the corresponding mean square is:

::<math>\begin{align}
M{{S}_{PE}}= & \frac{S{{S}_{PE}}}{degrees\text{ }of\text{ }freedom} \\
= & \frac{0.052}{5-1} \\
= & 0.013
\end{align}\,\!</math>

Alternatively, <math>M{{S}_{PE}}\,\!</math> can be directly obtained by calculating the variance of the response values at the center points:

::<math>\begin{align}
M{{S}_{PE}}= & {{s}^{2}} \\
= & \frac{\underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}}}{5-1}
\end{align}\,\!</math>

Once <math>M{{S}_{PE}}\,\!</math> is known, it can be used as the error mean square, <math>M{{S}_{E}}\,\!</math>, to carry out the test of significance for each effect. For example, to test the significance of the main effect of factor <math>A,\,\!</math> the sum of squares corresponding to this effect is obtained in the usual manner by considering only the four runs of the original <math>{2}^{2}\,\!</math> design.

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/4)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/4)J]y \\
= & 0.5625
\end{align}\,\!</math>

Then, the test statistic to test the significance of the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{A}}= & \frac{M{{S}_{A}}}{M{{S}_{E}}} \\
= & \frac{0.5625/1}{0.052/4} \\
= & 43.2692
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{A}}=43.2692\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{A}}) \\
= & 1-0.9972 \\
= & 0.0028
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value < 0.1, it can be concluded that the main effect of factor <math>A\,\!</math> significantly affects the response. This result is displayed in the ANOVA table as shown in the following figure. Test for the significance of other factors can be carried out in a similar manner.

[[Image:doe7_20.png|center|649px|Results for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

===Using Center Point Replicates to Test Curvature===

Center point replicates can also be used to check for curvature in replicated or unreplicated <math>{2}^{k}\,\!</math> designs. The test for curvature investigates whether the model between the response and the factors is linear. The way DOE++ handles center point replicates is similar to its handling of blocks. The center point replicates are treated as an additional factor in the model. The factor is labeled as Curvature in the results of the DOE folio. If Curvature turns out to be a significant factor in the results, then this indicates the presence of curvature in the model.

====Example: Use Center Point to Test Curvature====

To illustrate the use of center point replicates in testing for curvature, consider again the data of the single replicate <math>{2}^{2}\,\!</math> experiment from a preceding figure(labeled "<math>2^2</math> design augmented by five center point runs"). Let <math>{{x}_{1}}\,\!</math> be the indicator variable to indicate if the run is a center point:

::<math>\begin{matrix}
{{x}_{1}}=0 & {} & \text{Center point run} \\
{{x}_{1}}=1 & {} & \text{Other run} \\
\end{matrix}\,\!</math>

If <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> are the indicator variables representing factors <math>A\,\!</math> and <math>B\,\!</math>, respectively, then the model for this experiment is:

::<math>Y={{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}\,\!</math>

To investigate the presence of curvature, the following hypotheses need to be tested:

::<math>\begin{align}
& {{H}_{0}}: & {{\beta }_{1}}=0\text{ (Curvature is absent)} \\
& {{H}_{1}}: & {{\beta }_{1}}\ne 0
\end{align}\,\!</math>

The test statistic to be used for this test is:

::<math>{{({{F}_{0}})}_{curvature}}=\frac{M{{S}_{curvature}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{curvature}}\,\!</math> is the mean square for Curvature and <math>M{{S}_{E}}\,\!</math> is the error mean square.

'''Calculation of the Sum of Squares'''

The <math>X\,\!</math> matrix and <math>y\,\!</math> vector for this experiment are:

<center><math>X=\left[ \begin{matrix}
1 & 1 & -1 & -1 & 1 \\
1 & 1 & 1 & -1 & -1 \\
1 & 1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
\end{matrix} \right]\text{ }y=\left[ \begin{matrix}
24.6 \\
25.4 \\
25.0 \\
25.7 \\
25.2 \\
25.3 \\
25.4 \\
25.1 \\
25.3 \\
\end{matrix} \right]\,\!</math></center>

The sum of squares can now be calculated. For example, the error sum of squares is:

::<math>\begin{align}
& S{{S}_{E}}= & {{y}^{\prime }}[I-H]y \\
& = & 0.052
\end{align}\,\!</math>

where <math>I\,\!</math> is the identity matrix and <math>H\,\!</math> is the hat matrix. It can be seen that this is equal to <math>S{{S}_{PE\text{ }}}\,\!</math> (the sum of squares due to pure error) because of the replicates at the center point, as obtained in the [[Two_Level_Factorial_Experiments#Example_3| example]]. The number of degrees of freedom associated with <math>S{{S}_{E}}\,\!</math>, <math>dof(S{{S}_{E}})\,\!</math> is four. The extra sum of squares corresponding to the center point replicates (or Curvature) is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & Model\text{ }Sum\text{ }of\text{ }Squares- \\
& & Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }center\text{ }point \\
& = & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Curvature}}-(1/9)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }Curvature}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }Curvature}}={{X}_{\tilde{\ }Curv}}{{(X_{\tilde{\ }Curv}^{\prime }{{X}_{\tilde{\ }Curv}})}^{-1}}X_{\tilde{\ }Curv}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }Curv}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the center point. Thus, the extra sum of squares corresponding to Curvature is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Center}}-(1/9)J]y \\
& = & 0.7036-0.6875 \\
& = & 0.0161
\end{align}\,\!</math>

This extra sum of squares can be used to test for the significance of curvature. The corresponding mean square is:

::<math>\begin{align}
& M{{S}_{Curvature}}= & \frac{Sum\text{ }of\text{ }squares\text{ }corresponding\text{ }to\text{ }Curvature}{degrees\text{ }of\text{ }freedom} \\
& = & \frac{0.0161}{1} \\
& = & 0.0161
\end{align}\,\!</math>

'''Calculation of the Test Statistic'''

Knowing the mean squares, the statistic to check the significance of curvature can be calculated.

::<math>\begin{align}
& {{({{f}_{0}})}_{Curvature}}= & \frac{M{{S}_{Curvature}}}{M{{S}_{E}}} \\
& = & \frac{0.0161/1}{0.052/4} \\
& = & 1.24
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{Curvature}}=1.24\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and four degrees of freedom in the denominator is:

::<math>\begin{align}
& p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{Curvature}}) \\
& = & 1-0.6713 \\
& = & 0.3287
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that curvature does not exist for this design. This results is shown in the ANOVA table in the figure above. The surface of the fitted model based on these results, along with the observed response values, is shown in the figure below.

[[Image:doe7.21.png|center|400px|Model surface and observed response values for the design in the [[Two_Level_Factorial_Experiments#Example_4| example]].]]

 

==Blocking in 2k Designs==

Blocking can be used in the <math>{2}^{k}\,\!</math> designs to deal with cases when replicates cannot be run under identical conditions. Randomized complete block designs that were discussed in [[Randomization and Blocking in DOE]] for factorial experiments are also applicable here. At times, even with just two levels per factor, it is not possible to run all treatment combinations for one replicate of the experiment under homogeneous conditions. For example, each replicate of the <math>{2}^{2}\,\!</math> design requires four runs. If each run requires two hours and testing facilities are available for only four hours per day, two days of testing would be required to run one complete replicate. Blocking can be used to separate the treatment runs on the two different days. Blocks that do not contain all treatments of a replicate are called incomplete blocks. In incomplete block designs, the block effect is confounded with certain effect(s) under investigation. For the <math>{2}^{2}\,\!</math> design assume that treatments <math>(1)\,\!</math> and <math>ab\,\!</math> were run on the first day and treatments <math>a\,\!</math> and <math>b\,\!</math> were run on the second day. Then, the incomplete block design for this experiment is:

::<math>\begin{matrix}
\text{Block 1} & {} & \text{Block 2} \\
\left[ \begin{matrix}
(1) \\
ab \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
b \\
\end{matrix} \right] \\
\end{matrix}\,\!</math>

For this design the block effect may be calculated as:

::<math>\begin{align}
& Block\text{ }Effect= & Average\text{ }response\text{ }for\text{ }Block\text{ }1- \\
& & Average\text{ }response\text{ }for\text{ }Block\text{ }2 \\
& = & \frac{(1)+ab}{2}-\frac{a+b}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The <math>AB\,\!</math> interaction effect is:

::<math>\begin{align}
& AB= & Average\text{ }response\text{ }at\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{low}}}- \\
& & Average\text{ }response\text{ }at\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{low}}} \\
& = & \frac{ab+(1)}{2}-\frac{b+a}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The two equations given above show that, in this design, the <math>AB\,\!</math> interaction effect cannot be distinguished from the block effect because the formulas to calculate these effects are the same. In other words, the <math>AB\,\!</math> interaction is said to be confounded with the block effect and it is not possible to say if the effect calculated based on these equations is due to the <math>AB\,\!</math> interaction effect, the block effect or both. In incomplete block designs some effects are always confounded with the blocks. Therefore, it is important to design these experiments in such a way that the important effects are not confounded with the blocks. In most cases, the experimenter can assume that higher order interactions are unimportant. In this case, it would better to use incomplete block designs that confound these effects with the blocks.
One way to design incomplete block designs is to use defining contrasts as shown next:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}+...+{{\alpha }_{k}}{{q}_{k}}\,\!</math>

where the <math>{{\alpha }_{i}}\,\!</math> s are the exponents for the factors in the effect that is to be confounded with the block effect and the <math>{{q}_{i}}\,\!</math> s are values based on the level of the <math>i\,\!</math> the factor (in a treatment that is to be allocated to a block). For <math>{2}^{k}\,\!</math> designs the <math>{{\alpha }_{i}}\,\!</math> s are either 0 or 1 and the <math>{{q}_{i}}\,\!</math> s have a value of 0 for the low level of the <math>i\,\!</math> th factor and a value of 1 for the high level of the factor in the treatment under consideration. As an example, consider the <math>{2}^{2}\,\!</math> design where the interaction effect <math>AB\,\!</math> is confounded with the block. Since there are two factors, <math>k=2\,\!</math>, with <math>i=1\,\!</math> representing factor <math>A\,\!</math> and <math>i=2\,\!</math> representing factor <math>B\,\!</math>. Therefore:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}\,\!</math>

The value of <math>{{\alpha }_{1}}\,\!</math> is one because the exponent of factor <math>A\,\!</math> in the confounded interaction <math>AB\,\!</math> is one. Similarly, the value of <math>{{\alpha }_{2}}\,\!</math> is one because the exponent of factor <math>B\,\!</math> in the confounded interaction <math>AB\,\!</math> is also one. Therefore, the defining contrast for this design can be written as:

::<math>\begin{align}
& L= & {{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}} \\
& = & 1\cdot {{q}_{1}}+1\cdot {{q}_{2}} \\
& = & {{q}_{1}}+{{q}_{2}}
\end{align}\,\!</math>

Once the defining contrast is known, it can be used to allocate treatments to the blocks. For the <math>{2}^{2}\,\!</math> design, there are four treatments <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math> and <math>ab\,\!</math>. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. In order to decide which block the treatment <math>(1)\,\!</math> belongs to, the levels of factors <math>A\,\!</math> and <math>B\,\!</math> for this run are used. Since factor <math>A\,\!</math> is at the low level in this treatment, <math>{{q}_{1}}=0\,\!</math>. Similarly, since factor <math>B\,\!</math> is also at the low level in this treatment, <math>{{q}_{2}}=0\,\!</math>. Therefore:

::<math>\begin{align}
& L= & {{q}_{1}}+{{q}_{2}} \\
& = & 0+0=0\text{ (mod 2)}
\end{align}\,\!</math>

Note that the value of <math>L\,\!</math> used to decide the block allocation is "mod 2" of the original value. This value is obtained by taking the value of 1 for odd numbers and 0 otherwise. Based on the value of <math>L\,\!</math>, treatment <math>(1)\,\!</math> is assigned to block 1. Other treatments can be assigned using the following calculations:

::<math>\begin{align}
& (1): & \text{ }L=0+0=0=0\text{ (mod 2)} \\
& a: & \text{ }L=1+0=1=1\text{ (mod 2)} \\
& b: & \text{ }L=0+1=1=1\text{ (mod 2)} \\
& ab: & \text{ }L=1+1=2=0\text{ (mod 2)}
\end{align}\,\!</math>

Therefore, to confound the interaction <math>AB\,\!</math> with the block effect in the <math>{2}^{2}\,\!</math> incomplete block design, treatments <math>(1)\,\!</math> and <math>ab\,\!</math> (with <math>L=0\,\!</math>) should be assigned to block 2 and treatment combinations <math>a\,\!</math> and <math>b\,\!</math> (with <math>L=1\,\!</math>) should be assigned to block 1.

====Example: Two Level Factorial Design with Two Blocks====

This example illustrates how treatments can be allocated to two blocks for an unreplicated <math>{2}^{k}\,\!</math> design. Consider the unreplicated <math>{2}^{4}\,\!</math> design to investigate the four factors affecting the defects in automobile vinyl panels discussed in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments required for this experiment were run by two different operators with each operator conducting 8 runs. This experiment is an example of an incomplete block design. The analyst in charge of this experiment assumed that the interaction <math>ABCD\,\!</math> was not significant and decided to allocate treatments to the two operators so that the <math>ABCD\,\!</math> interaction was confounded with the block effect (the two operators are the blocks). The allocation scheme to assign treatments to the two operators can be obtained as follows.
 
The defining contrast for the <math>{2}^{4}\,\!</math> design where the <math>ABCD\,\!</math> interaction is confounded with the blocks is:

::<math>L={{q}_{1}}+{{q}_{2}}+{{q}_{3}}+{{q}_{4}}\,\!</math>

The treatments can be allocated to the two operators using the values of the defining contrast. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. Then the value of the defining contrast for treatment <math>a\,\!</math> is:

::<math>a\ \ :\ \ \text{ }L=1+0+0+0=1=1\text{ (mod 2)}\,\!</math>

Therefore, treatment <math>a\,\!</math> should be assigned to Block 1 or the first operator. Similarly, for treatment <math>ab\,\!</math> we have:

::<math>ab\ \ :\ \ \text{ }L=1+1+0+0=2=0\text{ (mod 2)}\,\!</math>

[[Image:doe7.22.png|center|200px| Allocation of treatments to two blocks for the <math>2^4</math> design in the example by confounding interaction of <math>ABCD</math> with the blocks.]]

Therefore, <math>ab\,\!</math> should be assigned to Block 2 or the second operator. Other treatments can be allocated to the two operators in a similar manner to arrive at the allocation scheme shown in the figure below.
In DOE++, to confound the <math>ABCD\,\!</math> interaction for the <math>{2}^{4}\,\!</math> design into two blocks, the number of blocks are specified as shown in the figure below. Then the interaction <math>ABCD\,\!</math> is entered in the Block Generator window (second following figure) which is available using the Block Generator button in the following figure. The design generated by the Weibull++ DOE folio is shown in the third of the following figures. This design matches the allocation scheme of the preceding figure.

[[Image:doe7_23.png|center|700px| Adding block properties for the experiment in the example.|link=]]

[[Image:doe7_24.png|center|700px|Specifying the interaction ABCD as the interaction to be confounded with the blocks for the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

[[Image:doe7_25.png|center|800px|Two block design for the experiment in the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

For the analysis of this design, the sum of squares for all effects are calculated assuming no blocking. Then, to account for blocking, the sum of squares corresponding to the <math>ABCD\,\!</math> interaction is considered as the sum of squares due to blocks and <math>ABCD\,\!</math>. In DOE++ this is done by displaying this sum of squares as the sum of squares due to the blocks. This is shown in the following figure where the sum of squares in question is obtained as 72.25 and is displayed against Block. The interaction ABCD, which is confounded with the blocks, is not displayed. Since the design is unreplicated, any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_26.png|center|793px|ANOVA table for the experiment of the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

===Unreplicated 2''k'' Designs in 2''p'' Blocks===

A single replicate of the <math>{2}^{k}\,\!</math> design can be run in up to <math>{2}^{p}\,\!</math> blocks where <math>p<k\,\!</math>. The number of effects confounded with the blocks equals the degrees of freedom associated with the block effect.

If two blocks are used (the block effect has two levels), then one (<math>2-1=1)\,\!</math> effect is confounded with the blocks. If four blocks are used, then three (<math>4-1=3\,\!</math>) effects are confounded with the blocks and so on. For example an unreplicated <math>{2}^{4}\,\!</math> design may be confounded in <math>{2}^{2}\,\!</math> (four) blocks using two contrasts, <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}}\,\!</math>. Let <math>AC\,\!</math> and <math>BD\,\!</math> be the effects to be confounded with the blocks. Corresponding to these two effects, the contrasts are respectively:

::<math>\begin{align}
& {{L}_{1}}= & {{q}_{1}}+{{q}_{3}} \\
& {{L}_{2}}= & {{q}_{2}}+{{q}_{4}}
\end{align}\,\!</math>

Based on the values of <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}},\,\!</math> the treatments can be assigned to the four blocks as follows:

<center><math>\begin{matrix}
\text{Block 4} & {} & \text{Block 3} & {} & \text{Block 2} & {} & \text{Block 1} \\
{{L}_{1}}=0,{{L}_{2}}=0 & {} & {{L}_{1}}=1,{{L}_{2}}=0 & {} & {{L}_{1}}=0,{{L}_{2}}=1 & {} & {{L}_{1}}=1,{{L}_{2}}=1 \\
{} & {} & {} & {} & {} & {} & {} \\
\left[ \begin{matrix}
(1) \\
ac \\
bd \\
abcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
c \\
abd \\
bcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
b \\
abc \\
d \\
acd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
ab \\
bc \\
ad \\
cd \\
\end{matrix} \right] \\
\end{matrix}\,\!</math></center>

Since the block effect has three degrees of freedom, three effects are confounded with the block effect. In addition to <math>AC\,\!</math> and <math>BD\,\!</math>, the third effect confounded with the block effect is their generalized interaction, <math>(AC)(BD)=ABCD\,\!</math>.
In general, when an unreplicated <math>{2}^{k}\,\!</math> design is confounded in <math>{2}^{p}\,\!</math> blocks, <math>p\,\!</math> contrasts are needed (<math>{{L}_{1}},{{L}_{2}}...{{L}_{p}}\,\!</math>). <math>p\,\!</math> effects are selected to define these contrasts such that none of these effects are the generalized interaction of the others. The <math>{2}^{p}\,\!</math> blocks can then be assigned the treatments using the <math>p\,\!</math> contrasts. <math>{{2}^{p}}-(p+1)\,\!</math> effects, that are also confounded with the blocks, are then obtained as the generalized interaction of the <math>p\,\!</math> effects. In the statistical analysis of these designs, the sum of squares are computed as if no blocking were used. Then the block sum of squares is obtained by adding the sum of squares for all the effects confounded with the blocks.

====Example: 2 Level Factorial Design with Four Blocks====

This example illustrates how a DOE folio obtains the sum of squares when treatments for an unreplicated <math>{2}^{k}\,\!</math> design are allocated among four blocks. Consider again the unreplicated <math>{2}^{4}\,\!</math> design used to investigate the defects in automobile vinyl panels presented in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments needed to complete the experiment were run by four operators. Therefore, there are four blocks. Assume that the treatments were allocated to the blocks using the generators mentioned in the previous section, i.e., treatments were allocated among the four operators by confounding the effects, <math>AC\,\!</math> and <math>BD,\,\!</math> with the blocks. These effects can be specified as Block Generators as shown in the following figure. (The generalized interaction of these two effects, interaction <math>ABCD\,\!</math>, will also get confounded with the blocks.) The resulting design is shown in the second following figure and matches the allocation scheme obtained in the previous section.

[[Image:doe7_27.png|center|700px|Specifying the interactions AC and BD as block generators for the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

The sum of squares in this case can be obtained by calculating the sum of squares for each of the effects assuming there is no blocking. Once the individual sum of squares have been obtained, the block sum of squares can be calculated. The block sum of squares is the sum of the sum of squares of effects, <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, since these effects are confounded with the block effect. As shown in the second following figure, this sum of squares is 92.25 and is displayed against Block. The interactions <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, which are confounded with the blocks, are not displayed. Since the present design is unreplicated any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_28.png|center|800px|Design for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

[[Image:doe7_29.png|center|793px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

==Variability Analysis==

For replicated two level factorial experiments, DOE++ provides the option of conducting variability analysis (using the Variability Analysis icon under the Data menu). The analysis is used to identify the treatment that results in the least amount of variation in the product or process being investigated. Variability analysis is conducted by treating the standard deviation of the response for each treatment of the experiment as an additional response. The standard deviation for a treatment is obtained by using the replicated response values at that treatment run. As an example, consider the <math>{2}^{3}\,\!</math> design shown in the following figure where each run is replicated four times. A variability analysis can be conducted for this design. DOE++ calculates eight standard deviation values corresponding to each treatment of the design (see second following figure). Then, the design is analyzed as an unreplicated <math>{2}^{3}\,\!</math> design with the standard deviations (displayed as Y Standard Deviation. in second following figure) as the response. The normal probability plot of effects identifies <math>AC\,\!</math> as the effect that influences variability (see third figure following). Based on the effect coefficients obtained in the fourth figure following, the model for Y Std. is:

::<math>\begin{align}
& \text{Y Std}\text{.}= & 0.6779+0.2491\cdot AC \\
& = & 0.6779+0.2491{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

Based on the model, the experimenter has two choices to minimize variability (by minimizing Y Std.). The first choice is that <math>{{x}_{1}}\,\!</math> should be <math>1\,\!</math> (i.e., <math>A\,\!</math> should be set at the high level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the low level). The second choice is that <math>{{x}_{1}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>A\,\!</math> should be set at the low level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the high level). The experimenter can select the most feasible choice.

[[Image:doe7.30.png|center|391px|A <math>2^3\,\!</math> design with four replicated response values that can be used to conduct a variability analysis.]]
 

[[Image:doe7_31.png|center|727px|Variability analysis in DOE++.|link=]]
 

[[Image:doe7_32.png|center|650px|Normal probability plot of effects for the variability analysis example.|link=]]
 

[[Image:doe7_33.png|center|727px|Effect coefficients for the variability analysis example.|link=]]

 

==Two Level Fractional Factorial Designs==

As the number of factors in a two level factorial design increases, the number of runs for even a single replicate of the <math>{2}^{k}\,\!</math> design becomes very large. For example, a single replicate of an eight factor two level experiment would require 256 runs. Fractional factorial designs can be used in these cases to draw out valuable conclusions from fewer runs. The basis of fractional factorial designs is the ''sparsity of effects'' principle.[[DOE_References|[Wu, 2000]]] The principle states that, most of the time, responses are affected by a small number of main effects and lower order interactions, while higher order interactions are relatively unimportant. Fractional factorial designs are used as screening experiments during the initial stages of experimentation. At these stages, a large number of factors have to be investigated and the focus is on the main effects and two factor interactions. These designs obtain information about main effects and lower order interactions with fewer experiment runs by confounding these effects with unimportant higher order interactions. As an example, consider a <math>{2}^{8}\,\!</math> design that requires 256 runs. This design allows for the investigation of 8 main effects and 28 two factor interactions. However, 219 degrees of freedom are devoted to three factor or higher order interactions. This full factorial design can prove to be very inefficient when these higher order interactions can be assumed to be unimportant. Instead, a fractional design can be used here to identify the important factors that can then be investigated more thoroughly in subsequent experiments. In unreplicated fractional factorial designs, no degrees of freedom are available to calculate the error sum of squares and the techniques mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] should be employed for the analysis of these designs.

==Half-fraction Designs==

A half-fraction of the <math>{2}^{k}\,\!</math> design involves running only half of the treatments of the full factorial design. For example, consider a <math>{2}^{3}\,\!</math> design that requires eight runs in all. The design matrix for this design is shown in the figure (a) below. A half-fraction of this design is the design in which only four of the eight treatments are run. The fraction is denoted as <math>{2}^{3-1}\,\!</math> with the "<math>-1\,\!</math>" in the index denoting a half-fraction. Assume that the treatments chosen for the half-fraction design are the ones where the interaction <math>ABC\,\!</math> is at the high level (i.e., only those rows are chosen from the following figure (a) where the column for <math>ABC\,\!</math> has entries of 1). The resulting <math>{2}^{3-1}\,\!</math> design has a design matrix as shown in figure (b) below.

[[Image:doe7.34.png|center|330px|Half-fractions of the <math>2^3\,\!</math> design. (a) shows the full factorial <math>2^3\,\!</math> design, (b) shows the <math>2^{3-1}\,\!</math> design with the defining relation <math>I=ABC\,\!</math> and (c) shows the <math>{2}^{3-1}\,\!</math> design with the defining relation <math>I=-ABC\,\!</math>.]]

In the <math>{2}^{3-1}\,\!</math> design of figure (b), since the interaction <math>ABC\,\!</math> is always included at the same level (the high level represented by 1), it is not possible to measure this interaction effect. The effect, <math>ABC\,\!</math>, is called the ''generator'' or ''word'' for this design. It can be noted that, in the design matrix of the following figure (b), the column corresponding to the intercept, <math>I\,\!</math>, and column corresponding to the interaction <math>ABC\,\!</math>, are identical. The identical columns are written as <math>I=ABC\,\!</math> and this equation is called the ''defining relation'' for the design. In a DOE folio, the present <math>{2}^{3-1}\,\!</math> design can be obtained by specifying the design properties as shown in the following figure.

[[Image:doe7_35.png|center|700px|Design properties for the <math>2^{3-1}\,\!</math> design.|link=]]

The defining relation, <math>I=ABC\,\!</math>, is entered in the Fraction Generator window as shown next.

[[Image:doe7_36.png|center|700px|Specifying the defining relation for the <math>2^{3-1}\,\!</math> design.|link=]]

Note that in the figure following that, the defining relation is specified as <math>C=AB\,\!</math>. This relation is obtained by multiplying the defining relation, <math>I=ABC\,\!</math>, by the last factor, <math>C\,\!</math>, of the design.

===Calculation of Effects===

Using the four runs of the <math>{2}^{3-1}\,\!</math> design in figure (b) discussed above, the main effects can be calculated as follows:

::<math>\begin{align}
& A= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& B= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& C= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

where <math>a\,\!</math>, <math>b\,\!</math>, <math>c\,\!</math> and <math>abc\,\!</math> are the treatments included in the <math>{2}^{3-1}\,\!</math> design.

Similarly, the two factor interactions can also be obtained as:

::<math>\begin{align}
& BC= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& AC= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& AB= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

The equations for <math>A\,\!</math> and <math>BC\,\!</math> above result in the same effect values showing that effects <math>A\,\!</math> and <math>BC\,\!</math> are confounded in the present <math>{2}^{3-1}\,\!</math> design. Thus, the quantity, <math>\tfrac{1}{2}(a-b-c+abc),\,\!</math> estimates <math>A+BC\,\!</math> (i.e., both the main effect <math>A\,\!</math> and the two-factor interaction <math>BC\,\!</math>). The effects, <math>A\,\!</math> and <math>BC,\,\!</math> are called ''aliases''. From the remaining equations given above, it can be seen that the other aliases for this design are <math>B\,\!</math> and <math>AC\,\!</math>, and <math>C\,\!</math> and <math>AB\,\!</math>. Therefore, the equations to calculate the effects in the present <math>{2}^{3-1}\,\!</math> design can be written as follows:

::<math>\begin{align}
& A+BC= & \frac{1}{2}(a-b-c+abc) \\
& B+AC= & \frac{1}{2}(-a+b-c+abc) \\
& C+AB= & \frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

===Calculation of Aliases===

Aliases for a fractional factorial design can be obtained using the defining relation for the design. The defining relation for the present <math>{2}^{3-1}\,\!</math> design is:

::<math>I=ABC\,\!</math>

Multiplying both sides of the previous equation by the main effect, <math>A,\,\!</math> gives the alias effect of <math>A\,\!</math> :

::<math>\begin{align}
& A\cdot I= & A\cdot ABC \\
& A= & {{A}^{2}}BC \\
& A= & BC
\end{align}\,\!</math>

Note that in calculating the alias effects, any effect multiplied by <math>I\,\!</math> remains the same (<math>A\cdot I=A\,\!</math>), while an effect multiplied by itself results in <math>I\,\!</math> (<math>{{A}^{2}}=I\,\!</math>). Other aliases can also be obtained:

::<math>\begin{align}
& B\cdot I= & B\cdot ABC \\
& B= & A{{B}^{2}}C \\
& B= & AC
\end{align}\,\!</math>

:and:

::<math>\begin{align}
& C\cdot I= & C\cdot ABC \\
& C= & AB{{C}^{2}} \\
& C= & AB
\end{align}\,\!</math>

===Fold-over Design===

If it can be assumed for this design that the two-factor interactions are unimportant, then in the absence of <math>BC\,\!</math>, <math>AC\,\!</math> and <math>AB\,\!</math>, the equations for (A+BC), (B+AC) and (C+AB) can be used to estimate the main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. However, if such an assumption is not applicable, then to uncouple the main effects from their two factor aliases, the alternate fraction that contains runs having <math>ABC\,\!</math> at the lower level should be run. The design matrix for this design is shown in the preceding figure (c). The defining relation for this design is <math>I=-ABC\,\!</math> because the four runs for this design are obtained by selecting the rows of the preceding figure (a) for which the value of the <math>ABC\,\!</math> column is <math>-1\,\!</math>. The aliases for this fraction can be obtained as explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]] as <math>A=-BC\,\!</math>, <math>B=-AC\,\!</math> and <math>C=-AB\,\!</math>. The effects for this design can be calculated as:

::<math>\begin{align}
& A-BC= & \frac{1}{2}(ab+ac-(1)-bc) \\
& B-AC= & \frac{1}{2}(ab-ac+(1)-bc) \\
& C-AB= & \frac{1}{2}(-ab+ac-(1)+bc)
\end{align}\,\!</math>

These equations can be combined with the equations for (A+BC), (B+AC) and (C+AB) to obtain the de-aliased main effects and two factor interactions. For example, adding equations (A+BC) and (A-BC) returns the main effect <math>A\,\!</math>.

::<math>\begin{align}
& 2A= & \frac{1}{2}(a-b-c+abc)+ \\
& & \frac{1}{2}(ab+ac-(1)-bc)
\end{align}\,\!</math>

The process of augmenting a fractional factorial design by a second fraction of the same size by simply reversing the signs (of all effect columns except <math>I\,\!</math>) is called ''folding over''. The combined design is referred to as a ''fold-over design''.

==Quarter and Smaller Fraction Designs==

At times, the number of runs even for a half-fraction design are very large. In these cases, smaller fractions are used. A quarter-fraction design, denoted as <math>{2}^{k-2}\,\!</math>, consists of a fourth of the runs of the full factorial design. Quarter-fraction designs require two defining relations. The first defining relation returns the half-fraction or the <math>{2}^{k-1}\,\!</math> design. The second defining relation selects half of the runs of the <math>{2}^{k-1}\,\!</math> design to give the quarter-fraction. For example, consider the <math>{2}^{4}\,\!</math> design. To obtain a <math>{2}^{4-2}\,\!</math> design from this design, first a half-fraction of this design is obtained by using a defining relation. Assume that the defining relation used is <math>I=ABCD\,\!</math>. The design matrix for the resulting <math>{2}^{4-1}\,\!</math> design is shown in figure (a) below. Now, a quarter-fraction can be obtained from the <math>{2}^{4-1}\,\!</math> design shown in figure (a) below using a second defining relation <math>I=AD\,\!</math>. The resulting <math>{2}^{4-2}\,\!</math> design obtained is shown in figure (b) below.

[[Image:doe7.37.png|center|465px|Fractions of the <math>2^4\,\!</math> design - Figure (a) shows the <math>2^{4-1}</math> design with the defining relation <math>I=ABCD\,\!</math> and (b) shows the <math>2^{4-2}\,\!</math> design with the defining relation <math>I=ABCD=AD=BC\,\!</math>.]]

The complete defining relation for this <math>{2}^{4-2}\,\!</math> design is:

::<math>I=ABCD=AD=BC\,\!</math>

Note that the effect, <math>BC,\,\!</math> in the defining relation is the generalized interaction of <math>ABCD\,\!</math> and <math>AD\,\!</math> and is obtained using <math>(ABCD)(AD)={{A}^{2}}BC{{D}^{2}}=BC\,\!</math>. In general, a <math>{2}^{k-p}\,\!</math> fractional factorial design requires <math>p\,\!</math> independent generators. The defining relation for the design consists of the <math>p\,\!</math> independent generators and their <math>{2}^{p}\,\!</math> - (<math>p\,\!</math> +1) generalized interactions.

===Calculation of Aliases===

The alias structure for the present <math>{2}^{4-2}\,\!</math> design can be obtained using the defining relation of equation (I=ABCD=AD=BC) following the procedure explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]]. For example, multiplying the defining relation by <math>A\,\!</math> returns the effects aliased with the main effect, <math>A\,\!</math>, as follows:

::<math>\begin{align}
& A\cdot I= & A\cdot ABCD=A\cdot AD=A\cdot BC \\
& A= & {{A}^{2}}BCD={{A}^{2}}D=ABC \\
& A= & BCD=D=ABC
\end{align}\,\!</math>

Therefore, in the present <math>{2}^{4-2}\,\!</math> design, it is not possible to distinguish between effects <math>A\,\!</math>, <math>D\,\!</math>, <math>BCD\,\!</math> and <math>ABC\,\!</math>. Similarly, multiplying the defining relation by <math>B\,\!</math> and <math>AB\,\!</math> returns the effects that are aliased with these effects:

::<math>\begin{align}
& B= & ACD=ABD=C \\
& AB= & CD=AD=AC
\end{align}\,\!</math>

Other aliases can be obtained in a similar way. It can be seen that each effect in this design has three aliases. In general, each effect in a <math>{2}^{k-p}\,\!</math> design has <math>{2}^{p-1}\,\!</math> aliases.
The aliases for the <math>{2}^{4-2}\,\!</math> design show that in this design the main effects are aliased with each other (<math>A\,\!</math> is aliased with <math>D\,\!</math> and <math>B\,\!</math> is aliased with <math>C\,\!</math>). Therefore, this design is not a useful design and is not available in a Weibull++ DOE folio. It is important to ensure that main effects and lower order interactions of interest are not aliased in a fractional factorial design. This is known by looking at the resolution of the fractional factorial design.

==Design Resolution==

The resolution of a fractional factorial design is defined as the number of factors in the lowest order effect in the defining relation. For example, in the defining relation <math>I=ABCD=AD=BC\,\!</math> of the previous <math>{2}^{4-2}\,\!</math> design, the lowest-order effect is either <math>AD\,\!</math> or <math>BC,\,\!</math> containing two factors. Therefore, the resolution of this design is equal to two. The resolution of a fractional factorial design is represented using Roman numerals. For example, the previously mentioned <math>{2}^{4-2}\,\!</math> design with a resolution of two can be represented as 2 <math>_{\text{II}}^{4-2}\,\!</math>. The resolution provides information about the confounding in the design as explained next:
 
 
#'''Resolution III Designs''' In these designs, the lowest order effect in the defining relation has three factors (e.g., a <math>{2}^{5-2}\,\!</math> design with the defining relation <math>I=ABDE=ABC=CDE\,\!</math>). In resolution III designs, no main effects are aliased with any other main effects, but main effects are aliased with two factor interactions. In addition, some two factor interactions are aliased with each other.
#'''Resolution IV Designs''' In these designs, the lowest order effect in the defining relation has four factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABDE\,\!</math>). In resolution IV designs, no main effects are aliased with any other main effects or two factor interactions. However, some main effects are aliased with three factor interactions and the two factor interactions are aliased with each other.
#'''Resolution V Designs''' In these designs the lowest order effect in the defining relation has five factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABCDE\,\!</math>). In resolution V designs, no main effects or two factor interactions are aliased with any other main effects or two factor interactions. However, some main effects are aliased with four factor interactions and the two factor interactions are aliased with three factor interactions.

Fractional factorial designs with the highest resolution possible should be selected because the higher the resolution of the design, the less severe the degree of confounding. In general, designs with a resolution less than III are never used because in these designs some of the main effects are aliased with each other. The table below shows fractional factorial designs with the highest available resolution for three to ten factor designs along with their defining relations.

[[Image:doet7.3.png|center|474px|Highest resolution designs available for fractional factorial designs with 3 to 10 factors.]]

All of the two level fractional factorial designs available in a DOE folio are shown next.

[[Image:doe7.38.png|center|471px|Two level fractional factorial designs available in Weibull++ and their resolutions.]]

===Minimum Aberration Designs===
At times, different designs with the same resolution but different aliasing may be available. The best design to select in such a case is the minimum aberration design. For example, all <math>{2}^{7-2}\,\!</math> designs in the fourth table have a resolution of four (since the generator with the minimum number of factors in each design has four factors). Design <math>1\,\!</math> has three generators of length four (<math>ABCF,\,\!</math> <math>BCDG,\,\!</math> <math>ADFG\,\!</math>). Design <math>2\,\!</math> has two generators of length four (<math>ABCF,\,\!</math> <math>ADEG\,\!</math>). Design <math>3\,\!</math> has one generator of length four (<math>CEFG\,\!</math>). Therefore, design <math>3\,\!</math> has the least number of generators with the minimum length of four. Design <math>3\,\!</math> is called the minimum aberration design. It can be seen that the alias structure for design <math>3\,\!</math> is less involved compared to the other designs. For details refer to [[DOE_References|[Wu, 2000]]].

[[Image:doet7.4.png|center|432px|Three <math>2_{IV}^{7-2}\,\!</math> designs with different defining relations.]]

====Example====

The design of an automobile fuel cone is thought to be affected by six factors in the manufacturing process: cavity temperature (factor <math>A\,\!</math>), core temperature (factor <math>B\,\!</math>), melt temperature (factor <math>C\,\!</math>), hold pressure (factor <math>D\,\!</math>), injection speed (factor <math>E\,\!</math>) and cool time (factor <math>F\,\!</math>). The manufacturer of the fuel cone is unable to run the <math>{2}^{6}=64\,\!</math> runs required to complete one replicate for a two level full factorial experiment with six factors. Instead, they decide to run a fractional factorial design. Considering that three factor and higher order interactions are likely to be inactive, the manufacturer selects a <math>{2}^{6-2}\,\!</math> design that will require only 16 runs. The manufacturer chooses the resolution IV design which will ensure that all main effects are free from aliasing (assuming three factor and higher order interactions are absent). However, in this design the two factor interactions may be aliased with each other. It is decided that, if important two factor interactions are found to be present, additional experiment trials may be conducted to separate the aliased effects. The performance of the fuel cone is measured on a scale of 1 to 15. In DOE++, the design for this experiment is set up using the properties shown in the following figure. The Fraction Generators for the design, <math>E=ABC\,\!</math> and <math>F=BCD\,\!</math>, are the same as the defaults used in DOE++. The resulting <math>{2}^{6-2}\,\!</math> design and the corresponding response values are shown in the following two figures.

[[Image:doe7_39.png|center|644px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

[[Image:doe7_40.png|center|534px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The complete alias structure for the 2 <math>_{\text{IV}}^{6-2}\,\!</math> design is shown next.

<center><math>I=ABCE=ADEF=BCDF\,\!</math></center>

 
<center><math>\begin{align}
& A= & BCE=DEF=ABCDF \\
& B= & ACE=CDF=ABDEF \\
& C= & ABE=BDF=ACDEF \\
& D= & AEF=BCF=ABCDE \\
& E= & ABC=ADF=BCDEF \\
& F= & ADE=BCD=ABCEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& AB= & CE=ACDF=BDEF \\
& AC= & BE=ABDF=CDEF \\
& AD= & EF=ABCF=BCDE \\
& AE= & BC=DF=ABCDEF \\
& AF= & DE=ABCD=BCEF \\
& BD= & CF=ABEF=ACDE \\
& BF= & CD=ABDE=ACEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& ABD= & ACF=BEF=CDE \\
& ABF= & ACD=BDE=CEF
\end{align}\,\!</math></center>

In DOE++, the alias structure is displayed in the Design Summary and as part of the Design Evaluation result, as shown next:

[[Image:doe7_41.png|center|700px|Alias structure for the experiment design in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The normal probability plot of effects for this unreplicated design shows the main effects of factors <math>C\,\!</math> and <math>D\,\!</math> and the interaction effect, <math>BF\,\!</math>, to be significant (see the following figure).

[[Image:doe7_42.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

From the alias structure, it can be seen that for the present design interaction effect, <math>BF,\,\!</math> is confounded with <math>CD\,\!</math>. Therefore, the actual source of this effect cannot be known on the basis of the present experiment. However because neither factor <math>B\,\!</math> nor <math>F\,\!</math> is found to be significant there is an indication the observed effect is likely due to interaction, <math>CD\,\!</math>. To confirm this, a follow-up <math>{2}^{2}\,\!</math> experiment is run involving only factors <math>B\,\!</math> and <math>F\,\!</math>. The interaction, <math>BF\,\!</math>, is found to be inactive, leading to the conclusion that the interaction effect in the original experiment is effect, <math>CD\,\!</math>. Given these results, the fitted regression model for the fuel cone design as per the coefficients obtained from DOE++ is shown next.

::<math>\hat{y}=7.6875+C+2\cdot D+2.1875\cdot CD\,\!</math>

[[Image:doe7_43.png|center|700px|Effect coefficients for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

==Projection==

Projection refers to the reduction of a fractional factorial design to a full factorial design by dropping out some of the factors of the design. Any fractional factorial design of resolution, <math>R,\,\!</math> can be reduced to complete factorial designs in any subset of <math>R-1\,\!</math> factors. For example, consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. The resolution of this design is four. Therefore, this design can be reduced to full factorial designs in any three (<math>4-1=3\,\!</math>) of the original seven factors (by dropping the remaining four of factors). Further, a fractional factorial design can also be reduced to a full factorial design in any <math>R\,\!</math> of the original factors, as long as these <math>R\,\!</math> factors are not part of the generator in the defining relation. Again consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. This design can be reduced to a full factorial design in four factors provided these four factors do not appear together as a generator in the defining relation. The complete defining relation for this design is:

::<math>\begin{align}
& I= & ABCE=BCDF=ACDG=ADEF=ABFG=BDEG=CEFG
\end{align}\,\!</math>

Therefore, there are seven four factor combinations out of the 35 (<math>(_{7}^{4})=35\,\!</math>) possible four-factor combinations that are used as generators in the defining relation. The designs with the remaining 28 four factor combinations would be full factorial 16-run designs. For example, factors <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math> do not occur as a generator in the defining relation of the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. If the remaining factors, <math>E\,\!</math>, <math>F\,\!</math> and <math>G\,\!</math>, are dropped, the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design will reduce to a full factorial design in <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math>.

==Resolution III Designs==

At times, the factors to be investigated in screening experiments are so large that even running a fractional factorial design is impractical. This can be partially solved by using resolution III fractional factorial designs in the cases where three factor and higher order interactions can be assumed to be unimportant. Resolution III designs, such as the 2 <math>_{\text{III}}^{3-1}\,\!</math> design, can be used to estimate <math>k\,\!</math> main effects using just <math>k+1\,\!</math> runs. In these designs, the main effects are aliased with two factor interactions. Once the results from these designs are obtained, and knowing that three factor and higher order interactions are unimportant, the experimenter can decide if there is a need to run a fold-over design to de-alias the main effects from the two factor interactions. Thus, the 2 <math>_{\text{III}}^{3-1}\,\!</math> design can be used to investigate three factors in four runs, the 2 <math>_{\text{III}}^{7-4}\,\!</math> design can be used to investigate seven factors in eight runs, the 2 <math>_{\text{III}}^{15-11}\,\!</math> design can be used to investigate fifteen factors in sixteen runs and so on.

====Example====

{{:Resolution_III_Design_Example}}

==Alias Matrix==
In [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction designs]] and [[Two_Level_Factorial_Experiments#Quarter_and_Smaller_Fraction_Designs| Quarter and Smaller Fraction Designs]], the alias structure for fractional factorial designs was obtained using the defining relation. However, this method of obtaining the alias structure is not very efficient when the alias structure is very complex or when partial aliasing is involved. One of the ways to obtain the alias structure for any design, regardless of its complexity, is to use the alias matrix. The alias matrix for a design is calculated using <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> where <math>{{X}_{1}}\,\!</math> is the portion of the design matrix, <math>X,\,\!</math> that contains the effects for which the aliases need to be calculated, and <math>{{X}_{2}}\,\!</math> contains the remaining columns of the design matrix, other than those included in <math>{{X}_{1}}\,\!</math>.

To illustrate the use of the alias matrix, consider the design matrix for the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design (using the defining relation <math>I=ABCD\,\!</math>) shown next:

[[Image:Chapter7__879.png|center|link=]]

The alias structure for this design can be obtained by defining <math>{{X}_{1}}\,\!</math> using eight columns since the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design estimates eight effects. If the first eight columns of <math>X\,\!</math> are used then <math>{{X}_{1}}\,\!</math> is:

[[Image:Chapter7__884.png|center|link=]]

<math>{{X}_{2}}\,\!</math> is obtained using the remaining columns as:

[[Image:Chapter7__886.png|center|link=]]

Then the alias matrix <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> is:

[[Image:Chapter7__888.png|center|link=]]

The alias relations can be easily obtained by observing the alias matrix as:

<center><math>\begin{align}
& I= & ABCD \\
& A= & BCD \\
& B= & ACD \\
& AB= & CD \\
& C= & ABD \\
& AC= & BD \\
& BC= & AD \\
& D= & ABC
\end{align}\,\!</math></center>

Two Level Factorial Experiments

2017-08-10T17:37:27Z

Richard House: /* Calculation of Aliases */

{{Template:Doebook|8}}
Two level factorial experiments are factorial experiments in which each factor is investigated at only two levels. The early stages of experimentation usually involve the investigation of a large number of potential factors to discover the "vital few" factors. Two level factorial experiments are used during these stages to quickly filter out unwanted effects so that attention can then be focused on the important ones.

==2''k'' Designs==
The factorial experiments, where all combination of the levels of the factors are run, are usually referred to as ''full factorial experiments''. Full factorial two level experiments are also referred to as <math>{2}^{k}\,\!</math> designs where <math>k\,\!</math> denotes the number of factors being investigated in the experiment. In Weibull++ DOE folios, these designs are referred to as 2 Level Factorial Designs as shown in the figure below.

[[Image:doe7_1.png|center|487px|Selection of full factorial experiments with two levels in Weibull++.|link=]]

A full factorial two level design with <math>k\,\!</math> factors requires <math>{{2}^{k}}\,\!</math> runs for a single replicate. For example, a two level experiment with three factors will require <math>2\times 2\times 2={{2}^{3}}=8\,\!</math> runs. The choice of the two levels of factors used in two level experiments depends on the factor; some factors naturally have two levels. For example, if gender is a factor, then male and female are the two levels. For other factors, the limits of the range of interest are usually used. For example, if temperature is a factor that varies from <math>{45}^{o}C\,\!</math> to <math>{90}^{o}C\,\!</math>, then the two levels used in the <math>{2}^{k}\,\!</math> design for this factor would be <math>{45}^{o}\,\!C\,\!</math> and <math>{90}^{o}\,\!C\,\!</math>.

The two levels of the factor in the <math>{2}^{k}\,\!</math> design are usually represented as <math>-1\,\!</math> (for the first level) and <math>1\,\!</math> (for the second level). Note that this representation is reversed from the coding used in [[General Full Factorial Designs]] for the indicator variables that represent two level factors in ANOVA models. For ANOVA models, the first level of the factor was represented using a value of <math>1\,\!</math> for the indicator variable, while the second level was represented using a value of <math>-1\,\!</math>. For details on the notation used for two level experiments refer to [[Two_Level_Factorial_Experiments#Notation| Notation]].

===The 22 Design===

The simplest of the two level factorial experiments is the <math>{2}^{2}\,\!</math> design where two factors (say factor <math>A\,\!</math> and factor <math>B\,\!</math>) are investigated at two levels. A single replicate of this design will require four runs (<math>{{2}^{2}}=2\times 2=4\,\!</math>) The effects investigated by this design are the two main effects, <math>A\,\!</math> and <math>B,\,\!</math> and the interaction effect <math>AB\,\!</math>. The treatments for this design are shown in figure (a) below. In figure (a), letters are used to represent the treatments. The presence of a letter indicates the high level of the corresponding factor and the absence indicates the low level. For example, (1) represents the treatment combination where all factors involved are at the low level or the level represented by <math>-1\,\!</math> ; <math>a\,\!</math> represents the treatment combination where factor <math>A\,\!</math> is at the high level or the level of <math>1\,\!</math>, while the remaining factors (in this case, factor <math>B\,\!</math>) are at the low level or the level of <math>-1\,\!</math>. Similarly, <math>b\,\!</math> represents the treatment combination where factor <math>B\,\!</math> is at the high level or the level of <math>1\,\!</math>, while factor <math>A\,\!</math> is at the low level and <math>ab\,\!</math> represents the treatment combination where factors <math>A\,\!</math> and <math>B\,\!</math> are at the high level or the level of the 1. Figure (b) below shows the design matrix for the <math>{2}^{2}\,\!</math> design. It can be noted that the sum of the terms resulting from the product of any two columns of the design matrix is zero. As a result the <math>{2}^{2}\,\!</math> design is an ''orthogonal design''. In fact, all <math>{2}^{k}\,\!</math> designs are orthogonal designs. This property of the <math>{2}^{k}\,\!</math> designs offers a great advantage in the analysis because of the simplifications that result from orthogonality. These simplifications are explained later on in this chapter.
The <math>{2}^{2}\,\!</math> design can also be represented geometrically using a square with the four treatment combinations lying at the four corners, as shown in figure (c) below.

[[Image:doe7.2.png|center|400px|The <math>2^2\,\!</math> design. Figure (a) displays the experiment design, (b) displays the design matrix and (c) displays the geometric representation for the design. In Figure (b), the column names I, A, B and AB are used. Column I represents the intercept term. Columns A and B represent the respective factor settings. Column AB represents the interaction and is the product of columns A and B.]]
 

===The 23 Design===

The <math>{2}^{3}\,\!</math> design is a two level factorial experiment design with three factors (say factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>). This design tests three (<math>k=3\,\!</math>) main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math> ; three (<math>(_{2}^{k})=\,\!</math> <math>(_{2}^{3})=3\,\!</math>) two factor interaction effects, <math>AB\,\!</math>, <math>BC\,\!</math>, <math>AC\,\!</math> ; and one (<math>(_{3}^{k})=\,\!</math> <math>(_{3}^{3})=1\,\!</math>) three factor interaction effect, <math>ABC\,\!</math>. The design requires eight runs per replicate. The eight treatment combinations corresponding to these runs are <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math>, <math>ab\,\!</math>, <math>c\,\!</math>, <math>ac\,\!</math>, <math>bc\,\!</math> and <math>abc\,\!</math>. Note that the treatment combinations are written in such an order that factors are introduced one by one with each new factor being combined with the preceding terms. This order of writing the treatments is called the ''standard order'' or ''Yates' order''. The <math>{2}^{3}\,\!</math> design is shown in figure (a) below. The design matrix for the <math>{2}^{3}\,\!</math> design is shown in figure (b). The design matrix can be constructed by following the standard order for the treatment combinations to obtain the columns for the main effects and then multiplying the main effects columns to obtain the interaction columns.

[[Image:doe7.3.png|center|324px|The <math>2^3\,\!</math> design. Figure (a) shows the experiment design and (b) shows the design matrix.]]

[[Image:doe7.4.png|center|290px|Geometric representation of the <math>2^3\,\!</math> design.]]

The <math>{2}^{3}\,\!</math> design can also be represented geometrically using a cube with the eight treatment combinations lying at the eight corners as shown in the figure above.

==Analysis of 2''k'' Designs==

The <math>{2}^{k}\,\!</math> designs are a special category of the factorial experiments where all the factors are at two levels. The fact that these designs contain factors at only two levels and are orthogonal greatly simplifies their analysis even when the number of factors is large. The use of <math>{2}^{k}\,\!</math> designs in investigating a large number of factors calls for a revision of the notation used previously for the ANOVA models. The case for revised notation is made stronger by the fact that the ANOVA and multiple linear regression models are identical for <math>{2}^{k}\,\!</math> designs because all factors are only at two levels. Therefore, the notation of the regression models is applied to the ANOVA models for these designs, as explained next.

===Notation===

Based on the notation used in [[General Full Factorial Designs]], the ANOVA model for a two level factorial experiment with three factors would be as follows:

::<math>\begin{align}
& Y= & \mu +{{\tau }_{1}}\cdot {{x}_{1}}+{{\delta }_{1}}\cdot {{x}_{2}}+{{(\tau \delta )}_{11}}\cdot {{x}_{1}}{{x}_{2}}+{{\gamma }_{1}}\cdot {{x}_{3}} \\
& & +{{(\tau \gamma )}_{11}}\cdot {{x}_{1}}{{x}_{3}}+{{(\delta \gamma )}_{11}}\cdot {{x}_{2}}{{x}_{3}}+{{(\tau \delta \gamma )}_{111}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where:
 
:• <math>\mu \,\!</math> represents the overall mean
:• <math>{{\tau }_{1}}\,\!</math> represents the independent effect of the first factor (factor <math>A\,\!</math>) out of the two effects <math>{{\tau }_{1}}\,\!</math> and <math>{{\tau }_{2}}\,\!</math>
:• <math>{{\delta }_{1}}\,\!</math> represents the independent effect of the second factor (factor <math>B\,\!</math>) out of the two effects <math>{{\delta }_{1}}\,\!</math> and <math>{{\delta }_{2}}\,\!</math>
:• <math>{{(\tau \delta )}_{11}}\,\!</math> represents the independent effect of the interaction <math>AB\,\!</math> out of the other interaction effects
:• <math>{{\gamma }_{1}}\,\!</math> represents the effect of the third factor (factor <math>C\,\!</math>) out of the two effects <math>{{\gamma }_{1}}\,\!</math> and <math>{{\gamma }_{2}}\,\!</math>
:• <math>{{(\tau \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>AC\,\!</math> out of the other interaction effects
:• <math>{{(\delta \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>BC\,\!</math> out of the other interaction effects
:• <math>{{(\tau \delta \gamma )}_{111}}\,\!</math> represents the effect of the interaction <math>ABC\,\!</math> out of the other interaction effects
and <math>\epsilon \,\!</math> is the random error term.

 
The notation for a linear regression model having three predictor variables with interactions is:

::<math>\begin{align}
& Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& & +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

The notation for the regression model is much more convenient, especially for the case when a large number of higher order interactions are present. In two level experiments, the ANOVA model requires only one indicator variable to represent each factor for both qualitative and quantitative factors. Therefore, the notation for the multiple linear regression model can be applied to the ANOVA model of the experiment that has all the factors at two levels. For example, for the experiment of the ANOVA model given above, <math>{{\beta }_{0}}\,\!</math> can represent the overall mean instead of <math>\mu \,\!</math>, and <math>{{\beta }_{1}}\,\!</math> can represent the independent effect, <math>{{\tau }_{1}}\,\!</math>, of factor <math>A\,\!</math>. Other main effects can be represented in a similar manner. The notation for the interaction effects is much more simplified (e.g., <math>{{\beta }_{123}}\,\!</math> can be used to represent the three factor interaction effect, <math>{{(\tau \beta \gamma )}_{111}}\,\!</math>).

As mentioned earlier, it is important to note that the coding for the indicator variables for the ANOVA models of two level factorial experiments is reversed from the coding followed in [[General Full Factorial Designs]]. Here <math>-1\,\!</math> represents the first level of the factor while <math>1\,\!</math> represents the second level. This is because for a two level factor a single variable is needed to represent the factor for both qualitative and quantitative factors. For quantitative factors, using <math>-1\,\!</math> for the first level (which is the low level) and 1 for the second level (which is the high level) keeps the coding consistent with the numerical value of the factors. The change in coding between the two coding schemes does not affect the analysis except that signs of the estimated effect coefficients will be reversed (i.e., numerical values of <math>{{\hat{\tau }}_{1}}\,\!</math>, obtained based on the coding of [[General Full Factorial Designs]], and <math>{{\hat{\beta }}_{1}}\,\!</math>, obtained based on the new coding, will be the same but their signs would be opposite).

::<math>\begin{align}
& & \text{Factor }A\text{ Coding (two level factor)} \\
& &
\end{align}\,\!</math>

::<math>\begin{matrix}
\text{Previous Coding} & {} & {} & {} & \text{Coding for }{{\text{2}}^{k}}\text{ Designs} \\
{} & {} & {} & {} & {} \\
Effect\text{ }{{\tau }_{1}}\ \ :\ \ {{x}_{1}}=1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{1}}\text{ (or }-{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=-1\text{ } \\
Effect\text{ }{{\tau }_{2}}\ \ :\ \ {{x}_{1}}=-1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{2}}\text{ (or }{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=1\text{ } \\
\end{matrix}\,\!</math>

In summary, the ANOVA model for the experiments with all factors at two levels is different from the ANOVA models for other experiments in terms of the notation in the following two ways:
 

:• The notation of the regression models is used for the effect coefficients.
:• The coding of the indicator variables is reversed.

===Special Features===

Consider the design matrix, <math>X\,\!</math>, for the <math>{2}^{3}\,\!</math> design discussed above. The (<math>{{X}^{\prime }}X\,\!</math>) <math>^{-1}\,\!</math> matrix is:

<center><math>{{({{X}^{\prime }}X)}^{-1}}=\left[ \begin{matrix}
0.125 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0.125 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0.125 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0.125 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0.125 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0.125 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0.125 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.125 \\
\end{matrix} \right]\,\!</math></center>

Notice that, due to the orthogonal design of the <math>X\,\!</math> matrix, the <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> has been simplified to a diagonal matrix which can be written as:

::<math>\begin{align}
{{({{X}^{\prime }}X)}^{-1}}= & 0.125\cdot I = & \frac{1}{8}\cdot I = & \frac{1}{{{2}^{3}}}\cdot I
\end{align}\,\!</math>

where <math>I\,\!</math> represents the identity matrix of the same order as the design matrix, <math>X\,\!</math>. Since there are eight observations per replicate of the <math>{2}^{3}\,\!</math> design, the <math>(X\,\!</math> ' <math>X{{)}^{-1}}\,\!</math> matrix for <math>m\,\!</math> replicates of this design can be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{3}}\cdot m)}\cdot I\,\!</math>

The <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> matrix for any <math>{2}^{k}\,\!</math> design can now be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{k}}\cdot m)}\cdot I\,\!</math>

Then the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is:

::<math>\begin{align}
C= & {{{\hat{\sigma }}}^{2}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & M{{S}_{E}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & \frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}\cdot I
\end{align}\,\!</math>

Note that the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is also a diagonal matrix. Therefore, the estimated effect coefficients (<math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math>, <math>{{\beta }_{12}},\,\!</math> etc.) for these designs are uncorrelated. This implies that the terms in the <math>{2}^{k}\,\!</math> design (main effects, interactions) are independent of each other. Consequently, the extra sum of squares for each of the terms in these designs is independent of the sequence of terms in the model, and also independent of the presence of other terms in the model. As a result the sequential and partial sum of squares for the terms are identical for these designs and will always add up to the model sum of squares. Multicollinearity is also not an issue for these designs.

It can also be noted from the equation given above, that in addition to the <math>C\,\!</math> matrix being diagonal, all diagonal elements of the <math>C\,\!</math> matrix are identical. This means that the variance (or its square root, the standard error) of all estimated effect coefficients are the same. The standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>, for all the coefficients is:

::<math>\begin{align}
se({{{\hat{\beta }}}_{j}})= & \sqrt{{{C}_{jj}}} = & \sqrt{\frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}}\text{ }for\text{ }all\text{ }j
\end{align}\,\!</math>

This property is used to construct the normal probability plot of effects in <math>{2}^{k}\,\!</math> designs and identify significant effects using graphical techniques. For details on the normal probability plot of effects in a Weibull++ DOE folio, refer to [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]].

====Example====
To illustrate the analysis of a full factorial <math>{2}^{k}\,\!</math> design, consider a three factor experiment to investigate the effect of honing pressure, number of strokes and cycle time on the surface finish of automobile brake drums. Each of these factors is investigated at two levels. The honing pressure is investigated at levels of 200 <math>psi\,\!</math> and 400 <math>psi\,\!</math>, the number of strokes used is 3 and 5 and the two levels of the cycle time are 3 and 5 seconds. The design for this experiment is set up in a Weibull++ DOE folio as shown in the first two following figures. It is decided to run two replicates for this experiment. The surface finish data collected from each run (using randomization) and the complete design is shown in the third following figure. The analysis of the experiment data is explained next.

[[Image:doe7_5.png|center|877px|Design properties for the experiment in the example.|link=]]

[[Image:doe7_6.png|center|859px|Design summary for the experiment in the example.|link=]]

[[Image:doe7_7.png|center|778px|Experiment design for the example to investigate the surface finish of automobile brake drums.|link=]]

The applicable model using the notation for <math>{2}^{k}\,\!</math> designs is:

::<math>\begin{align}
Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where the indicator variable, <math>{{x}_{1,}}\,\!</math> represents factor <math>A\,\!</math> (honing pressure), <math>{{x}_{1}}=-1\,\!</math> represents the low level of 200 <math>psi\,\!</math> and <math>{{x}_{1}}=1\,\!</math> represents the high level of 400 <math>psi\,\!</math>. Similarly, <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> represent factors <math>B\,\!</math> (number of strokes) and <math>C\,\!</math> (cycle time), respectively. <math>{{\beta }_{0}}\,\!</math> is the overall mean, while <math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math> and <math>{{\beta }_{3}}\,\!</math> are the effect coefficients for the main effects of factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. <math>{{\beta }_{12}}\,\!</math>, <math>{{\beta }_{13}}\,\!</math> and <math>{{\beta }_{23}}\,\!</math> are the effect coefficients for the <math>AB\,\!</math>, <math>AC\,\!</math> and <math>BC\,\!</math> interactions, while <math>{{\beta }_{123}}\,\!</math> represents the <math>ABC\,\!</math> interaction.

 
If the subscripts for the run (<math>i\,\!</math> ; <math>i=\,\!</math> 1 to 8) and replicates (<math>j\,\!</math> ; <math>j=\,\!</math> 1,2) are included, then the model can be written as:

::<math>\begin{align}
{{Y}_{ij}}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{ij1}}+{{\beta }_{2}}\cdot {{x}_{ij2}}+{{\beta }_{12}}\cdot {{x}_{ij1}}{{x}_{ij2}}+{{\beta }_{3}}\cdot {{x}_{ij3}} \\
& +{{\beta }_{13}}\cdot {{x}_{ij1}}{{x}_{ij3}}+{{\beta }_{23}}\cdot {{x}_{ij2}}{{x}_{ij3}}+{{\beta }_{123}}\cdot {{x}_{ij1}}{{x}_{ij2}}{{x}_{ij3}}+{{\epsilon }_{ij}}
\end{align}\,\!</math>

To investigate how the given factors affect the response, the following hypothesis tests need to be carried:

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{1}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{1}}\ne 0\,\!</math>

This test investigates the main effect of factor <math>A\,\!</math> (honing pressure). The statistic for this test is:

::<math>{{({{F}_{0}})}_{A}}=\frac{M{{S}_{A}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{A}}\,\!</math> is the mean square for factor <math>A\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other main effects, <math>B\,\!</math> and <math>C\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{12}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{12}}\ne 0\,\!</math>

This test investigates the two factor interaction <math>AB\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{AB}}=\frac{M{{S}_{AB}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the interaction <math>AB\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other two factor interactions, <math>AC\,\!</math> and <math>BC\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{123}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{123}}\ne 0\,\!</math>

This test investigates the three factor interaction <math>ABC\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{ABC}}=\frac{M{{S}_{ABC}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{ABC}}\,\!</math> is the mean square for the interaction <math>ABC\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square.
To calculate the test statistics, it is convenient to express the ANOVA model in the form <math>y=X\beta +\epsilon \,\!</math>.

====Expression of the ANOVA Model as <math>y=X\beta +\epsilon \,\!</math>====

In matrix notation, the ANOVA model can be expressed as:

::<math>y=X\beta +\epsilon \,\!</math>

where:

<center><math>y=\left[ \begin{matrix}
{{Y}_{11}} \\
{{Y}_{21}} \\
. \\
{{Y}_{81}} \\
{{Y}_{12}} \\
. \\
{{Y}_{82}} \\
\end{matrix} \right]=\left[ \begin{matrix}
90 \\
90 \\
. \\
90 \\
86 \\
. \\
80 \\
\end{matrix} \right]\text{ }X=\left[ \begin{matrix}
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\end{matrix} \right]\,\!</math></center>

<center><math>\beta =\left[ \begin{matrix}
{{\beta }_{0}} \\
{{\beta }_{1}} \\
{{\beta }_{2}} \\
{{\beta }_{12}} \\
{{\beta }_{3}} \\
{{\beta }_{13}} \\
{{\beta }_{23}} \\
{{\beta }_{123}} \\
\end{matrix} \right]\text{ }\epsilon =\left[ \begin{matrix}
{{\epsilon }_{11}} \\
{{\epsilon }_{21}} \\
. \\
{{\epsilon }_{81}} \\
{{\epsilon }_{12}} \\
. \\
. \\
{{\epsilon }_{82}} \\
\end{matrix} \right]\,\!</math></center>

====Calculation of the Extra Sum of Squares for the Factors====

Knowing the matrices <math>y\,\!</math>, <math>X\,\!</math> and <math>\beta \,\!</math>, the extra sum of squares for the factors can be calculated. These are used to calculate the mean squares that are used to obtain the test statistics. Since the experiment design is orthogonal, the partial and sequential extra sum of squares are identical. The extra sum of squares for each effect can be calculated as shown next. As an example, the extra sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & Model\text{ }Sum\text{ }of\text{ }Squares - Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }main\text{ }effect\text{ }of\text{ }A \\
= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }A}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }A}}={{X}_{\tilde{\ }A}}{{(X_{\tilde{\ }A}^{\prime }{{X}_{\tilde{\ }A}})}^{-1}}X_{\tilde{\ }A}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }A}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the main effect of factor <math>A\,\!</math>. Thus, the sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y \\
= & 654.4375-549.375 \\
= & 105.0625
\end{align}\,\!</math>

Similarly, the extra sum of squares for the interaction effect <math>AB\,\!</math> is:

::<math>\begin{align}
S{{S}_{AB}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }AB}}-(1/16)J]y \\
= & 654.4375-636.375 \\
= & 18.0625
\end{align}\,\!</math>

The extra sum of squares for other effects can be obtained in a similar manner.

====Calculation of the Test Statistics====

Knowing the extra sum of squares, the test statistic for the effects can be calculated. For example, the test statistic for the interaction <math>AB\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{AB}}= & \frac{M{{S}_{AB}}}{M{{S}_{E}}} \\
= & \frac{S{{S}_{AB}}/dof(S{{S}_{AB}})}{S{{S}_{E}}/dof(S{{S}_{E}})} \\
= & \frac{18.0625/1}{147.5/8} \\
= & 0.9797
\end{align}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the <math>AB\,\!</math> interaction and <math>M{{S}_{E}}\,\!</math> is the error mean square. The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{AB}}=0.9797\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{AB}}) \\
= & 1-0.6487 \\
= & 0.3513
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that the interaction between honing pressure and number of strokes does not affect the surface finish of the brake drums. Tests for other effects can be carried out in a similar manner. The results are shown in the ANOVA Table in the following figure. The values S, R-sq and R-sq(adj) in the figure indicate how well the model fits the data. The value of S represents the standard error of the model, R-sq represents the coefficient of multiple determination and R-sq(adj) represents the adjusted coefficient of multiple determination. For details on these values refer to [[Multiple_Linear_Regression_Analysis|Multiple Linear Regression Analysis]].

[[Image:doe7_8.png|center|799px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

====Calculation of Effect Coefficients====

The estimate of effect coefficients can also be obtained:

<center><math>\begin{align}
\hat{\beta }= & {{({{X}^{\prime }}X)}^{-1}}{{X}^{\prime }}y \\
= & \left[ \begin{matrix}
86.4375 \\
2.5625 \\
-4.9375 \\
1.0625 \\
-1.0625 \\
2.4375 \\
-1.3125 \\
-0.1875 \\
\end{matrix} \right]
\end{align}\,\!</math></center>

[[Image:doe7_9.png|center|800px|Regression Information table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

The coefficients and related results are shown in the Regression Information table above. In the table, the Effect column displays the effects, which are simply twice the coefficients. The Standard Error column displays the standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>. The Low CI and High CI columns display the confidence interval on the coefficients. The interval shown is the 90% interval as the significance is chosen as 0.1. The T Value column displays the <math>t\,\!</math> statistic, <math>{{t}_{0}}\,\!</math>, corresponding to the coefficients. The P Value column displays the <math>p\,\!</math> value corresponding to the <math>t\,\!</math> statistic. (For details on how these results are calculated, refer to [[General Full Factorial Designs]]). Plots of residuals can also be obtained from the DOE folio to ensure that the assumptions related to the ANOVA model are not violated.

====Model Equation====

From the analysis results in the above figure within [[Two_Level_Factorial_Experiments#Calculation_of_Effect_Coefficients|calculation of effect coefficients]] section, it is seen that effects <math>A\,\!</math>, <math>B\,\!</math> and <math>AC\,\!</math> are significant. In a DOE folio, the <math>p\,\!</math> values for the significant effects are displayed in red in the ANOVA Table for easy identification. Using the values of the estimated effect coefficients, the model for the present <math>{2}^{3}\,\!</math> design in terms of the coded values can be written as:

::<math>\begin{align}
\hat{y}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}} \\
= & 86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+2.4375{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

To make the model hierarchical, the main effect, <math>C\,\!</math>, needs to be included in the model (because the interaction <math>AC\,\!</math> is included in the model). The resulting model is:

::<math>\hat{y}=86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+1.0625{{x}_{3}}+2.4375{{x}_{1}}{{x}_{3}}\,\!</math>

This equation can be viewed in a DOE folio, as shown in the following figure, using the Show Analysis Summary icon in the Control Panel. The equation shown in the figure will match the hierarchical model once the required terms are selected using the Select Effects icon.

[[Image:doe7_10.png|center|557px|The model equation for the experiment of the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

==Replicated and Repeated Runs==

In the case of replicated experiments, it is important to note the difference between replicated runs and repeated runs. Both repeated and replicated runs are multiple response readings taken at the same factor levels. However, repeated runs are response observations taken at the same time or in succession. Replicated runs are response observations recorded in a random order. Therefore, replicated runs include more variation than repeated runs. For example, a baker, who wants to investigate the effect of two factors on the quality of cakes, will have to bake four cakes to complete one replicate of a <math>{2}^{2}\,\!</math> design. Assume that the baker bakes eight cakes in all. If, for each of the four treatments of the <math>{2}^{2}\,\!</math> design, the baker selects one treatment at random and then bakes two cakes for this treatment at the same time then this is a case of two repeated runs. If, however, the baker bakes all the eight cakes randomly, then the eight cakes represent two sets of replicated runs.
For repeated measurements, the average values of the response for each treatment should be entered into a DOE folio as shown in the following figure (a) when the two cakes for a particular treatment are baked together. For replicated measurements, when all the cakes are baked randomly, the data is entered as shown in the following figure (b).

[[Image:doe7.11.png|center|300px|Data entry for repeated and replicated runs. Figure (a) shows repeated runs and (b) shows replicated runs.]]

==Unreplicated 2''k'' Designs==

If a factorial experiment is run only for a single replicate then it is not possible to test hypotheses about the main effects and interactions as the error sum of squares cannot be obtained. This is because the number of observations in a single replicate equals the number of terms in the ANOVA model. Hence the model fits the data perfectly and no degrees of freedom are available to obtain the error sum of squares.

However, sometimes it is only possible to run a single replicate of the <math>{2}^{k}\,\!</math> design because of constraints on resources and time. In the absence of the error sum of squares, hypothesis tests to identify significant factors cannot be conducted. A number of methods of analyzing information obtained from unreplicated <math>{2}^{k}\,\!</math> designs are available. These include pooling higher order interactions, using the normal probability plot of effects or including center point replicates in the design.

===Pooling Higher Order Interactions===

One of the ways to deal with unreplicated <math>{2}^{k}\,\!</math> designs is to use the sum of squares of some of the higher order interactions as the error sum of squares provided these higher order interactions can be assumed to be insignificant. By dropping some of the higher order interactions from the model, the degrees of freedom corresponding to these interactions can be used to estimate the error mean square. Once the error mean square is known, the test statistics to conduct hypothesis tests on the factors can be calculated.

===Normal Probability Plot of Effects===
Another way to use unreplicated <math>{2}^{k}\,\!</math> designs to identify significant effects is to construct the normal probability plot of the effects. As mentioned in [[Two_Level_Factorial_Experiments#Special_Features| Special Features]], the standard error for all effect coefficients in the <math>{2}^{k}\,\!</math> designs is the same. Therefore, on a normal probability plot of effect coefficients, all non-significant effect coefficients (with <math>\beta =0\,\!</math>) will fall along the straight line representative of the normal distribution, N(<math>0,{{\sigma }^{2}}/({{2}^{k}}\cdot m)\,\!</math>). Effect coefficients that show large deviations from this line will be significant since they do not come from this normal distribution. Similarly, since effects <math>=2\times \,\!</math> effect coefficients, all non-significant effects will also follow a straight line on the normal probability plot of effects. For replicated designs, the Effects Probability plot of a DOE folio plots the normalized effect values (or the T Values) on the standard normal probability line, N(0,1). However, in the case of unreplicated <math>{2}^{k}\,\!</math> designs, <math>{{\sigma }^{2}}\,\!</math> remains unknown since <math>M{{S}_{E}}\,\!</math> cannot be obtained. Lenth's method is used in this case to estimate the variance of the effects. For details on Lenth's method, please refer to [[DOE References| Montgomery (2001)]]. The DOE folio then uses this variance value to plot effects along the N(0, Lenth's effect variance) line. The
method is illustrated in the following example.

====Example====

Vinyl panels, used as instrument panels in a certain automobile, are seen to develop defects after a certain amount of time. To investigate the issue, it is decided to carry out a two level factorial experiment. Potential factors to be investigated in the experiment are vacuum rate (factor <math>A\,\!</math>), material temperature (factor <math>B\,\!</math>), element intensity (factor <math>C\,\!</math>) and pre-stretch (factor <math>D\,\!</math>). The two levels of the factors used in the experiment are as shown in below.

[[Image:doet7.1.png|center|300px|Factors to investigate defects in vinyl panels.]]

With a <math>{2}^{4}\,\!</math> design requiring 16 runs per replicate it is only feasible for the manufacturer to run a single replicate.

The experiment design and data, collected as percent defects, are shown in the following figure. Since the present experiment design contains only a single replicate, it is not possible to obtain an estimate of the error sum of squares, <math>S{{S}_{E}}\,\!</math>. It is decided to use the normal probability plot of effects to identify the significant effects. The effect values for each term are obtained as shown in the following figure.

[[Image:doe7_13.png|center|650px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

Lenth's method uses these values to estimate the variance. As described in [[DOE_References|[Lenth, 1989]]], if all effects are arranged in ascending order, using their absolute values, then <math>{{s}_{0}}\,\!</math> is defined as 1.5 times the median value:

::<math>\begin{align}
{{s}_{0}}= & 1.5\cdot median(\left| effect \right|) \\
= & 1.5\cdot 2 \\
= & 3
\end{align}\,\!</math>

Using <math>{{s}_{0}}\,\!</math>, the "pseudo standard error" (<math>PSE\,\!</math>) is calculated as 1.5 times the median value of all effects that are less than 2.5 <math>{{s}_{0}}\,\!</math> :

::<math>\begin{align}
PSE= & 1.5\cdot median(\left| effect \right|\ \ :\ \ \left| effect \right|<2.5{{s}_{0}}) \\
= & 1.5\cdot 1.5 \\
= & 2.25
\end{align}\,\!</math>

Using <math>PSE\,\!</math> as an estimate of the effect variance, the effect variance is 2.25. Knowing the effect variance, the normal probability plot of effects for the present unreplicated experiment can be constructed as shown in the following figure. The line on this plot is the line N(0, 2.25). The plot shows that the effects <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math> do not follow the distribution represented by this line. Therefore, these effects are significant.

The significant effects can also be identified by comparing individual effect values to the margin of error or the threshold value using the pareto chart (see the third following figure). If the required significance is 0.1, then:

::<math>margin\text{ }of\text{ }error={{t}_{\alpha /2,d}}\cdot PSE\,\!</math>

The <math>t\,\!</math> statistic, <math>{{t}_{\alpha /2,d}}\,\!</math>, is calculated at a significance of <math>\alpha /2\,\!</math> (for the two-sided hypothesis) and degrees of freedom <math>d=(\,\!</math> number of effects <math>)/3\,\!</math>. Thus:

::<math>\begin{align}
margin\text{ }of\text{ }error= & {{t}_{0.05,5}}\cdot PSE \\
= & 2.015\cdot 2.25 \\
= & 4.534
\end{align}\,\!</math>

The value of 4.534 is shown as the critical value line in the third following figure. All effects with absolute values greater than the margin of error can be considered to be significant. These effects are <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math>. Therefore, the vacuum rate, the pre-stretch and their interaction have a significant effect on the defects of the vinyl panels.

[[Image:doe7_14.png|center|800px|Effect values for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_15.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_16.png|center|650px|Pareto chart for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

===Center Point Replicates===

Another method of dealing with unreplicated <math>{2}^{k}\,\!</math> designs that only have quantitative factors is to use replicated runs at the center point. The center point is the response corresponding to the treatment exactly midway between the two levels of all factors. Running multiple replicates at this point provides an estimate of pure error. Although running multiple replicates at any treatment level can provide an estimate of pure error, the other advantage of running center point replicates in the <math>{2}^{k}\,\!</math> design is in checking for the presence of curvature. The test for curvature investigates whether the model between the response and the factors is linear and is discussed in [[Two_Level_Factorial_Experiments#Using_Center_Point_Replicates_to_Test_Curvature| Center Pt. Replicates to Test Curvature]].

====Example: Use Center Point to Get Pure Error====

Consider a <math>{2}^{2}\,\!</math> experiment design to investigate the effect of two factors, <math>A\,\!</math> and <math>B\,\!</math>, on a certain response. The energy consumed when the treatments of the <math>{2}^{2}\,\!</math> design are run is considerably larger than the energy consumed for the center point run (because at the center point the factors are at their middle levels). Therefore, the analyst decides to run only a single replicate of the design and augment the design by five replicated runs at the center point as shown in the following figure. The design properties for this experiment are shown in the second following figure. The complete experiment design is shown in the third following figure. The center points can be used in the identification of significant effects as shown next.

[[Image:doe7.17.png||center|300px|<math>2^2\,\!</math> design augmented by five center point runs.]]
[[Image:doe7_18.png|center|537px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

[[Image:doe7_19.png|center|630px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

Since the present <math>{2}^{2}\,\!</math> design is unreplicated, there are no degrees of freedom available to calculate the error sum of squares. By augmenting this design with five center points, the response values at the center points, <math>y_{i}^{c}\,\!</math>, can be used to obtain an estimate of pure error, <math>S{{S}_{PE}}\,\!</math>. Let <math>{{\bar{y}}^{c}}\,\!</math> represent the average response for the five replicates at the center. Then:

::<math>S{{S}_{PE}}=Sum\text{ }of\text{ }Squares\text{ }for\text{ }center\text{ }points\,\!</math>

::<math>\begin{align}
S{{S}_{PE}}= & \underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}} \\
= & {{(25.2-25.26)}^{2}}+...+{{(25.3-25.26)}^{2}} \\
= & 0.052
\end{align}\,\!</math>

Then the corresponding mean square is:

::<math>\begin{align}
M{{S}_{PE}}= & \frac{S{{S}_{PE}}}{degrees\text{ }of\text{ }freedom} \\
= & \frac{0.052}{5-1} \\
= & 0.013
\end{align}\,\!</math>

Alternatively, <math>M{{S}_{PE}}\,\!</math> can be directly obtained by calculating the variance of the response values at the center points:

::<math>\begin{align}
M{{S}_{PE}}= & {{s}^{2}} \\
= & \frac{\underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}}}{5-1}
\end{align}\,\!</math>

Once <math>M{{S}_{PE}}\,\!</math> is known, it can be used as the error mean square, <math>M{{S}_{E}}\,\!</math>, to carry out the test of significance for each effect. For example, to test the significance of the main effect of factor <math>A,\,\!</math> the sum of squares corresponding to this effect is obtained in the usual manner by considering only the four runs of the original <math>{2}^{2}\,\!</math> design.

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/4)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/4)J]y \\
= & 0.5625
\end{align}\,\!</math>

Then, the test statistic to test the significance of the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{A}}= & \frac{M{{S}_{A}}}{M{{S}_{E}}} \\
= & \frac{0.5625/1}{0.052/4} \\
= & 43.2692
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{A}}=43.2692\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{A}}) \\
= & 1-0.9972 \\
= & 0.0028
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value < 0.1, it can be concluded that the main effect of factor <math>A\,\!</math> significantly affects the response. This result is displayed in the ANOVA table as shown in the following figure. Test for the significance of other factors can be carried out in a similar manner.

[[Image:doe7_20.png|center|649px|Results for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

===Using Center Point Replicates to Test Curvature===

Center point replicates can also be used to check for curvature in replicated or unreplicated <math>{2}^{k}\,\!</math> designs. The test for curvature investigates whether the model between the response and the factors is linear. The way DOE++ handles center point replicates is similar to its handling of blocks. The center point replicates are treated as an additional factor in the model. The factor is labeled as Curvature in the results of the DOE folio. If Curvature turns out to be a significant factor in the results, then this indicates the presence of curvature in the model.

====Example: Use Center Point to Test Curvature====

To illustrate the use of center point replicates in testing for curvature, consider again the data of the single replicate <math>{2}^{2}\,\!</math> experiment from a preceding figure(labeled "<math>2^2</math> design augmented by five center point runs"). Let <math>{{x}_{1}}\,\!</math> be the indicator variable to indicate if the run is a center point:

::<math>\begin{matrix}
{{x}_{1}}=0 & {} & \text{Center point run} \\
{{x}_{1}}=1 & {} & \text{Other run} \\
\end{matrix}\,\!</math>

If <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> are the indicator variables representing factors <math>A\,\!</math> and <math>B\,\!</math>, respectively, then the model for this experiment is:

::<math>Y={{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}\,\!</math>

To investigate the presence of curvature, the following hypotheses need to be tested:

::<math>\begin{align}
& {{H}_{0}}: & {{\beta }_{1}}=0\text{ (Curvature is absent)} \\
& {{H}_{1}}: & {{\beta }_{1}}\ne 0
\end{align}\,\!</math>

The test statistic to be used for this test is:

::<math>{{({{F}_{0}})}_{curvature}}=\frac{M{{S}_{curvature}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{curvature}}\,\!</math> is the mean square for Curvature and <math>M{{S}_{E}}\,\!</math> is the error mean square.

'''Calculation of the Sum of Squares'''

The <math>X\,\!</math> matrix and <math>y\,\!</math> vector for this experiment are:

<center><math>X=\left[ \begin{matrix}
1 & 1 & -1 & -1 & 1 \\
1 & 1 & 1 & -1 & -1 \\
1 & 1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
\end{matrix} \right]\text{ }y=\left[ \begin{matrix}
24.6 \\
25.4 \\
25.0 \\
25.7 \\
25.2 \\
25.3 \\
25.4 \\
25.1 \\
25.3 \\
\end{matrix} \right]\,\!</math></center>

The sum of squares can now be calculated. For example, the error sum of squares is:

::<math>\begin{align}
& S{{S}_{E}}= & {{y}^{\prime }}[I-H]y \\
& = & 0.052
\end{align}\,\!</math>

where <math>I\,\!</math> is the identity matrix and <math>H\,\!</math> is the hat matrix. It can be seen that this is equal to <math>S{{S}_{PE\text{ }}}\,\!</math> (the sum of squares due to pure error) because of the replicates at the center point, as obtained in the [[Two_Level_Factorial_Experiments#Example_3| example]]. The number of degrees of freedom associated with <math>S{{S}_{E}}\,\!</math>, <math>dof(S{{S}_{E}})\,\!</math> is four. The extra sum of squares corresponding to the center point replicates (or Curvature) is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & Model\text{ }Sum\text{ }of\text{ }Squares- \\
& & Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }center\text{ }point \\
& = & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Curvature}}-(1/9)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }Curvature}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }Curvature}}={{X}_{\tilde{\ }Curv}}{{(X_{\tilde{\ }Curv}^{\prime }{{X}_{\tilde{\ }Curv}})}^{-1}}X_{\tilde{\ }Curv}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }Curv}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the center point. Thus, the extra sum of squares corresponding to Curvature is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Center}}-(1/9)J]y \\
& = & 0.7036-0.6875 \\
& = & 0.0161
\end{align}\,\!</math>

This extra sum of squares can be used to test for the significance of curvature. The corresponding mean square is:

::<math>\begin{align}
& M{{S}_{Curvature}}= & \frac{Sum\text{ }of\text{ }squares\text{ }corresponding\text{ }to\text{ }Curvature}{degrees\text{ }of\text{ }freedom} \\
& = & \frac{0.0161}{1} \\
& = & 0.0161
\end{align}\,\!</math>

'''Calculation of the Test Statistic'''

Knowing the mean squares, the statistic to check the significance of curvature can be calculated.

::<math>\begin{align}
& {{({{f}_{0}})}_{Curvature}}= & \frac{M{{S}_{Curvature}}}{M{{S}_{E}}} \\
& = & \frac{0.0161/1}{0.052/4} \\
& = & 1.24
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{Curvature}}=1.24\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and four degrees of freedom in the denominator is:

::<math>\begin{align}
& p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{Curvature}}) \\
& = & 1-0.6713 \\
& = & 0.3287
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that curvature does not exist for this design. This results is shown in the ANOVA table in the figure above. The surface of the fitted model based on these results, along with the observed response values, is shown in the figure below.

[[Image:doe7.21.png|center|400px|Model surface and observed response values for the design in the [[Two_Level_Factorial_Experiments#Example_4| example]].]]

 

==Blocking in 2k Designs==

Blocking can be used in the <math>{2}^{k}\,\!</math> designs to deal with cases when replicates cannot be run under identical conditions. Randomized complete block designs that were discussed in [[Randomization and Blocking in DOE]] for factorial experiments are also applicable here. At times, even with just two levels per factor, it is not possible to run all treatment combinations for one replicate of the experiment under homogeneous conditions. For example, each replicate of the <math>{2}^{2}\,\!</math> design requires four runs. If each run requires two hours and testing facilities are available for only four hours per day, two days of testing would be required to run one complete replicate. Blocking can be used to separate the treatment runs on the two different days. Blocks that do not contain all treatments of a replicate are called incomplete blocks. In incomplete block designs, the block effect is confounded with certain effect(s) under investigation. For the <math>{2}^{2}\,\!</math> design assume that treatments <math>(1)\,\!</math> and <math>ab\,\!</math> were run on the first day and treatments <math>a\,\!</math> and <math>b\,\!</math> were run on the second day. Then, the incomplete block design for this experiment is:

::<math>\begin{matrix}
\text{Block 1} & {} & \text{Block 2} \\
\left[ \begin{matrix}
(1) \\
ab \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
b \\
\end{matrix} \right] \\
\end{matrix}\,\!</math>

For this design the block effect may be calculated as:

::<math>\begin{align}
& Block\text{ }Effect= & Average\text{ }response\text{ }for\text{ }Block\text{ }1- \\
& & Average\text{ }response\text{ }for\text{ }Block\text{ }2 \\
& = & \frac{(1)+ab}{2}-\frac{a+b}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The <math>AB\,\!</math> interaction effect is:

::<math>\begin{align}
& AB= & Average\text{ }response\text{ }at\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{low}}}- \\
& & Average\text{ }response\text{ }at\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{low}}} \\
& = & \frac{ab+(1)}{2}-\frac{b+a}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The two equations given above show that, in this design, the <math>AB\,\!</math> interaction effect cannot be distinguished from the block effect because the formulas to calculate these effects are the same. In other words, the <math>AB\,\!</math> interaction is said to be confounded with the block effect and it is not possible to say if the effect calculated based on these equations is due to the <math>AB\,\!</math> interaction effect, the block effect or both. In incomplete block designs some effects are always confounded with the blocks. Therefore, it is important to design these experiments in such a way that the important effects are not confounded with the blocks. In most cases, the experimenter can assume that higher order interactions are unimportant. In this case, it would better to use incomplete block designs that confound these effects with the blocks.
One way to design incomplete block designs is to use defining contrasts as shown next:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}+...+{{\alpha }_{k}}{{q}_{k}}\,\!</math>

where the <math>{{\alpha }_{i}}\,\!</math> s are the exponents for the factors in the effect that is to be confounded with the block effect and the <math>{{q}_{i}}\,\!</math> s are values based on the level of the <math>i\,\!</math> the factor (in a treatment that is to be allocated to a block). For <math>{2}^{k}\,\!</math> designs the <math>{{\alpha }_{i}}\,\!</math> s are either 0 or 1 and the <math>{{q}_{i}}\,\!</math> s have a value of 0 for the low level of the <math>i\,\!</math> th factor and a value of 1 for the high level of the factor in the treatment under consideration. As an example, consider the <math>{2}^{2}\,\!</math> design where the interaction effect <math>AB\,\!</math> is confounded with the block. Since there are two factors, <math>k=2\,\!</math>, with <math>i=1\,\!</math> representing factor <math>A\,\!</math> and <math>i=2\,\!</math> representing factor <math>B\,\!</math>. Therefore:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}\,\!</math>

The value of <math>{{\alpha }_{1}}\,\!</math> is one because the exponent of factor <math>A\,\!</math> in the confounded interaction <math>AB\,\!</math> is one. Similarly, the value of <math>{{\alpha }_{2}}\,\!</math> is one because the exponent of factor <math>B\,\!</math> in the confounded interaction <math>AB\,\!</math> is also one. Therefore, the defining contrast for this design can be written as:

::<math>\begin{align}
& L= & {{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}} \\
& = & 1\cdot {{q}_{1}}+1\cdot {{q}_{2}} \\
& = & {{q}_{1}}+{{q}_{2}}
\end{align}\,\!</math>

Once the defining contrast is known, it can be used to allocate treatments to the blocks. For the <math>{2}^{2}\,\!</math> design, there are four treatments <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math> and <math>ab\,\!</math>. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. In order to decide which block the treatment <math>(1)\,\!</math> belongs to, the levels of factors <math>A\,\!</math> and <math>B\,\!</math> for this run are used. Since factor <math>A\,\!</math> is at the low level in this treatment, <math>{{q}_{1}}=0\,\!</math>. Similarly, since factor <math>B\,\!</math> is also at the low level in this treatment, <math>{{q}_{2}}=0\,\!</math>. Therefore:

::<math>\begin{align}
& L= & {{q}_{1}}+{{q}_{2}} \\
& = & 0+0=0\text{ (mod 2)}
\end{align}\,\!</math>

Note that the value of <math>L\,\!</math> used to decide the block allocation is "mod 2" of the original value. This value is obtained by taking the value of 1 for odd numbers and 0 otherwise. Based on the value of <math>L\,\!</math>, treatment <math>(1)\,\!</math> is assigned to block 1. Other treatments can be assigned using the following calculations:

::<math>\begin{align}
& (1): & \text{ }L=0+0=0=0\text{ (mod 2)} \\
& a: & \text{ }L=1+0=1=1\text{ (mod 2)} \\
& b: & \text{ }L=0+1=1=1\text{ (mod 2)} \\
& ab: & \text{ }L=1+1=2=0\text{ (mod 2)}
\end{align}\,\!</math>

Therefore, to confound the interaction <math>AB\,\!</math> with the block effect in the <math>{2}^{2}\,\!</math> incomplete block design, treatments <math>(1)\,\!</math> and <math>ab\,\!</math> (with <math>L=0\,\!</math>) should be assigned to block 2 and treatment combinations <math>a\,\!</math> and <math>b\,\!</math> (with <math>L=1\,\!</math>) should be assigned to block 1.

====Example: Two Level Factorial Design with Two Blocks====

This example illustrates how treatments can be allocated to two blocks for an unreplicated <math>{2}^{k}\,\!</math> design. Consider the unreplicated <math>{2}^{4}\,\!</math> design to investigate the four factors affecting the defects in automobile vinyl panels discussed in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments required for this experiment were run by two different operators with each operator conducting 8 runs. This experiment is an example of an incomplete block design. The analyst in charge of this experiment assumed that the interaction <math>ABCD\,\!</math> was not significant and decided to allocate treatments to the two operators so that the <math>ABCD\,\!</math> interaction was confounded with the block effect (the two operators are the blocks). The allocation scheme to assign treatments to the two operators can be obtained as follows.
 
The defining contrast for the <math>{2}^{4}\,\!</math> design where the <math>ABCD\,\!</math> interaction is confounded with the blocks is:

::<math>L={{q}_{1}}+{{q}_{2}}+{{q}_{3}}+{{q}_{4}}\,\!</math>

The treatments can be allocated to the two operators using the values of the defining contrast. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. Then the value of the defining contrast for treatment <math>a\,\!</math> is:

::<math>a\ \ :\ \ \text{ }L=1+0+0+0=1=1\text{ (mod 2)}\,\!</math>

Therefore, treatment <math>a\,\!</math> should be assigned to Block 1 or the first operator. Similarly, for treatment <math>ab\,\!</math> we have:

::<math>ab\ \ :\ \ \text{ }L=1+1+0+0=2=0\text{ (mod 2)}\,\!</math>

[[Image:doe7.22.png|center|200px| Allocation of treatments to two blocks for the <math>2^4</math> design in the example by confounding interaction of <math>ABCD</math> with the blocks.]]

Therefore, <math>ab\,\!</math> should be assigned to Block 2 or the second operator. Other treatments can be allocated to the two operators in a similar manner to arrive at the allocation scheme shown in the figure below.
In DOE++, to confound the <math>ABCD\,\!</math> interaction for the <math>{2}^{4}\,\!</math> design into two blocks, the number of blocks are specified as shown in the figure below. Then the interaction <math>ABCD\,\!</math> is entered in the Block Generator window (second following figure) which is available using the Block Generator button in the following figure. The design generated by the Weibull++ DOE folio is shown in the third of the following figures. This design matches the allocation scheme of the preceding figure.

[[Image:doe7_23.png|center|700px| Adding block properties for the experiment in the example.|link=]]

[[Image:doe7_24.png|center|700px|Specifying the interaction ABCD as the interaction to be confounded with the blocks for the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

[[Image:doe7_25.png|center|800px|Two block design for the experiment in the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

For the analysis of this design, the sum of squares for all effects are calculated assuming no blocking. Then, to account for blocking, the sum of squares corresponding to the <math>ABCD\,\!</math> interaction is considered as the sum of squares due to blocks and <math>ABCD\,\!</math>. In DOE++ this is done by displaying this sum of squares as the sum of squares due to the blocks. This is shown in the following figure where the sum of squares in question is obtained as 72.25 and is displayed against Block. The interaction ABCD, which is confounded with the blocks, is not displayed. Since the design is unreplicated, any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_26.png|center|793px|ANOVA table for the experiment of the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

===Unreplicated 2''k'' Designs in 2''p'' Blocks===

A single replicate of the <math>{2}^{k}\,\!</math> design can be run in up to <math>{2}^{p}\,\!</math> blocks where <math>p<k\,\!</math>. The number of effects confounded with the blocks equals the degrees of freedom associated with the block effect.

If two blocks are used (the block effect has two levels), then one (<math>2-1=1)\,\!</math> effect is confounded with the blocks. If four blocks are used, then three (<math>4-1=3\,\!</math>) effects are confounded with the blocks and so on. For example an unreplicated <math>{2}^{4}\,\!</math> design may be confounded in <math>{2}^{2}\,\!</math> (four) blocks using two contrasts, <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}}\,\!</math>. Let <math>AC\,\!</math> and <math>BD\,\!</math> be the effects to be confounded with the blocks. Corresponding to these two effects, the contrasts are respectively:

::<math>\begin{align}
& {{L}_{1}}= & {{q}_{1}}+{{q}_{3}} \\
& {{L}_{2}}= & {{q}_{2}}+{{q}_{4}}
\end{align}\,\!</math>

Based on the values of <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}},\,\!</math> the treatments can be assigned to the four blocks as follows:

<center><math>\begin{matrix}
\text{Block 4} & {} & \text{Block 3} & {} & \text{Block 2} & {} & \text{Block 1} \\
{{L}_{1}}=0,{{L}_{2}}=0 & {} & {{L}_{1}}=1,{{L}_{2}}=0 & {} & {{L}_{1}}=0,{{L}_{2}}=1 & {} & {{L}_{1}}=1,{{L}_{2}}=1 \\
{} & {} & {} & {} & {} & {} & {} \\
\left[ \begin{matrix}
(1) \\
ac \\
bd \\
abcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
c \\
abd \\
bcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
b \\
abc \\
d \\
acd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
ab \\
bc \\
ad \\
cd \\
\end{matrix} \right] \\
\end{matrix}\,\!</math></center>

Since the block effect has three degrees of freedom, three effects are confounded with the block effect. In addition to <math>AC\,\!</math> and <math>BD\,\!</math>, the third effect confounded with the block effect is their generalized interaction, <math>(AC)(BD)=ABCD\,\!</math>.
In general, when an unreplicated <math>{2}^{k}\,\!</math> design is confounded in <math>{2}^{p}\,\!</math> blocks, <math>p\,\!</math> contrasts are needed (<math>{{L}_{1}},{{L}_{2}}...{{L}_{p}}\,\!</math>). <math>p\,\!</math> effects are selected to define these contrasts such that none of these effects are the generalized interaction of the others. The <math>{2}^{p}\,\!</math> blocks can then be assigned the treatments using the <math>p\,\!</math> contrasts. <math>{{2}^{p}}-(p+1)\,\!</math> effects, that are also confounded with the blocks, are then obtained as the generalized interaction of the <math>p\,\!</math> effects. In the statistical analysis of these designs, the sum of squares are computed as if no blocking were used. Then the block sum of squares is obtained by adding the sum of squares for all the effects confounded with the blocks.

====Example: 2 Level Factorial Design with Four Blocks====

This example illustrates how a DOE folio obtains the sum of squares when treatments for an unreplicated <math>{2}^{k}\,\!</math> design are allocated among four blocks. Consider again the unreplicated <math>{2}^{4}\,\!</math> design used to investigate the defects in automobile vinyl panels presented in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments needed to complete the experiment were run by four operators. Therefore, there are four blocks. Assume that the treatments were allocated to the blocks using the generators mentioned in the previous section, i.e., treatments were allocated among the four operators by confounding the effects, <math>AC\,\!</math> and <math>BD,\,\!</math> with the blocks. These effects can be specified as Block Generators as shown in the following figure. (The generalized interaction of these two effects, interaction <math>ABCD\,\!</math>, will also get confounded with the blocks.) The resulting design is shown in the second following figure and matches the allocation scheme obtained in the previous section.

[[Image:doe7_27.png|center|700px|Specifying the interactions AC and BD as block generators for the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

The sum of squares in this case can be obtained by calculating the sum of squares for each of the effects assuming there is no blocking. Once the individual sum of squares have been obtained, the block sum of squares can be calculated. The block sum of squares is the sum of the sum of squares of effects, <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, since these effects are confounded with the block effect. As shown in the second following figure, this sum of squares is 92.25 and is displayed against Block. The interactions <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, which are confounded with the blocks, are not displayed. Since the present design is unreplicated any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_28.png|center|800px|Design for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

[[Image:doe7_29.png|center|793px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

==Variability Analysis==

For replicated two level factorial experiments, DOE++ provides the option of conducting variability analysis (using the Variability Analysis icon under the Data menu). The analysis is used to identify the treatment that results in the least amount of variation in the product or process being investigated. Variability analysis is conducted by treating the standard deviation of the response for each treatment of the experiment as an additional response. The standard deviation for a treatment is obtained by using the replicated response values at that treatment run. As an example, consider the <math>{2}^{3}\,\!</math> design shown in the following figure where each run is replicated four times. A variability analysis can be conducted for this design. DOE++ calculates eight standard deviation values corresponding to each treatment of the design (see second following figure). Then, the design is analyzed as an unreplicated <math>{2}^{3}\,\!</math> design with the standard deviations (displayed as Y Standard Deviation. in second following figure) as the response. The normal probability plot of effects identifies <math>AC\,\!</math> as the effect that influences variability (see third figure following). Based on the effect coefficients obtained in the fourth figure following, the model for Y Std. is:

::<math>\begin{align}
& \text{Y Std}\text{.}= & 0.6779+0.2491\cdot AC \\
& = & 0.6779+0.2491{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

Based on the model, the experimenter has two choices to minimize variability (by minimizing Y Std.). The first choice is that <math>{{x}_{1}}\,\!</math> should be <math>1\,\!</math> (i.e., <math>A\,\!</math> should be set at the high level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the low level). The second choice is that <math>{{x}_{1}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>A\,\!</math> should be set at the low level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the high level). The experimenter can select the most feasible choice.

[[Image:doe7.30.png|center|391px|A <math>2^3\,\!</math> design with four replicated response values that can be used to conduct a variability analysis.]]
 

[[Image:doe7_31.png|center|727px|Variability analysis in DOE++.|link=]]
 

[[Image:doe7_32.png|center|650px|Normal probability plot of effects for the variability analysis example.|link=]]
 

[[Image:doe7_33.png|center|727px|Effect coefficients for the variability analysis example.|link=]]

 

==Two Level Fractional Factorial Designs==

As the number of factors in a two level factorial design increases, the number of runs for even a single replicate of the <math>{2}^{k}\,\!</math> design becomes very large. For example, a single replicate of an eight factor two level experiment would require 256 runs. Fractional factorial designs can be used in these cases to draw out valuable conclusions from fewer runs. The basis of fractional factorial designs is the ''sparsity of effects'' principle.[[DOE_References|[Wu, 2000]]] The principle states that, most of the time, responses are affected by a small number of main effects and lower order interactions, while higher order interactions are relatively unimportant. Fractional factorial designs are used as screening experiments during the initial stages of experimentation. At these stages, a large number of factors have to be investigated and the focus is on the main effects and two factor interactions. These designs obtain information about main effects and lower order interactions with fewer experiment runs by confounding these effects with unimportant higher order interactions. As an example, consider a <math>{2}^{8}\,\!</math> design that requires 256 runs. This design allows for the investigation of 8 main effects and 28 two factor interactions. However, 219 degrees of freedom are devoted to three factor or higher order interactions. This full factorial design can prove to be very inefficient when these higher order interactions can be assumed to be unimportant. Instead, a fractional design can be used here to identify the important factors that can then be investigated more thoroughly in subsequent experiments. In unreplicated fractional factorial designs, no degrees of freedom are available to calculate the error sum of squares and the techniques mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] should be employed for the analysis of these designs.

==Half-fraction Designs==

A half-fraction of the <math>{2}^{k}\,\!</math> design involves running only half of the treatments of the full factorial design. For example, consider a <math>{2}^{3}\,\!</math> design that requires eight runs in all. The design matrix for this design is shown in the figure (a) below. A half-fraction of this design is the design in which only four of the eight treatments are run. The fraction is denoted as <math>{2}^{3-1}\,\!</math> with the "<math>-1\,\!</math>" in the index denoting a half-fraction. Assume that the treatments chosen for the half-fraction design are the ones where the interaction <math>ABC\,\!</math> is at the high level (i.e., only those rows are chosen from the following figure (a) where the column for <math>ABC\,\!</math> has entries of 1). The resulting <math>{2}^{3-1}\,\!</math> design has a design matrix as shown in figure (b) below.

[[Image:doe7.34.png|center|330px|Half-fractions of the <math>2^3\,\!</math> design. (a) shows the full factorial <math>2^3\,\!</math> design, (b) shows the <math>2^{3-1}\,\!</math> design with the defining relation <math>I=ABC\,\!</math> and (c) shows the <math>{2}^{3-1}\,\!</math> design with the defining relation <math>I=-ABC\,\!</math>.]]

In the <math>{2}^{3-1}\,\!</math> design of figure (b), since the interaction <math>ABC\,\!</math> is always included at the same level (the high level represented by 1), it is not possible to measure this interaction effect. The effect, <math>ABC\,\!</math>, is called the ''generator'' or ''word'' for this design. It can be noted that, in the design matrix of the following figure (b), the column corresponding to the intercept, <math>I\,\!</math>, and column corresponding to the interaction <math>ABC\,\!</math>, are identical. The identical columns are written as <math>I=ABC\,\!</math> and this equation is called the ''defining relation'' for the design. In a DOE folio, the present <math>{2}^{3-1}\,\!</math> design can be obtained by specifying the design properties as shown in the following figure.

[[Image:doe7_35.png|center|700px|Design properties for the <math>2^{3-1}\,\!</math> design.|link=]]

The defining relation, <math>I=ABC\,\!</math>, is entered in the Fraction Generator window as shown next.

[[Image:doe7_36.png|center|700px|Specifying the defining relation for the <math>2^{3-1}\,\!</math> design.|link=]]

Note that in the figure following that, the defining relation is specified as <math>C=AB\,\!</math>. This relation is obtained by multiplying the defining relation, <math>I=ABC\,\!</math>, by the last factor, <math>C\,\!</math>, of the design.

===Calculation of Effects===

Using the four runs of the <math>{2}^{3-1}\,\!</math> design in figure (b) discussed above, the main effects can be calculated as follows:

::<math>\begin{align}
& A= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& B= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& C= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

where <math>a\,\!</math>, <math>b\,\!</math>, <math>c\,\!</math> and <math>abc\,\!</math> are the treatments included in the <math>{2}^{3-1}\,\!</math> design.

Similarly, the two factor interactions can also be obtained as:

::<math>\begin{align}
& BC= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& AC= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& AB= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

The equations for <math>A\,\!</math> and <math>BC\,\!</math> above result in the same effect values showing that effects <math>A\,\!</math> and <math>BC\,\!</math> are confounded in the present <math>{2}^{3-1}\,\!</math> design. Thus, the quantity, <math>\tfrac{1}{2}(a-b-c+abc),\,\!</math> estimates <math>A+BC\,\!</math> (i.e., both the main effect <math>A\,\!</math> and the two-factor interaction <math>BC\,\!</math>). The effects, <math>A\,\!</math> and <math>BC,\,\!</math> are called ''aliases''. From the remaining equations given above, it can be seen that the other aliases for this design are <math>B\,\!</math> and <math>AC\,\!</math>, and <math>C\,\!</math> and <math>AB\,\!</math>. Therefore, the equations to calculate the effects in the present <math>{2}^{3-1}\,\!</math> design can be written as follows:

::<math>\begin{align}
& A+BC= & \frac{1}{2}(a-b-c+abc) \\
& B+AC= & \frac{1}{2}(-a+b-c+abc) \\
& C+AB= & \frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

===Calculation of Aliases===

Aliases for a fractional factorial design can be obtained using the defining relation for the design. The defining relation for the present <math>{2}^{3-1}\,\!</math> design is:

::<math>I=ABC\,\!</math>

Multiplying both sides of the previous equation by the main effect, <math>A,\,\!</math> gives the alias effect of <math>A\,\!</math> :

::<math>\begin{align}
& A\cdot I= & A\cdot ABC \\
& A= & {{A}^{2}}BC \\
& A= & BC
\end{align}\,\!</math>

Note that in calculating the alias effects, any effect multiplied by <math>I\,\!</math> remains the same (<math>A\cdot I=A\,\!</math>), while an effect multiplied by itself results in <math>I\,\!</math> (<math>{{A}^{2}}=I\,\!</math>). Other aliases can also be obtained:

::<math>\begin{align}
& B\cdot I= & B\cdot ABC \\
& B= & A{{B}^{2}}C \\
& B= & AC
\end{align}\,\!</math>

:and:

::<math>\begin{align}
& C\cdot I= & C\cdot ABC \\
& C= & AB{{C}^{2}} \\
& C= & AB
\end{align}\,\!</math>

===Fold-over Design===

If it can be assumed for this design that the two-factor interactions are unimportant, then in the absence of <math>BC\,\!</math>, <math>AC\,\!</math> and <math>AB\,\!</math>, the equations for (A+BC), (B+AC) and (C+AB) can be used to estimate the main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. However, if such an assumption is not applicable, then to uncouple the main effects from their two factor aliases, the alternate fraction that contains runs having <math>ABC\,\!</math> at the lower level should be run. The design matrix for this design is shown in the preceding figure (c). The defining relation for this design is <math>I=-ABC\,\!</math> because the four runs for this design are obtained by selecting the rows of the preceding figure (a) for which the value of the <math>ABC\,\!</math> column is <math>-1\,\!</math>. The aliases for this fraction can be obtained as explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]] as <math>A=-BC\,\!</math>, <math>B=-AC\,\!</math> and <math>C=-AB\,\!</math>. The effects for this design can be calculated as:

::<math>\begin{align}
& A-BC= & \frac{1}{2}(ab+ac-(1)-bc) \\
& B-AC= & \frac{1}{2}(ab-ac+(1)-bc) \\
& C-AB= & \frac{1}{2}(-ab+ac-(1)+bc)
\end{align}\,\!</math>

These equations can be combined with the equations for (A+BC), (B+AC) and (C+AB) to obtain the de-aliased main effects and two factor interactions. For example, adding equations (A+BC) and (A-BC) returns the main effect <math>A\,\!</math>.

::<math>\begin{align}
& 2A= & \frac{1}{2}(a-b-c+abc)+ \\
& & \frac{1}{2}(ab+ac-(1)-bc)
\end{align}\,\!</math>

The process of augmenting a fractional factorial design by a second fraction of the same size by simply reversing the signs (of all effect columns except <math>I\,\!</math>) is called ''folding over''. The combined design is referred to as a ''fold-over design''.

==Quarter and Smaller Fraction Designs==

At times, the number of runs even for a half-fraction design are very large. In these cases, smaller fractions are used. A quarter-fraction design, denoted as <math>{2}^{k-2}\,\!</math>, consists of a fourth of the runs of the full factorial design. Quarter-fraction designs require two defining relations. The first defining relation returns the half-fraction or the <math>{2}^{k-1}\,\!</math> design. The second defining relation selects half of the runs of the <math>{2}^{k-1}\,\!</math> design to give the quarter-fraction. For example, consider the <math>{2}^{4}\,\!</math> design. To obtain a <math>{2}^{4-2}\,\!</math> design from this design, first a half-fraction of this design is obtained by using a defining relation. Assume that the defining relation used is <math>I=ABCD\,\!</math>. The design matrix for the resulting <math>{2}^{4-1}\,\!</math> design is shown in figure (a) below. Now, a quarter-fraction can be obtained from the <math>{2}^{4-1}\,\!</math> design shown in figure (a) below using a second defining relation <math>I=AD\,\!</math>. The resulting <math>{2}^{4-2}\,\!</math> design obtained is shown in figure (b) below.

[[Image:doe7.37.png|center|465px|Fractions of the <math>2^4\,\!</math> design - Figure (a) shows the <math>2^{4-1}</math> design with the defining relation <math>I=ABCD\,\!</math> and (b) shows the <math>2^{4-2}\,\!</math> design with the defining relation <math>I=ABCD=AD=BC\,\!</math>.]]

The complete defining relation for this <math>{2}^{4-2}\,\!</math> design is:

::<math>I=ABCD=AD=BC\,\!</math>

Note that the effect, <math>BC,\,\!</math> in the defining relation is the generalized interaction of <math>ABCD\,\!</math> and <math>AD\,\!</math> and is obtained using <math>(ABCD)(AD)={{A}^{2}}BC{{D}^{2}}=BC\,\!</math>. In general, a <math>{2}^{k-p}\,\!</math> fractional factorial design requires <math>p\,\!</math> independent generators. The defining relation for the design consists of the <math>p\,\!</math> independent generators and their <math>{2}^{p}\,\!</math> - (<math>p\,\!</math> +1) generalized interactions.

===Calculation of Aliases===

The alias structure for the present <math>{2}^{4-2}\,\!</math> design can be obtained using the defining relation of equation (I=ABCD=AD=BC) following the procedure explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]]. For example, multiplying the defining relation by <math>A\,\!</math> returns the effects aliased with the main effect, <math>A\,\!</math>, as follows:

::<math>\begin{align}
& A\cdot I= & A\cdot ABCD=A\cdot AD=A\cdot BC \\
& A= & {{A}^{2}}BCD={{A}^{2}}D=ABC \\
& A= & BCD=D=ABC
\end{align}\,\!</math>

Therefore, in the present <math>{2}^{4-2}\,\!</math> design, it is not possible to distinguish between effects <math>A\,\!</math>, <math>D\,\!</math>, <math>BCD\,\!</math> and <math>ABC\,\!</math>. Similarly, multiplying the defining relation by <math>B\,\!</math> and <math>AB\,\!</math> returns the effects that are aliased with these effects:

::<math>\begin{align}
& B= & ACD=ABD=C \\
& AB= & CD=AD=AC
\end{align}\,\!</math>

Other aliases can be obtained in a similar way. It can be seen that each effect in this design has three aliases. In general, each effect in a <math>{2}^{k-p}\,\!</math> design has <math>{2}^{p-1}\,\!</math> aliases.
The aliases for the <math>{2}^{4-2}\,\!</math> design show that in this design the main effects are aliased with each other (<math>A\,\!</math> is aliased with <math>D\,\!</math> and <math>B\,\!</math> is aliased with <math>C\,\!</math>). Therefore, this design is not a useful design and is not available in a Weibull++ DOE folio. It is important to ensure that main effects and lower order interactions of interest are not aliased in a fractional factorial design. This is known by looking at the resolution of the fractional factorial design.

==Design Resolution==

The resolution of a fractional factorial design is defined as the number of factors in the lowest order effect in the defining relation. For example, in the defining relation <math>I=ABCD=AD=BC\,\!</math> of the previous <math>{2}^{4-2}\,\!</math> design, the lowest-order effect is either <math>AD\,\!</math> or <math>BC,\,\!</math> containing two factors. Therefore, the resolution of this design is equal to two. The resolution of a fractional factorial design is represented using Roman numerals. For example, the previously mentioned <math>{2}^{4-2}\,\!</math> design with a resolution of two can be represented as 2 <math>_{\text{II}}^{4-2}\,\!</math>. The resolution provides information about the confounding in the design as explained next:
 
 
#'''Resolution III Designs''' In these designs, the lowest order effect in the defining relation has three factors (e.g., a <math>{2}^{5-2}\,\!</math> design with the defining relation <math>I=ABDE=ABC=CDE\,\!</math>). In resolution III designs, no main effects are aliased with any other main effects, but main effects are aliased with two factor interactions. In addition, some two factor interactions are aliased with each other.
#'''Resolution IV Designs''' In these designs, the lowest order effect in the defining relation has four factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABDE\,\!</math>). In resolution IV designs, no main effects are aliased with any other main effects or two factor interactions. However, some main effects are aliased with three factor interactions and the two factor interactions are aliased with each other.
#'''Resolution V Designs''' In these designs the lowest order effect in the defining relation has five factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABCDE\,\!</math>). In resolution V designs, no main effects or two factor interactions are aliased with any other main effects or two factor interactions. However, some main effects are aliased with four factor interactions and the two factor interactions are aliased with three factor interactions.

Fractional factorial designs with the highest resolution possible should be selected because the higher the resolution of the design, the less severe the degree of confounding. In general, designs with a resolution less than III are never used because in these designs some of the main effects are aliased with each other. The table below shows fractional factorial designs with the highest available resolution for three to ten factor designs along with their defining relations.

[[Image:doet7.3.png|center|474px|Highest resolution designs available for fractional factorial designs with 3 to 10 factors.]]

All of the two level fractional factorial designs available in DOE++ are shown next.

[[Image:doe7.38.png|center|471px|Two level fractional factorial designs available in DOE++ and their resolutions.]]

===Minimum Aberration Designs===
At times, different designs with the same resolution but different aliasing may be available. The best design to select in such a case is the minimum aberration design. For example, all <math>{2}^{7-2}\,\!</math> designs in the fourth table have a resolution of four (since the generator with the minimum number of factors in each design has four factors). Design <math>1\,\!</math> has three generators of length four (<math>ABCF,\,\!</math> <math>BCDG,\,\!</math> <math>ADFG\,\!</math>). Design <math>2\,\!</math> has two generators of length four (<math>ABCF,\,\!</math> <math>ADEG\,\!</math>). Design <math>3\,\!</math> has one generator of length four (<math>CEFG\,\!</math>). Therefore, design <math>3\,\!</math> has the least number of generators with the minimum length of four. Design <math>3\,\!</math> is called the minimum aberration design. It can be seen that the alias structure for design <math>3\,\!</math> is less involved compared to the other designs. For details refer to [[DOE_References|[Wu, 2000]]].

[[Image:doet7.4.png|center|432px|Three <math>2_{IV}^{7-2}\,\!</math> designs with different defining relations.]]

====Example====

The design of an automobile fuel cone is thought to be affected by six factors in the manufacturing process: cavity temperature (factor <math>A\,\!</math>), core temperature (factor <math>B\,\!</math>), melt temperature (factor <math>C\,\!</math>), hold pressure (factor <math>D\,\!</math>), injection speed (factor <math>E\,\!</math>) and cool time (factor <math>F\,\!</math>). The manufacturer of the fuel cone is unable to run the <math>{2}^{6}=64\,\!</math> runs required to complete one replicate for a two level full factorial experiment with six factors. Instead, they decide to run a fractional factorial design. Considering that three factor and higher order interactions are likely to be inactive, the manufacturer selects a <math>{2}^{6-2}\,\!</math> design that will require only 16 runs. The manufacturer chooses the resolution IV design which will ensure that all main effects are free from aliasing (assuming three factor and higher order interactions are absent). However, in this design the two factor interactions may be aliased with each other. It is decided that, if important two factor interactions are found to be present, additional experiment trials may be conducted to separate the aliased effects. The performance of the fuel cone is measured on a scale of 1 to 15. In DOE++, the design for this experiment is set up using the properties shown in the following figure. The Fraction Generators for the design, <math>E=ABC\,\!</math> and <math>F=BCD\,\!</math>, are the same as the defaults used in DOE++. The resulting <math>{2}^{6-2}\,\!</math> design and the corresponding response values are shown in the following two figures.

[[Image:doe7_39.png|center|644px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

[[Image:doe7_40.png|center|534px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The complete alias structure for the 2 <math>_{\text{IV}}^{6-2}\,\!</math> design is shown next.

<center><math>I=ABCE=ADEF=BCDF\,\!</math></center>

 
<center><math>\begin{align}
& A= & BCE=DEF=ABCDF \\
& B= & ACE=CDF=ABDEF \\
& C= & ABE=BDF=ACDEF \\
& D= & AEF=BCF=ABCDE \\
& E= & ABC=ADF=BCDEF \\
& F= & ADE=BCD=ABCEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& AB= & CE=ACDF=BDEF \\
& AC= & BE=ABDF=CDEF \\
& AD= & EF=ABCF=BCDE \\
& AE= & BC=DF=ABCDEF \\
& AF= & DE=ABCD=BCEF \\
& BD= & CF=ABEF=ACDE \\
& BF= & CD=ABDE=ACEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& ABD= & ACF=BEF=CDE \\
& ABF= & ACD=BDE=CEF
\end{align}\,\!</math></center>

In DOE++, the alias structure is displayed in the Design Summary and as part of the Design Evaluation result, as shown next:

[[Image:doe7_41.png|center|700px|Alias structure for the experiment design in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The normal probability plot of effects for this unreplicated design shows the main effects of factors <math>C\,\!</math> and <math>D\,\!</math> and the interaction effect, <math>BF\,\!</math>, to be significant (see the following figure).

[[Image:doe7_42.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

From the alias structure, it can be seen that for the present design interaction effect, <math>BF,\,\!</math> is confounded with <math>CD\,\!</math>. Therefore, the actual source of this effect cannot be known on the basis of the present experiment. However because neither factor <math>B\,\!</math> nor <math>F\,\!</math> is found to be significant there is an indication the observed effect is likely due to interaction, <math>CD\,\!</math>. To confirm this, a follow-up <math>{2}^{2}\,\!</math> experiment is run involving only factors <math>B\,\!</math> and <math>F\,\!</math>. The interaction, <math>BF\,\!</math>, is found to be inactive, leading to the conclusion that the interaction effect in the original experiment is effect, <math>CD\,\!</math>. Given these results, the fitted regression model for the fuel cone design as per the coefficients obtained from DOE++ is shown next.

::<math>\hat{y}=7.6875+C+2\cdot D+2.1875\cdot CD\,\!</math>

[[Image:doe7_43.png|center|700px|Effect coefficients for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

==Projection==

Projection refers to the reduction of a fractional factorial design to a full factorial design by dropping out some of the factors of the design. Any fractional factorial design of resolution, <math>R,\,\!</math> can be reduced to complete factorial designs in any subset of <math>R-1\,\!</math> factors. For example, consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. The resolution of this design is four. Therefore, this design can be reduced to full factorial designs in any three (<math>4-1=3\,\!</math>) of the original seven factors (by dropping the remaining four of factors). Further, a fractional factorial design can also be reduced to a full factorial design in any <math>R\,\!</math> of the original factors, as long as these <math>R\,\!</math> factors are not part of the generator in the defining relation. Again consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. This design can be reduced to a full factorial design in four factors provided these four factors do not appear together as a generator in the defining relation. The complete defining relation for this design is:

::<math>\begin{align}
& I= & ABCE=BCDF=ACDG=ADEF=ABFG=BDEG=CEFG
\end{align}\,\!</math>

Therefore, there are seven four factor combinations out of the 35 (<math>(_{7}^{4})=35\,\!</math>) possible four-factor combinations that are used as generators in the defining relation. The designs with the remaining 28 four factor combinations would be full factorial 16-run designs. For example, factors <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math> do not occur as a generator in the defining relation of the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. If the remaining factors, <math>E\,\!</math>, <math>F\,\!</math> and <math>G\,\!</math>, are dropped, the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design will reduce to a full factorial design in <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math>.

==Resolution III Designs==

At times, the factors to be investigated in screening experiments are so large that even running a fractional factorial design is impractical. This can be partially solved by using resolution III fractional factorial designs in the cases where three factor and higher order interactions can be assumed to be unimportant. Resolution III designs, such as the 2 <math>_{\text{III}}^{3-1}\,\!</math> design, can be used to estimate <math>k\,\!</math> main effects using just <math>k+1\,\!</math> runs. In these designs, the main effects are aliased with two factor interactions. Once the results from these designs are obtained, and knowing that three factor and higher order interactions are unimportant, the experimenter can decide if there is a need to run a fold-over design to de-alias the main effects from the two factor interactions. Thus, the 2 <math>_{\text{III}}^{3-1}\,\!</math> design can be used to investigate three factors in four runs, the 2 <math>_{\text{III}}^{7-4}\,\!</math> design can be used to investigate seven factors in eight runs, the 2 <math>_{\text{III}}^{15-11}\,\!</math> design can be used to investigate fifteen factors in sixteen runs and so on.

====Example====

{{:Resolution_III_Design_Example}}

==Alias Matrix==
In [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction designs]] and [[Two_Level_Factorial_Experiments#Quarter_and_Smaller_Fraction_Designs| Quarter and Smaller Fraction Designs]], the alias structure for fractional factorial designs was obtained using the defining relation. However, this method of obtaining the alias structure is not very efficient when the alias structure is very complex or when partial aliasing is involved. One of the ways to obtain the alias structure for any design, regardless of its complexity, is to use the alias matrix. The alias matrix for a design is calculated using <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> where <math>{{X}_{1}}\,\!</math> is the portion of the design matrix, <math>X,\,\!</math> that contains the effects for which the aliases need to be calculated, and <math>{{X}_{2}}\,\!</math> contains the remaining columns of the design matrix, other than those included in <math>{{X}_{1}}\,\!</math>.

To illustrate the use of the alias matrix, consider the design matrix for the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design (using the defining relation <math>I=ABCD\,\!</math>) shown next:

[[Image:Chapter7__879.png|center|link=]]

The alias structure for this design can be obtained by defining <math>{{X}_{1}}\,\!</math> using eight columns since the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design estimates eight effects. If the first eight columns of <math>X\,\!</math> are used then <math>{{X}_{1}}\,\!</math> is:

[[Image:Chapter7__884.png|center|link=]]

<math>{{X}_{2}}\,\!</math> is obtained using the remaining columns as:

[[Image:Chapter7__886.png|center|link=]]

Then the alias matrix <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> is:

[[Image:Chapter7__888.png|center|link=]]

The alias relations can be easily obtained by observing the alias matrix as:

<center><math>\begin{align}
& I= & ABCD \\
& A= & BCD \\
& B= & ACD \\
& AB= & CD \\
& C= & ABD \\
& AC= & BD \\
& BC= & AD \\
& D= & ABC
\end{align}\,\!</math></center>

Two Level Factorial Experiments

2017-08-10T17:36:34Z

Richard House: /* Half-fraction Designs */

{{Template:Doebook|8}}
Two level factorial experiments are factorial experiments in which each factor is investigated at only two levels. The early stages of experimentation usually involve the investigation of a large number of potential factors to discover the "vital few" factors. Two level factorial experiments are used during these stages to quickly filter out unwanted effects so that attention can then be focused on the important ones.

==2''k'' Designs==
The factorial experiments, where all combination of the levels of the factors are run, are usually referred to as ''full factorial experiments''. Full factorial two level experiments are also referred to as <math>{2}^{k}\,\!</math> designs where <math>k\,\!</math> denotes the number of factors being investigated in the experiment. In Weibull++ DOE folios, these designs are referred to as 2 Level Factorial Designs as shown in the figure below.

[[Image:doe7_1.png|center|487px|Selection of full factorial experiments with two levels in Weibull++.|link=]]

A full factorial two level design with <math>k\,\!</math> factors requires <math>{{2}^{k}}\,\!</math> runs for a single replicate. For example, a two level experiment with three factors will require <math>2\times 2\times 2={{2}^{3}}=8\,\!</math> runs. The choice of the two levels of factors used in two level experiments depends on the factor; some factors naturally have two levels. For example, if gender is a factor, then male and female are the two levels. For other factors, the limits of the range of interest are usually used. For example, if temperature is a factor that varies from <math>{45}^{o}C\,\!</math> to <math>{90}^{o}C\,\!</math>, then the two levels used in the <math>{2}^{k}\,\!</math> design for this factor would be <math>{45}^{o}\,\!C\,\!</math> and <math>{90}^{o}\,\!C\,\!</math>.

The two levels of the factor in the <math>{2}^{k}\,\!</math> design are usually represented as <math>-1\,\!</math> (for the first level) and <math>1\,\!</math> (for the second level). Note that this representation is reversed from the coding used in [[General Full Factorial Designs]] for the indicator variables that represent two level factors in ANOVA models. For ANOVA models, the first level of the factor was represented using a value of <math>1\,\!</math> for the indicator variable, while the second level was represented using a value of <math>-1\,\!</math>. For details on the notation used for two level experiments refer to [[Two_Level_Factorial_Experiments#Notation| Notation]].

===The 22 Design===

The simplest of the two level factorial experiments is the <math>{2}^{2}\,\!</math> design where two factors (say factor <math>A\,\!</math> and factor <math>B\,\!</math>) are investigated at two levels. A single replicate of this design will require four runs (<math>{{2}^{2}}=2\times 2=4\,\!</math>) The effects investigated by this design are the two main effects, <math>A\,\!</math> and <math>B,\,\!</math> and the interaction effect <math>AB\,\!</math>. The treatments for this design are shown in figure (a) below. In figure (a), letters are used to represent the treatments. The presence of a letter indicates the high level of the corresponding factor and the absence indicates the low level. For example, (1) represents the treatment combination where all factors involved are at the low level or the level represented by <math>-1\,\!</math> ; <math>a\,\!</math> represents the treatment combination where factor <math>A\,\!</math> is at the high level or the level of <math>1\,\!</math>, while the remaining factors (in this case, factor <math>B\,\!</math>) are at the low level or the level of <math>-1\,\!</math>. Similarly, <math>b\,\!</math> represents the treatment combination where factor <math>B\,\!</math> is at the high level or the level of <math>1\,\!</math>, while factor <math>A\,\!</math> is at the low level and <math>ab\,\!</math> represents the treatment combination where factors <math>A\,\!</math> and <math>B\,\!</math> are at the high level or the level of the 1. Figure (b) below shows the design matrix for the <math>{2}^{2}\,\!</math> design. It can be noted that the sum of the terms resulting from the product of any two columns of the design matrix is zero. As a result the <math>{2}^{2}\,\!</math> design is an ''orthogonal design''. In fact, all <math>{2}^{k}\,\!</math> designs are orthogonal designs. This property of the <math>{2}^{k}\,\!</math> designs offers a great advantage in the analysis because of the simplifications that result from orthogonality. These simplifications are explained later on in this chapter.
The <math>{2}^{2}\,\!</math> design can also be represented geometrically using a square with the four treatment combinations lying at the four corners, as shown in figure (c) below.

[[Image:doe7.2.png|center|400px|The <math>2^2\,\!</math> design. Figure (a) displays the experiment design, (b) displays the design matrix and (c) displays the geometric representation for the design. In Figure (b), the column names I, A, B and AB are used. Column I represents the intercept term. Columns A and B represent the respective factor settings. Column AB represents the interaction and is the product of columns A and B.]]
 

===The 23 Design===

The <math>{2}^{3}\,\!</math> design is a two level factorial experiment design with three factors (say factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>). This design tests three (<math>k=3\,\!</math>) main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math> ; three (<math>(_{2}^{k})=\,\!</math> <math>(_{2}^{3})=3\,\!</math>) two factor interaction effects, <math>AB\,\!</math>, <math>BC\,\!</math>, <math>AC\,\!</math> ; and one (<math>(_{3}^{k})=\,\!</math> <math>(_{3}^{3})=1\,\!</math>) three factor interaction effect, <math>ABC\,\!</math>. The design requires eight runs per replicate. The eight treatment combinations corresponding to these runs are <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math>, <math>ab\,\!</math>, <math>c\,\!</math>, <math>ac\,\!</math>, <math>bc\,\!</math> and <math>abc\,\!</math>. Note that the treatment combinations are written in such an order that factors are introduced one by one with each new factor being combined with the preceding terms. This order of writing the treatments is called the ''standard order'' or ''Yates' order''. The <math>{2}^{3}\,\!</math> design is shown in figure (a) below. The design matrix for the <math>{2}^{3}\,\!</math> design is shown in figure (b). The design matrix can be constructed by following the standard order for the treatment combinations to obtain the columns for the main effects and then multiplying the main effects columns to obtain the interaction columns.

[[Image:doe7.3.png|center|324px|The <math>2^3\,\!</math> design. Figure (a) shows the experiment design and (b) shows the design matrix.]]

[[Image:doe7.4.png|center|290px|Geometric representation of the <math>2^3\,\!</math> design.]]

The <math>{2}^{3}\,\!</math> design can also be represented geometrically using a cube with the eight treatment combinations lying at the eight corners as shown in the figure above.

==Analysis of 2''k'' Designs==

The <math>{2}^{k}\,\!</math> designs are a special category of the factorial experiments where all the factors are at two levels. The fact that these designs contain factors at only two levels and are orthogonal greatly simplifies their analysis even when the number of factors is large. The use of <math>{2}^{k}\,\!</math> designs in investigating a large number of factors calls for a revision of the notation used previously for the ANOVA models. The case for revised notation is made stronger by the fact that the ANOVA and multiple linear regression models are identical for <math>{2}^{k}\,\!</math> designs because all factors are only at two levels. Therefore, the notation of the regression models is applied to the ANOVA models for these designs, as explained next.

===Notation===

Based on the notation used in [[General Full Factorial Designs]], the ANOVA model for a two level factorial experiment with three factors would be as follows:

::<math>\begin{align}
& Y= & \mu +{{\tau }_{1}}\cdot {{x}_{1}}+{{\delta }_{1}}\cdot {{x}_{2}}+{{(\tau \delta )}_{11}}\cdot {{x}_{1}}{{x}_{2}}+{{\gamma }_{1}}\cdot {{x}_{3}} \\
& & +{{(\tau \gamma )}_{11}}\cdot {{x}_{1}}{{x}_{3}}+{{(\delta \gamma )}_{11}}\cdot {{x}_{2}}{{x}_{3}}+{{(\tau \delta \gamma )}_{111}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where:
 
:• <math>\mu \,\!</math> represents the overall mean
:• <math>{{\tau }_{1}}\,\!</math> represents the independent effect of the first factor (factor <math>A\,\!</math>) out of the two effects <math>{{\tau }_{1}}\,\!</math> and <math>{{\tau }_{2}}\,\!</math>
:• <math>{{\delta }_{1}}\,\!</math> represents the independent effect of the second factor (factor <math>B\,\!</math>) out of the two effects <math>{{\delta }_{1}}\,\!</math> and <math>{{\delta }_{2}}\,\!</math>
:• <math>{{(\tau \delta )}_{11}}\,\!</math> represents the independent effect of the interaction <math>AB\,\!</math> out of the other interaction effects
:• <math>{{\gamma }_{1}}\,\!</math> represents the effect of the third factor (factor <math>C\,\!</math>) out of the two effects <math>{{\gamma }_{1}}\,\!</math> and <math>{{\gamma }_{2}}\,\!</math>
:• <math>{{(\tau \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>AC\,\!</math> out of the other interaction effects
:• <math>{{(\delta \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>BC\,\!</math> out of the other interaction effects
:• <math>{{(\tau \delta \gamma )}_{111}}\,\!</math> represents the effect of the interaction <math>ABC\,\!</math> out of the other interaction effects
and <math>\epsilon \,\!</math> is the random error term.

 
The notation for a linear regression model having three predictor variables with interactions is:

::<math>\begin{align}
& Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& & +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

The notation for the regression model is much more convenient, especially for the case when a large number of higher order interactions are present. In two level experiments, the ANOVA model requires only one indicator variable to represent each factor for both qualitative and quantitative factors. Therefore, the notation for the multiple linear regression model can be applied to the ANOVA model of the experiment that has all the factors at two levels. For example, for the experiment of the ANOVA model given above, <math>{{\beta }_{0}}\,\!</math> can represent the overall mean instead of <math>\mu \,\!</math>, and <math>{{\beta }_{1}}\,\!</math> can represent the independent effect, <math>{{\tau }_{1}}\,\!</math>, of factor <math>A\,\!</math>. Other main effects can be represented in a similar manner. The notation for the interaction effects is much more simplified (e.g., <math>{{\beta }_{123}}\,\!</math> can be used to represent the three factor interaction effect, <math>{{(\tau \beta \gamma )}_{111}}\,\!</math>).

As mentioned earlier, it is important to note that the coding for the indicator variables for the ANOVA models of two level factorial experiments is reversed from the coding followed in [[General Full Factorial Designs]]. Here <math>-1\,\!</math> represents the first level of the factor while <math>1\,\!</math> represents the second level. This is because for a two level factor a single variable is needed to represent the factor for both qualitative and quantitative factors. For quantitative factors, using <math>-1\,\!</math> for the first level (which is the low level) and 1 for the second level (which is the high level) keeps the coding consistent with the numerical value of the factors. The change in coding between the two coding schemes does not affect the analysis except that signs of the estimated effect coefficients will be reversed (i.e., numerical values of <math>{{\hat{\tau }}_{1}}\,\!</math>, obtained based on the coding of [[General Full Factorial Designs]], and <math>{{\hat{\beta }}_{1}}\,\!</math>, obtained based on the new coding, will be the same but their signs would be opposite).

::<math>\begin{align}
& & \text{Factor }A\text{ Coding (two level factor)} \\
& &
\end{align}\,\!</math>

::<math>\begin{matrix}
\text{Previous Coding} & {} & {} & {} & \text{Coding for }{{\text{2}}^{k}}\text{ Designs} \\
{} & {} & {} & {} & {} \\
Effect\text{ }{{\tau }_{1}}\ \ :\ \ {{x}_{1}}=1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{1}}\text{ (or }-{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=-1\text{ } \\
Effect\text{ }{{\tau }_{2}}\ \ :\ \ {{x}_{1}}=-1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{2}}\text{ (or }{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=1\text{ } \\
\end{matrix}\,\!</math>

In summary, the ANOVA model for the experiments with all factors at two levels is different from the ANOVA models for other experiments in terms of the notation in the following two ways:
 

:• The notation of the regression models is used for the effect coefficients.
:• The coding of the indicator variables is reversed.

===Special Features===

Consider the design matrix, <math>X\,\!</math>, for the <math>{2}^{3}\,\!</math> design discussed above. The (<math>{{X}^{\prime }}X\,\!</math>) <math>^{-1}\,\!</math> matrix is:

<center><math>{{({{X}^{\prime }}X)}^{-1}}=\left[ \begin{matrix}
0.125 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0.125 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0.125 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0.125 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0.125 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0.125 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0.125 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.125 \\
\end{matrix} \right]\,\!</math></center>

Notice that, due to the orthogonal design of the <math>X\,\!</math> matrix, the <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> has been simplified to a diagonal matrix which can be written as:

::<math>\begin{align}
{{({{X}^{\prime }}X)}^{-1}}= & 0.125\cdot I = & \frac{1}{8}\cdot I = & \frac{1}{{{2}^{3}}}\cdot I
\end{align}\,\!</math>

where <math>I\,\!</math> represents the identity matrix of the same order as the design matrix, <math>X\,\!</math>. Since there are eight observations per replicate of the <math>{2}^{3}\,\!</math> design, the <math>(X\,\!</math> ' <math>X{{)}^{-1}}\,\!</math> matrix for <math>m\,\!</math> replicates of this design can be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{3}}\cdot m)}\cdot I\,\!</math>

The <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> matrix for any <math>{2}^{k}\,\!</math> design can now be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{k}}\cdot m)}\cdot I\,\!</math>

Then the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is:

::<math>\begin{align}
C= & {{{\hat{\sigma }}}^{2}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & M{{S}_{E}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & \frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}\cdot I
\end{align}\,\!</math>

Note that the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is also a diagonal matrix. Therefore, the estimated effect coefficients (<math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math>, <math>{{\beta }_{12}},\,\!</math> etc.) for these designs are uncorrelated. This implies that the terms in the <math>{2}^{k}\,\!</math> design (main effects, interactions) are independent of each other. Consequently, the extra sum of squares for each of the terms in these designs is independent of the sequence of terms in the model, and also independent of the presence of other terms in the model. As a result the sequential and partial sum of squares for the terms are identical for these designs and will always add up to the model sum of squares. Multicollinearity is also not an issue for these designs.

It can also be noted from the equation given above, that in addition to the <math>C\,\!</math> matrix being diagonal, all diagonal elements of the <math>C\,\!</math> matrix are identical. This means that the variance (or its square root, the standard error) of all estimated effect coefficients are the same. The standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>, for all the coefficients is:

::<math>\begin{align}
se({{{\hat{\beta }}}_{j}})= & \sqrt{{{C}_{jj}}} = & \sqrt{\frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}}\text{ }for\text{ }all\text{ }j
\end{align}\,\!</math>

This property is used to construct the normal probability plot of effects in <math>{2}^{k}\,\!</math> designs and identify significant effects using graphical techniques. For details on the normal probability plot of effects in a Weibull++ DOE folio, refer to [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]].

====Example====
To illustrate the analysis of a full factorial <math>{2}^{k}\,\!</math> design, consider a three factor experiment to investigate the effect of honing pressure, number of strokes and cycle time on the surface finish of automobile brake drums. Each of these factors is investigated at two levels. The honing pressure is investigated at levels of 200 <math>psi\,\!</math> and 400 <math>psi\,\!</math>, the number of strokes used is 3 and 5 and the two levels of the cycle time are 3 and 5 seconds. The design for this experiment is set up in a Weibull++ DOE folio as shown in the first two following figures. It is decided to run two replicates for this experiment. The surface finish data collected from each run (using randomization) and the complete design is shown in the third following figure. The analysis of the experiment data is explained next.

[[Image:doe7_5.png|center|877px|Design properties for the experiment in the example.|link=]]

[[Image:doe7_6.png|center|859px|Design summary for the experiment in the example.|link=]]

[[Image:doe7_7.png|center|778px|Experiment design for the example to investigate the surface finish of automobile brake drums.|link=]]

The applicable model using the notation for <math>{2}^{k}\,\!</math> designs is:

::<math>\begin{align}
Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where the indicator variable, <math>{{x}_{1,}}\,\!</math> represents factor <math>A\,\!</math> (honing pressure), <math>{{x}_{1}}=-1\,\!</math> represents the low level of 200 <math>psi\,\!</math> and <math>{{x}_{1}}=1\,\!</math> represents the high level of 400 <math>psi\,\!</math>. Similarly, <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> represent factors <math>B\,\!</math> (number of strokes) and <math>C\,\!</math> (cycle time), respectively. <math>{{\beta }_{0}}\,\!</math> is the overall mean, while <math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math> and <math>{{\beta }_{3}}\,\!</math> are the effect coefficients for the main effects of factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. <math>{{\beta }_{12}}\,\!</math>, <math>{{\beta }_{13}}\,\!</math> and <math>{{\beta }_{23}}\,\!</math> are the effect coefficients for the <math>AB\,\!</math>, <math>AC\,\!</math> and <math>BC\,\!</math> interactions, while <math>{{\beta }_{123}}\,\!</math> represents the <math>ABC\,\!</math> interaction.

 
If the subscripts for the run (<math>i\,\!</math> ; <math>i=\,\!</math> 1 to 8) and replicates (<math>j\,\!</math> ; <math>j=\,\!</math> 1,2) are included, then the model can be written as:

::<math>\begin{align}
{{Y}_{ij}}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{ij1}}+{{\beta }_{2}}\cdot {{x}_{ij2}}+{{\beta }_{12}}\cdot {{x}_{ij1}}{{x}_{ij2}}+{{\beta }_{3}}\cdot {{x}_{ij3}} \\
& +{{\beta }_{13}}\cdot {{x}_{ij1}}{{x}_{ij3}}+{{\beta }_{23}}\cdot {{x}_{ij2}}{{x}_{ij3}}+{{\beta }_{123}}\cdot {{x}_{ij1}}{{x}_{ij2}}{{x}_{ij3}}+{{\epsilon }_{ij}}
\end{align}\,\!</math>

To investigate how the given factors affect the response, the following hypothesis tests need to be carried:

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{1}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{1}}\ne 0\,\!</math>

This test investigates the main effect of factor <math>A\,\!</math> (honing pressure). The statistic for this test is:

::<math>{{({{F}_{0}})}_{A}}=\frac{M{{S}_{A}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{A}}\,\!</math> is the mean square for factor <math>A\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other main effects, <math>B\,\!</math> and <math>C\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{12}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{12}}\ne 0\,\!</math>

This test investigates the two factor interaction <math>AB\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{AB}}=\frac{M{{S}_{AB}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the interaction <math>AB\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other two factor interactions, <math>AC\,\!</math> and <math>BC\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{123}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{123}}\ne 0\,\!</math>

This test investigates the three factor interaction <math>ABC\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{ABC}}=\frac{M{{S}_{ABC}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{ABC}}\,\!</math> is the mean square for the interaction <math>ABC\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square.
To calculate the test statistics, it is convenient to express the ANOVA model in the form <math>y=X\beta +\epsilon \,\!</math>.

====Expression of the ANOVA Model as <math>y=X\beta +\epsilon \,\!</math>====

In matrix notation, the ANOVA model can be expressed as:

::<math>y=X\beta +\epsilon \,\!</math>

where:

<center><math>y=\left[ \begin{matrix}
{{Y}_{11}} \\
{{Y}_{21}} \\
. \\
{{Y}_{81}} \\
{{Y}_{12}} \\
. \\
{{Y}_{82}} \\
\end{matrix} \right]=\left[ \begin{matrix}
90 \\
90 \\
. \\
90 \\
86 \\
. \\
80 \\
\end{matrix} \right]\text{ }X=\left[ \begin{matrix}
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\end{matrix} \right]\,\!</math></center>

<center><math>\beta =\left[ \begin{matrix}
{{\beta }_{0}} \\
{{\beta }_{1}} \\
{{\beta }_{2}} \\
{{\beta }_{12}} \\
{{\beta }_{3}} \\
{{\beta }_{13}} \\
{{\beta }_{23}} \\
{{\beta }_{123}} \\
\end{matrix} \right]\text{ }\epsilon =\left[ \begin{matrix}
{{\epsilon }_{11}} \\
{{\epsilon }_{21}} \\
. \\
{{\epsilon }_{81}} \\
{{\epsilon }_{12}} \\
. \\
. \\
{{\epsilon }_{82}} \\
\end{matrix} \right]\,\!</math></center>

====Calculation of the Extra Sum of Squares for the Factors====

Knowing the matrices <math>y\,\!</math>, <math>X\,\!</math> and <math>\beta \,\!</math>, the extra sum of squares for the factors can be calculated. These are used to calculate the mean squares that are used to obtain the test statistics. Since the experiment design is orthogonal, the partial and sequential extra sum of squares are identical. The extra sum of squares for each effect can be calculated as shown next. As an example, the extra sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & Model\text{ }Sum\text{ }of\text{ }Squares - Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }main\text{ }effect\text{ }of\text{ }A \\
= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }A}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }A}}={{X}_{\tilde{\ }A}}{{(X_{\tilde{\ }A}^{\prime }{{X}_{\tilde{\ }A}})}^{-1}}X_{\tilde{\ }A}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }A}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the main effect of factor <math>A\,\!</math>. Thus, the sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y \\
= & 654.4375-549.375 \\
= & 105.0625
\end{align}\,\!</math>

Similarly, the extra sum of squares for the interaction effect <math>AB\,\!</math> is:

::<math>\begin{align}
S{{S}_{AB}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }AB}}-(1/16)J]y \\
= & 654.4375-636.375 \\
= & 18.0625
\end{align}\,\!</math>

The extra sum of squares for other effects can be obtained in a similar manner.

====Calculation of the Test Statistics====

Knowing the extra sum of squares, the test statistic for the effects can be calculated. For example, the test statistic for the interaction <math>AB\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{AB}}= & \frac{M{{S}_{AB}}}{M{{S}_{E}}} \\
= & \frac{S{{S}_{AB}}/dof(S{{S}_{AB}})}{S{{S}_{E}}/dof(S{{S}_{E}})} \\
= & \frac{18.0625/1}{147.5/8} \\
= & 0.9797
\end{align}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the <math>AB\,\!</math> interaction and <math>M{{S}_{E}}\,\!</math> is the error mean square. The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{AB}}=0.9797\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{AB}}) \\
= & 1-0.6487 \\
= & 0.3513
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that the interaction between honing pressure and number of strokes does not affect the surface finish of the brake drums. Tests for other effects can be carried out in a similar manner. The results are shown in the ANOVA Table in the following figure. The values S, R-sq and R-sq(adj) in the figure indicate how well the model fits the data. The value of S represents the standard error of the model, R-sq represents the coefficient of multiple determination and R-sq(adj) represents the adjusted coefficient of multiple determination. For details on these values refer to [[Multiple_Linear_Regression_Analysis|Multiple Linear Regression Analysis]].

[[Image:doe7_8.png|center|799px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

====Calculation of Effect Coefficients====

The estimate of effect coefficients can also be obtained:

<center><math>\begin{align}
\hat{\beta }= & {{({{X}^{\prime }}X)}^{-1}}{{X}^{\prime }}y \\
= & \left[ \begin{matrix}
86.4375 \\
2.5625 \\
-4.9375 \\
1.0625 \\
-1.0625 \\
2.4375 \\
-1.3125 \\
-0.1875 \\
\end{matrix} \right]
\end{align}\,\!</math></center>

[[Image:doe7_9.png|center|800px|Regression Information table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

The coefficients and related results are shown in the Regression Information table above. In the table, the Effect column displays the effects, which are simply twice the coefficients. The Standard Error column displays the standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>. The Low CI and High CI columns display the confidence interval on the coefficients. The interval shown is the 90% interval as the significance is chosen as 0.1. The T Value column displays the <math>t\,\!</math> statistic, <math>{{t}_{0}}\,\!</math>, corresponding to the coefficients. The P Value column displays the <math>p\,\!</math> value corresponding to the <math>t\,\!</math> statistic. (For details on how these results are calculated, refer to [[General Full Factorial Designs]]). Plots of residuals can also be obtained from the DOE folio to ensure that the assumptions related to the ANOVA model are not violated.

====Model Equation====

From the analysis results in the above figure within [[Two_Level_Factorial_Experiments#Calculation_of_Effect_Coefficients|calculation of effect coefficients]] section, it is seen that effects <math>A\,\!</math>, <math>B\,\!</math> and <math>AC\,\!</math> are significant. In a DOE folio, the <math>p\,\!</math> values for the significant effects are displayed in red in the ANOVA Table for easy identification. Using the values of the estimated effect coefficients, the model for the present <math>{2}^{3}\,\!</math> design in terms of the coded values can be written as:

::<math>\begin{align}
\hat{y}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}} \\
= & 86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+2.4375{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

To make the model hierarchical, the main effect, <math>C\,\!</math>, needs to be included in the model (because the interaction <math>AC\,\!</math> is included in the model). The resulting model is:

::<math>\hat{y}=86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+1.0625{{x}_{3}}+2.4375{{x}_{1}}{{x}_{3}}\,\!</math>

This equation can be viewed in a DOE folio, as shown in the following figure, using the Show Analysis Summary icon in the Control Panel. The equation shown in the figure will match the hierarchical model once the required terms are selected using the Select Effects icon.

[[Image:doe7_10.png|center|557px|The model equation for the experiment of the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

==Replicated and Repeated Runs==

In the case of replicated experiments, it is important to note the difference between replicated runs and repeated runs. Both repeated and replicated runs are multiple response readings taken at the same factor levels. However, repeated runs are response observations taken at the same time or in succession. Replicated runs are response observations recorded in a random order. Therefore, replicated runs include more variation than repeated runs. For example, a baker, who wants to investigate the effect of two factors on the quality of cakes, will have to bake four cakes to complete one replicate of a <math>{2}^{2}\,\!</math> design. Assume that the baker bakes eight cakes in all. If, for each of the four treatments of the <math>{2}^{2}\,\!</math> design, the baker selects one treatment at random and then bakes two cakes for this treatment at the same time then this is a case of two repeated runs. If, however, the baker bakes all the eight cakes randomly, then the eight cakes represent two sets of replicated runs.
For repeated measurements, the average values of the response for each treatment should be entered into a DOE folio as shown in the following figure (a) when the two cakes for a particular treatment are baked together. For replicated measurements, when all the cakes are baked randomly, the data is entered as shown in the following figure (b).

[[Image:doe7.11.png|center|300px|Data entry for repeated and replicated runs. Figure (a) shows repeated runs and (b) shows replicated runs.]]

==Unreplicated 2''k'' Designs==

If a factorial experiment is run only for a single replicate then it is not possible to test hypotheses about the main effects and interactions as the error sum of squares cannot be obtained. This is because the number of observations in a single replicate equals the number of terms in the ANOVA model. Hence the model fits the data perfectly and no degrees of freedom are available to obtain the error sum of squares.

However, sometimes it is only possible to run a single replicate of the <math>{2}^{k}\,\!</math> design because of constraints on resources and time. In the absence of the error sum of squares, hypothesis tests to identify significant factors cannot be conducted. A number of methods of analyzing information obtained from unreplicated <math>{2}^{k}\,\!</math> designs are available. These include pooling higher order interactions, using the normal probability plot of effects or including center point replicates in the design.

===Pooling Higher Order Interactions===

One of the ways to deal with unreplicated <math>{2}^{k}\,\!</math> designs is to use the sum of squares of some of the higher order interactions as the error sum of squares provided these higher order interactions can be assumed to be insignificant. By dropping some of the higher order interactions from the model, the degrees of freedom corresponding to these interactions can be used to estimate the error mean square. Once the error mean square is known, the test statistics to conduct hypothesis tests on the factors can be calculated.

===Normal Probability Plot of Effects===
Another way to use unreplicated <math>{2}^{k}\,\!</math> designs to identify significant effects is to construct the normal probability plot of the effects. As mentioned in [[Two_Level_Factorial_Experiments#Special_Features| Special Features]], the standard error for all effect coefficients in the <math>{2}^{k}\,\!</math> designs is the same. Therefore, on a normal probability plot of effect coefficients, all non-significant effect coefficients (with <math>\beta =0\,\!</math>) will fall along the straight line representative of the normal distribution, N(<math>0,{{\sigma }^{2}}/({{2}^{k}}\cdot m)\,\!</math>). Effect coefficients that show large deviations from this line will be significant since they do not come from this normal distribution. Similarly, since effects <math>=2\times \,\!</math> effect coefficients, all non-significant effects will also follow a straight line on the normal probability plot of effects. For replicated designs, the Effects Probability plot of a DOE folio plots the normalized effect values (or the T Values) on the standard normal probability line, N(0,1). However, in the case of unreplicated <math>{2}^{k}\,\!</math> designs, <math>{{\sigma }^{2}}\,\!</math> remains unknown since <math>M{{S}_{E}}\,\!</math> cannot be obtained. Lenth's method is used in this case to estimate the variance of the effects. For details on Lenth's method, please refer to [[DOE References| Montgomery (2001)]]. The DOE folio then uses this variance value to plot effects along the N(0, Lenth's effect variance) line. The
method is illustrated in the following example.

====Example====

Vinyl panels, used as instrument panels in a certain automobile, are seen to develop defects after a certain amount of time. To investigate the issue, it is decided to carry out a two level factorial experiment. Potential factors to be investigated in the experiment are vacuum rate (factor <math>A\,\!</math>), material temperature (factor <math>B\,\!</math>), element intensity (factor <math>C\,\!</math>) and pre-stretch (factor <math>D\,\!</math>). The two levels of the factors used in the experiment are as shown in below.

[[Image:doet7.1.png|center|300px|Factors to investigate defects in vinyl panels.]]

With a <math>{2}^{4}\,\!</math> design requiring 16 runs per replicate it is only feasible for the manufacturer to run a single replicate.

The experiment design and data, collected as percent defects, are shown in the following figure. Since the present experiment design contains only a single replicate, it is not possible to obtain an estimate of the error sum of squares, <math>S{{S}_{E}}\,\!</math>. It is decided to use the normal probability plot of effects to identify the significant effects. The effect values for each term are obtained as shown in the following figure.

[[Image:doe7_13.png|center|650px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

Lenth's method uses these values to estimate the variance. As described in [[DOE_References|[Lenth, 1989]]], if all effects are arranged in ascending order, using their absolute values, then <math>{{s}_{0}}\,\!</math> is defined as 1.5 times the median value:

::<math>\begin{align}
{{s}_{0}}= & 1.5\cdot median(\left| effect \right|) \\
= & 1.5\cdot 2 \\
= & 3
\end{align}\,\!</math>

Using <math>{{s}_{0}}\,\!</math>, the "pseudo standard error" (<math>PSE\,\!</math>) is calculated as 1.5 times the median value of all effects that are less than 2.5 <math>{{s}_{0}}\,\!</math> :

::<math>\begin{align}
PSE= & 1.5\cdot median(\left| effect \right|\ \ :\ \ \left| effect \right|<2.5{{s}_{0}}) \\
= & 1.5\cdot 1.5 \\
= & 2.25
\end{align}\,\!</math>

Using <math>PSE\,\!</math> as an estimate of the effect variance, the effect variance is 2.25. Knowing the effect variance, the normal probability plot of effects for the present unreplicated experiment can be constructed as shown in the following figure. The line on this plot is the line N(0, 2.25). The plot shows that the effects <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math> do not follow the distribution represented by this line. Therefore, these effects are significant.

The significant effects can also be identified by comparing individual effect values to the margin of error or the threshold value using the pareto chart (see the third following figure). If the required significance is 0.1, then:

::<math>margin\text{ }of\text{ }error={{t}_{\alpha /2,d}}\cdot PSE\,\!</math>

The <math>t\,\!</math> statistic, <math>{{t}_{\alpha /2,d}}\,\!</math>, is calculated at a significance of <math>\alpha /2\,\!</math> (for the two-sided hypothesis) and degrees of freedom <math>d=(\,\!</math> number of effects <math>)/3\,\!</math>. Thus:

::<math>\begin{align}
margin\text{ }of\text{ }error= & {{t}_{0.05,5}}\cdot PSE \\
= & 2.015\cdot 2.25 \\
= & 4.534
\end{align}\,\!</math>

The value of 4.534 is shown as the critical value line in the third following figure. All effects with absolute values greater than the margin of error can be considered to be significant. These effects are <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math>. Therefore, the vacuum rate, the pre-stretch and their interaction have a significant effect on the defects of the vinyl panels.

[[Image:doe7_14.png|center|800px|Effect values for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_15.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_16.png|center|650px|Pareto chart for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

===Center Point Replicates===

Another method of dealing with unreplicated <math>{2}^{k}\,\!</math> designs that only have quantitative factors is to use replicated runs at the center point. The center point is the response corresponding to the treatment exactly midway between the two levels of all factors. Running multiple replicates at this point provides an estimate of pure error. Although running multiple replicates at any treatment level can provide an estimate of pure error, the other advantage of running center point replicates in the <math>{2}^{k}\,\!</math> design is in checking for the presence of curvature. The test for curvature investigates whether the model between the response and the factors is linear and is discussed in [[Two_Level_Factorial_Experiments#Using_Center_Point_Replicates_to_Test_Curvature| Center Pt. Replicates to Test Curvature]].

====Example: Use Center Point to Get Pure Error====

Consider a <math>{2}^{2}\,\!</math> experiment design to investigate the effect of two factors, <math>A\,\!</math> and <math>B\,\!</math>, on a certain response. The energy consumed when the treatments of the <math>{2}^{2}\,\!</math> design are run is considerably larger than the energy consumed for the center point run (because at the center point the factors are at their middle levels). Therefore, the analyst decides to run only a single replicate of the design and augment the design by five replicated runs at the center point as shown in the following figure. The design properties for this experiment are shown in the second following figure. The complete experiment design is shown in the third following figure. The center points can be used in the identification of significant effects as shown next.

[[Image:doe7.17.png||center|300px|<math>2^2\,\!</math> design augmented by five center point runs.]]
[[Image:doe7_18.png|center|537px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

[[Image:doe7_19.png|center|630px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

Since the present <math>{2}^{2}\,\!</math> design is unreplicated, there are no degrees of freedom available to calculate the error sum of squares. By augmenting this design with five center points, the response values at the center points, <math>y_{i}^{c}\,\!</math>, can be used to obtain an estimate of pure error, <math>S{{S}_{PE}}\,\!</math>. Let <math>{{\bar{y}}^{c}}\,\!</math> represent the average response for the five replicates at the center. Then:

::<math>S{{S}_{PE}}=Sum\text{ }of\text{ }Squares\text{ }for\text{ }center\text{ }points\,\!</math>

::<math>\begin{align}
S{{S}_{PE}}= & \underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}} \\
= & {{(25.2-25.26)}^{2}}+...+{{(25.3-25.26)}^{2}} \\
= & 0.052
\end{align}\,\!</math>

Then the corresponding mean square is:

::<math>\begin{align}
M{{S}_{PE}}= & \frac{S{{S}_{PE}}}{degrees\text{ }of\text{ }freedom} \\
= & \frac{0.052}{5-1} \\
= & 0.013
\end{align}\,\!</math>

Alternatively, <math>M{{S}_{PE}}\,\!</math> can be directly obtained by calculating the variance of the response values at the center points:

::<math>\begin{align}
M{{S}_{PE}}= & {{s}^{2}} \\
= & \frac{\underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}}}{5-1}
\end{align}\,\!</math>

Once <math>M{{S}_{PE}}\,\!</math> is known, it can be used as the error mean square, <math>M{{S}_{E}}\,\!</math>, to carry out the test of significance for each effect. For example, to test the significance of the main effect of factor <math>A,\,\!</math> the sum of squares corresponding to this effect is obtained in the usual manner by considering only the four runs of the original <math>{2}^{2}\,\!</math> design.

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/4)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/4)J]y \\
= & 0.5625
\end{align}\,\!</math>

Then, the test statistic to test the significance of the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{A}}= & \frac{M{{S}_{A}}}{M{{S}_{E}}} \\
= & \frac{0.5625/1}{0.052/4} \\
= & 43.2692
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{A}}=43.2692\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{A}}) \\
= & 1-0.9972 \\
= & 0.0028
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value < 0.1, it can be concluded that the main effect of factor <math>A\,\!</math> significantly affects the response. This result is displayed in the ANOVA table as shown in the following figure. Test for the significance of other factors can be carried out in a similar manner.

[[Image:doe7_20.png|center|649px|Results for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

===Using Center Point Replicates to Test Curvature===

Center point replicates can also be used to check for curvature in replicated or unreplicated <math>{2}^{k}\,\!</math> designs. The test for curvature investigates whether the model between the response and the factors is linear. The way DOE++ handles center point replicates is similar to its handling of blocks. The center point replicates are treated as an additional factor in the model. The factor is labeled as Curvature in the results of the DOE folio. If Curvature turns out to be a significant factor in the results, then this indicates the presence of curvature in the model.

====Example: Use Center Point to Test Curvature====

To illustrate the use of center point replicates in testing for curvature, consider again the data of the single replicate <math>{2}^{2}\,\!</math> experiment from a preceding figure(labeled "<math>2^2</math> design augmented by five center point runs"). Let <math>{{x}_{1}}\,\!</math> be the indicator variable to indicate if the run is a center point:

::<math>\begin{matrix}
{{x}_{1}}=0 & {} & \text{Center point run} \\
{{x}_{1}}=1 & {} & \text{Other run} \\
\end{matrix}\,\!</math>

If <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> are the indicator variables representing factors <math>A\,\!</math> and <math>B\,\!</math>, respectively, then the model for this experiment is:

::<math>Y={{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}\,\!</math>

To investigate the presence of curvature, the following hypotheses need to be tested:

::<math>\begin{align}
& {{H}_{0}}: & {{\beta }_{1}}=0\text{ (Curvature is absent)} \\
& {{H}_{1}}: & {{\beta }_{1}}\ne 0
\end{align}\,\!</math>

The test statistic to be used for this test is:

::<math>{{({{F}_{0}})}_{curvature}}=\frac{M{{S}_{curvature}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{curvature}}\,\!</math> is the mean square for Curvature and <math>M{{S}_{E}}\,\!</math> is the error mean square.

'''Calculation of the Sum of Squares'''

The <math>X\,\!</math> matrix and <math>y\,\!</math> vector for this experiment are:

<center><math>X=\left[ \begin{matrix}
1 & 1 & -1 & -1 & 1 \\
1 & 1 & 1 & -1 & -1 \\
1 & 1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
\end{matrix} \right]\text{ }y=\left[ \begin{matrix}
24.6 \\
25.4 \\
25.0 \\
25.7 \\
25.2 \\
25.3 \\
25.4 \\
25.1 \\
25.3 \\
\end{matrix} \right]\,\!</math></center>

The sum of squares can now be calculated. For example, the error sum of squares is:

::<math>\begin{align}
& S{{S}_{E}}= & {{y}^{\prime }}[I-H]y \\
& = & 0.052
\end{align}\,\!</math>

where <math>I\,\!</math> is the identity matrix and <math>H\,\!</math> is the hat matrix. It can be seen that this is equal to <math>S{{S}_{PE\text{ }}}\,\!</math> (the sum of squares due to pure error) because of the replicates at the center point, as obtained in the [[Two_Level_Factorial_Experiments#Example_3| example]]. The number of degrees of freedom associated with <math>S{{S}_{E}}\,\!</math>, <math>dof(S{{S}_{E}})\,\!</math> is four. The extra sum of squares corresponding to the center point replicates (or Curvature) is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & Model\text{ }Sum\text{ }of\text{ }Squares- \\
& & Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }center\text{ }point \\
& = & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Curvature}}-(1/9)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }Curvature}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }Curvature}}={{X}_{\tilde{\ }Curv}}{{(X_{\tilde{\ }Curv}^{\prime }{{X}_{\tilde{\ }Curv}})}^{-1}}X_{\tilde{\ }Curv}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }Curv}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the center point. Thus, the extra sum of squares corresponding to Curvature is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Center}}-(1/9)J]y \\
& = & 0.7036-0.6875 \\
& = & 0.0161
\end{align}\,\!</math>

This extra sum of squares can be used to test for the significance of curvature. The corresponding mean square is:

::<math>\begin{align}
& M{{S}_{Curvature}}= & \frac{Sum\text{ }of\text{ }squares\text{ }corresponding\text{ }to\text{ }Curvature}{degrees\text{ }of\text{ }freedom} \\
& = & \frac{0.0161}{1} \\
& = & 0.0161
\end{align}\,\!</math>

'''Calculation of the Test Statistic'''

Knowing the mean squares, the statistic to check the significance of curvature can be calculated.

::<math>\begin{align}
& {{({{f}_{0}})}_{Curvature}}= & \frac{M{{S}_{Curvature}}}{M{{S}_{E}}} \\
& = & \frac{0.0161/1}{0.052/4} \\
& = & 1.24
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{Curvature}}=1.24\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and four degrees of freedom in the denominator is:

::<math>\begin{align}
& p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{Curvature}}) \\
& = & 1-0.6713 \\
& = & 0.3287
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that curvature does not exist for this design. This results is shown in the ANOVA table in the figure above. The surface of the fitted model based on these results, along with the observed response values, is shown in the figure below.

[[Image:doe7.21.png|center|400px|Model surface and observed response values for the design in the [[Two_Level_Factorial_Experiments#Example_4| example]].]]

 

==Blocking in 2k Designs==

Blocking can be used in the <math>{2}^{k}\,\!</math> designs to deal with cases when replicates cannot be run under identical conditions. Randomized complete block designs that were discussed in [[Randomization and Blocking in DOE]] for factorial experiments are also applicable here. At times, even with just two levels per factor, it is not possible to run all treatment combinations for one replicate of the experiment under homogeneous conditions. For example, each replicate of the <math>{2}^{2}\,\!</math> design requires four runs. If each run requires two hours and testing facilities are available for only four hours per day, two days of testing would be required to run one complete replicate. Blocking can be used to separate the treatment runs on the two different days. Blocks that do not contain all treatments of a replicate are called incomplete blocks. In incomplete block designs, the block effect is confounded with certain effect(s) under investigation. For the <math>{2}^{2}\,\!</math> design assume that treatments <math>(1)\,\!</math> and <math>ab\,\!</math> were run on the first day and treatments <math>a\,\!</math> and <math>b\,\!</math> were run on the second day. Then, the incomplete block design for this experiment is:

::<math>\begin{matrix}
\text{Block 1} & {} & \text{Block 2} \\
\left[ \begin{matrix}
(1) \\
ab \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
b \\
\end{matrix} \right] \\
\end{matrix}\,\!</math>

For this design the block effect may be calculated as:

::<math>\begin{align}
& Block\text{ }Effect= & Average\text{ }response\text{ }for\text{ }Block\text{ }1- \\
& & Average\text{ }response\text{ }for\text{ }Block\text{ }2 \\
& = & \frac{(1)+ab}{2}-\frac{a+b}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The <math>AB\,\!</math> interaction effect is:

::<math>\begin{align}
& AB= & Average\text{ }response\text{ }at\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{low}}}- \\
& & Average\text{ }response\text{ }at\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{low}}} \\
& = & \frac{ab+(1)}{2}-\frac{b+a}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The two equations given above show that, in this design, the <math>AB\,\!</math> interaction effect cannot be distinguished from the block effect because the formulas to calculate these effects are the same. In other words, the <math>AB\,\!</math> interaction is said to be confounded with the block effect and it is not possible to say if the effect calculated based on these equations is due to the <math>AB\,\!</math> interaction effect, the block effect or both. In incomplete block designs some effects are always confounded with the blocks. Therefore, it is important to design these experiments in such a way that the important effects are not confounded with the blocks. In most cases, the experimenter can assume that higher order interactions are unimportant. In this case, it would better to use incomplete block designs that confound these effects with the blocks.
One way to design incomplete block designs is to use defining contrasts as shown next:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}+...+{{\alpha }_{k}}{{q}_{k}}\,\!</math>

where the <math>{{\alpha }_{i}}\,\!</math> s are the exponents for the factors in the effect that is to be confounded with the block effect and the <math>{{q}_{i}}\,\!</math> s are values based on the level of the <math>i\,\!</math> the factor (in a treatment that is to be allocated to a block). For <math>{2}^{k}\,\!</math> designs the <math>{{\alpha }_{i}}\,\!</math> s are either 0 or 1 and the <math>{{q}_{i}}\,\!</math> s have a value of 0 for the low level of the <math>i\,\!</math> th factor and a value of 1 for the high level of the factor in the treatment under consideration. As an example, consider the <math>{2}^{2}\,\!</math> design where the interaction effect <math>AB\,\!</math> is confounded with the block. Since there are two factors, <math>k=2\,\!</math>, with <math>i=1\,\!</math> representing factor <math>A\,\!</math> and <math>i=2\,\!</math> representing factor <math>B\,\!</math>. Therefore:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}\,\!</math>

The value of <math>{{\alpha }_{1}}\,\!</math> is one because the exponent of factor <math>A\,\!</math> in the confounded interaction <math>AB\,\!</math> is one. Similarly, the value of <math>{{\alpha }_{2}}\,\!</math> is one because the exponent of factor <math>B\,\!</math> in the confounded interaction <math>AB\,\!</math> is also one. Therefore, the defining contrast for this design can be written as:

::<math>\begin{align}
& L= & {{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}} \\
& = & 1\cdot {{q}_{1}}+1\cdot {{q}_{2}} \\
& = & {{q}_{1}}+{{q}_{2}}
\end{align}\,\!</math>

Once the defining contrast is known, it can be used to allocate treatments to the blocks. For the <math>{2}^{2}\,\!</math> design, there are four treatments <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math> and <math>ab\,\!</math>. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. In order to decide which block the treatment <math>(1)\,\!</math> belongs to, the levels of factors <math>A\,\!</math> and <math>B\,\!</math> for this run are used. Since factor <math>A\,\!</math> is at the low level in this treatment, <math>{{q}_{1}}=0\,\!</math>. Similarly, since factor <math>B\,\!</math> is also at the low level in this treatment, <math>{{q}_{2}}=0\,\!</math>. Therefore:

::<math>\begin{align}
& L= & {{q}_{1}}+{{q}_{2}} \\
& = & 0+0=0\text{ (mod 2)}
\end{align}\,\!</math>

Note that the value of <math>L\,\!</math> used to decide the block allocation is "mod 2" of the original value. This value is obtained by taking the value of 1 for odd numbers and 0 otherwise. Based on the value of <math>L\,\!</math>, treatment <math>(1)\,\!</math> is assigned to block 1. Other treatments can be assigned using the following calculations:

::<math>\begin{align}
& (1): & \text{ }L=0+0=0=0\text{ (mod 2)} \\
& a: & \text{ }L=1+0=1=1\text{ (mod 2)} \\
& b: & \text{ }L=0+1=1=1\text{ (mod 2)} \\
& ab: & \text{ }L=1+1=2=0\text{ (mod 2)}
\end{align}\,\!</math>

Therefore, to confound the interaction <math>AB\,\!</math> with the block effect in the <math>{2}^{2}\,\!</math> incomplete block design, treatments <math>(1)\,\!</math> and <math>ab\,\!</math> (with <math>L=0\,\!</math>) should be assigned to block 2 and treatment combinations <math>a\,\!</math> and <math>b\,\!</math> (with <math>L=1\,\!</math>) should be assigned to block 1.

====Example: Two Level Factorial Design with Two Blocks====

This example illustrates how treatments can be allocated to two blocks for an unreplicated <math>{2}^{k}\,\!</math> design. Consider the unreplicated <math>{2}^{4}\,\!</math> design to investigate the four factors affecting the defects in automobile vinyl panels discussed in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments required for this experiment were run by two different operators with each operator conducting 8 runs. This experiment is an example of an incomplete block design. The analyst in charge of this experiment assumed that the interaction <math>ABCD\,\!</math> was not significant and decided to allocate treatments to the two operators so that the <math>ABCD\,\!</math> interaction was confounded with the block effect (the two operators are the blocks). The allocation scheme to assign treatments to the two operators can be obtained as follows.
 
The defining contrast for the <math>{2}^{4}\,\!</math> design where the <math>ABCD\,\!</math> interaction is confounded with the blocks is:

::<math>L={{q}_{1}}+{{q}_{2}}+{{q}_{3}}+{{q}_{4}}\,\!</math>

The treatments can be allocated to the two operators using the values of the defining contrast. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. Then the value of the defining contrast for treatment <math>a\,\!</math> is:

::<math>a\ \ :\ \ \text{ }L=1+0+0+0=1=1\text{ (mod 2)}\,\!</math>

Therefore, treatment <math>a\,\!</math> should be assigned to Block 1 or the first operator. Similarly, for treatment <math>ab\,\!</math> we have:

::<math>ab\ \ :\ \ \text{ }L=1+1+0+0=2=0\text{ (mod 2)}\,\!</math>

[[Image:doe7.22.png|center|200px| Allocation of treatments to two blocks for the <math>2^4</math> design in the example by confounding interaction of <math>ABCD</math> with the blocks.]]

Therefore, <math>ab\,\!</math> should be assigned to Block 2 or the second operator. Other treatments can be allocated to the two operators in a similar manner to arrive at the allocation scheme shown in the figure below.
In DOE++, to confound the <math>ABCD\,\!</math> interaction for the <math>{2}^{4}\,\!</math> design into two blocks, the number of blocks are specified as shown in the figure below. Then the interaction <math>ABCD\,\!</math> is entered in the Block Generator window (second following figure) which is available using the Block Generator button in the following figure. The design generated by the Weibull++ DOE folio is shown in the third of the following figures. This design matches the allocation scheme of the preceding figure.

[[Image:doe7_23.png|center|700px| Adding block properties for the experiment in the example.|link=]]

[[Image:doe7_24.png|center|700px|Specifying the interaction ABCD as the interaction to be confounded with the blocks for the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

[[Image:doe7_25.png|center|800px|Two block design for the experiment in the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

For the analysis of this design, the sum of squares for all effects are calculated assuming no blocking. Then, to account for blocking, the sum of squares corresponding to the <math>ABCD\,\!</math> interaction is considered as the sum of squares due to blocks and <math>ABCD\,\!</math>. In DOE++ this is done by displaying this sum of squares as the sum of squares due to the blocks. This is shown in the following figure where the sum of squares in question is obtained as 72.25 and is displayed against Block. The interaction ABCD, which is confounded with the blocks, is not displayed. Since the design is unreplicated, any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_26.png|center|793px|ANOVA table for the experiment of the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

===Unreplicated 2''k'' Designs in 2''p'' Blocks===

A single replicate of the <math>{2}^{k}\,\!</math> design can be run in up to <math>{2}^{p}\,\!</math> blocks where <math>p<k\,\!</math>. The number of effects confounded with the blocks equals the degrees of freedom associated with the block effect.

If two blocks are used (the block effect has two levels), then one (<math>2-1=1)\,\!</math> effect is confounded with the blocks. If four blocks are used, then three (<math>4-1=3\,\!</math>) effects are confounded with the blocks and so on. For example an unreplicated <math>{2}^{4}\,\!</math> design may be confounded in <math>{2}^{2}\,\!</math> (four) blocks using two contrasts, <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}}\,\!</math>. Let <math>AC\,\!</math> and <math>BD\,\!</math> be the effects to be confounded with the blocks. Corresponding to these two effects, the contrasts are respectively:

::<math>\begin{align}
& {{L}_{1}}= & {{q}_{1}}+{{q}_{3}} \\
& {{L}_{2}}= & {{q}_{2}}+{{q}_{4}}
\end{align}\,\!</math>

Based on the values of <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}},\,\!</math> the treatments can be assigned to the four blocks as follows:

<center><math>\begin{matrix}
\text{Block 4} & {} & \text{Block 3} & {} & \text{Block 2} & {} & \text{Block 1} \\
{{L}_{1}}=0,{{L}_{2}}=0 & {} & {{L}_{1}}=1,{{L}_{2}}=0 & {} & {{L}_{1}}=0,{{L}_{2}}=1 & {} & {{L}_{1}}=1,{{L}_{2}}=1 \\
{} & {} & {} & {} & {} & {} & {} \\
\left[ \begin{matrix}
(1) \\
ac \\
bd \\
abcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
c \\
abd \\
bcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
b \\
abc \\
d \\
acd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
ab \\
bc \\
ad \\
cd \\
\end{matrix} \right] \\
\end{matrix}\,\!</math></center>

Since the block effect has three degrees of freedom, three effects are confounded with the block effect. In addition to <math>AC\,\!</math> and <math>BD\,\!</math>, the third effect confounded with the block effect is their generalized interaction, <math>(AC)(BD)=ABCD\,\!</math>.
In general, when an unreplicated <math>{2}^{k}\,\!</math> design is confounded in <math>{2}^{p}\,\!</math> blocks, <math>p\,\!</math> contrasts are needed (<math>{{L}_{1}},{{L}_{2}}...{{L}_{p}}\,\!</math>). <math>p\,\!</math> effects are selected to define these contrasts such that none of these effects are the generalized interaction of the others. The <math>{2}^{p}\,\!</math> blocks can then be assigned the treatments using the <math>p\,\!</math> contrasts. <math>{{2}^{p}}-(p+1)\,\!</math> effects, that are also confounded with the blocks, are then obtained as the generalized interaction of the <math>p\,\!</math> effects. In the statistical analysis of these designs, the sum of squares are computed as if no blocking were used. Then the block sum of squares is obtained by adding the sum of squares for all the effects confounded with the blocks.

====Example: 2 Level Factorial Design with Four Blocks====

This example illustrates how a DOE folio obtains the sum of squares when treatments for an unreplicated <math>{2}^{k}\,\!</math> design are allocated among four blocks. Consider again the unreplicated <math>{2}^{4}\,\!</math> design used to investigate the defects in automobile vinyl panels presented in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments needed to complete the experiment were run by four operators. Therefore, there are four blocks. Assume that the treatments were allocated to the blocks using the generators mentioned in the previous section, i.e., treatments were allocated among the four operators by confounding the effects, <math>AC\,\!</math> and <math>BD,\,\!</math> with the blocks. These effects can be specified as Block Generators as shown in the following figure. (The generalized interaction of these two effects, interaction <math>ABCD\,\!</math>, will also get confounded with the blocks.) The resulting design is shown in the second following figure and matches the allocation scheme obtained in the previous section.

[[Image:doe7_27.png|center|700px|Specifying the interactions AC and BD as block generators for the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

The sum of squares in this case can be obtained by calculating the sum of squares for each of the effects assuming there is no blocking. Once the individual sum of squares have been obtained, the block sum of squares can be calculated. The block sum of squares is the sum of the sum of squares of effects, <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, since these effects are confounded with the block effect. As shown in the second following figure, this sum of squares is 92.25 and is displayed against Block. The interactions <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, which are confounded with the blocks, are not displayed. Since the present design is unreplicated any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_28.png|center|800px|Design for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

[[Image:doe7_29.png|center|793px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

==Variability Analysis==

For replicated two level factorial experiments, DOE++ provides the option of conducting variability analysis (using the Variability Analysis icon under the Data menu). The analysis is used to identify the treatment that results in the least amount of variation in the product or process being investigated. Variability analysis is conducted by treating the standard deviation of the response for each treatment of the experiment as an additional response. The standard deviation for a treatment is obtained by using the replicated response values at that treatment run. As an example, consider the <math>{2}^{3}\,\!</math> design shown in the following figure where each run is replicated four times. A variability analysis can be conducted for this design. DOE++ calculates eight standard deviation values corresponding to each treatment of the design (see second following figure). Then, the design is analyzed as an unreplicated <math>{2}^{3}\,\!</math> design with the standard deviations (displayed as Y Standard Deviation. in second following figure) as the response. The normal probability plot of effects identifies <math>AC\,\!</math> as the effect that influences variability (see third figure following). Based on the effect coefficients obtained in the fourth figure following, the model for Y Std. is:

::<math>\begin{align}
& \text{Y Std}\text{.}= & 0.6779+0.2491\cdot AC \\
& = & 0.6779+0.2491{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

Based on the model, the experimenter has two choices to minimize variability (by minimizing Y Std.). The first choice is that <math>{{x}_{1}}\,\!</math> should be <math>1\,\!</math> (i.e., <math>A\,\!</math> should be set at the high level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the low level). The second choice is that <math>{{x}_{1}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>A\,\!</math> should be set at the low level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the high level). The experimenter can select the most feasible choice.

[[Image:doe7.30.png|center|391px|A <math>2^3\,\!</math> design with four replicated response values that can be used to conduct a variability analysis.]]
 

[[Image:doe7_31.png|center|727px|Variability analysis in DOE++.|link=]]
 

[[Image:doe7_32.png|center|650px|Normal probability plot of effects for the variability analysis example.|link=]]
 

[[Image:doe7_33.png|center|727px|Effect coefficients for the variability analysis example.|link=]]

 

==Two Level Fractional Factorial Designs==

As the number of factors in a two level factorial design increases, the number of runs for even a single replicate of the <math>{2}^{k}\,\!</math> design becomes very large. For example, a single replicate of an eight factor two level experiment would require 256 runs. Fractional factorial designs can be used in these cases to draw out valuable conclusions from fewer runs. The basis of fractional factorial designs is the ''sparsity of effects'' principle.[[DOE_References|[Wu, 2000]]] The principle states that, most of the time, responses are affected by a small number of main effects and lower order interactions, while higher order interactions are relatively unimportant. Fractional factorial designs are used as screening experiments during the initial stages of experimentation. At these stages, a large number of factors have to be investigated and the focus is on the main effects and two factor interactions. These designs obtain information about main effects and lower order interactions with fewer experiment runs by confounding these effects with unimportant higher order interactions. As an example, consider a <math>{2}^{8}\,\!</math> design that requires 256 runs. This design allows for the investigation of 8 main effects and 28 two factor interactions. However, 219 degrees of freedom are devoted to three factor or higher order interactions. This full factorial design can prove to be very inefficient when these higher order interactions can be assumed to be unimportant. Instead, a fractional design can be used here to identify the important factors that can then be investigated more thoroughly in subsequent experiments. In unreplicated fractional factorial designs, no degrees of freedom are available to calculate the error sum of squares and the techniques mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] should be employed for the analysis of these designs.

==Half-fraction Designs==

A half-fraction of the <math>{2}^{k}\,\!</math> design involves running only half of the treatments of the full factorial design. For example, consider a <math>{2}^{3}\,\!</math> design that requires eight runs in all. The design matrix for this design is shown in the figure (a) below. A half-fraction of this design is the design in which only four of the eight treatments are run. The fraction is denoted as <math>{2}^{3-1}\,\!</math> with the "<math>-1\,\!</math>" in the index denoting a half-fraction. Assume that the treatments chosen for the half-fraction design are the ones where the interaction <math>ABC\,\!</math> is at the high level (i.e., only those rows are chosen from the following figure (a) where the column for <math>ABC\,\!</math> has entries of 1). The resulting <math>{2}^{3-1}\,\!</math> design has a design matrix as shown in figure (b) below.

[[Image:doe7.34.png|center|330px|Half-fractions of the <math>2^3\,\!</math> design. (a) shows the full factorial <math>2^3\,\!</math> design, (b) shows the <math>2^{3-1}\,\!</math> design with the defining relation <math>I=ABC\,\!</math> and (c) shows the <math>{2}^{3-1}\,\!</math> design with the defining relation <math>I=-ABC\,\!</math>.]]

In the <math>{2}^{3-1}\,\!</math> design of figure (b), since the interaction <math>ABC\,\!</math> is always included at the same level (the high level represented by 1), it is not possible to measure this interaction effect. The effect, <math>ABC\,\!</math>, is called the ''generator'' or ''word'' for this design. It can be noted that, in the design matrix of the following figure (b), the column corresponding to the intercept, <math>I\,\!</math>, and column corresponding to the interaction <math>ABC\,\!</math>, are identical. The identical columns are written as <math>I=ABC\,\!</math> and this equation is called the ''defining relation'' for the design. In a DOE folio, the present <math>{2}^{3-1}\,\!</math> design can be obtained by specifying the design properties as shown in the following figure.

[[Image:doe7_35.png|center|700px|Design properties for the <math>2^{3-1}\,\!</math> design.|link=]]

The defining relation, <math>I=ABC\,\!</math>, is entered in the Fraction Generator window as shown next.

[[Image:doe7_36.png|center|700px|Specifying the defining relation for the <math>2^{3-1}\,\!</math> design.|link=]]

Note that in the figure following that, the defining relation is specified as <math>C=AB\,\!</math>. This relation is obtained by multiplying the defining relation, <math>I=ABC\,\!</math>, by the last factor, <math>C\,\!</math>, of the design.

===Calculation of Effects===

Using the four runs of the <math>{2}^{3-1}\,\!</math> design in figure (b) discussed above, the main effects can be calculated as follows:

::<math>\begin{align}
& A= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& B= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& C= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

where <math>a\,\!</math>, <math>b\,\!</math>, <math>c\,\!</math> and <math>abc\,\!</math> are the treatments included in the <math>{2}^{3-1}\,\!</math> design.

Similarly, the two factor interactions can also be obtained as:

::<math>\begin{align}
& BC= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& AC= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& AB= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

The equations for <math>A\,\!</math> and <math>BC\,\!</math> above result in the same effect values showing that effects <math>A\,\!</math> and <math>BC\,\!</math> are confounded in the present <math>{2}^{3-1}\,\!</math> design. Thus, the quantity, <math>\tfrac{1}{2}(a-b-c+abc),\,\!</math> estimates <math>A+BC\,\!</math> (i.e., both the main effect <math>A\,\!</math> and the two-factor interaction <math>BC\,\!</math>). The effects, <math>A\,\!</math> and <math>BC,\,\!</math> are called ''aliases''. From the remaining equations given above, it can be seen that the other aliases for this design are <math>B\,\!</math> and <math>AC\,\!</math>, and <math>C\,\!</math> and <math>AB\,\!</math>. Therefore, the equations to calculate the effects in the present <math>{2}^{3-1}\,\!</math> design can be written as follows:

::<math>\begin{align}
& A+BC= & \frac{1}{2}(a-b-c+abc) \\
& B+AC= & \frac{1}{2}(-a+b-c+abc) \\
& C+AB= & \frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

===Calculation of Aliases===

Aliases for a fractional factorial design can be obtained using the defining relation for the design. The defining relation for the present <math>{2}^{3-1}\,\!</math> design is:

::<math>I=ABC\,\!</math>

Multiplying both sides of the previous equation by the main effect, <math>A,\,\!</math> gives the alias effect of <math>A\,\!</math> :

::<math>\begin{align}
& A\cdot I= & A\cdot ABC \\
& A= & {{A}^{2}}BC \\
& A= & BC
\end{align}\,\!</math>

Note that in calculating the alias effects, any effect multiplied by <math>I\,\!</math> remains the same (<math>A\cdot I=A\,\!</math>), while an effect multiplied by itself results in <math>I\,\!</math> (<math>{{A}^{2}}=I\,\!</math>). Other aliases can also be obtained:

::<math>\begin{align}
& B\cdot I= & B\cdot ABC \\
& B= & A{{B}^{2}}C \\
& B= & AC
\end{align}\,\!</math>

:and:

::<math>\begin{align}
& C\cdot I= & C\cdot ABC \\
& C= & AB{{C}^{2}} \\
& C= & AB
\end{align}\,\!</math>

===Fold-over Design===

If it can be assumed for this design that the two-factor interactions are unimportant, then in the absence of <math>BC\,\!</math>, <math>AC\,\!</math> and <math>AB\,\!</math>, the equations for (A+BC), (B+AC) and (C+AB) can be used to estimate the main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. However, if such an assumption is not applicable, then to uncouple the main effects from their two factor aliases, the alternate fraction that contains runs having <math>ABC\,\!</math> at the lower level should be run. The design matrix for this design is shown in the preceding figure (c). The defining relation for this design is <math>I=-ABC\,\!</math> because the four runs for this design are obtained by selecting the rows of the preceding figure (a) for which the value of the <math>ABC\,\!</math> column is <math>-1\,\!</math>. The aliases for this fraction can be obtained as explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]] as <math>A=-BC\,\!</math>, <math>B=-AC\,\!</math> and <math>C=-AB\,\!</math>. The effects for this design can be calculated as:

::<math>\begin{align}
& A-BC= & \frac{1}{2}(ab+ac-(1)-bc) \\
& B-AC= & \frac{1}{2}(ab-ac+(1)-bc) \\
& C-AB= & \frac{1}{2}(-ab+ac-(1)+bc)
\end{align}\,\!</math>

These equations can be combined with the equations for (A+BC), (B+AC) and (C+AB) to obtain the de-aliased main effects and two factor interactions. For example, adding equations (A+BC) and (A-BC) returns the main effect <math>A\,\!</math>.

::<math>\begin{align}
& 2A= & \frac{1}{2}(a-b-c+abc)+ \\
& & \frac{1}{2}(ab+ac-(1)-bc)
\end{align}\,\!</math>

The process of augmenting a fractional factorial design by a second fraction of the same size by simply reversing the signs (of all effect columns except <math>I\,\!</math>) is called ''folding over''. The combined design is referred to as a ''fold-over design''.

==Quarter and Smaller Fraction Designs==

At times, the number of runs even for a half-fraction design are very large. In these cases, smaller fractions are used. A quarter-fraction design, denoted as <math>{2}^{k-2}\,\!</math>, consists of a fourth of the runs of the full factorial design. Quarter-fraction designs require two defining relations. The first defining relation returns the half-fraction or the <math>{2}^{k-1}\,\!</math> design. The second defining relation selects half of the runs of the <math>{2}^{k-1}\,\!</math> design to give the quarter-fraction. For example, consider the <math>{2}^{4}\,\!</math> design. To obtain a <math>{2}^{4-2}\,\!</math> design from this design, first a half-fraction of this design is obtained by using a defining relation. Assume that the defining relation used is <math>I=ABCD\,\!</math>. The design matrix for the resulting <math>{2}^{4-1}\,\!</math> design is shown in figure (a) below. Now, a quarter-fraction can be obtained from the <math>{2}^{4-1}\,\!</math> design shown in figure (a) below using a second defining relation <math>I=AD\,\!</math>. The resulting <math>{2}^{4-2}\,\!</math> design obtained is shown in figure (b) below.

[[Image:doe7.37.png|center|465px|Fractions of the <math>2^4\,\!</math> design - Figure (a) shows the <math>2^{4-1}</math> design with the defining relation <math>I=ABCD\,\!</math> and (b) shows the <math>2^{4-2}\,\!</math> design with the defining relation <math>I=ABCD=AD=BC\,\!</math>.]]

The complete defining relation for this <math>{2}^{4-2}\,\!</math> design is:

::<math>I=ABCD=AD=BC\,\!</math>

Note that the effect, <math>BC,\,\!</math> in the defining relation is the generalized interaction of <math>ABCD\,\!</math> and <math>AD\,\!</math> and is obtained using <math>(ABCD)(AD)={{A}^{2}}BC{{D}^{2}}=BC\,\!</math>. In general, a <math>{2}^{k-p}\,\!</math> fractional factorial design requires <math>p\,\!</math> independent generators. The defining relation for the design consists of the <math>p\,\!</math> independent generators and their <math>{2}^{p}\,\!</math> - (<math>p\,\!</math> +1) generalized interactions.

===Calculation of Aliases===

The alias structure for the present <math>{2}^{4-2}\,\!</math> design can be obtained using the defining relation of equation (I=ABCD=AD=BC) following the procedure explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]]. For example, multiplying the defining relation by <math>A\,\!</math> returns the effects aliased with the main effect, <math>A\,\!</math>, as follows:

::<math>\begin{align}
& A\cdot I= & A\cdot ABCD=A\cdot AD=A\cdot BC \\
& A= & {{A}^{2}}BCD={{A}^{2}}D=ABC \\
& A= & BCD=D=ABC
\end{align}\,\!</math>

Therefore, in the present <math>{2}^{4-2}\,\!</math> design, it is not possible to distinguish between effects <math>A\,\!</math>, <math>D\,\!</math>, <math>BCD\,\!</math> and <math>ABC\,\!</math>. Similarly, multiplying the defining relation by <math>B\,\!</math> and <math>AB\,\!</math> returns the effects that are aliased with these effects:

::<math>\begin{align}
& B= & ACD=ABD=C \\
& AB= & CD=AD=AC
\end{align}\,\!</math>

Other aliases can be obtained in a similar way. It can be seen that each effect in this design has three aliases. In general, each effect in a <math>{2}^{k-p}\,\!</math> design has <math>{2}^{p-1}\,\!</math> aliases.
The aliases for the <math>{2}^{4-2}\,\!</math> design show that in this design the main effects are aliased with each other (<math>A\,\!</math> is aliased with <math>D\,\!</math> and <math>B\,\!</math> is aliased with <math>C\,\!</math>). Therefore, this design is not a useful design and is not available in DOE++. It is important to ensure that main effects and lower order interactions of interest are not aliased in a fractional factorial design. This is known by looking at the resolution of the fractional factorial design.

==Design Resolution==

The resolution of a fractional factorial design is defined as the number of factors in the lowest order effect in the defining relation. For example, in the defining relation <math>I=ABCD=AD=BC\,\!</math> of the previous <math>{2}^{4-2}\,\!</math> design, the lowest-order effect is either <math>AD\,\!</math> or <math>BC,\,\!</math> containing two factors. Therefore, the resolution of this design is equal to two. The resolution of a fractional factorial design is represented using Roman numerals. For example, the previously mentioned <math>{2}^{4-2}\,\!</math> design with a resolution of two can be represented as 2 <math>_{\text{II}}^{4-2}\,\!</math>. The resolution provides information about the confounding in the design as explained next:
 
 
#'''Resolution III Designs''' In these designs, the lowest order effect in the defining relation has three factors (e.g., a <math>{2}^{5-2}\,\!</math> design with the defining relation <math>I=ABDE=ABC=CDE\,\!</math>). In resolution III designs, no main effects are aliased with any other main effects, but main effects are aliased with two factor interactions. In addition, some two factor interactions are aliased with each other.
#'''Resolution IV Designs''' In these designs, the lowest order effect in the defining relation has four factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABDE\,\!</math>). In resolution IV designs, no main effects are aliased with any other main effects or two factor interactions. However, some main effects are aliased with three factor interactions and the two factor interactions are aliased with each other.
#'''Resolution V Designs''' In these designs the lowest order effect in the defining relation has five factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABCDE\,\!</math>). In resolution V designs, no main effects or two factor interactions are aliased with any other main effects or two factor interactions. However, some main effects are aliased with four factor interactions and the two factor interactions are aliased with three factor interactions.

Fractional factorial designs with the highest resolution possible should be selected because the higher the resolution of the design, the less severe the degree of confounding. In general, designs with a resolution less than III are never used because in these designs some of the main effects are aliased with each other. The table below shows fractional factorial designs with the highest available resolution for three to ten factor designs along with their defining relations.

[[Image:doet7.3.png|center|474px|Highest resolution designs available for fractional factorial designs with 3 to 10 factors.]]

All of the two level fractional factorial designs available in DOE++ are shown next.

[[Image:doe7.38.png|center|471px|Two level fractional factorial designs available in DOE++ and their resolutions.]]

===Minimum Aberration Designs===
At times, different designs with the same resolution but different aliasing may be available. The best design to select in such a case is the minimum aberration design. For example, all <math>{2}^{7-2}\,\!</math> designs in the fourth table have a resolution of four (since the generator with the minimum number of factors in each design has four factors). Design <math>1\,\!</math> has three generators of length four (<math>ABCF,\,\!</math> <math>BCDG,\,\!</math> <math>ADFG\,\!</math>). Design <math>2\,\!</math> has two generators of length four (<math>ABCF,\,\!</math> <math>ADEG\,\!</math>). Design <math>3\,\!</math> has one generator of length four (<math>CEFG\,\!</math>). Therefore, design <math>3\,\!</math> has the least number of generators with the minimum length of four. Design <math>3\,\!</math> is called the minimum aberration design. It can be seen that the alias structure for design <math>3\,\!</math> is less involved compared to the other designs. For details refer to [[DOE_References|[Wu, 2000]]].

[[Image:doet7.4.png|center|432px|Three <math>2_{IV}^{7-2}\,\!</math> designs with different defining relations.]]

====Example====

The design of an automobile fuel cone is thought to be affected by six factors in the manufacturing process: cavity temperature (factor <math>A\,\!</math>), core temperature (factor <math>B\,\!</math>), melt temperature (factor <math>C\,\!</math>), hold pressure (factor <math>D\,\!</math>), injection speed (factor <math>E\,\!</math>) and cool time (factor <math>F\,\!</math>). The manufacturer of the fuel cone is unable to run the <math>{2}^{6}=64\,\!</math> runs required to complete one replicate for a two level full factorial experiment with six factors. Instead, they decide to run a fractional factorial design. Considering that three factor and higher order interactions are likely to be inactive, the manufacturer selects a <math>{2}^{6-2}\,\!</math> design that will require only 16 runs. The manufacturer chooses the resolution IV design which will ensure that all main effects are free from aliasing (assuming three factor and higher order interactions are absent). However, in this design the two factor interactions may be aliased with each other. It is decided that, if important two factor interactions are found to be present, additional experiment trials may be conducted to separate the aliased effects. The performance of the fuel cone is measured on a scale of 1 to 15. In DOE++, the design for this experiment is set up using the properties shown in the following figure. The Fraction Generators for the design, <math>E=ABC\,\!</math> and <math>F=BCD\,\!</math>, are the same as the defaults used in DOE++. The resulting <math>{2}^{6-2}\,\!</math> design and the corresponding response values are shown in the following two figures.

[[Image:doe7_39.png|center|644px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

[[Image:doe7_40.png|center|534px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The complete alias structure for the 2 <math>_{\text{IV}}^{6-2}\,\!</math> design is shown next.

<center><math>I=ABCE=ADEF=BCDF\,\!</math></center>

 
<center><math>\begin{align}
& A= & BCE=DEF=ABCDF \\
& B= & ACE=CDF=ABDEF \\
& C= & ABE=BDF=ACDEF \\
& D= & AEF=BCF=ABCDE \\
& E= & ABC=ADF=BCDEF \\
& F= & ADE=BCD=ABCEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& AB= & CE=ACDF=BDEF \\
& AC= & BE=ABDF=CDEF \\
& AD= & EF=ABCF=BCDE \\
& AE= & BC=DF=ABCDEF \\
& AF= & DE=ABCD=BCEF \\
& BD= & CF=ABEF=ACDE \\
& BF= & CD=ABDE=ACEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& ABD= & ACF=BEF=CDE \\
& ABF= & ACD=BDE=CEF
\end{align}\,\!</math></center>

In DOE++, the alias structure is displayed in the Design Summary and as part of the Design Evaluation result, as shown next:

[[Image:doe7_41.png|center|700px|Alias structure for the experiment design in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The normal probability plot of effects for this unreplicated design shows the main effects of factors <math>C\,\!</math> and <math>D\,\!</math> and the interaction effect, <math>BF\,\!</math>, to be significant (see the following figure).

[[Image:doe7_42.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

From the alias structure, it can be seen that for the present design interaction effect, <math>BF,\,\!</math> is confounded with <math>CD\,\!</math>. Therefore, the actual source of this effect cannot be known on the basis of the present experiment. However because neither factor <math>B\,\!</math> nor <math>F\,\!</math> is found to be significant there is an indication the observed effect is likely due to interaction, <math>CD\,\!</math>. To confirm this, a follow-up <math>{2}^{2}\,\!</math> experiment is run involving only factors <math>B\,\!</math> and <math>F\,\!</math>. The interaction, <math>BF\,\!</math>, is found to be inactive, leading to the conclusion that the interaction effect in the original experiment is effect, <math>CD\,\!</math>. Given these results, the fitted regression model for the fuel cone design as per the coefficients obtained from DOE++ is shown next.

::<math>\hat{y}=7.6875+C+2\cdot D+2.1875\cdot CD\,\!</math>

[[Image:doe7_43.png|center|700px|Effect coefficients for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

==Projection==

Projection refers to the reduction of a fractional factorial design to a full factorial design by dropping out some of the factors of the design. Any fractional factorial design of resolution, <math>R,\,\!</math> can be reduced to complete factorial designs in any subset of <math>R-1\,\!</math> factors. For example, consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. The resolution of this design is four. Therefore, this design can be reduced to full factorial designs in any three (<math>4-1=3\,\!</math>) of the original seven factors (by dropping the remaining four of factors). Further, a fractional factorial design can also be reduced to a full factorial design in any <math>R\,\!</math> of the original factors, as long as these <math>R\,\!</math> factors are not part of the generator in the defining relation. Again consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. This design can be reduced to a full factorial design in four factors provided these four factors do not appear together as a generator in the defining relation. The complete defining relation for this design is:

::<math>\begin{align}
& I= & ABCE=BCDF=ACDG=ADEF=ABFG=BDEG=CEFG
\end{align}\,\!</math>

Therefore, there are seven four factor combinations out of the 35 (<math>(_{7}^{4})=35\,\!</math>) possible four-factor combinations that are used as generators in the defining relation. The designs with the remaining 28 four factor combinations would be full factorial 16-run designs. For example, factors <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math> do not occur as a generator in the defining relation of the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. If the remaining factors, <math>E\,\!</math>, <math>F\,\!</math> and <math>G\,\!</math>, are dropped, the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design will reduce to a full factorial design in <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math>.

==Resolution III Designs==

At times, the factors to be investigated in screening experiments are so large that even running a fractional factorial design is impractical. This can be partially solved by using resolution III fractional factorial designs in the cases where three factor and higher order interactions can be assumed to be unimportant. Resolution III designs, such as the 2 <math>_{\text{III}}^{3-1}\,\!</math> design, can be used to estimate <math>k\,\!</math> main effects using just <math>k+1\,\!</math> runs. In these designs, the main effects are aliased with two factor interactions. Once the results from these designs are obtained, and knowing that three factor and higher order interactions are unimportant, the experimenter can decide if there is a need to run a fold-over design to de-alias the main effects from the two factor interactions. Thus, the 2 <math>_{\text{III}}^{3-1}\,\!</math> design can be used to investigate three factors in four runs, the 2 <math>_{\text{III}}^{7-4}\,\!</math> design can be used to investigate seven factors in eight runs, the 2 <math>_{\text{III}}^{15-11}\,\!</math> design can be used to investigate fifteen factors in sixteen runs and so on.

====Example====

{{:Resolution_III_Design_Example}}

==Alias Matrix==
In [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction designs]] and [[Two_Level_Factorial_Experiments#Quarter_and_Smaller_Fraction_Designs| Quarter and Smaller Fraction Designs]], the alias structure for fractional factorial designs was obtained using the defining relation. However, this method of obtaining the alias structure is not very efficient when the alias structure is very complex or when partial aliasing is involved. One of the ways to obtain the alias structure for any design, regardless of its complexity, is to use the alias matrix. The alias matrix for a design is calculated using <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> where <math>{{X}_{1}}\,\!</math> is the portion of the design matrix, <math>X,\,\!</math> that contains the effects for which the aliases need to be calculated, and <math>{{X}_{2}}\,\!</math> contains the remaining columns of the design matrix, other than those included in <math>{{X}_{1}}\,\!</math>.

To illustrate the use of the alias matrix, consider the design matrix for the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design (using the defining relation <math>I=ABCD\,\!</math>) shown next:

[[Image:Chapter7__879.png|center|link=]]

The alias structure for this design can be obtained by defining <math>{{X}_{1}}\,\!</math> using eight columns since the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design estimates eight effects. If the first eight columns of <math>X\,\!</math> are used then <math>{{X}_{1}}\,\!</math> is:

[[Image:Chapter7__884.png|center|link=]]

<math>{{X}_{2}}\,\!</math> is obtained using the remaining columns as:

[[Image:Chapter7__886.png|center|link=]]

Then the alias matrix <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> is:

[[Image:Chapter7__888.png|center|link=]]

The alias relations can be easily obtained by observing the alias matrix as:

<center><math>\begin{align}
& I= & ABCD \\
& A= & BCD \\
& B= & ACD \\
& AB= & CD \\
& C= & ABD \\
& AC= & BD \\
& BC= & AD \\
& D= & ABC
\end{align}\,\!</math></center>

Two Level Factorial Experiments

2017-08-10T17:35:07Z

Richard House: /* Example: 2 Level Factorial Design with Four Blocks */

{{Template:Doebook|8}}
Two level factorial experiments are factorial experiments in which each factor is investigated at only two levels. The early stages of experimentation usually involve the investigation of a large number of potential factors to discover the "vital few" factors. Two level factorial experiments are used during these stages to quickly filter out unwanted effects so that attention can then be focused on the important ones.

==2''k'' Designs==
The factorial experiments, where all combination of the levels of the factors are run, are usually referred to as ''full factorial experiments''. Full factorial two level experiments are also referred to as <math>{2}^{k}\,\!</math> designs where <math>k\,\!</math> denotes the number of factors being investigated in the experiment. In Weibull++ DOE folios, these designs are referred to as 2 Level Factorial Designs as shown in the figure below.

[[Image:doe7_1.png|center|487px|Selection of full factorial experiments with two levels in Weibull++.|link=]]

A full factorial two level design with <math>k\,\!</math> factors requires <math>{{2}^{k}}\,\!</math> runs for a single replicate. For example, a two level experiment with three factors will require <math>2\times 2\times 2={{2}^{3}}=8\,\!</math> runs. The choice of the two levels of factors used in two level experiments depends on the factor; some factors naturally have two levels. For example, if gender is a factor, then male and female are the two levels. For other factors, the limits of the range of interest are usually used. For example, if temperature is a factor that varies from <math>{45}^{o}C\,\!</math> to <math>{90}^{o}C\,\!</math>, then the two levels used in the <math>{2}^{k}\,\!</math> design for this factor would be <math>{45}^{o}\,\!C\,\!</math> and <math>{90}^{o}\,\!C\,\!</math>.

The two levels of the factor in the <math>{2}^{k}\,\!</math> design are usually represented as <math>-1\,\!</math> (for the first level) and <math>1\,\!</math> (for the second level). Note that this representation is reversed from the coding used in [[General Full Factorial Designs]] for the indicator variables that represent two level factors in ANOVA models. For ANOVA models, the first level of the factor was represented using a value of <math>1\,\!</math> for the indicator variable, while the second level was represented using a value of <math>-1\,\!</math>. For details on the notation used for two level experiments refer to [[Two_Level_Factorial_Experiments#Notation| Notation]].

===The 22 Design===

The simplest of the two level factorial experiments is the <math>{2}^{2}\,\!</math> design where two factors (say factor <math>A\,\!</math> and factor <math>B\,\!</math>) are investigated at two levels. A single replicate of this design will require four runs (<math>{{2}^{2}}=2\times 2=4\,\!</math>) The effects investigated by this design are the two main effects, <math>A\,\!</math> and <math>B,\,\!</math> and the interaction effect <math>AB\,\!</math>. The treatments for this design are shown in figure (a) below. In figure (a), letters are used to represent the treatments. The presence of a letter indicates the high level of the corresponding factor and the absence indicates the low level. For example, (1) represents the treatment combination where all factors involved are at the low level or the level represented by <math>-1\,\!</math> ; <math>a\,\!</math> represents the treatment combination where factor <math>A\,\!</math> is at the high level or the level of <math>1\,\!</math>, while the remaining factors (in this case, factor <math>B\,\!</math>) are at the low level or the level of <math>-1\,\!</math>. Similarly, <math>b\,\!</math> represents the treatment combination where factor <math>B\,\!</math> is at the high level or the level of <math>1\,\!</math>, while factor <math>A\,\!</math> is at the low level and <math>ab\,\!</math> represents the treatment combination where factors <math>A\,\!</math> and <math>B\,\!</math> are at the high level or the level of the 1. Figure (b) below shows the design matrix for the <math>{2}^{2}\,\!</math> design. It can be noted that the sum of the terms resulting from the product of any two columns of the design matrix is zero. As a result the <math>{2}^{2}\,\!</math> design is an ''orthogonal design''. In fact, all <math>{2}^{k}\,\!</math> designs are orthogonal designs. This property of the <math>{2}^{k}\,\!</math> designs offers a great advantage in the analysis because of the simplifications that result from orthogonality. These simplifications are explained later on in this chapter.
The <math>{2}^{2}\,\!</math> design can also be represented geometrically using a square with the four treatment combinations lying at the four corners, as shown in figure (c) below.

[[Image:doe7.2.png|center|400px|The <math>2^2\,\!</math> design. Figure (a) displays the experiment design, (b) displays the design matrix and (c) displays the geometric representation for the design. In Figure (b), the column names I, A, B and AB are used. Column I represents the intercept term. Columns A and B represent the respective factor settings. Column AB represents the interaction and is the product of columns A and B.]]
 

===The 23 Design===

The <math>{2}^{3}\,\!</math> design is a two level factorial experiment design with three factors (say factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>). This design tests three (<math>k=3\,\!</math>) main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math> ; three (<math>(_{2}^{k})=\,\!</math> <math>(_{2}^{3})=3\,\!</math>) two factor interaction effects, <math>AB\,\!</math>, <math>BC\,\!</math>, <math>AC\,\!</math> ; and one (<math>(_{3}^{k})=\,\!</math> <math>(_{3}^{3})=1\,\!</math>) three factor interaction effect, <math>ABC\,\!</math>. The design requires eight runs per replicate. The eight treatment combinations corresponding to these runs are <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math>, <math>ab\,\!</math>, <math>c\,\!</math>, <math>ac\,\!</math>, <math>bc\,\!</math> and <math>abc\,\!</math>. Note that the treatment combinations are written in such an order that factors are introduced one by one with each new factor being combined with the preceding terms. This order of writing the treatments is called the ''standard order'' or ''Yates' order''. The <math>{2}^{3}\,\!</math> design is shown in figure (a) below. The design matrix for the <math>{2}^{3}\,\!</math> design is shown in figure (b). The design matrix can be constructed by following the standard order for the treatment combinations to obtain the columns for the main effects and then multiplying the main effects columns to obtain the interaction columns.

[[Image:doe7.3.png|center|324px|The <math>2^3\,\!</math> design. Figure (a) shows the experiment design and (b) shows the design matrix.]]

[[Image:doe7.4.png|center|290px|Geometric representation of the <math>2^3\,\!</math> design.]]

The <math>{2}^{3}\,\!</math> design can also be represented geometrically using a cube with the eight treatment combinations lying at the eight corners as shown in the figure above.

==Analysis of 2''k'' Designs==

The <math>{2}^{k}\,\!</math> designs are a special category of the factorial experiments where all the factors are at two levels. The fact that these designs contain factors at only two levels and are orthogonal greatly simplifies their analysis even when the number of factors is large. The use of <math>{2}^{k}\,\!</math> designs in investigating a large number of factors calls for a revision of the notation used previously for the ANOVA models. The case for revised notation is made stronger by the fact that the ANOVA and multiple linear regression models are identical for <math>{2}^{k}\,\!</math> designs because all factors are only at two levels. Therefore, the notation of the regression models is applied to the ANOVA models for these designs, as explained next.

===Notation===

Based on the notation used in [[General Full Factorial Designs]], the ANOVA model for a two level factorial experiment with three factors would be as follows:

::<math>\begin{align}
& Y= & \mu +{{\tau }_{1}}\cdot {{x}_{1}}+{{\delta }_{1}}\cdot {{x}_{2}}+{{(\tau \delta )}_{11}}\cdot {{x}_{1}}{{x}_{2}}+{{\gamma }_{1}}\cdot {{x}_{3}} \\
& & +{{(\tau \gamma )}_{11}}\cdot {{x}_{1}}{{x}_{3}}+{{(\delta \gamma )}_{11}}\cdot {{x}_{2}}{{x}_{3}}+{{(\tau \delta \gamma )}_{111}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where:
 
:• <math>\mu \,\!</math> represents the overall mean
:• <math>{{\tau }_{1}}\,\!</math> represents the independent effect of the first factor (factor <math>A\,\!</math>) out of the two effects <math>{{\tau }_{1}}\,\!</math> and <math>{{\tau }_{2}}\,\!</math>
:• <math>{{\delta }_{1}}\,\!</math> represents the independent effect of the second factor (factor <math>B\,\!</math>) out of the two effects <math>{{\delta }_{1}}\,\!</math> and <math>{{\delta }_{2}}\,\!</math>
:• <math>{{(\tau \delta )}_{11}}\,\!</math> represents the independent effect of the interaction <math>AB\,\!</math> out of the other interaction effects
:• <math>{{\gamma }_{1}}\,\!</math> represents the effect of the third factor (factor <math>C\,\!</math>) out of the two effects <math>{{\gamma }_{1}}\,\!</math> and <math>{{\gamma }_{2}}\,\!</math>
:• <math>{{(\tau \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>AC\,\!</math> out of the other interaction effects
:• <math>{{(\delta \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>BC\,\!</math> out of the other interaction effects
:• <math>{{(\tau \delta \gamma )}_{111}}\,\!</math> represents the effect of the interaction <math>ABC\,\!</math> out of the other interaction effects
and <math>\epsilon \,\!</math> is the random error term.

 
The notation for a linear regression model having three predictor variables with interactions is:

::<math>\begin{align}
& Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& & +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

The notation for the regression model is much more convenient, especially for the case when a large number of higher order interactions are present. In two level experiments, the ANOVA model requires only one indicator variable to represent each factor for both qualitative and quantitative factors. Therefore, the notation for the multiple linear regression model can be applied to the ANOVA model of the experiment that has all the factors at two levels. For example, for the experiment of the ANOVA model given above, <math>{{\beta }_{0}}\,\!</math> can represent the overall mean instead of <math>\mu \,\!</math>, and <math>{{\beta }_{1}}\,\!</math> can represent the independent effect, <math>{{\tau }_{1}}\,\!</math>, of factor <math>A\,\!</math>. Other main effects can be represented in a similar manner. The notation for the interaction effects is much more simplified (e.g., <math>{{\beta }_{123}}\,\!</math> can be used to represent the three factor interaction effect, <math>{{(\tau \beta \gamma )}_{111}}\,\!</math>).

As mentioned earlier, it is important to note that the coding for the indicator variables for the ANOVA models of two level factorial experiments is reversed from the coding followed in [[General Full Factorial Designs]]. Here <math>-1\,\!</math> represents the first level of the factor while <math>1\,\!</math> represents the second level. This is because for a two level factor a single variable is needed to represent the factor for both qualitative and quantitative factors. For quantitative factors, using <math>-1\,\!</math> for the first level (which is the low level) and 1 for the second level (which is the high level) keeps the coding consistent with the numerical value of the factors. The change in coding between the two coding schemes does not affect the analysis except that signs of the estimated effect coefficients will be reversed (i.e., numerical values of <math>{{\hat{\tau }}_{1}}\,\!</math>, obtained based on the coding of [[General Full Factorial Designs]], and <math>{{\hat{\beta }}_{1}}\,\!</math>, obtained based on the new coding, will be the same but their signs would be opposite).

::<math>\begin{align}
& & \text{Factor }A\text{ Coding (two level factor)} \\
& &
\end{align}\,\!</math>

::<math>\begin{matrix}
\text{Previous Coding} & {} & {} & {} & \text{Coding for }{{\text{2}}^{k}}\text{ Designs} \\
{} & {} & {} & {} & {} \\
Effect\text{ }{{\tau }_{1}}\ \ :\ \ {{x}_{1}}=1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{1}}\text{ (or }-{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=-1\text{ } \\
Effect\text{ }{{\tau }_{2}}\ \ :\ \ {{x}_{1}}=-1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{2}}\text{ (or }{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=1\text{ } \\
\end{matrix}\,\!</math>

In summary, the ANOVA model for the experiments with all factors at two levels is different from the ANOVA models for other experiments in terms of the notation in the following two ways:
 

:• The notation of the regression models is used for the effect coefficients.
:• The coding of the indicator variables is reversed.

===Special Features===

Consider the design matrix, <math>X\,\!</math>, for the <math>{2}^{3}\,\!</math> design discussed above. The (<math>{{X}^{\prime }}X\,\!</math>) <math>^{-1}\,\!</math> matrix is:

<center><math>{{({{X}^{\prime }}X)}^{-1}}=\left[ \begin{matrix}
0.125 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0.125 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0.125 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0.125 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0.125 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0.125 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0.125 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.125 \\
\end{matrix} \right]\,\!</math></center>

Notice that, due to the orthogonal design of the <math>X\,\!</math> matrix, the <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> has been simplified to a diagonal matrix which can be written as:

::<math>\begin{align}
{{({{X}^{\prime }}X)}^{-1}}= & 0.125\cdot I = & \frac{1}{8}\cdot I = & \frac{1}{{{2}^{3}}}\cdot I
\end{align}\,\!</math>

where <math>I\,\!</math> represents the identity matrix of the same order as the design matrix, <math>X\,\!</math>. Since there are eight observations per replicate of the <math>{2}^{3}\,\!</math> design, the <math>(X\,\!</math> ' <math>X{{)}^{-1}}\,\!</math> matrix for <math>m\,\!</math> replicates of this design can be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{3}}\cdot m)}\cdot I\,\!</math>

The <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> matrix for any <math>{2}^{k}\,\!</math> design can now be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{k}}\cdot m)}\cdot I\,\!</math>

Then the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is:

::<math>\begin{align}
C= & {{{\hat{\sigma }}}^{2}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & M{{S}_{E}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & \frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}\cdot I
\end{align}\,\!</math>

Note that the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is also a diagonal matrix. Therefore, the estimated effect coefficients (<math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math>, <math>{{\beta }_{12}},\,\!</math> etc.) for these designs are uncorrelated. This implies that the terms in the <math>{2}^{k}\,\!</math> design (main effects, interactions) are independent of each other. Consequently, the extra sum of squares for each of the terms in these designs is independent of the sequence of terms in the model, and also independent of the presence of other terms in the model. As a result the sequential and partial sum of squares for the terms are identical for these designs and will always add up to the model sum of squares. Multicollinearity is also not an issue for these designs.

It can also be noted from the equation given above, that in addition to the <math>C\,\!</math> matrix being diagonal, all diagonal elements of the <math>C\,\!</math> matrix are identical. This means that the variance (or its square root, the standard error) of all estimated effect coefficients are the same. The standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>, for all the coefficients is:

::<math>\begin{align}
se({{{\hat{\beta }}}_{j}})= & \sqrt{{{C}_{jj}}} = & \sqrt{\frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}}\text{ }for\text{ }all\text{ }j
\end{align}\,\!</math>

This property is used to construct the normal probability plot of effects in <math>{2}^{k}\,\!</math> designs and identify significant effects using graphical techniques. For details on the normal probability plot of effects in a Weibull++ DOE folio, refer to [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]].

====Example====
To illustrate the analysis of a full factorial <math>{2}^{k}\,\!</math> design, consider a three factor experiment to investigate the effect of honing pressure, number of strokes and cycle time on the surface finish of automobile brake drums. Each of these factors is investigated at two levels. The honing pressure is investigated at levels of 200 <math>psi\,\!</math> and 400 <math>psi\,\!</math>, the number of strokes used is 3 and 5 and the two levels of the cycle time are 3 and 5 seconds. The design for this experiment is set up in a Weibull++ DOE folio as shown in the first two following figures. It is decided to run two replicates for this experiment. The surface finish data collected from each run (using randomization) and the complete design is shown in the third following figure. The analysis of the experiment data is explained next.

[[Image:doe7_5.png|center|877px|Design properties for the experiment in the example.|link=]]

[[Image:doe7_6.png|center|859px|Design summary for the experiment in the example.|link=]]

[[Image:doe7_7.png|center|778px|Experiment design for the example to investigate the surface finish of automobile brake drums.|link=]]

The applicable model using the notation for <math>{2}^{k}\,\!</math> designs is:

::<math>\begin{align}
Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where the indicator variable, <math>{{x}_{1,}}\,\!</math> represents factor <math>A\,\!</math> (honing pressure), <math>{{x}_{1}}=-1\,\!</math> represents the low level of 200 <math>psi\,\!</math> and <math>{{x}_{1}}=1\,\!</math> represents the high level of 400 <math>psi\,\!</math>. Similarly, <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> represent factors <math>B\,\!</math> (number of strokes) and <math>C\,\!</math> (cycle time), respectively. <math>{{\beta }_{0}}\,\!</math> is the overall mean, while <math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math> and <math>{{\beta }_{3}}\,\!</math> are the effect coefficients for the main effects of factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. <math>{{\beta }_{12}}\,\!</math>, <math>{{\beta }_{13}}\,\!</math> and <math>{{\beta }_{23}}\,\!</math> are the effect coefficients for the <math>AB\,\!</math>, <math>AC\,\!</math> and <math>BC\,\!</math> interactions, while <math>{{\beta }_{123}}\,\!</math> represents the <math>ABC\,\!</math> interaction.

 
If the subscripts for the run (<math>i\,\!</math> ; <math>i=\,\!</math> 1 to 8) and replicates (<math>j\,\!</math> ; <math>j=\,\!</math> 1,2) are included, then the model can be written as:

::<math>\begin{align}
{{Y}_{ij}}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{ij1}}+{{\beta }_{2}}\cdot {{x}_{ij2}}+{{\beta }_{12}}\cdot {{x}_{ij1}}{{x}_{ij2}}+{{\beta }_{3}}\cdot {{x}_{ij3}} \\
& +{{\beta }_{13}}\cdot {{x}_{ij1}}{{x}_{ij3}}+{{\beta }_{23}}\cdot {{x}_{ij2}}{{x}_{ij3}}+{{\beta }_{123}}\cdot {{x}_{ij1}}{{x}_{ij2}}{{x}_{ij3}}+{{\epsilon }_{ij}}
\end{align}\,\!</math>

To investigate how the given factors affect the response, the following hypothesis tests need to be carried:

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{1}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{1}}\ne 0\,\!</math>

This test investigates the main effect of factor <math>A\,\!</math> (honing pressure). The statistic for this test is:

::<math>{{({{F}_{0}})}_{A}}=\frac{M{{S}_{A}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{A}}\,\!</math> is the mean square for factor <math>A\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other main effects, <math>B\,\!</math> and <math>C\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{12}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{12}}\ne 0\,\!</math>

This test investigates the two factor interaction <math>AB\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{AB}}=\frac{M{{S}_{AB}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the interaction <math>AB\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other two factor interactions, <math>AC\,\!</math> and <math>BC\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{123}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{123}}\ne 0\,\!</math>

This test investigates the three factor interaction <math>ABC\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{ABC}}=\frac{M{{S}_{ABC}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{ABC}}\,\!</math> is the mean square for the interaction <math>ABC\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square.
To calculate the test statistics, it is convenient to express the ANOVA model in the form <math>y=X\beta +\epsilon \,\!</math>.

====Expression of the ANOVA Model as <math>y=X\beta +\epsilon \,\!</math>====

In matrix notation, the ANOVA model can be expressed as:

::<math>y=X\beta +\epsilon \,\!</math>

where:

<center><math>y=\left[ \begin{matrix}
{{Y}_{11}} \\
{{Y}_{21}} \\
. \\
{{Y}_{81}} \\
{{Y}_{12}} \\
. \\
{{Y}_{82}} \\
\end{matrix} \right]=\left[ \begin{matrix}
90 \\
90 \\
. \\
90 \\
86 \\
. \\
80 \\
\end{matrix} \right]\text{ }X=\left[ \begin{matrix}
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\end{matrix} \right]\,\!</math></center>

<center><math>\beta =\left[ \begin{matrix}
{{\beta }_{0}} \\
{{\beta }_{1}} \\
{{\beta }_{2}} \\
{{\beta }_{12}} \\
{{\beta }_{3}} \\
{{\beta }_{13}} \\
{{\beta }_{23}} \\
{{\beta }_{123}} \\
\end{matrix} \right]\text{ }\epsilon =\left[ \begin{matrix}
{{\epsilon }_{11}} \\
{{\epsilon }_{21}} \\
. \\
{{\epsilon }_{81}} \\
{{\epsilon }_{12}} \\
. \\
. \\
{{\epsilon }_{82}} \\
\end{matrix} \right]\,\!</math></center>

====Calculation of the Extra Sum of Squares for the Factors====

Knowing the matrices <math>y\,\!</math>, <math>X\,\!</math> and <math>\beta \,\!</math>, the extra sum of squares for the factors can be calculated. These are used to calculate the mean squares that are used to obtain the test statistics. Since the experiment design is orthogonal, the partial and sequential extra sum of squares are identical. The extra sum of squares for each effect can be calculated as shown next. As an example, the extra sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & Model\text{ }Sum\text{ }of\text{ }Squares - Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }main\text{ }effect\text{ }of\text{ }A \\
= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }A}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }A}}={{X}_{\tilde{\ }A}}{{(X_{\tilde{\ }A}^{\prime }{{X}_{\tilde{\ }A}})}^{-1}}X_{\tilde{\ }A}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }A}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the main effect of factor <math>A\,\!</math>. Thus, the sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y \\
= & 654.4375-549.375 \\
= & 105.0625
\end{align}\,\!</math>

Similarly, the extra sum of squares for the interaction effect <math>AB\,\!</math> is:

::<math>\begin{align}
S{{S}_{AB}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }AB}}-(1/16)J]y \\
= & 654.4375-636.375 \\
= & 18.0625
\end{align}\,\!</math>

The extra sum of squares for other effects can be obtained in a similar manner.

====Calculation of the Test Statistics====

Knowing the extra sum of squares, the test statistic for the effects can be calculated. For example, the test statistic for the interaction <math>AB\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{AB}}= & \frac{M{{S}_{AB}}}{M{{S}_{E}}} \\
= & \frac{S{{S}_{AB}}/dof(S{{S}_{AB}})}{S{{S}_{E}}/dof(S{{S}_{E}})} \\
= & \frac{18.0625/1}{147.5/8} \\
= & 0.9797
\end{align}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the <math>AB\,\!</math> interaction and <math>M{{S}_{E}}\,\!</math> is the error mean square. The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{AB}}=0.9797\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{AB}}) \\
= & 1-0.6487 \\
= & 0.3513
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that the interaction between honing pressure and number of strokes does not affect the surface finish of the brake drums. Tests for other effects can be carried out in a similar manner. The results are shown in the ANOVA Table in the following figure. The values S, R-sq and R-sq(adj) in the figure indicate how well the model fits the data. The value of S represents the standard error of the model, R-sq represents the coefficient of multiple determination and R-sq(adj) represents the adjusted coefficient of multiple determination. For details on these values refer to [[Multiple_Linear_Regression_Analysis|Multiple Linear Regression Analysis]].

[[Image:doe7_8.png|center|799px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

====Calculation of Effect Coefficients====

The estimate of effect coefficients can also be obtained:

<center><math>\begin{align}
\hat{\beta }= & {{({{X}^{\prime }}X)}^{-1}}{{X}^{\prime }}y \\
= & \left[ \begin{matrix}
86.4375 \\
2.5625 \\
-4.9375 \\
1.0625 \\
-1.0625 \\
2.4375 \\
-1.3125 \\
-0.1875 \\
\end{matrix} \right]
\end{align}\,\!</math></center>

[[Image:doe7_9.png|center|800px|Regression Information table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

The coefficients and related results are shown in the Regression Information table above. In the table, the Effect column displays the effects, which are simply twice the coefficients. The Standard Error column displays the standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>. The Low CI and High CI columns display the confidence interval on the coefficients. The interval shown is the 90% interval as the significance is chosen as 0.1. The T Value column displays the <math>t\,\!</math> statistic, <math>{{t}_{0}}\,\!</math>, corresponding to the coefficients. The P Value column displays the <math>p\,\!</math> value corresponding to the <math>t\,\!</math> statistic. (For details on how these results are calculated, refer to [[General Full Factorial Designs]]). Plots of residuals can also be obtained from the DOE folio to ensure that the assumptions related to the ANOVA model are not violated.

====Model Equation====

From the analysis results in the above figure within [[Two_Level_Factorial_Experiments#Calculation_of_Effect_Coefficients|calculation of effect coefficients]] section, it is seen that effects <math>A\,\!</math>, <math>B\,\!</math> and <math>AC\,\!</math> are significant. In a DOE folio, the <math>p\,\!</math> values for the significant effects are displayed in red in the ANOVA Table for easy identification. Using the values of the estimated effect coefficients, the model for the present <math>{2}^{3}\,\!</math> design in terms of the coded values can be written as:

::<math>\begin{align}
\hat{y}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}} \\
= & 86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+2.4375{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

To make the model hierarchical, the main effect, <math>C\,\!</math>, needs to be included in the model (because the interaction <math>AC\,\!</math> is included in the model). The resulting model is:

::<math>\hat{y}=86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+1.0625{{x}_{3}}+2.4375{{x}_{1}}{{x}_{3}}\,\!</math>

This equation can be viewed in a DOE folio, as shown in the following figure, using the Show Analysis Summary icon in the Control Panel. The equation shown in the figure will match the hierarchical model once the required terms are selected using the Select Effects icon.

[[Image:doe7_10.png|center|557px|The model equation for the experiment of the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

==Replicated and Repeated Runs==

In the case of replicated experiments, it is important to note the difference between replicated runs and repeated runs. Both repeated and replicated runs are multiple response readings taken at the same factor levels. However, repeated runs are response observations taken at the same time or in succession. Replicated runs are response observations recorded in a random order. Therefore, replicated runs include more variation than repeated runs. For example, a baker, who wants to investigate the effect of two factors on the quality of cakes, will have to bake four cakes to complete one replicate of a <math>{2}^{2}\,\!</math> design. Assume that the baker bakes eight cakes in all. If, for each of the four treatments of the <math>{2}^{2}\,\!</math> design, the baker selects one treatment at random and then bakes two cakes for this treatment at the same time then this is a case of two repeated runs. If, however, the baker bakes all the eight cakes randomly, then the eight cakes represent two sets of replicated runs.
For repeated measurements, the average values of the response for each treatment should be entered into a DOE folio as shown in the following figure (a) when the two cakes for a particular treatment are baked together. For replicated measurements, when all the cakes are baked randomly, the data is entered as shown in the following figure (b).

[[Image:doe7.11.png|center|300px|Data entry for repeated and replicated runs. Figure (a) shows repeated runs and (b) shows replicated runs.]]

==Unreplicated 2''k'' Designs==

If a factorial experiment is run only for a single replicate then it is not possible to test hypotheses about the main effects and interactions as the error sum of squares cannot be obtained. This is because the number of observations in a single replicate equals the number of terms in the ANOVA model. Hence the model fits the data perfectly and no degrees of freedom are available to obtain the error sum of squares.

However, sometimes it is only possible to run a single replicate of the <math>{2}^{k}\,\!</math> design because of constraints on resources and time. In the absence of the error sum of squares, hypothesis tests to identify significant factors cannot be conducted. A number of methods of analyzing information obtained from unreplicated <math>{2}^{k}\,\!</math> designs are available. These include pooling higher order interactions, using the normal probability plot of effects or including center point replicates in the design.

===Pooling Higher Order Interactions===

One of the ways to deal with unreplicated <math>{2}^{k}\,\!</math> designs is to use the sum of squares of some of the higher order interactions as the error sum of squares provided these higher order interactions can be assumed to be insignificant. By dropping some of the higher order interactions from the model, the degrees of freedom corresponding to these interactions can be used to estimate the error mean square. Once the error mean square is known, the test statistics to conduct hypothesis tests on the factors can be calculated.

===Normal Probability Plot of Effects===
Another way to use unreplicated <math>{2}^{k}\,\!</math> designs to identify significant effects is to construct the normal probability plot of the effects. As mentioned in [[Two_Level_Factorial_Experiments#Special_Features| Special Features]], the standard error for all effect coefficients in the <math>{2}^{k}\,\!</math> designs is the same. Therefore, on a normal probability plot of effect coefficients, all non-significant effect coefficients (with <math>\beta =0\,\!</math>) will fall along the straight line representative of the normal distribution, N(<math>0,{{\sigma }^{2}}/({{2}^{k}}\cdot m)\,\!</math>). Effect coefficients that show large deviations from this line will be significant since they do not come from this normal distribution. Similarly, since effects <math>=2\times \,\!</math> effect coefficients, all non-significant effects will also follow a straight line on the normal probability plot of effects. For replicated designs, the Effects Probability plot of a DOE folio plots the normalized effect values (or the T Values) on the standard normal probability line, N(0,1). However, in the case of unreplicated <math>{2}^{k}\,\!</math> designs, <math>{{\sigma }^{2}}\,\!</math> remains unknown since <math>M{{S}_{E}}\,\!</math> cannot be obtained. Lenth's method is used in this case to estimate the variance of the effects. For details on Lenth's method, please refer to [[DOE References| Montgomery (2001)]]. The DOE folio then uses this variance value to plot effects along the N(0, Lenth's effect variance) line. The
method is illustrated in the following example.

====Example====

Vinyl panels, used as instrument panels in a certain automobile, are seen to develop defects after a certain amount of time. To investigate the issue, it is decided to carry out a two level factorial experiment. Potential factors to be investigated in the experiment are vacuum rate (factor <math>A\,\!</math>), material temperature (factor <math>B\,\!</math>), element intensity (factor <math>C\,\!</math>) and pre-stretch (factor <math>D\,\!</math>). The two levels of the factors used in the experiment are as shown in below.

[[Image:doet7.1.png|center|300px|Factors to investigate defects in vinyl panels.]]

With a <math>{2}^{4}\,\!</math> design requiring 16 runs per replicate it is only feasible for the manufacturer to run a single replicate.

The experiment design and data, collected as percent defects, are shown in the following figure. Since the present experiment design contains only a single replicate, it is not possible to obtain an estimate of the error sum of squares, <math>S{{S}_{E}}\,\!</math>. It is decided to use the normal probability plot of effects to identify the significant effects. The effect values for each term are obtained as shown in the following figure.

[[Image:doe7_13.png|center|650px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

Lenth's method uses these values to estimate the variance. As described in [[DOE_References|[Lenth, 1989]]], if all effects are arranged in ascending order, using their absolute values, then <math>{{s}_{0}}\,\!</math> is defined as 1.5 times the median value:

::<math>\begin{align}
{{s}_{0}}= & 1.5\cdot median(\left| effect \right|) \\
= & 1.5\cdot 2 \\
= & 3
\end{align}\,\!</math>

Using <math>{{s}_{0}}\,\!</math>, the "pseudo standard error" (<math>PSE\,\!</math>) is calculated as 1.5 times the median value of all effects that are less than 2.5 <math>{{s}_{0}}\,\!</math> :

::<math>\begin{align}
PSE= & 1.5\cdot median(\left| effect \right|\ \ :\ \ \left| effect \right|<2.5{{s}_{0}}) \\
= & 1.5\cdot 1.5 \\
= & 2.25
\end{align}\,\!</math>

Using <math>PSE\,\!</math> as an estimate of the effect variance, the effect variance is 2.25. Knowing the effect variance, the normal probability plot of effects for the present unreplicated experiment can be constructed as shown in the following figure. The line on this plot is the line N(0, 2.25). The plot shows that the effects <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math> do not follow the distribution represented by this line. Therefore, these effects are significant.

The significant effects can also be identified by comparing individual effect values to the margin of error or the threshold value using the pareto chart (see the third following figure). If the required significance is 0.1, then:

::<math>margin\text{ }of\text{ }error={{t}_{\alpha /2,d}}\cdot PSE\,\!</math>

The <math>t\,\!</math> statistic, <math>{{t}_{\alpha /2,d}}\,\!</math>, is calculated at a significance of <math>\alpha /2\,\!</math> (for the two-sided hypothesis) and degrees of freedom <math>d=(\,\!</math> number of effects <math>)/3\,\!</math>. Thus:

::<math>\begin{align}
margin\text{ }of\text{ }error= & {{t}_{0.05,5}}\cdot PSE \\
= & 2.015\cdot 2.25 \\
= & 4.534
\end{align}\,\!</math>

The value of 4.534 is shown as the critical value line in the third following figure. All effects with absolute values greater than the margin of error can be considered to be significant. These effects are <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math>. Therefore, the vacuum rate, the pre-stretch and their interaction have a significant effect on the defects of the vinyl panels.

[[Image:doe7_14.png|center|800px|Effect values for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_15.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_16.png|center|650px|Pareto chart for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

===Center Point Replicates===

Another method of dealing with unreplicated <math>{2}^{k}\,\!</math> designs that only have quantitative factors is to use replicated runs at the center point. The center point is the response corresponding to the treatment exactly midway between the two levels of all factors. Running multiple replicates at this point provides an estimate of pure error. Although running multiple replicates at any treatment level can provide an estimate of pure error, the other advantage of running center point replicates in the <math>{2}^{k}\,\!</math> design is in checking for the presence of curvature. The test for curvature investigates whether the model between the response and the factors is linear and is discussed in [[Two_Level_Factorial_Experiments#Using_Center_Point_Replicates_to_Test_Curvature| Center Pt. Replicates to Test Curvature]].

====Example: Use Center Point to Get Pure Error====

Consider a <math>{2}^{2}\,\!</math> experiment design to investigate the effect of two factors, <math>A\,\!</math> and <math>B\,\!</math>, on a certain response. The energy consumed when the treatments of the <math>{2}^{2}\,\!</math> design are run is considerably larger than the energy consumed for the center point run (because at the center point the factors are at their middle levels). Therefore, the analyst decides to run only a single replicate of the design and augment the design by five replicated runs at the center point as shown in the following figure. The design properties for this experiment are shown in the second following figure. The complete experiment design is shown in the third following figure. The center points can be used in the identification of significant effects as shown next.

[[Image:doe7.17.png||center|300px|<math>2^2\,\!</math> design augmented by five center point runs.]]
[[Image:doe7_18.png|center|537px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

[[Image:doe7_19.png|center|630px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

Since the present <math>{2}^{2}\,\!</math> design is unreplicated, there are no degrees of freedom available to calculate the error sum of squares. By augmenting this design with five center points, the response values at the center points, <math>y_{i}^{c}\,\!</math>, can be used to obtain an estimate of pure error, <math>S{{S}_{PE}}\,\!</math>. Let <math>{{\bar{y}}^{c}}\,\!</math> represent the average response for the five replicates at the center. Then:

::<math>S{{S}_{PE}}=Sum\text{ }of\text{ }Squares\text{ }for\text{ }center\text{ }points\,\!</math>

::<math>\begin{align}
S{{S}_{PE}}= & \underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}} \\
= & {{(25.2-25.26)}^{2}}+...+{{(25.3-25.26)}^{2}} \\
= & 0.052
\end{align}\,\!</math>

Then the corresponding mean square is:

::<math>\begin{align}
M{{S}_{PE}}= & \frac{S{{S}_{PE}}}{degrees\text{ }of\text{ }freedom} \\
= & \frac{0.052}{5-1} \\
= & 0.013
\end{align}\,\!</math>

Alternatively, <math>M{{S}_{PE}}\,\!</math> can be directly obtained by calculating the variance of the response values at the center points:

::<math>\begin{align}
M{{S}_{PE}}= & {{s}^{2}} \\
= & \frac{\underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}}}{5-1}
\end{align}\,\!</math>

Once <math>M{{S}_{PE}}\,\!</math> is known, it can be used as the error mean square, <math>M{{S}_{E}}\,\!</math>, to carry out the test of significance for each effect. For example, to test the significance of the main effect of factor <math>A,\,\!</math> the sum of squares corresponding to this effect is obtained in the usual manner by considering only the four runs of the original <math>{2}^{2}\,\!</math> design.

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/4)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/4)J]y \\
= & 0.5625
\end{align}\,\!</math>

Then, the test statistic to test the significance of the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{A}}= & \frac{M{{S}_{A}}}{M{{S}_{E}}} \\
= & \frac{0.5625/1}{0.052/4} \\
= & 43.2692
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{A}}=43.2692\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{A}}) \\
= & 1-0.9972 \\
= & 0.0028
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value < 0.1, it can be concluded that the main effect of factor <math>A\,\!</math> significantly affects the response. This result is displayed in the ANOVA table as shown in the following figure. Test for the significance of other factors can be carried out in a similar manner.

[[Image:doe7_20.png|center|649px|Results for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

===Using Center Point Replicates to Test Curvature===

Center point replicates can also be used to check for curvature in replicated or unreplicated <math>{2}^{k}\,\!</math> designs. The test for curvature investigates whether the model between the response and the factors is linear. The way DOE++ handles center point replicates is similar to its handling of blocks. The center point replicates are treated as an additional factor in the model. The factor is labeled as Curvature in the results of the DOE folio. If Curvature turns out to be a significant factor in the results, then this indicates the presence of curvature in the model.

====Example: Use Center Point to Test Curvature====

To illustrate the use of center point replicates in testing for curvature, consider again the data of the single replicate <math>{2}^{2}\,\!</math> experiment from a preceding figure(labeled "<math>2^2</math> design augmented by five center point runs"). Let <math>{{x}_{1}}\,\!</math> be the indicator variable to indicate if the run is a center point:

::<math>\begin{matrix}
{{x}_{1}}=0 & {} & \text{Center point run} \\
{{x}_{1}}=1 & {} & \text{Other run} \\
\end{matrix}\,\!</math>

If <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> are the indicator variables representing factors <math>A\,\!</math> and <math>B\,\!</math>, respectively, then the model for this experiment is:

::<math>Y={{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}\,\!</math>

To investigate the presence of curvature, the following hypotheses need to be tested:

::<math>\begin{align}
& {{H}_{0}}: & {{\beta }_{1}}=0\text{ (Curvature is absent)} \\
& {{H}_{1}}: & {{\beta }_{1}}\ne 0
\end{align}\,\!</math>

The test statistic to be used for this test is:

::<math>{{({{F}_{0}})}_{curvature}}=\frac{M{{S}_{curvature}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{curvature}}\,\!</math> is the mean square for Curvature and <math>M{{S}_{E}}\,\!</math> is the error mean square.

'''Calculation of the Sum of Squares'''

The <math>X\,\!</math> matrix and <math>y\,\!</math> vector for this experiment are:

<center><math>X=\left[ \begin{matrix}
1 & 1 & -1 & -1 & 1 \\
1 & 1 & 1 & -1 & -1 \\
1 & 1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
\end{matrix} \right]\text{ }y=\left[ \begin{matrix}
24.6 \\
25.4 \\
25.0 \\
25.7 \\
25.2 \\
25.3 \\
25.4 \\
25.1 \\
25.3 \\
\end{matrix} \right]\,\!</math></center>

The sum of squares can now be calculated. For example, the error sum of squares is:

::<math>\begin{align}
& S{{S}_{E}}= & {{y}^{\prime }}[I-H]y \\
& = & 0.052
\end{align}\,\!</math>

where <math>I\,\!</math> is the identity matrix and <math>H\,\!</math> is the hat matrix. It can be seen that this is equal to <math>S{{S}_{PE\text{ }}}\,\!</math> (the sum of squares due to pure error) because of the replicates at the center point, as obtained in the [[Two_Level_Factorial_Experiments#Example_3| example]]. The number of degrees of freedom associated with <math>S{{S}_{E}}\,\!</math>, <math>dof(S{{S}_{E}})\,\!</math> is four. The extra sum of squares corresponding to the center point replicates (or Curvature) is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & Model\text{ }Sum\text{ }of\text{ }Squares- \\
& & Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }center\text{ }point \\
& = & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Curvature}}-(1/9)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }Curvature}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }Curvature}}={{X}_{\tilde{\ }Curv}}{{(X_{\tilde{\ }Curv}^{\prime }{{X}_{\tilde{\ }Curv}})}^{-1}}X_{\tilde{\ }Curv}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }Curv}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the center point. Thus, the extra sum of squares corresponding to Curvature is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Center}}-(1/9)J]y \\
& = & 0.7036-0.6875 \\
& = & 0.0161
\end{align}\,\!</math>

This extra sum of squares can be used to test for the significance of curvature. The corresponding mean square is:

::<math>\begin{align}
& M{{S}_{Curvature}}= & \frac{Sum\text{ }of\text{ }squares\text{ }corresponding\text{ }to\text{ }Curvature}{degrees\text{ }of\text{ }freedom} \\
& = & \frac{0.0161}{1} \\
& = & 0.0161
\end{align}\,\!</math>

'''Calculation of the Test Statistic'''

Knowing the mean squares, the statistic to check the significance of curvature can be calculated.

::<math>\begin{align}
& {{({{f}_{0}})}_{Curvature}}= & \frac{M{{S}_{Curvature}}}{M{{S}_{E}}} \\
& = & \frac{0.0161/1}{0.052/4} \\
& = & 1.24
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{Curvature}}=1.24\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and four degrees of freedom in the denominator is:

::<math>\begin{align}
& p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{Curvature}}) \\
& = & 1-0.6713 \\
& = & 0.3287
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that curvature does not exist for this design. This results is shown in the ANOVA table in the figure above. The surface of the fitted model based on these results, along with the observed response values, is shown in the figure below.

[[Image:doe7.21.png|center|400px|Model surface and observed response values for the design in the [[Two_Level_Factorial_Experiments#Example_4| example]].]]

 

==Blocking in 2k Designs==

Blocking can be used in the <math>{2}^{k}\,\!</math> designs to deal with cases when replicates cannot be run under identical conditions. Randomized complete block designs that were discussed in [[Randomization and Blocking in DOE]] for factorial experiments are also applicable here. At times, even with just two levels per factor, it is not possible to run all treatment combinations for one replicate of the experiment under homogeneous conditions. For example, each replicate of the <math>{2}^{2}\,\!</math> design requires four runs. If each run requires two hours and testing facilities are available for only four hours per day, two days of testing would be required to run one complete replicate. Blocking can be used to separate the treatment runs on the two different days. Blocks that do not contain all treatments of a replicate are called incomplete blocks. In incomplete block designs, the block effect is confounded with certain effect(s) under investigation. For the <math>{2}^{2}\,\!</math> design assume that treatments <math>(1)\,\!</math> and <math>ab\,\!</math> were run on the first day and treatments <math>a\,\!</math> and <math>b\,\!</math> were run on the second day. Then, the incomplete block design for this experiment is:

::<math>\begin{matrix}
\text{Block 1} & {} & \text{Block 2} \\
\left[ \begin{matrix}
(1) \\
ab \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
b \\
\end{matrix} \right] \\
\end{matrix}\,\!</math>

For this design the block effect may be calculated as:

::<math>\begin{align}
& Block\text{ }Effect= & Average\text{ }response\text{ }for\text{ }Block\text{ }1- \\
& & Average\text{ }response\text{ }for\text{ }Block\text{ }2 \\
& = & \frac{(1)+ab}{2}-\frac{a+b}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The <math>AB\,\!</math> interaction effect is:

::<math>\begin{align}
& AB= & Average\text{ }response\text{ }at\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{low}}}- \\
& & Average\text{ }response\text{ }at\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{low}}} \\
& = & \frac{ab+(1)}{2}-\frac{b+a}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The two equations given above show that, in this design, the <math>AB\,\!</math> interaction effect cannot be distinguished from the block effect because the formulas to calculate these effects are the same. In other words, the <math>AB\,\!</math> interaction is said to be confounded with the block effect and it is not possible to say if the effect calculated based on these equations is due to the <math>AB\,\!</math> interaction effect, the block effect or both. In incomplete block designs some effects are always confounded with the blocks. Therefore, it is important to design these experiments in such a way that the important effects are not confounded with the blocks. In most cases, the experimenter can assume that higher order interactions are unimportant. In this case, it would better to use incomplete block designs that confound these effects with the blocks.
One way to design incomplete block designs is to use defining contrasts as shown next:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}+...+{{\alpha }_{k}}{{q}_{k}}\,\!</math>

where the <math>{{\alpha }_{i}}\,\!</math> s are the exponents for the factors in the effect that is to be confounded with the block effect and the <math>{{q}_{i}}\,\!</math> s are values based on the level of the <math>i\,\!</math> the factor (in a treatment that is to be allocated to a block). For <math>{2}^{k}\,\!</math> designs the <math>{{\alpha }_{i}}\,\!</math> s are either 0 or 1 and the <math>{{q}_{i}}\,\!</math> s have a value of 0 for the low level of the <math>i\,\!</math> th factor and a value of 1 for the high level of the factor in the treatment under consideration. As an example, consider the <math>{2}^{2}\,\!</math> design where the interaction effect <math>AB\,\!</math> is confounded with the block. Since there are two factors, <math>k=2\,\!</math>, with <math>i=1\,\!</math> representing factor <math>A\,\!</math> and <math>i=2\,\!</math> representing factor <math>B\,\!</math>. Therefore:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}\,\!</math>

The value of <math>{{\alpha }_{1}}\,\!</math> is one because the exponent of factor <math>A\,\!</math> in the confounded interaction <math>AB\,\!</math> is one. Similarly, the value of <math>{{\alpha }_{2}}\,\!</math> is one because the exponent of factor <math>B\,\!</math> in the confounded interaction <math>AB\,\!</math> is also one. Therefore, the defining contrast for this design can be written as:

::<math>\begin{align}
& L= & {{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}} \\
& = & 1\cdot {{q}_{1}}+1\cdot {{q}_{2}} \\
& = & {{q}_{1}}+{{q}_{2}}
\end{align}\,\!</math>

Once the defining contrast is known, it can be used to allocate treatments to the blocks. For the <math>{2}^{2}\,\!</math> design, there are four treatments <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math> and <math>ab\,\!</math>. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. In order to decide which block the treatment <math>(1)\,\!</math> belongs to, the levels of factors <math>A\,\!</math> and <math>B\,\!</math> for this run are used. Since factor <math>A\,\!</math> is at the low level in this treatment, <math>{{q}_{1}}=0\,\!</math>. Similarly, since factor <math>B\,\!</math> is also at the low level in this treatment, <math>{{q}_{2}}=0\,\!</math>. Therefore:

::<math>\begin{align}
& L= & {{q}_{1}}+{{q}_{2}} \\
& = & 0+0=0\text{ (mod 2)}
\end{align}\,\!</math>

Note that the value of <math>L\,\!</math> used to decide the block allocation is "mod 2" of the original value. This value is obtained by taking the value of 1 for odd numbers and 0 otherwise. Based on the value of <math>L\,\!</math>, treatment <math>(1)\,\!</math> is assigned to block 1. Other treatments can be assigned using the following calculations:

::<math>\begin{align}
& (1): & \text{ }L=0+0=0=0\text{ (mod 2)} \\
& a: & \text{ }L=1+0=1=1\text{ (mod 2)} \\
& b: & \text{ }L=0+1=1=1\text{ (mod 2)} \\
& ab: & \text{ }L=1+1=2=0\text{ (mod 2)}
\end{align}\,\!</math>

Therefore, to confound the interaction <math>AB\,\!</math> with the block effect in the <math>{2}^{2}\,\!</math> incomplete block design, treatments <math>(1)\,\!</math> and <math>ab\,\!</math> (with <math>L=0\,\!</math>) should be assigned to block 2 and treatment combinations <math>a\,\!</math> and <math>b\,\!</math> (with <math>L=1\,\!</math>) should be assigned to block 1.

====Example: Two Level Factorial Design with Two Blocks====

This example illustrates how treatments can be allocated to two blocks for an unreplicated <math>{2}^{k}\,\!</math> design. Consider the unreplicated <math>{2}^{4}\,\!</math> design to investigate the four factors affecting the defects in automobile vinyl panels discussed in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments required for this experiment were run by two different operators with each operator conducting 8 runs. This experiment is an example of an incomplete block design. The analyst in charge of this experiment assumed that the interaction <math>ABCD\,\!</math> was not significant and decided to allocate treatments to the two operators so that the <math>ABCD\,\!</math> interaction was confounded with the block effect (the two operators are the blocks). The allocation scheme to assign treatments to the two operators can be obtained as follows.
 
The defining contrast for the <math>{2}^{4}\,\!</math> design where the <math>ABCD\,\!</math> interaction is confounded with the blocks is:

::<math>L={{q}_{1}}+{{q}_{2}}+{{q}_{3}}+{{q}_{4}}\,\!</math>

The treatments can be allocated to the two operators using the values of the defining contrast. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. Then the value of the defining contrast for treatment <math>a\,\!</math> is:

::<math>a\ \ :\ \ \text{ }L=1+0+0+0=1=1\text{ (mod 2)}\,\!</math>

Therefore, treatment <math>a\,\!</math> should be assigned to Block 1 or the first operator. Similarly, for treatment <math>ab\,\!</math> we have:

::<math>ab\ \ :\ \ \text{ }L=1+1+0+0=2=0\text{ (mod 2)}\,\!</math>

[[Image:doe7.22.png|center|200px| Allocation of treatments to two blocks for the <math>2^4</math> design in the example by confounding interaction of <math>ABCD</math> with the blocks.]]

Therefore, <math>ab\,\!</math> should be assigned to Block 2 or the second operator. Other treatments can be allocated to the two operators in a similar manner to arrive at the allocation scheme shown in the figure below.
In DOE++, to confound the <math>ABCD\,\!</math> interaction for the <math>{2}^{4}\,\!</math> design into two blocks, the number of blocks are specified as shown in the figure below. Then the interaction <math>ABCD\,\!</math> is entered in the Block Generator window (second following figure) which is available using the Block Generator button in the following figure. The design generated by the Weibull++ DOE folio is shown in the third of the following figures. This design matches the allocation scheme of the preceding figure.

[[Image:doe7_23.png|center|700px| Adding block properties for the experiment in the example.|link=]]

[[Image:doe7_24.png|center|700px|Specifying the interaction ABCD as the interaction to be confounded with the blocks for the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

[[Image:doe7_25.png|center|800px|Two block design for the experiment in the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

For the analysis of this design, the sum of squares for all effects are calculated assuming no blocking. Then, to account for blocking, the sum of squares corresponding to the <math>ABCD\,\!</math> interaction is considered as the sum of squares due to blocks and <math>ABCD\,\!</math>. In DOE++ this is done by displaying this sum of squares as the sum of squares due to the blocks. This is shown in the following figure where the sum of squares in question is obtained as 72.25 and is displayed against Block. The interaction ABCD, which is confounded with the blocks, is not displayed. Since the design is unreplicated, any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_26.png|center|793px|ANOVA table for the experiment of the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

===Unreplicated 2''k'' Designs in 2''p'' Blocks===

A single replicate of the <math>{2}^{k}\,\!</math> design can be run in up to <math>{2}^{p}\,\!</math> blocks where <math>p<k\,\!</math>. The number of effects confounded with the blocks equals the degrees of freedom associated with the block effect.

If two blocks are used (the block effect has two levels), then one (<math>2-1=1)\,\!</math> effect is confounded with the blocks. If four blocks are used, then three (<math>4-1=3\,\!</math>) effects are confounded with the blocks and so on. For example an unreplicated <math>{2}^{4}\,\!</math> design may be confounded in <math>{2}^{2}\,\!</math> (four) blocks using two contrasts, <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}}\,\!</math>. Let <math>AC\,\!</math> and <math>BD\,\!</math> be the effects to be confounded with the blocks. Corresponding to these two effects, the contrasts are respectively:

::<math>\begin{align}
& {{L}_{1}}= & {{q}_{1}}+{{q}_{3}} \\
& {{L}_{2}}= & {{q}_{2}}+{{q}_{4}}
\end{align}\,\!</math>

Based on the values of <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}},\,\!</math> the treatments can be assigned to the four blocks as follows:

<center><math>\begin{matrix}
\text{Block 4} & {} & \text{Block 3} & {} & \text{Block 2} & {} & \text{Block 1} \\
{{L}_{1}}=0,{{L}_{2}}=0 & {} & {{L}_{1}}=1,{{L}_{2}}=0 & {} & {{L}_{1}}=0,{{L}_{2}}=1 & {} & {{L}_{1}}=1,{{L}_{2}}=1 \\
{} & {} & {} & {} & {} & {} & {} \\
\left[ \begin{matrix}
(1) \\
ac \\
bd \\
abcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
c \\
abd \\
bcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
b \\
abc \\
d \\
acd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
ab \\
bc \\
ad \\
cd \\
\end{matrix} \right] \\
\end{matrix}\,\!</math></center>

Since the block effect has three degrees of freedom, three effects are confounded with the block effect. In addition to <math>AC\,\!</math> and <math>BD\,\!</math>, the third effect confounded with the block effect is their generalized interaction, <math>(AC)(BD)=ABCD\,\!</math>.
In general, when an unreplicated <math>{2}^{k}\,\!</math> design is confounded in <math>{2}^{p}\,\!</math> blocks, <math>p\,\!</math> contrasts are needed (<math>{{L}_{1}},{{L}_{2}}...{{L}_{p}}\,\!</math>). <math>p\,\!</math> effects are selected to define these contrasts such that none of these effects are the generalized interaction of the others. The <math>{2}^{p}\,\!</math> blocks can then be assigned the treatments using the <math>p\,\!</math> contrasts. <math>{{2}^{p}}-(p+1)\,\!</math> effects, that are also confounded with the blocks, are then obtained as the generalized interaction of the <math>p\,\!</math> effects. In the statistical analysis of these designs, the sum of squares are computed as if no blocking were used. Then the block sum of squares is obtained by adding the sum of squares for all the effects confounded with the blocks.

====Example: 2 Level Factorial Design with Four Blocks====

This example illustrates how a DOE folio obtains the sum of squares when treatments for an unreplicated <math>{2}^{k}\,\!</math> design are allocated among four blocks. Consider again the unreplicated <math>{2}^{4}\,\!</math> design used to investigate the defects in automobile vinyl panels presented in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments needed to complete the experiment were run by four operators. Therefore, there are four blocks. Assume that the treatments were allocated to the blocks using the generators mentioned in the previous section, i.e., treatments were allocated among the four operators by confounding the effects, <math>AC\,\!</math> and <math>BD,\,\!</math> with the blocks. These effects can be specified as Block Generators as shown in the following figure. (The generalized interaction of these two effects, interaction <math>ABCD\,\!</math>, will also get confounded with the blocks.) The resulting design is shown in the second following figure and matches the allocation scheme obtained in the previous section.

[[Image:doe7_27.png|center|700px|Specifying the interactions AC and BD as block generators for the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

The sum of squares in this case can be obtained by calculating the sum of squares for each of the effects assuming there is no blocking. Once the individual sum of squares have been obtained, the block sum of squares can be calculated. The block sum of squares is the sum of the sum of squares of effects, <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, since these effects are confounded with the block effect. As shown in the second following figure, this sum of squares is 92.25 and is displayed against Block. The interactions <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, which are confounded with the blocks, are not displayed. Since the present design is unreplicated any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_28.png|center|800px|Design for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

[[Image:doe7_29.png|center|793px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

==Variability Analysis==

For replicated two level factorial experiments, DOE++ provides the option of conducting variability analysis (using the Variability Analysis icon under the Data menu). The analysis is used to identify the treatment that results in the least amount of variation in the product or process being investigated. Variability analysis is conducted by treating the standard deviation of the response for each treatment of the experiment as an additional response. The standard deviation for a treatment is obtained by using the replicated response values at that treatment run. As an example, consider the <math>{2}^{3}\,\!</math> design shown in the following figure where each run is replicated four times. A variability analysis can be conducted for this design. DOE++ calculates eight standard deviation values corresponding to each treatment of the design (see second following figure). Then, the design is analyzed as an unreplicated <math>{2}^{3}\,\!</math> design with the standard deviations (displayed as Y Standard Deviation. in second following figure) as the response. The normal probability plot of effects identifies <math>AC\,\!</math> as the effect that influences variability (see third figure following). Based on the effect coefficients obtained in the fourth figure following, the model for Y Std. is:

::<math>\begin{align}
& \text{Y Std}\text{.}= & 0.6779+0.2491\cdot AC \\
& = & 0.6779+0.2491{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

Based on the model, the experimenter has two choices to minimize variability (by minimizing Y Std.). The first choice is that <math>{{x}_{1}}\,\!</math> should be <math>1\,\!</math> (i.e., <math>A\,\!</math> should be set at the high level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the low level). The second choice is that <math>{{x}_{1}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>A\,\!</math> should be set at the low level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the high level). The experimenter can select the most feasible choice.

[[Image:doe7.30.png|center|391px|A <math>2^3\,\!</math> design with four replicated response values that can be used to conduct a variability analysis.]]
 

[[Image:doe7_31.png|center|727px|Variability analysis in DOE++.|link=]]
 

[[Image:doe7_32.png|center|650px|Normal probability plot of effects for the variability analysis example.|link=]]
 

[[Image:doe7_33.png|center|727px|Effect coefficients for the variability analysis example.|link=]]

 

==Two Level Fractional Factorial Designs==

As the number of factors in a two level factorial design increases, the number of runs for even a single replicate of the <math>{2}^{k}\,\!</math> design becomes very large. For example, a single replicate of an eight factor two level experiment would require 256 runs. Fractional factorial designs can be used in these cases to draw out valuable conclusions from fewer runs. The basis of fractional factorial designs is the ''sparsity of effects'' principle.[[DOE_References|[Wu, 2000]]] The principle states that, most of the time, responses are affected by a small number of main effects and lower order interactions, while higher order interactions are relatively unimportant. Fractional factorial designs are used as screening experiments during the initial stages of experimentation. At these stages, a large number of factors have to be investigated and the focus is on the main effects and two factor interactions. These designs obtain information about main effects and lower order interactions with fewer experiment runs by confounding these effects with unimportant higher order interactions. As an example, consider a <math>{2}^{8}\,\!</math> design that requires 256 runs. This design allows for the investigation of 8 main effects and 28 two factor interactions. However, 219 degrees of freedom are devoted to three factor or higher order interactions. This full factorial design can prove to be very inefficient when these higher order interactions can be assumed to be unimportant. Instead, a fractional design can be used here to identify the important factors that can then be investigated more thoroughly in subsequent experiments. In unreplicated fractional factorial designs, no degrees of freedom are available to calculate the error sum of squares and the techniques mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] should be employed for the analysis of these designs.

==Half-fraction Designs==

A half-fraction of the <math>{2}^{k}\,\!</math> design involves running only half of the treatments of the full factorial design. For example, consider a <math>{2}^{3}\,\!</math> design that requires eight runs in all. The design matrix for this design is shown in the figure (a) below. A half-fraction of this design is the design in which only four of the eight treatments are run. The fraction is denoted as <math>{2}^{3-1}\,\!</math> with the "<math>-1\,\!</math>" in the index denoting a half-fraction. Assume that the treatments chosen for the half-fraction design are the ones where the interaction <math>ABC\,\!</math> is at the high level (i.e., only those rows are chosen from the following figure (a) where the column for <math>ABC\,\!</math> has entries of 1). The resulting <math>{2}^{3-1}\,\!</math> design has a design matrix as shown in figure (b) below.

[[Image:doe7.34.png|center|330px|Half-fractions of the <math>2^3\,\!</math> design. (a) shows the full factorial <math>2^3\,\!</math> design, (b) shows the <math>2^{3-1}\,\!</math> design with the defining relation <math>I=ABC\,\!</math> and (c) shows the <math>{2}^{3-1}\,\!</math> design with the defining relation <math>I=-ABC\,\!</math>.]]

In the <math>{2}^{3-1}\,\!</math> design of figure (b), since the interaction <math>ABC\,\!</math> is always included at the same level (the high level represented by 1), it is not possible to measure this interaction effect. The effect, <math>ABC\,\!</math>, is called the ''generator'' or ''word'' for this design. It can be noted that, in the design matrix of the following figure (b), the column corresponding to the intercept, <math>I\,\!</math>, and column corresponding to the interaction <math>ABC\,\!</math>, are identical. The identical columns are written as <math>I=ABC\,\!</math> and this equation is called the ''defining relation'' for the design. In DOE++, the present <math>{2}^{3-1}\,\!</math> design can be obtained by specifying the design properties as shown in the following figure.

[[Image:doe7_35.png|center|700px|Design properties for the <math>2^{3-1}\,\!</math> design.|link=]]

The defining relation, <math>I=ABC\,\!</math>, is entered in the Fraction Generator window as shown next.

[[Image:doe7_36.png|center|700px|Specifying the defining relation for the <math>2^{3-1}\,\!</math> design.|link=]]

Note that in the figure following that, the defining relation is specified as <math>C=AB\,\!</math>. This relation is obtained by multiplying the defining relation, <math>I=ABC\,\!</math>, by the last factor, <math>C\,\!</math>, of the design.

===Calculation of Effects===

Using the four runs of the <math>{2}^{3-1}\,\!</math> design in figure (b) discussed above, the main effects can be calculated as follows:

::<math>\begin{align}
& A= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& B= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& C= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

where <math>a\,\!</math>, <math>b\,\!</math>, <math>c\,\!</math> and <math>abc\,\!</math> are the treatments included in the <math>{2}^{3-1}\,\!</math> design.

Similarly, the two factor interactions can also be obtained as:

::<math>\begin{align}
& BC= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& AC= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& AB= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

The equations for <math>A\,\!</math> and <math>BC\,\!</math> above result in the same effect values showing that effects <math>A\,\!</math> and <math>BC\,\!</math> are confounded in the present <math>{2}^{3-1}\,\!</math> design. Thus, the quantity, <math>\tfrac{1}{2}(a-b-c+abc),\,\!</math> estimates <math>A+BC\,\!</math> (i.e., both the main effect <math>A\,\!</math> and the two-factor interaction <math>BC\,\!</math>). The effects, <math>A\,\!</math> and <math>BC,\,\!</math> are called ''aliases''. From the remaining equations given above, it can be seen that the other aliases for this design are <math>B\,\!</math> and <math>AC\,\!</math>, and <math>C\,\!</math> and <math>AB\,\!</math>. Therefore, the equations to calculate the effects in the present <math>{2}^{3-1}\,\!</math> design can be written as follows:

::<math>\begin{align}
& A+BC= & \frac{1}{2}(a-b-c+abc) \\
& B+AC= & \frac{1}{2}(-a+b-c+abc) \\
& C+AB= & \frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

===Calculation of Aliases===

Aliases for a fractional factorial design can be obtained using the defining relation for the design. The defining relation for the present <math>{2}^{3-1}\,\!</math> design is:

::<math>I=ABC\,\!</math>

Multiplying both sides of the previous equation by the main effect, <math>A,\,\!</math> gives the alias effect of <math>A\,\!</math> :

::<math>\begin{align}
& A\cdot I= & A\cdot ABC \\
& A= & {{A}^{2}}BC \\
& A= & BC
\end{align}\,\!</math>

Note that in calculating the alias effects, any effect multiplied by <math>I\,\!</math> remains the same (<math>A\cdot I=A\,\!</math>), while an effect multiplied by itself results in <math>I\,\!</math> (<math>{{A}^{2}}=I\,\!</math>). Other aliases can also be obtained:

::<math>\begin{align}
& B\cdot I= & B\cdot ABC \\
& B= & A{{B}^{2}}C \\
& B= & AC
\end{align}\,\!</math>

:and:

::<math>\begin{align}
& C\cdot I= & C\cdot ABC \\
& C= & AB{{C}^{2}} \\
& C= & AB
\end{align}\,\!</math>

===Fold-over Design===

If it can be assumed for this design that the two-factor interactions are unimportant, then in the absence of <math>BC\,\!</math>, <math>AC\,\!</math> and <math>AB\,\!</math>, the equations for (A+BC), (B+AC) and (C+AB) can be used to estimate the main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. However, if such an assumption is not applicable, then to uncouple the main effects from their two factor aliases, the alternate fraction that contains runs having <math>ABC\,\!</math> at the lower level should be run. The design matrix for this design is shown in the preceding figure (c). The defining relation for this design is <math>I=-ABC\,\!</math> because the four runs for this design are obtained by selecting the rows of the preceding figure (a) for which the value of the <math>ABC\,\!</math> column is <math>-1\,\!</math>. The aliases for this fraction can be obtained as explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]] as <math>A=-BC\,\!</math>, <math>B=-AC\,\!</math> and <math>C=-AB\,\!</math>. The effects for this design can be calculated as:

::<math>\begin{align}
& A-BC= & \frac{1}{2}(ab+ac-(1)-bc) \\
& B-AC= & \frac{1}{2}(ab-ac+(1)-bc) \\
& C-AB= & \frac{1}{2}(-ab+ac-(1)+bc)
\end{align}\,\!</math>

These equations can be combined with the equations for (A+BC), (B+AC) and (C+AB) to obtain the de-aliased main effects and two factor interactions. For example, adding equations (A+BC) and (A-BC) returns the main effect <math>A\,\!</math>.

::<math>\begin{align}
& 2A= & \frac{1}{2}(a-b-c+abc)+ \\
& & \frac{1}{2}(ab+ac-(1)-bc)
\end{align}\,\!</math>

The process of augmenting a fractional factorial design by a second fraction of the same size by simply reversing the signs (of all effect columns except <math>I\,\!</math>) is called ''folding over''. The combined design is referred to as a ''fold-over design''.

==Quarter and Smaller Fraction Designs==

At times, the number of runs even for a half-fraction design are very large. In these cases, smaller fractions are used. A quarter-fraction design, denoted as <math>{2}^{k-2}\,\!</math>, consists of a fourth of the runs of the full factorial design. Quarter-fraction designs require two defining relations. The first defining relation returns the half-fraction or the <math>{2}^{k-1}\,\!</math> design. The second defining relation selects half of the runs of the <math>{2}^{k-1}\,\!</math> design to give the quarter-fraction. For example, consider the <math>{2}^{4}\,\!</math> design. To obtain a <math>{2}^{4-2}\,\!</math> design from this design, first a half-fraction of this design is obtained by using a defining relation. Assume that the defining relation used is <math>I=ABCD\,\!</math>. The design matrix for the resulting <math>{2}^{4-1}\,\!</math> design is shown in figure (a) below. Now, a quarter-fraction can be obtained from the <math>{2}^{4-1}\,\!</math> design shown in figure (a) below using a second defining relation <math>I=AD\,\!</math>. The resulting <math>{2}^{4-2}\,\!</math> design obtained is shown in figure (b) below.

[[Image:doe7.37.png|center|465px|Fractions of the <math>2^4\,\!</math> design - Figure (a) shows the <math>2^{4-1}</math> design with the defining relation <math>I=ABCD\,\!</math> and (b) shows the <math>2^{4-2}\,\!</math> design with the defining relation <math>I=ABCD=AD=BC\,\!</math>.]]

The complete defining relation for this <math>{2}^{4-2}\,\!</math> design is:

::<math>I=ABCD=AD=BC\,\!</math>

Note that the effect, <math>BC,\,\!</math> in the defining relation is the generalized interaction of <math>ABCD\,\!</math> and <math>AD\,\!</math> and is obtained using <math>(ABCD)(AD)={{A}^{2}}BC{{D}^{2}}=BC\,\!</math>. In general, a <math>{2}^{k-p}\,\!</math> fractional factorial design requires <math>p\,\!</math> independent generators. The defining relation for the design consists of the <math>p\,\!</math> independent generators and their <math>{2}^{p}\,\!</math> - (<math>p\,\!</math> +1) generalized interactions.

===Calculation of Aliases===

The alias structure for the present <math>{2}^{4-2}\,\!</math> design can be obtained using the defining relation of equation (I=ABCD=AD=BC) following the procedure explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]]. For example, multiplying the defining relation by <math>A\,\!</math> returns the effects aliased with the main effect, <math>A\,\!</math>, as follows:

::<math>\begin{align}
& A\cdot I= & A\cdot ABCD=A\cdot AD=A\cdot BC \\
& A= & {{A}^{2}}BCD={{A}^{2}}D=ABC \\
& A= & BCD=D=ABC
\end{align}\,\!</math>

Therefore, in the present <math>{2}^{4-2}\,\!</math> design, it is not possible to distinguish between effects <math>A\,\!</math>, <math>D\,\!</math>, <math>BCD\,\!</math> and <math>ABC\,\!</math>. Similarly, multiplying the defining relation by <math>B\,\!</math> and <math>AB\,\!</math> returns the effects that are aliased with these effects:

::<math>\begin{align}
& B= & ACD=ABD=C \\
& AB= & CD=AD=AC
\end{align}\,\!</math>

Other aliases can be obtained in a similar way. It can be seen that each effect in this design has three aliases. In general, each effect in a <math>{2}^{k-p}\,\!</math> design has <math>{2}^{p-1}\,\!</math> aliases.
The aliases for the <math>{2}^{4-2}\,\!</math> design show that in this design the main effects are aliased with each other (<math>A\,\!</math> is aliased with <math>D\,\!</math> and <math>B\,\!</math> is aliased with <math>C\,\!</math>). Therefore, this design is not a useful design and is not available in DOE++. It is important to ensure that main effects and lower order interactions of interest are not aliased in a fractional factorial design. This is known by looking at the resolution of the fractional factorial design.

==Design Resolution==

The resolution of a fractional factorial design is defined as the number of factors in the lowest order effect in the defining relation. For example, in the defining relation <math>I=ABCD=AD=BC\,\!</math> of the previous <math>{2}^{4-2}\,\!</math> design, the lowest-order effect is either <math>AD\,\!</math> or <math>BC,\,\!</math> containing two factors. Therefore, the resolution of this design is equal to two. The resolution of a fractional factorial design is represented using Roman numerals. For example, the previously mentioned <math>{2}^{4-2}\,\!</math> design with a resolution of two can be represented as 2 <math>_{\text{II}}^{4-2}\,\!</math>. The resolution provides information about the confounding in the design as explained next:
 
 
#'''Resolution III Designs''' In these designs, the lowest order effect in the defining relation has three factors (e.g., a <math>{2}^{5-2}\,\!</math> design with the defining relation <math>I=ABDE=ABC=CDE\,\!</math>). In resolution III designs, no main effects are aliased with any other main effects, but main effects are aliased with two factor interactions. In addition, some two factor interactions are aliased with each other.
#'''Resolution IV Designs''' In these designs, the lowest order effect in the defining relation has four factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABDE\,\!</math>). In resolution IV designs, no main effects are aliased with any other main effects or two factor interactions. However, some main effects are aliased with three factor interactions and the two factor interactions are aliased with each other.
#'''Resolution V Designs''' In these designs the lowest order effect in the defining relation has five factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABCDE\,\!</math>). In resolution V designs, no main effects or two factor interactions are aliased with any other main effects or two factor interactions. However, some main effects are aliased with four factor interactions and the two factor interactions are aliased with three factor interactions.

Fractional factorial designs with the highest resolution possible should be selected because the higher the resolution of the design, the less severe the degree of confounding. In general, designs with a resolution less than III are never used because in these designs some of the main effects are aliased with each other. The table below shows fractional factorial designs with the highest available resolution for three to ten factor designs along with their defining relations.

[[Image:doet7.3.png|center|474px|Highest resolution designs available for fractional factorial designs with 3 to 10 factors.]]

All of the two level fractional factorial designs available in DOE++ are shown next.

[[Image:doe7.38.png|center|471px|Two level fractional factorial designs available in DOE++ and their resolutions.]]

===Minimum Aberration Designs===
At times, different designs with the same resolution but different aliasing may be available. The best design to select in such a case is the minimum aberration design. For example, all <math>{2}^{7-2}\,\!</math> designs in the fourth table have a resolution of four (since the generator with the minimum number of factors in each design has four factors). Design <math>1\,\!</math> has three generators of length four (<math>ABCF,\,\!</math> <math>BCDG,\,\!</math> <math>ADFG\,\!</math>). Design <math>2\,\!</math> has two generators of length four (<math>ABCF,\,\!</math> <math>ADEG\,\!</math>). Design <math>3\,\!</math> has one generator of length four (<math>CEFG\,\!</math>). Therefore, design <math>3\,\!</math> has the least number of generators with the minimum length of four. Design <math>3\,\!</math> is called the minimum aberration design. It can be seen that the alias structure for design <math>3\,\!</math> is less involved compared to the other designs. For details refer to [[DOE_References|[Wu, 2000]]].

[[Image:doet7.4.png|center|432px|Three <math>2_{IV}^{7-2}\,\!</math> designs with different defining relations.]]

====Example====

The design of an automobile fuel cone is thought to be affected by six factors in the manufacturing process: cavity temperature (factor <math>A\,\!</math>), core temperature (factor <math>B\,\!</math>), melt temperature (factor <math>C\,\!</math>), hold pressure (factor <math>D\,\!</math>), injection speed (factor <math>E\,\!</math>) and cool time (factor <math>F\,\!</math>). The manufacturer of the fuel cone is unable to run the <math>{2}^{6}=64\,\!</math> runs required to complete one replicate for a two level full factorial experiment with six factors. Instead, they decide to run a fractional factorial design. Considering that three factor and higher order interactions are likely to be inactive, the manufacturer selects a <math>{2}^{6-2}\,\!</math> design that will require only 16 runs. The manufacturer chooses the resolution IV design which will ensure that all main effects are free from aliasing (assuming three factor and higher order interactions are absent). However, in this design the two factor interactions may be aliased with each other. It is decided that, if important two factor interactions are found to be present, additional experiment trials may be conducted to separate the aliased effects. The performance of the fuel cone is measured on a scale of 1 to 15. In DOE++, the design for this experiment is set up using the properties shown in the following figure. The Fraction Generators for the design, <math>E=ABC\,\!</math> and <math>F=BCD\,\!</math>, are the same as the defaults used in DOE++. The resulting <math>{2}^{6-2}\,\!</math> design and the corresponding response values are shown in the following two figures.

[[Image:doe7_39.png|center|644px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

[[Image:doe7_40.png|center|534px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The complete alias structure for the 2 <math>_{\text{IV}}^{6-2}\,\!</math> design is shown next.

<center><math>I=ABCE=ADEF=BCDF\,\!</math></center>

 
<center><math>\begin{align}
& A= & BCE=DEF=ABCDF \\
& B= & ACE=CDF=ABDEF \\
& C= & ABE=BDF=ACDEF \\
& D= & AEF=BCF=ABCDE \\
& E= & ABC=ADF=BCDEF \\
& F= & ADE=BCD=ABCEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& AB= & CE=ACDF=BDEF \\
& AC= & BE=ABDF=CDEF \\
& AD= & EF=ABCF=BCDE \\
& AE= & BC=DF=ABCDEF \\
& AF= & DE=ABCD=BCEF \\
& BD= & CF=ABEF=ACDE \\
& BF= & CD=ABDE=ACEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& ABD= & ACF=BEF=CDE \\
& ABF= & ACD=BDE=CEF
\end{align}\,\!</math></center>

In DOE++, the alias structure is displayed in the Design Summary and as part of the Design Evaluation result, as shown next:

[[Image:doe7_41.png|center|700px|Alias structure for the experiment design in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The normal probability plot of effects for this unreplicated design shows the main effects of factors <math>C\,\!</math> and <math>D\,\!</math> and the interaction effect, <math>BF\,\!</math>, to be significant (see the following figure).

[[Image:doe7_42.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

From the alias structure, it can be seen that for the present design interaction effect, <math>BF,\,\!</math> is confounded with <math>CD\,\!</math>. Therefore, the actual source of this effect cannot be known on the basis of the present experiment. However because neither factor <math>B\,\!</math> nor <math>F\,\!</math> is found to be significant there is an indication the observed effect is likely due to interaction, <math>CD\,\!</math>. To confirm this, a follow-up <math>{2}^{2}\,\!</math> experiment is run involving only factors <math>B\,\!</math> and <math>F\,\!</math>. The interaction, <math>BF\,\!</math>, is found to be inactive, leading to the conclusion that the interaction effect in the original experiment is effect, <math>CD\,\!</math>. Given these results, the fitted regression model for the fuel cone design as per the coefficients obtained from DOE++ is shown next.

::<math>\hat{y}=7.6875+C+2\cdot D+2.1875\cdot CD\,\!</math>

[[Image:doe7_43.png|center|700px|Effect coefficients for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

==Projection==

Projection refers to the reduction of a fractional factorial design to a full factorial design by dropping out some of the factors of the design. Any fractional factorial design of resolution, <math>R,\,\!</math> can be reduced to complete factorial designs in any subset of <math>R-1\,\!</math> factors. For example, consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. The resolution of this design is four. Therefore, this design can be reduced to full factorial designs in any three (<math>4-1=3\,\!</math>) of the original seven factors (by dropping the remaining four of factors). Further, a fractional factorial design can also be reduced to a full factorial design in any <math>R\,\!</math> of the original factors, as long as these <math>R\,\!</math> factors are not part of the generator in the defining relation. Again consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. This design can be reduced to a full factorial design in four factors provided these four factors do not appear together as a generator in the defining relation. The complete defining relation for this design is:

::<math>\begin{align}
& I= & ABCE=BCDF=ACDG=ADEF=ABFG=BDEG=CEFG
\end{align}\,\!</math>

Therefore, there are seven four factor combinations out of the 35 (<math>(_{7}^{4})=35\,\!</math>) possible four-factor combinations that are used as generators in the defining relation. The designs with the remaining 28 four factor combinations would be full factorial 16-run designs. For example, factors <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math> do not occur as a generator in the defining relation of the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. If the remaining factors, <math>E\,\!</math>, <math>F\,\!</math> and <math>G\,\!</math>, are dropped, the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design will reduce to a full factorial design in <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math>.

==Resolution III Designs==

At times, the factors to be investigated in screening experiments are so large that even running a fractional factorial design is impractical. This can be partially solved by using resolution III fractional factorial designs in the cases where three factor and higher order interactions can be assumed to be unimportant. Resolution III designs, such as the 2 <math>_{\text{III}}^{3-1}\,\!</math> design, can be used to estimate <math>k\,\!</math> main effects using just <math>k+1\,\!</math> runs. In these designs, the main effects are aliased with two factor interactions. Once the results from these designs are obtained, and knowing that three factor and higher order interactions are unimportant, the experimenter can decide if there is a need to run a fold-over design to de-alias the main effects from the two factor interactions. Thus, the 2 <math>_{\text{III}}^{3-1}\,\!</math> design can be used to investigate three factors in four runs, the 2 <math>_{\text{III}}^{7-4}\,\!</math> design can be used to investigate seven factors in eight runs, the 2 <math>_{\text{III}}^{15-11}\,\!</math> design can be used to investigate fifteen factors in sixteen runs and so on.

====Example====

{{:Resolution_III_Design_Example}}

==Alias Matrix==
In [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction designs]] and [[Two_Level_Factorial_Experiments#Quarter_and_Smaller_Fraction_Designs| Quarter and Smaller Fraction Designs]], the alias structure for fractional factorial designs was obtained using the defining relation. However, this method of obtaining the alias structure is not very efficient when the alias structure is very complex or when partial aliasing is involved. One of the ways to obtain the alias structure for any design, regardless of its complexity, is to use the alias matrix. The alias matrix for a design is calculated using <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> where <math>{{X}_{1}}\,\!</math> is the portion of the design matrix, <math>X,\,\!</math> that contains the effects for which the aliases need to be calculated, and <math>{{X}_{2}}\,\!</math> contains the remaining columns of the design matrix, other than those included in <math>{{X}_{1}}\,\!</math>.

To illustrate the use of the alias matrix, consider the design matrix for the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design (using the defining relation <math>I=ABCD\,\!</math>) shown next:

[[Image:Chapter7__879.png|center|link=]]

The alias structure for this design can be obtained by defining <math>{{X}_{1}}\,\!</math> using eight columns since the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design estimates eight effects. If the first eight columns of <math>X\,\!</math> are used then <math>{{X}_{1}}\,\!</math> is:

[[Image:Chapter7__884.png|center|link=]]

<math>{{X}_{2}}\,\!</math> is obtained using the remaining columns as:

[[Image:Chapter7__886.png|center|link=]]

Then the alias matrix <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> is:

[[Image:Chapter7__888.png|center|link=]]

The alias relations can be easily obtained by observing the alias matrix as:

<center><math>\begin{align}
& I= & ABCD \\
& A= & BCD \\
& B= & ACD \\
& AB= & CD \\
& C= & ABD \\
& AC= & BD \\
& BC= & AD \\
& D= & ABC
\end{align}\,\!</math></center>

Two Level Factorial Experiments

2017-08-10T17:33:33Z

Richard House: /* Example: Two Level Factorial Design with Two Blocks */

{{Template:Doebook|8}}
Two level factorial experiments are factorial experiments in which each factor is investigated at only two levels. The early stages of experimentation usually involve the investigation of a large number of potential factors to discover the "vital few" factors. Two level factorial experiments are used during these stages to quickly filter out unwanted effects so that attention can then be focused on the important ones.

==2''k'' Designs==
The factorial experiments, where all combination of the levels of the factors are run, are usually referred to as ''full factorial experiments''. Full factorial two level experiments are also referred to as <math>{2}^{k}\,\!</math> designs where <math>k\,\!</math> denotes the number of factors being investigated in the experiment. In Weibull++ DOE folios, these designs are referred to as 2 Level Factorial Designs as shown in the figure below.

[[Image:doe7_1.png|center|487px|Selection of full factorial experiments with two levels in Weibull++.|link=]]

A full factorial two level design with <math>k\,\!</math> factors requires <math>{{2}^{k}}\,\!</math> runs for a single replicate. For example, a two level experiment with three factors will require <math>2\times 2\times 2={{2}^{3}}=8\,\!</math> runs. The choice of the two levels of factors used in two level experiments depends on the factor; some factors naturally have two levels. For example, if gender is a factor, then male and female are the two levels. For other factors, the limits of the range of interest are usually used. For example, if temperature is a factor that varies from <math>{45}^{o}C\,\!</math> to <math>{90}^{o}C\,\!</math>, then the two levels used in the <math>{2}^{k}\,\!</math> design for this factor would be <math>{45}^{o}\,\!C\,\!</math> and <math>{90}^{o}\,\!C\,\!</math>.

The two levels of the factor in the <math>{2}^{k}\,\!</math> design are usually represented as <math>-1\,\!</math> (for the first level) and <math>1\,\!</math> (for the second level). Note that this representation is reversed from the coding used in [[General Full Factorial Designs]] for the indicator variables that represent two level factors in ANOVA models. For ANOVA models, the first level of the factor was represented using a value of <math>1\,\!</math> for the indicator variable, while the second level was represented using a value of <math>-1\,\!</math>. For details on the notation used for two level experiments refer to [[Two_Level_Factorial_Experiments#Notation| Notation]].

===The 22 Design===

The simplest of the two level factorial experiments is the <math>{2}^{2}\,\!</math> design where two factors (say factor <math>A\,\!</math> and factor <math>B\,\!</math>) are investigated at two levels. A single replicate of this design will require four runs (<math>{{2}^{2}}=2\times 2=4\,\!</math>) The effects investigated by this design are the two main effects, <math>A\,\!</math> and <math>B,\,\!</math> and the interaction effect <math>AB\,\!</math>. The treatments for this design are shown in figure (a) below. In figure (a), letters are used to represent the treatments. The presence of a letter indicates the high level of the corresponding factor and the absence indicates the low level. For example, (1) represents the treatment combination where all factors involved are at the low level or the level represented by <math>-1\,\!</math> ; <math>a\,\!</math> represents the treatment combination where factor <math>A\,\!</math> is at the high level or the level of <math>1\,\!</math>, while the remaining factors (in this case, factor <math>B\,\!</math>) are at the low level or the level of <math>-1\,\!</math>. Similarly, <math>b\,\!</math> represents the treatment combination where factor <math>B\,\!</math> is at the high level or the level of <math>1\,\!</math>, while factor <math>A\,\!</math> is at the low level and <math>ab\,\!</math> represents the treatment combination where factors <math>A\,\!</math> and <math>B\,\!</math> are at the high level or the level of the 1. Figure (b) below shows the design matrix for the <math>{2}^{2}\,\!</math> design. It can be noted that the sum of the terms resulting from the product of any two columns of the design matrix is zero. As a result the <math>{2}^{2}\,\!</math> design is an ''orthogonal design''. In fact, all <math>{2}^{k}\,\!</math> designs are orthogonal designs. This property of the <math>{2}^{k}\,\!</math> designs offers a great advantage in the analysis because of the simplifications that result from orthogonality. These simplifications are explained later on in this chapter.
The <math>{2}^{2}\,\!</math> design can also be represented geometrically using a square with the four treatment combinations lying at the four corners, as shown in figure (c) below.

[[Image:doe7.2.png|center|400px|The <math>2^2\,\!</math> design. Figure (a) displays the experiment design, (b) displays the design matrix and (c) displays the geometric representation for the design. In Figure (b), the column names I, A, B and AB are used. Column I represents the intercept term. Columns A and B represent the respective factor settings. Column AB represents the interaction and is the product of columns A and B.]]
 

===The 23 Design===

The <math>{2}^{3}\,\!</math> design is a two level factorial experiment design with three factors (say factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>). This design tests three (<math>k=3\,\!</math>) main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math> ; three (<math>(_{2}^{k})=\,\!</math> <math>(_{2}^{3})=3\,\!</math>) two factor interaction effects, <math>AB\,\!</math>, <math>BC\,\!</math>, <math>AC\,\!</math> ; and one (<math>(_{3}^{k})=\,\!</math> <math>(_{3}^{3})=1\,\!</math>) three factor interaction effect, <math>ABC\,\!</math>. The design requires eight runs per replicate. The eight treatment combinations corresponding to these runs are <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math>, <math>ab\,\!</math>, <math>c\,\!</math>, <math>ac\,\!</math>, <math>bc\,\!</math> and <math>abc\,\!</math>. Note that the treatment combinations are written in such an order that factors are introduced one by one with each new factor being combined with the preceding terms. This order of writing the treatments is called the ''standard order'' or ''Yates' order''. The <math>{2}^{3}\,\!</math> design is shown in figure (a) below. The design matrix for the <math>{2}^{3}\,\!</math> design is shown in figure (b). The design matrix can be constructed by following the standard order for the treatment combinations to obtain the columns for the main effects and then multiplying the main effects columns to obtain the interaction columns.

[[Image:doe7.3.png|center|324px|The <math>2^3\,\!</math> design. Figure (a) shows the experiment design and (b) shows the design matrix.]]

[[Image:doe7.4.png|center|290px|Geometric representation of the <math>2^3\,\!</math> design.]]

The <math>{2}^{3}\,\!</math> design can also be represented geometrically using a cube with the eight treatment combinations lying at the eight corners as shown in the figure above.

==Analysis of 2''k'' Designs==

The <math>{2}^{k}\,\!</math> designs are a special category of the factorial experiments where all the factors are at two levels. The fact that these designs contain factors at only two levels and are orthogonal greatly simplifies their analysis even when the number of factors is large. The use of <math>{2}^{k}\,\!</math> designs in investigating a large number of factors calls for a revision of the notation used previously for the ANOVA models. The case for revised notation is made stronger by the fact that the ANOVA and multiple linear regression models are identical for <math>{2}^{k}\,\!</math> designs because all factors are only at two levels. Therefore, the notation of the regression models is applied to the ANOVA models for these designs, as explained next.

===Notation===

Based on the notation used in [[General Full Factorial Designs]], the ANOVA model for a two level factorial experiment with three factors would be as follows:

::<math>\begin{align}
& Y= & \mu +{{\tau }_{1}}\cdot {{x}_{1}}+{{\delta }_{1}}\cdot {{x}_{2}}+{{(\tau \delta )}_{11}}\cdot {{x}_{1}}{{x}_{2}}+{{\gamma }_{1}}\cdot {{x}_{3}} \\
& & +{{(\tau \gamma )}_{11}}\cdot {{x}_{1}}{{x}_{3}}+{{(\delta \gamma )}_{11}}\cdot {{x}_{2}}{{x}_{3}}+{{(\tau \delta \gamma )}_{111}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where:
 
:• <math>\mu \,\!</math> represents the overall mean
:• <math>{{\tau }_{1}}\,\!</math> represents the independent effect of the first factor (factor <math>A\,\!</math>) out of the two effects <math>{{\tau }_{1}}\,\!</math> and <math>{{\tau }_{2}}\,\!</math>
:• <math>{{\delta }_{1}}\,\!</math> represents the independent effect of the second factor (factor <math>B\,\!</math>) out of the two effects <math>{{\delta }_{1}}\,\!</math> and <math>{{\delta }_{2}}\,\!</math>
:• <math>{{(\tau \delta )}_{11}}\,\!</math> represents the independent effect of the interaction <math>AB\,\!</math> out of the other interaction effects
:• <math>{{\gamma }_{1}}\,\!</math> represents the effect of the third factor (factor <math>C\,\!</math>) out of the two effects <math>{{\gamma }_{1}}\,\!</math> and <math>{{\gamma }_{2}}\,\!</math>
:• <math>{{(\tau \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>AC\,\!</math> out of the other interaction effects
:• <math>{{(\delta \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>BC\,\!</math> out of the other interaction effects
:• <math>{{(\tau \delta \gamma )}_{111}}\,\!</math> represents the effect of the interaction <math>ABC\,\!</math> out of the other interaction effects
and <math>\epsilon \,\!</math> is the random error term.

 
The notation for a linear regression model having three predictor variables with interactions is:

::<math>\begin{align}
& Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& & +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

The notation for the regression model is much more convenient, especially for the case when a large number of higher order interactions are present. In two level experiments, the ANOVA model requires only one indicator variable to represent each factor for both qualitative and quantitative factors. Therefore, the notation for the multiple linear regression model can be applied to the ANOVA model of the experiment that has all the factors at two levels. For example, for the experiment of the ANOVA model given above, <math>{{\beta }_{0}}\,\!</math> can represent the overall mean instead of <math>\mu \,\!</math>, and <math>{{\beta }_{1}}\,\!</math> can represent the independent effect, <math>{{\tau }_{1}}\,\!</math>, of factor <math>A\,\!</math>. Other main effects can be represented in a similar manner. The notation for the interaction effects is much more simplified (e.g., <math>{{\beta }_{123}}\,\!</math> can be used to represent the three factor interaction effect, <math>{{(\tau \beta \gamma )}_{111}}\,\!</math>).

As mentioned earlier, it is important to note that the coding for the indicator variables for the ANOVA models of two level factorial experiments is reversed from the coding followed in [[General Full Factorial Designs]]. Here <math>-1\,\!</math> represents the first level of the factor while <math>1\,\!</math> represents the second level. This is because for a two level factor a single variable is needed to represent the factor for both qualitative and quantitative factors. For quantitative factors, using <math>-1\,\!</math> for the first level (which is the low level) and 1 for the second level (which is the high level) keeps the coding consistent with the numerical value of the factors. The change in coding between the two coding schemes does not affect the analysis except that signs of the estimated effect coefficients will be reversed (i.e., numerical values of <math>{{\hat{\tau }}_{1}}\,\!</math>, obtained based on the coding of [[General Full Factorial Designs]], and <math>{{\hat{\beta }}_{1}}\,\!</math>, obtained based on the new coding, will be the same but their signs would be opposite).

::<math>\begin{align}
& & \text{Factor }A\text{ Coding (two level factor)} \\
& &
\end{align}\,\!</math>

::<math>\begin{matrix}
\text{Previous Coding} & {} & {} & {} & \text{Coding for }{{\text{2}}^{k}}\text{ Designs} \\
{} & {} & {} & {} & {} \\
Effect\text{ }{{\tau }_{1}}\ \ :\ \ {{x}_{1}}=1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{1}}\text{ (or }-{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=-1\text{ } \\
Effect\text{ }{{\tau }_{2}}\ \ :\ \ {{x}_{1}}=-1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{2}}\text{ (or }{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=1\text{ } \\
\end{matrix}\,\!</math>

In summary, the ANOVA model for the experiments with all factors at two levels is different from the ANOVA models for other experiments in terms of the notation in the following two ways:
 

:• The notation of the regression models is used for the effect coefficients.
:• The coding of the indicator variables is reversed.

===Special Features===

Consider the design matrix, <math>X\,\!</math>, for the <math>{2}^{3}\,\!</math> design discussed above. The (<math>{{X}^{\prime }}X\,\!</math>) <math>^{-1}\,\!</math> matrix is:

<center><math>{{({{X}^{\prime }}X)}^{-1}}=\left[ \begin{matrix}
0.125 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0.125 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0.125 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0.125 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0.125 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0.125 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0.125 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.125 \\
\end{matrix} \right]\,\!</math></center>

Notice that, due to the orthogonal design of the <math>X\,\!</math> matrix, the <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> has been simplified to a diagonal matrix which can be written as:

::<math>\begin{align}
{{({{X}^{\prime }}X)}^{-1}}= & 0.125\cdot I = & \frac{1}{8}\cdot I = & \frac{1}{{{2}^{3}}}\cdot I
\end{align}\,\!</math>

where <math>I\,\!</math> represents the identity matrix of the same order as the design matrix, <math>X\,\!</math>. Since there are eight observations per replicate of the <math>{2}^{3}\,\!</math> design, the <math>(X\,\!</math> ' <math>X{{)}^{-1}}\,\!</math> matrix for <math>m\,\!</math> replicates of this design can be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{3}}\cdot m)}\cdot I\,\!</math>

The <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> matrix for any <math>{2}^{k}\,\!</math> design can now be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{k}}\cdot m)}\cdot I\,\!</math>

Then the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is:

::<math>\begin{align}
C= & {{{\hat{\sigma }}}^{2}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & M{{S}_{E}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & \frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}\cdot I
\end{align}\,\!</math>

Note that the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is also a diagonal matrix. Therefore, the estimated effect coefficients (<math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math>, <math>{{\beta }_{12}},\,\!</math> etc.) for these designs are uncorrelated. This implies that the terms in the <math>{2}^{k}\,\!</math> design (main effects, interactions) are independent of each other. Consequently, the extra sum of squares for each of the terms in these designs is independent of the sequence of terms in the model, and also independent of the presence of other terms in the model. As a result the sequential and partial sum of squares for the terms are identical for these designs and will always add up to the model sum of squares. Multicollinearity is also not an issue for these designs.

It can also be noted from the equation given above, that in addition to the <math>C\,\!</math> matrix being diagonal, all diagonal elements of the <math>C\,\!</math> matrix are identical. This means that the variance (or its square root, the standard error) of all estimated effect coefficients are the same. The standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>, for all the coefficients is:

::<math>\begin{align}
se({{{\hat{\beta }}}_{j}})= & \sqrt{{{C}_{jj}}} = & \sqrt{\frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}}\text{ }for\text{ }all\text{ }j
\end{align}\,\!</math>

This property is used to construct the normal probability plot of effects in <math>{2}^{k}\,\!</math> designs and identify significant effects using graphical techniques. For details on the normal probability plot of effects in a Weibull++ DOE folio, refer to [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]].

====Example====
To illustrate the analysis of a full factorial <math>{2}^{k}\,\!</math> design, consider a three factor experiment to investigate the effect of honing pressure, number of strokes and cycle time on the surface finish of automobile brake drums. Each of these factors is investigated at two levels. The honing pressure is investigated at levels of 200 <math>psi\,\!</math> and 400 <math>psi\,\!</math>, the number of strokes used is 3 and 5 and the two levels of the cycle time are 3 and 5 seconds. The design for this experiment is set up in a Weibull++ DOE folio as shown in the first two following figures. It is decided to run two replicates for this experiment. The surface finish data collected from each run (using randomization) and the complete design is shown in the third following figure. The analysis of the experiment data is explained next.

[[Image:doe7_5.png|center|877px|Design properties for the experiment in the example.|link=]]

[[Image:doe7_6.png|center|859px|Design summary for the experiment in the example.|link=]]

[[Image:doe7_7.png|center|778px|Experiment design for the example to investigate the surface finish of automobile brake drums.|link=]]

The applicable model using the notation for <math>{2}^{k}\,\!</math> designs is:

::<math>\begin{align}
Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where the indicator variable, <math>{{x}_{1,}}\,\!</math> represents factor <math>A\,\!</math> (honing pressure), <math>{{x}_{1}}=-1\,\!</math> represents the low level of 200 <math>psi\,\!</math> and <math>{{x}_{1}}=1\,\!</math> represents the high level of 400 <math>psi\,\!</math>. Similarly, <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> represent factors <math>B\,\!</math> (number of strokes) and <math>C\,\!</math> (cycle time), respectively. <math>{{\beta }_{0}}\,\!</math> is the overall mean, while <math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math> and <math>{{\beta }_{3}}\,\!</math> are the effect coefficients for the main effects of factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. <math>{{\beta }_{12}}\,\!</math>, <math>{{\beta }_{13}}\,\!</math> and <math>{{\beta }_{23}}\,\!</math> are the effect coefficients for the <math>AB\,\!</math>, <math>AC\,\!</math> and <math>BC\,\!</math> interactions, while <math>{{\beta }_{123}}\,\!</math> represents the <math>ABC\,\!</math> interaction.

 
If the subscripts for the run (<math>i\,\!</math> ; <math>i=\,\!</math> 1 to 8) and replicates (<math>j\,\!</math> ; <math>j=\,\!</math> 1,2) are included, then the model can be written as:

::<math>\begin{align}
{{Y}_{ij}}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{ij1}}+{{\beta }_{2}}\cdot {{x}_{ij2}}+{{\beta }_{12}}\cdot {{x}_{ij1}}{{x}_{ij2}}+{{\beta }_{3}}\cdot {{x}_{ij3}} \\
& +{{\beta }_{13}}\cdot {{x}_{ij1}}{{x}_{ij3}}+{{\beta }_{23}}\cdot {{x}_{ij2}}{{x}_{ij3}}+{{\beta }_{123}}\cdot {{x}_{ij1}}{{x}_{ij2}}{{x}_{ij3}}+{{\epsilon }_{ij}}
\end{align}\,\!</math>

To investigate how the given factors affect the response, the following hypothesis tests need to be carried:

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{1}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{1}}\ne 0\,\!</math>

This test investigates the main effect of factor <math>A\,\!</math> (honing pressure). The statistic for this test is:

::<math>{{({{F}_{0}})}_{A}}=\frac{M{{S}_{A}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{A}}\,\!</math> is the mean square for factor <math>A\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other main effects, <math>B\,\!</math> and <math>C\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{12}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{12}}\ne 0\,\!</math>

This test investigates the two factor interaction <math>AB\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{AB}}=\frac{M{{S}_{AB}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the interaction <math>AB\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other two factor interactions, <math>AC\,\!</math> and <math>BC\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{123}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{123}}\ne 0\,\!</math>

This test investigates the three factor interaction <math>ABC\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{ABC}}=\frac{M{{S}_{ABC}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{ABC}}\,\!</math> is the mean square for the interaction <math>ABC\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square.
To calculate the test statistics, it is convenient to express the ANOVA model in the form <math>y=X\beta +\epsilon \,\!</math>.

====Expression of the ANOVA Model as <math>y=X\beta +\epsilon \,\!</math>====

In matrix notation, the ANOVA model can be expressed as:

::<math>y=X\beta +\epsilon \,\!</math>

where:

<center><math>y=\left[ \begin{matrix}
{{Y}_{11}} \\
{{Y}_{21}} \\
. \\
{{Y}_{81}} \\
{{Y}_{12}} \\
. \\
{{Y}_{82}} \\
\end{matrix} \right]=\left[ \begin{matrix}
90 \\
90 \\
. \\
90 \\
86 \\
. \\
80 \\
\end{matrix} \right]\text{ }X=\left[ \begin{matrix}
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\end{matrix} \right]\,\!</math></center>

<center><math>\beta =\left[ \begin{matrix}
{{\beta }_{0}} \\
{{\beta }_{1}} \\
{{\beta }_{2}} \\
{{\beta }_{12}} \\
{{\beta }_{3}} \\
{{\beta }_{13}} \\
{{\beta }_{23}} \\
{{\beta }_{123}} \\
\end{matrix} \right]\text{ }\epsilon =\left[ \begin{matrix}
{{\epsilon }_{11}} \\
{{\epsilon }_{21}} \\
. \\
{{\epsilon }_{81}} \\
{{\epsilon }_{12}} \\
. \\
. \\
{{\epsilon }_{82}} \\
\end{matrix} \right]\,\!</math></center>

====Calculation of the Extra Sum of Squares for the Factors====

Knowing the matrices <math>y\,\!</math>, <math>X\,\!</math> and <math>\beta \,\!</math>, the extra sum of squares for the factors can be calculated. These are used to calculate the mean squares that are used to obtain the test statistics. Since the experiment design is orthogonal, the partial and sequential extra sum of squares are identical. The extra sum of squares for each effect can be calculated as shown next. As an example, the extra sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & Model\text{ }Sum\text{ }of\text{ }Squares - Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }main\text{ }effect\text{ }of\text{ }A \\
= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }A}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }A}}={{X}_{\tilde{\ }A}}{{(X_{\tilde{\ }A}^{\prime }{{X}_{\tilde{\ }A}})}^{-1}}X_{\tilde{\ }A}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }A}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the main effect of factor <math>A\,\!</math>. Thus, the sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y \\
= & 654.4375-549.375 \\
= & 105.0625
\end{align}\,\!</math>

Similarly, the extra sum of squares for the interaction effect <math>AB\,\!</math> is:

::<math>\begin{align}
S{{S}_{AB}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }AB}}-(1/16)J]y \\
= & 654.4375-636.375 \\
= & 18.0625
\end{align}\,\!</math>

The extra sum of squares for other effects can be obtained in a similar manner.

====Calculation of the Test Statistics====

Knowing the extra sum of squares, the test statistic for the effects can be calculated. For example, the test statistic for the interaction <math>AB\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{AB}}= & \frac{M{{S}_{AB}}}{M{{S}_{E}}} \\
= & \frac{S{{S}_{AB}}/dof(S{{S}_{AB}})}{S{{S}_{E}}/dof(S{{S}_{E}})} \\
= & \frac{18.0625/1}{147.5/8} \\
= & 0.9797
\end{align}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the <math>AB\,\!</math> interaction and <math>M{{S}_{E}}\,\!</math> is the error mean square. The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{AB}}=0.9797\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{AB}}) \\
= & 1-0.6487 \\
= & 0.3513
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that the interaction between honing pressure and number of strokes does not affect the surface finish of the brake drums. Tests for other effects can be carried out in a similar manner. The results are shown in the ANOVA Table in the following figure. The values S, R-sq and R-sq(adj) in the figure indicate how well the model fits the data. The value of S represents the standard error of the model, R-sq represents the coefficient of multiple determination and R-sq(adj) represents the adjusted coefficient of multiple determination. For details on these values refer to [[Multiple_Linear_Regression_Analysis|Multiple Linear Regression Analysis]].

[[Image:doe7_8.png|center|799px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

====Calculation of Effect Coefficients====

The estimate of effect coefficients can also be obtained:

<center><math>\begin{align}
\hat{\beta }= & {{({{X}^{\prime }}X)}^{-1}}{{X}^{\prime }}y \\
= & \left[ \begin{matrix}
86.4375 \\
2.5625 \\
-4.9375 \\
1.0625 \\
-1.0625 \\
2.4375 \\
-1.3125 \\
-0.1875 \\
\end{matrix} \right]
\end{align}\,\!</math></center>

[[Image:doe7_9.png|center|800px|Regression Information table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

The coefficients and related results are shown in the Regression Information table above. In the table, the Effect column displays the effects, which are simply twice the coefficients. The Standard Error column displays the standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>. The Low CI and High CI columns display the confidence interval on the coefficients. The interval shown is the 90% interval as the significance is chosen as 0.1. The T Value column displays the <math>t\,\!</math> statistic, <math>{{t}_{0}}\,\!</math>, corresponding to the coefficients. The P Value column displays the <math>p\,\!</math> value corresponding to the <math>t\,\!</math> statistic. (For details on how these results are calculated, refer to [[General Full Factorial Designs]]). Plots of residuals can also be obtained from the DOE folio to ensure that the assumptions related to the ANOVA model are not violated.

====Model Equation====

From the analysis results in the above figure within [[Two_Level_Factorial_Experiments#Calculation_of_Effect_Coefficients|calculation of effect coefficients]] section, it is seen that effects <math>A\,\!</math>, <math>B\,\!</math> and <math>AC\,\!</math> are significant. In a DOE folio, the <math>p\,\!</math> values for the significant effects are displayed in red in the ANOVA Table for easy identification. Using the values of the estimated effect coefficients, the model for the present <math>{2}^{3}\,\!</math> design in terms of the coded values can be written as:

::<math>\begin{align}
\hat{y}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}} \\
= & 86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+2.4375{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

To make the model hierarchical, the main effect, <math>C\,\!</math>, needs to be included in the model (because the interaction <math>AC\,\!</math> is included in the model). The resulting model is:

::<math>\hat{y}=86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+1.0625{{x}_{3}}+2.4375{{x}_{1}}{{x}_{3}}\,\!</math>

This equation can be viewed in a DOE folio, as shown in the following figure, using the Show Analysis Summary icon in the Control Panel. The equation shown in the figure will match the hierarchical model once the required terms are selected using the Select Effects icon.

[[Image:doe7_10.png|center|557px|The model equation for the experiment of the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

==Replicated and Repeated Runs==

In the case of replicated experiments, it is important to note the difference between replicated runs and repeated runs. Both repeated and replicated runs are multiple response readings taken at the same factor levels. However, repeated runs are response observations taken at the same time or in succession. Replicated runs are response observations recorded in a random order. Therefore, replicated runs include more variation than repeated runs. For example, a baker, who wants to investigate the effect of two factors on the quality of cakes, will have to bake four cakes to complete one replicate of a <math>{2}^{2}\,\!</math> design. Assume that the baker bakes eight cakes in all. If, for each of the four treatments of the <math>{2}^{2}\,\!</math> design, the baker selects one treatment at random and then bakes two cakes for this treatment at the same time then this is a case of two repeated runs. If, however, the baker bakes all the eight cakes randomly, then the eight cakes represent two sets of replicated runs.
For repeated measurements, the average values of the response for each treatment should be entered into a DOE folio as shown in the following figure (a) when the two cakes for a particular treatment are baked together. For replicated measurements, when all the cakes are baked randomly, the data is entered as shown in the following figure (b).

[[Image:doe7.11.png|center|300px|Data entry for repeated and replicated runs. Figure (a) shows repeated runs and (b) shows replicated runs.]]

==Unreplicated 2''k'' Designs==

If a factorial experiment is run only for a single replicate then it is not possible to test hypotheses about the main effects and interactions as the error sum of squares cannot be obtained. This is because the number of observations in a single replicate equals the number of terms in the ANOVA model. Hence the model fits the data perfectly and no degrees of freedom are available to obtain the error sum of squares.

However, sometimes it is only possible to run a single replicate of the <math>{2}^{k}\,\!</math> design because of constraints on resources and time. In the absence of the error sum of squares, hypothesis tests to identify significant factors cannot be conducted. A number of methods of analyzing information obtained from unreplicated <math>{2}^{k}\,\!</math> designs are available. These include pooling higher order interactions, using the normal probability plot of effects or including center point replicates in the design.

===Pooling Higher Order Interactions===

One of the ways to deal with unreplicated <math>{2}^{k}\,\!</math> designs is to use the sum of squares of some of the higher order interactions as the error sum of squares provided these higher order interactions can be assumed to be insignificant. By dropping some of the higher order interactions from the model, the degrees of freedom corresponding to these interactions can be used to estimate the error mean square. Once the error mean square is known, the test statistics to conduct hypothesis tests on the factors can be calculated.

===Normal Probability Plot of Effects===
Another way to use unreplicated <math>{2}^{k}\,\!</math> designs to identify significant effects is to construct the normal probability plot of the effects. As mentioned in [[Two_Level_Factorial_Experiments#Special_Features| Special Features]], the standard error for all effect coefficients in the <math>{2}^{k}\,\!</math> designs is the same. Therefore, on a normal probability plot of effect coefficients, all non-significant effect coefficients (with <math>\beta =0\,\!</math>) will fall along the straight line representative of the normal distribution, N(<math>0,{{\sigma }^{2}}/({{2}^{k}}\cdot m)\,\!</math>). Effect coefficients that show large deviations from this line will be significant since they do not come from this normal distribution. Similarly, since effects <math>=2\times \,\!</math> effect coefficients, all non-significant effects will also follow a straight line on the normal probability plot of effects. For replicated designs, the Effects Probability plot of a DOE folio plots the normalized effect values (or the T Values) on the standard normal probability line, N(0,1). However, in the case of unreplicated <math>{2}^{k}\,\!</math> designs, <math>{{\sigma }^{2}}\,\!</math> remains unknown since <math>M{{S}_{E}}\,\!</math> cannot be obtained. Lenth's method is used in this case to estimate the variance of the effects. For details on Lenth's method, please refer to [[DOE References| Montgomery (2001)]]. The DOE folio then uses this variance value to plot effects along the N(0, Lenth's effect variance) line. The
method is illustrated in the following example.

====Example====

Vinyl panels, used as instrument panels in a certain automobile, are seen to develop defects after a certain amount of time. To investigate the issue, it is decided to carry out a two level factorial experiment. Potential factors to be investigated in the experiment are vacuum rate (factor <math>A\,\!</math>), material temperature (factor <math>B\,\!</math>), element intensity (factor <math>C\,\!</math>) and pre-stretch (factor <math>D\,\!</math>). The two levels of the factors used in the experiment are as shown in below.

[[Image:doet7.1.png|center|300px|Factors to investigate defects in vinyl panels.]]

With a <math>{2}^{4}\,\!</math> design requiring 16 runs per replicate it is only feasible for the manufacturer to run a single replicate.

The experiment design and data, collected as percent defects, are shown in the following figure. Since the present experiment design contains only a single replicate, it is not possible to obtain an estimate of the error sum of squares, <math>S{{S}_{E}}\,\!</math>. It is decided to use the normal probability plot of effects to identify the significant effects. The effect values for each term are obtained as shown in the following figure.

[[Image:doe7_13.png|center|650px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

Lenth's method uses these values to estimate the variance. As described in [[DOE_References|[Lenth, 1989]]], if all effects are arranged in ascending order, using their absolute values, then <math>{{s}_{0}}\,\!</math> is defined as 1.5 times the median value:

::<math>\begin{align}
{{s}_{0}}= & 1.5\cdot median(\left| effect \right|) \\
= & 1.5\cdot 2 \\
= & 3
\end{align}\,\!</math>

Using <math>{{s}_{0}}\,\!</math>, the "pseudo standard error" (<math>PSE\,\!</math>) is calculated as 1.5 times the median value of all effects that are less than 2.5 <math>{{s}_{0}}\,\!</math> :

::<math>\begin{align}
PSE= & 1.5\cdot median(\left| effect \right|\ \ :\ \ \left| effect \right|<2.5{{s}_{0}}) \\
= & 1.5\cdot 1.5 \\
= & 2.25
\end{align}\,\!</math>

Using <math>PSE\,\!</math> as an estimate of the effect variance, the effect variance is 2.25. Knowing the effect variance, the normal probability plot of effects for the present unreplicated experiment can be constructed as shown in the following figure. The line on this plot is the line N(0, 2.25). The plot shows that the effects <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math> do not follow the distribution represented by this line. Therefore, these effects are significant.

The significant effects can also be identified by comparing individual effect values to the margin of error or the threshold value using the pareto chart (see the third following figure). If the required significance is 0.1, then:

::<math>margin\text{ }of\text{ }error={{t}_{\alpha /2,d}}\cdot PSE\,\!</math>

The <math>t\,\!</math> statistic, <math>{{t}_{\alpha /2,d}}\,\!</math>, is calculated at a significance of <math>\alpha /2\,\!</math> (for the two-sided hypothesis) and degrees of freedom <math>d=(\,\!</math> number of effects <math>)/3\,\!</math>. Thus:

::<math>\begin{align}
margin\text{ }of\text{ }error= & {{t}_{0.05,5}}\cdot PSE \\
= & 2.015\cdot 2.25 \\
= & 4.534
\end{align}\,\!</math>

The value of 4.534 is shown as the critical value line in the third following figure. All effects with absolute values greater than the margin of error can be considered to be significant. These effects are <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math>. Therefore, the vacuum rate, the pre-stretch and their interaction have a significant effect on the defects of the vinyl panels.

[[Image:doe7_14.png|center|800px|Effect values for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_15.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_16.png|center|650px|Pareto chart for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

===Center Point Replicates===

Another method of dealing with unreplicated <math>{2}^{k}\,\!</math> designs that only have quantitative factors is to use replicated runs at the center point. The center point is the response corresponding to the treatment exactly midway between the two levels of all factors. Running multiple replicates at this point provides an estimate of pure error. Although running multiple replicates at any treatment level can provide an estimate of pure error, the other advantage of running center point replicates in the <math>{2}^{k}\,\!</math> design is in checking for the presence of curvature. The test for curvature investigates whether the model between the response and the factors is linear and is discussed in [[Two_Level_Factorial_Experiments#Using_Center_Point_Replicates_to_Test_Curvature| Center Pt. Replicates to Test Curvature]].

====Example: Use Center Point to Get Pure Error====

Consider a <math>{2}^{2}\,\!</math> experiment design to investigate the effect of two factors, <math>A\,\!</math> and <math>B\,\!</math>, on a certain response. The energy consumed when the treatments of the <math>{2}^{2}\,\!</math> design are run is considerably larger than the energy consumed for the center point run (because at the center point the factors are at their middle levels). Therefore, the analyst decides to run only a single replicate of the design and augment the design by five replicated runs at the center point as shown in the following figure. The design properties for this experiment are shown in the second following figure. The complete experiment design is shown in the third following figure. The center points can be used in the identification of significant effects as shown next.

[[Image:doe7.17.png||center|300px|<math>2^2\,\!</math> design augmented by five center point runs.]]
[[Image:doe7_18.png|center|537px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

[[Image:doe7_19.png|center|630px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

Since the present <math>{2}^{2}\,\!</math> design is unreplicated, there are no degrees of freedom available to calculate the error sum of squares. By augmenting this design with five center points, the response values at the center points, <math>y_{i}^{c}\,\!</math>, can be used to obtain an estimate of pure error, <math>S{{S}_{PE}}\,\!</math>. Let <math>{{\bar{y}}^{c}}\,\!</math> represent the average response for the five replicates at the center. Then:

::<math>S{{S}_{PE}}=Sum\text{ }of\text{ }Squares\text{ }for\text{ }center\text{ }points\,\!</math>

::<math>\begin{align}
S{{S}_{PE}}= & \underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}} \\
= & {{(25.2-25.26)}^{2}}+...+{{(25.3-25.26)}^{2}} \\
= & 0.052
\end{align}\,\!</math>

Then the corresponding mean square is:

::<math>\begin{align}
M{{S}_{PE}}= & \frac{S{{S}_{PE}}}{degrees\text{ }of\text{ }freedom} \\
= & \frac{0.052}{5-1} \\
= & 0.013
\end{align}\,\!</math>

Alternatively, <math>M{{S}_{PE}}\,\!</math> can be directly obtained by calculating the variance of the response values at the center points:

::<math>\begin{align}
M{{S}_{PE}}= & {{s}^{2}} \\
= & \frac{\underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}}}{5-1}
\end{align}\,\!</math>

Once <math>M{{S}_{PE}}\,\!</math> is known, it can be used as the error mean square, <math>M{{S}_{E}}\,\!</math>, to carry out the test of significance for each effect. For example, to test the significance of the main effect of factor <math>A,\,\!</math> the sum of squares corresponding to this effect is obtained in the usual manner by considering only the four runs of the original <math>{2}^{2}\,\!</math> design.

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/4)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/4)J]y \\
= & 0.5625
\end{align}\,\!</math>

Then, the test statistic to test the significance of the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{A}}= & \frac{M{{S}_{A}}}{M{{S}_{E}}} \\
= & \frac{0.5625/1}{0.052/4} \\
= & 43.2692
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{A}}=43.2692\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{A}}) \\
= & 1-0.9972 \\
= & 0.0028
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value < 0.1, it can be concluded that the main effect of factor <math>A\,\!</math> significantly affects the response. This result is displayed in the ANOVA table as shown in the following figure. Test for the significance of other factors can be carried out in a similar manner.

[[Image:doe7_20.png|center|649px|Results for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

===Using Center Point Replicates to Test Curvature===

Center point replicates can also be used to check for curvature in replicated or unreplicated <math>{2}^{k}\,\!</math> designs. The test for curvature investigates whether the model between the response and the factors is linear. The way DOE++ handles center point replicates is similar to its handling of blocks. The center point replicates are treated as an additional factor in the model. The factor is labeled as Curvature in the results of the DOE folio. If Curvature turns out to be a significant factor in the results, then this indicates the presence of curvature in the model.

====Example: Use Center Point to Test Curvature====

To illustrate the use of center point replicates in testing for curvature, consider again the data of the single replicate <math>{2}^{2}\,\!</math> experiment from a preceding figure(labeled "<math>2^2</math> design augmented by five center point runs"). Let <math>{{x}_{1}}\,\!</math> be the indicator variable to indicate if the run is a center point:

::<math>\begin{matrix}
{{x}_{1}}=0 & {} & \text{Center point run} \\
{{x}_{1}}=1 & {} & \text{Other run} \\
\end{matrix}\,\!</math>

If <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> are the indicator variables representing factors <math>A\,\!</math> and <math>B\,\!</math>, respectively, then the model for this experiment is:

::<math>Y={{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}\,\!</math>

To investigate the presence of curvature, the following hypotheses need to be tested:

::<math>\begin{align}
& {{H}_{0}}: & {{\beta }_{1}}=0\text{ (Curvature is absent)} \\
& {{H}_{1}}: & {{\beta }_{1}}\ne 0
\end{align}\,\!</math>

The test statistic to be used for this test is:

::<math>{{({{F}_{0}})}_{curvature}}=\frac{M{{S}_{curvature}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{curvature}}\,\!</math> is the mean square for Curvature and <math>M{{S}_{E}}\,\!</math> is the error mean square.

'''Calculation of the Sum of Squares'''

The <math>X\,\!</math> matrix and <math>y\,\!</math> vector for this experiment are:

<center><math>X=\left[ \begin{matrix}
1 & 1 & -1 & -1 & 1 \\
1 & 1 & 1 & -1 & -1 \\
1 & 1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
\end{matrix} \right]\text{ }y=\left[ \begin{matrix}
24.6 \\
25.4 \\
25.0 \\
25.7 \\
25.2 \\
25.3 \\
25.4 \\
25.1 \\
25.3 \\
\end{matrix} \right]\,\!</math></center>

The sum of squares can now be calculated. For example, the error sum of squares is:

::<math>\begin{align}
& S{{S}_{E}}= & {{y}^{\prime }}[I-H]y \\
& = & 0.052
\end{align}\,\!</math>

where <math>I\,\!</math> is the identity matrix and <math>H\,\!</math> is the hat matrix. It can be seen that this is equal to <math>S{{S}_{PE\text{ }}}\,\!</math> (the sum of squares due to pure error) because of the replicates at the center point, as obtained in the [[Two_Level_Factorial_Experiments#Example_3| example]]. The number of degrees of freedom associated with <math>S{{S}_{E}}\,\!</math>, <math>dof(S{{S}_{E}})\,\!</math> is four. The extra sum of squares corresponding to the center point replicates (or Curvature) is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & Model\text{ }Sum\text{ }of\text{ }Squares- \\
& & Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }center\text{ }point \\
& = & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Curvature}}-(1/9)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }Curvature}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }Curvature}}={{X}_{\tilde{\ }Curv}}{{(X_{\tilde{\ }Curv}^{\prime }{{X}_{\tilde{\ }Curv}})}^{-1}}X_{\tilde{\ }Curv}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }Curv}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the center point. Thus, the extra sum of squares corresponding to Curvature is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Center}}-(1/9)J]y \\
& = & 0.7036-0.6875 \\
& = & 0.0161
\end{align}\,\!</math>

This extra sum of squares can be used to test for the significance of curvature. The corresponding mean square is:

::<math>\begin{align}
& M{{S}_{Curvature}}= & \frac{Sum\text{ }of\text{ }squares\text{ }corresponding\text{ }to\text{ }Curvature}{degrees\text{ }of\text{ }freedom} \\
& = & \frac{0.0161}{1} \\
& = & 0.0161
\end{align}\,\!</math>

'''Calculation of the Test Statistic'''

Knowing the mean squares, the statistic to check the significance of curvature can be calculated.

::<math>\begin{align}
& {{({{f}_{0}})}_{Curvature}}= & \frac{M{{S}_{Curvature}}}{M{{S}_{E}}} \\
& = & \frac{0.0161/1}{0.052/4} \\
& = & 1.24
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{Curvature}}=1.24\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and four degrees of freedom in the denominator is:

::<math>\begin{align}
& p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{Curvature}}) \\
& = & 1-0.6713 \\
& = & 0.3287
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that curvature does not exist for this design. This results is shown in the ANOVA table in the figure above. The surface of the fitted model based on these results, along with the observed response values, is shown in the figure below.

[[Image:doe7.21.png|center|400px|Model surface and observed response values for the design in the [[Two_Level_Factorial_Experiments#Example_4| example]].]]

 

==Blocking in 2k Designs==

Blocking can be used in the <math>{2}^{k}\,\!</math> designs to deal with cases when replicates cannot be run under identical conditions. Randomized complete block designs that were discussed in [[Randomization and Blocking in DOE]] for factorial experiments are also applicable here. At times, even with just two levels per factor, it is not possible to run all treatment combinations for one replicate of the experiment under homogeneous conditions. For example, each replicate of the <math>{2}^{2}\,\!</math> design requires four runs. If each run requires two hours and testing facilities are available for only four hours per day, two days of testing would be required to run one complete replicate. Blocking can be used to separate the treatment runs on the two different days. Blocks that do not contain all treatments of a replicate are called incomplete blocks. In incomplete block designs, the block effect is confounded with certain effect(s) under investigation. For the <math>{2}^{2}\,\!</math> design assume that treatments <math>(1)\,\!</math> and <math>ab\,\!</math> were run on the first day and treatments <math>a\,\!</math> and <math>b\,\!</math> were run on the second day. Then, the incomplete block design for this experiment is:

::<math>\begin{matrix}
\text{Block 1} & {} & \text{Block 2} \\
\left[ \begin{matrix}
(1) \\
ab \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
b \\
\end{matrix} \right] \\
\end{matrix}\,\!</math>

For this design the block effect may be calculated as:

::<math>\begin{align}
& Block\text{ }Effect= & Average\text{ }response\text{ }for\text{ }Block\text{ }1- \\
& & Average\text{ }response\text{ }for\text{ }Block\text{ }2 \\
& = & \frac{(1)+ab}{2}-\frac{a+b}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The <math>AB\,\!</math> interaction effect is:

::<math>\begin{align}
& AB= & Average\text{ }response\text{ }at\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{low}}}- \\
& & Average\text{ }response\text{ }at\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{low}}} \\
& = & \frac{ab+(1)}{2}-\frac{b+a}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The two equations given above show that, in this design, the <math>AB\,\!</math> interaction effect cannot be distinguished from the block effect because the formulas to calculate these effects are the same. In other words, the <math>AB\,\!</math> interaction is said to be confounded with the block effect and it is not possible to say if the effect calculated based on these equations is due to the <math>AB\,\!</math> interaction effect, the block effect or both. In incomplete block designs some effects are always confounded with the blocks. Therefore, it is important to design these experiments in such a way that the important effects are not confounded with the blocks. In most cases, the experimenter can assume that higher order interactions are unimportant. In this case, it would better to use incomplete block designs that confound these effects with the blocks.
One way to design incomplete block designs is to use defining contrasts as shown next:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}+...+{{\alpha }_{k}}{{q}_{k}}\,\!</math>

where the <math>{{\alpha }_{i}}\,\!</math> s are the exponents for the factors in the effect that is to be confounded with the block effect and the <math>{{q}_{i}}\,\!</math> s are values based on the level of the <math>i\,\!</math> the factor (in a treatment that is to be allocated to a block). For <math>{2}^{k}\,\!</math> designs the <math>{{\alpha }_{i}}\,\!</math> s are either 0 or 1 and the <math>{{q}_{i}}\,\!</math> s have a value of 0 for the low level of the <math>i\,\!</math> th factor and a value of 1 for the high level of the factor in the treatment under consideration. As an example, consider the <math>{2}^{2}\,\!</math> design where the interaction effect <math>AB\,\!</math> is confounded with the block. Since there are two factors, <math>k=2\,\!</math>, with <math>i=1\,\!</math> representing factor <math>A\,\!</math> and <math>i=2\,\!</math> representing factor <math>B\,\!</math>. Therefore:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}\,\!</math>

The value of <math>{{\alpha }_{1}}\,\!</math> is one because the exponent of factor <math>A\,\!</math> in the confounded interaction <math>AB\,\!</math> is one. Similarly, the value of <math>{{\alpha }_{2}}\,\!</math> is one because the exponent of factor <math>B\,\!</math> in the confounded interaction <math>AB\,\!</math> is also one. Therefore, the defining contrast for this design can be written as:

::<math>\begin{align}
& L= & {{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}} \\
& = & 1\cdot {{q}_{1}}+1\cdot {{q}_{2}} \\
& = & {{q}_{1}}+{{q}_{2}}
\end{align}\,\!</math>

Once the defining contrast is known, it can be used to allocate treatments to the blocks. For the <math>{2}^{2}\,\!</math> design, there are four treatments <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math> and <math>ab\,\!</math>. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. In order to decide which block the treatment <math>(1)\,\!</math> belongs to, the levels of factors <math>A\,\!</math> and <math>B\,\!</math> for this run are used. Since factor <math>A\,\!</math> is at the low level in this treatment, <math>{{q}_{1}}=0\,\!</math>. Similarly, since factor <math>B\,\!</math> is also at the low level in this treatment, <math>{{q}_{2}}=0\,\!</math>. Therefore:

::<math>\begin{align}
& L= & {{q}_{1}}+{{q}_{2}} \\
& = & 0+0=0\text{ (mod 2)}
\end{align}\,\!</math>

Note that the value of <math>L\,\!</math> used to decide the block allocation is "mod 2" of the original value. This value is obtained by taking the value of 1 for odd numbers and 0 otherwise. Based on the value of <math>L\,\!</math>, treatment <math>(1)\,\!</math> is assigned to block 1. Other treatments can be assigned using the following calculations:

::<math>\begin{align}
& (1): & \text{ }L=0+0=0=0\text{ (mod 2)} \\
& a: & \text{ }L=1+0=1=1\text{ (mod 2)} \\
& b: & \text{ }L=0+1=1=1\text{ (mod 2)} \\
& ab: & \text{ }L=1+1=2=0\text{ (mod 2)}
\end{align}\,\!</math>

Therefore, to confound the interaction <math>AB\,\!</math> with the block effect in the <math>{2}^{2}\,\!</math> incomplete block design, treatments <math>(1)\,\!</math> and <math>ab\,\!</math> (with <math>L=0\,\!</math>) should be assigned to block 2 and treatment combinations <math>a\,\!</math> and <math>b\,\!</math> (with <math>L=1\,\!</math>) should be assigned to block 1.

====Example: Two Level Factorial Design with Two Blocks====

This example illustrates how treatments can be allocated to two blocks for an unreplicated <math>{2}^{k}\,\!</math> design. Consider the unreplicated <math>{2}^{4}\,\!</math> design to investigate the four factors affecting the defects in automobile vinyl panels discussed in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments required for this experiment were run by two different operators with each operator conducting 8 runs. This experiment is an example of an incomplete block design. The analyst in charge of this experiment assumed that the interaction <math>ABCD\,\!</math> was not significant and decided to allocate treatments to the two operators so that the <math>ABCD\,\!</math> interaction was confounded with the block effect (the two operators are the blocks). The allocation scheme to assign treatments to the two operators can be obtained as follows.
 
The defining contrast for the <math>{2}^{4}\,\!</math> design where the <math>ABCD\,\!</math> interaction is confounded with the blocks is:

::<math>L={{q}_{1}}+{{q}_{2}}+{{q}_{3}}+{{q}_{4}}\,\!</math>

The treatments can be allocated to the two operators using the values of the defining contrast. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. Then the value of the defining contrast for treatment <math>a\,\!</math> is:

::<math>a\ \ :\ \ \text{ }L=1+0+0+0=1=1\text{ (mod 2)}\,\!</math>

Therefore, treatment <math>a\,\!</math> should be assigned to Block 1 or the first operator. Similarly, for treatment <math>ab\,\!</math> we have:

::<math>ab\ \ :\ \ \text{ }L=1+1+0+0=2=0\text{ (mod 2)}\,\!</math>

[[Image:doe7.22.png|center|200px| Allocation of treatments to two blocks for the <math>2^4</math> design in the example by confounding interaction of <math>ABCD</math> with the blocks.]]

Therefore, <math>ab\,\!</math> should be assigned to Block 2 or the second operator. Other treatments can be allocated to the two operators in a similar manner to arrive at the allocation scheme shown in the figure below.
In DOE++, to confound the <math>ABCD\,\!</math> interaction for the <math>{2}^{4}\,\!</math> design into two blocks, the number of blocks are specified as shown in the figure below. Then the interaction <math>ABCD\,\!</math> is entered in the Block Generator window (second following figure) which is available using the Block Generator button in the following figure. The design generated by the Weibull++ DOE folio is shown in the third of the following figures. This design matches the allocation scheme of the preceding figure.

[[Image:doe7_23.png|center|700px| Adding block properties for the experiment in the example.|link=]]

[[Image:doe7_24.png|center|700px|Specifying the interaction ABCD as the interaction to be confounded with the blocks for the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

[[Image:doe7_25.png|center|800px|Two block design for the experiment in the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

For the analysis of this design, the sum of squares for all effects are calculated assuming no blocking. Then, to account for blocking, the sum of squares corresponding to the <math>ABCD\,\!</math> interaction is considered as the sum of squares due to blocks and <math>ABCD\,\!</math>. In DOE++ this is done by displaying this sum of squares as the sum of squares due to the blocks. This is shown in the following figure where the sum of squares in question is obtained as 72.25 and is displayed against Block. The interaction ABCD, which is confounded with the blocks, is not displayed. Since the design is unreplicated, any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_26.png|center|793px|ANOVA table for the experiment of the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

===Unreplicated 2''k'' Designs in 2''p'' Blocks===

A single replicate of the <math>{2}^{k}\,\!</math> design can be run in up to <math>{2}^{p}\,\!</math> blocks where <math>p<k\,\!</math>. The number of effects confounded with the blocks equals the degrees of freedom associated with the block effect.

If two blocks are used (the block effect has two levels), then one (<math>2-1=1)\,\!</math> effect is confounded with the blocks. If four blocks are used, then three (<math>4-1=3\,\!</math>) effects are confounded with the blocks and so on. For example an unreplicated <math>{2}^{4}\,\!</math> design may be confounded in <math>{2}^{2}\,\!</math> (four) blocks using two contrasts, <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}}\,\!</math>. Let <math>AC\,\!</math> and <math>BD\,\!</math> be the effects to be confounded with the blocks. Corresponding to these two effects, the contrasts are respectively:

::<math>\begin{align}
& {{L}_{1}}= & {{q}_{1}}+{{q}_{3}} \\
& {{L}_{2}}= & {{q}_{2}}+{{q}_{4}}
\end{align}\,\!</math>

Based on the values of <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}},\,\!</math> the treatments can be assigned to the four blocks as follows:

<center><math>\begin{matrix}
\text{Block 4} & {} & \text{Block 3} & {} & \text{Block 2} & {} & \text{Block 1} \\
{{L}_{1}}=0,{{L}_{2}}=0 & {} & {{L}_{1}}=1,{{L}_{2}}=0 & {} & {{L}_{1}}=0,{{L}_{2}}=1 & {} & {{L}_{1}}=1,{{L}_{2}}=1 \\
{} & {} & {} & {} & {} & {} & {} \\
\left[ \begin{matrix}
(1) \\
ac \\
bd \\
abcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
c \\
abd \\
bcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
b \\
abc \\
d \\
acd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
ab \\
bc \\
ad \\
cd \\
\end{matrix} \right] \\
\end{matrix}\,\!</math></center>

Since the block effect has three degrees of freedom, three effects are confounded with the block effect. In addition to <math>AC\,\!</math> and <math>BD\,\!</math>, the third effect confounded with the block effect is their generalized interaction, <math>(AC)(BD)=ABCD\,\!</math>.
In general, when an unreplicated <math>{2}^{k}\,\!</math> design is confounded in <math>{2}^{p}\,\!</math> blocks, <math>p\,\!</math> contrasts are needed (<math>{{L}_{1}},{{L}_{2}}...{{L}_{p}}\,\!</math>). <math>p\,\!</math> effects are selected to define these contrasts such that none of these effects are the generalized interaction of the others. The <math>{2}^{p}\,\!</math> blocks can then be assigned the treatments using the <math>p\,\!</math> contrasts. <math>{{2}^{p}}-(p+1)\,\!</math> effects, that are also confounded with the blocks, are then obtained as the generalized interaction of the <math>p\,\!</math> effects. In the statistical analysis of these designs, the sum of squares are computed as if no blocking were used. Then the block sum of squares is obtained by adding the sum of squares for all the effects confounded with the blocks.

====Example: 2 Level Factorial Design with Four Blocks====

This example illustrates how DOE++ obtains the sum of squares when treatments for an unreplicated <math>{2}^{k}\,\!</math> design are allocated among four blocks. Consider again the unreplicated <math>{2}^{4}\,\!</math> design used to investigate the defects in automobile vinyl panels presented in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments needed to complete the experiment were run by four operators. Therefore, there are four blocks. Assume that the treatments were allocated to the blocks using the generators mentioned in the previous section, i.e., treatments were allocated among the four operators by confounding the effects, <math>AC\,\!</math> and <math>BD,\,\!</math> with the blocks. These effects can be specified as Block Generators as shown in the following figure. (The generalized interaction of these two effects, interaction <math>ABCD\,\!</math>, will also get confounded with the blocks.) The resulting design is shown in the second following figure and matches the allocation scheme obtained in the previous section.

[[Image:doe7_27.png|center|700px|Specifying the interactions AC and BD as block generators for the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

The sum of squares in this case can be obtained by calculating the sum of squares for each of the effects assuming there is no blocking. Once the individual sum of squares have been obtained, the block sum of squares can be calculated. The block sum of squares is the sum of the sum of squares of effects, <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, since these effects are confounded with the block effect. As shown in the second following figure, this sum of squares is 92.25 and is displayed against Block. The interactions <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, which are confounded with the blocks, are not displayed. Since the present design is unreplicated any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_28.png|center|800px|Design for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

[[Image:doe7_29.png|center|793px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

==Variability Analysis==

For replicated two level factorial experiments, DOE++ provides the option of conducting variability analysis (using the Variability Analysis icon under the Data menu). The analysis is used to identify the treatment that results in the least amount of variation in the product or process being investigated. Variability analysis is conducted by treating the standard deviation of the response for each treatment of the experiment as an additional response. The standard deviation for a treatment is obtained by using the replicated response values at that treatment run. As an example, consider the <math>{2}^{3}\,\!</math> design shown in the following figure where each run is replicated four times. A variability analysis can be conducted for this design. DOE++ calculates eight standard deviation values corresponding to each treatment of the design (see second following figure). Then, the design is analyzed as an unreplicated <math>{2}^{3}\,\!</math> design with the standard deviations (displayed as Y Standard Deviation. in second following figure) as the response. The normal probability plot of effects identifies <math>AC\,\!</math> as the effect that influences variability (see third figure following). Based on the effect coefficients obtained in the fourth figure following, the model for Y Std. is:

::<math>\begin{align}
& \text{Y Std}\text{.}= & 0.6779+0.2491\cdot AC \\
& = & 0.6779+0.2491{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

Based on the model, the experimenter has two choices to minimize variability (by minimizing Y Std.). The first choice is that <math>{{x}_{1}}\,\!</math> should be <math>1\,\!</math> (i.e., <math>A\,\!</math> should be set at the high level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the low level). The second choice is that <math>{{x}_{1}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>A\,\!</math> should be set at the low level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the high level). The experimenter can select the most feasible choice.

[[Image:doe7.30.png|center|391px|A <math>2^3\,\!</math> design with four replicated response values that can be used to conduct a variability analysis.]]
 

[[Image:doe7_31.png|center|727px|Variability analysis in DOE++.|link=]]
 

[[Image:doe7_32.png|center|650px|Normal probability plot of effects for the variability analysis example.|link=]]
 

[[Image:doe7_33.png|center|727px|Effect coefficients for the variability analysis example.|link=]]

 

==Two Level Fractional Factorial Designs==

As the number of factors in a two level factorial design increases, the number of runs for even a single replicate of the <math>{2}^{k}\,\!</math> design becomes very large. For example, a single replicate of an eight factor two level experiment would require 256 runs. Fractional factorial designs can be used in these cases to draw out valuable conclusions from fewer runs. The basis of fractional factorial designs is the ''sparsity of effects'' principle.[[DOE_References|[Wu, 2000]]] The principle states that, most of the time, responses are affected by a small number of main effects and lower order interactions, while higher order interactions are relatively unimportant. Fractional factorial designs are used as screening experiments during the initial stages of experimentation. At these stages, a large number of factors have to be investigated and the focus is on the main effects and two factor interactions. These designs obtain information about main effects and lower order interactions with fewer experiment runs by confounding these effects with unimportant higher order interactions. As an example, consider a <math>{2}^{8}\,\!</math> design that requires 256 runs. This design allows for the investigation of 8 main effects and 28 two factor interactions. However, 219 degrees of freedom are devoted to three factor or higher order interactions. This full factorial design can prove to be very inefficient when these higher order interactions can be assumed to be unimportant. Instead, a fractional design can be used here to identify the important factors that can then be investigated more thoroughly in subsequent experiments. In unreplicated fractional factorial designs, no degrees of freedom are available to calculate the error sum of squares and the techniques mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] should be employed for the analysis of these designs.

==Half-fraction Designs==

A half-fraction of the <math>{2}^{k}\,\!</math> design involves running only half of the treatments of the full factorial design. For example, consider a <math>{2}^{3}\,\!</math> design that requires eight runs in all. The design matrix for this design is shown in the figure (a) below. A half-fraction of this design is the design in which only four of the eight treatments are run. The fraction is denoted as <math>{2}^{3-1}\,\!</math> with the "<math>-1\,\!</math>" in the index denoting a half-fraction. Assume that the treatments chosen for the half-fraction design are the ones where the interaction <math>ABC\,\!</math> is at the high level (i.e., only those rows are chosen from the following figure (a) where the column for <math>ABC\,\!</math> has entries of 1). The resulting <math>{2}^{3-1}\,\!</math> design has a design matrix as shown in figure (b) below.

[[Image:doe7.34.png|center|330px|Half-fractions of the <math>2^3\,\!</math> design. (a) shows the full factorial <math>2^3\,\!</math> design, (b) shows the <math>2^{3-1}\,\!</math> design with the defining relation <math>I=ABC\,\!</math> and (c) shows the <math>{2}^{3-1}\,\!</math> design with the defining relation <math>I=-ABC\,\!</math>.]]

In the <math>{2}^{3-1}\,\!</math> design of figure (b), since the interaction <math>ABC\,\!</math> is always included at the same level (the high level represented by 1), it is not possible to measure this interaction effect. The effect, <math>ABC\,\!</math>, is called the ''generator'' or ''word'' for this design. It can be noted that, in the design matrix of the following figure (b), the column corresponding to the intercept, <math>I\,\!</math>, and column corresponding to the interaction <math>ABC\,\!</math>, are identical. The identical columns are written as <math>I=ABC\,\!</math> and this equation is called the ''defining relation'' for the design. In DOE++, the present <math>{2}^{3-1}\,\!</math> design can be obtained by specifying the design properties as shown in the following figure.

[[Image:doe7_35.png|center|700px|Design properties for the <math>2^{3-1}\,\!</math> design.|link=]]

The defining relation, <math>I=ABC\,\!</math>, is entered in the Fraction Generator window as shown next.

[[Image:doe7_36.png|center|700px|Specifying the defining relation for the <math>2^{3-1}\,\!</math> design.|link=]]

Note that in the figure following that, the defining relation is specified as <math>C=AB\,\!</math>. This relation is obtained by multiplying the defining relation, <math>I=ABC\,\!</math>, by the last factor, <math>C\,\!</math>, of the design.

===Calculation of Effects===

Using the four runs of the <math>{2}^{3-1}\,\!</math> design in figure (b) discussed above, the main effects can be calculated as follows:

::<math>\begin{align}
& A= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& B= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& C= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

where <math>a\,\!</math>, <math>b\,\!</math>, <math>c\,\!</math> and <math>abc\,\!</math> are the treatments included in the <math>{2}^{3-1}\,\!</math> design.

Similarly, the two factor interactions can also be obtained as:

::<math>\begin{align}
& BC= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& AC= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& AB= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

The equations for <math>A\,\!</math> and <math>BC\,\!</math> above result in the same effect values showing that effects <math>A\,\!</math> and <math>BC\,\!</math> are confounded in the present <math>{2}^{3-1}\,\!</math> design. Thus, the quantity, <math>\tfrac{1}{2}(a-b-c+abc),\,\!</math> estimates <math>A+BC\,\!</math> (i.e., both the main effect <math>A\,\!</math> and the two-factor interaction <math>BC\,\!</math>). The effects, <math>A\,\!</math> and <math>BC,\,\!</math> are called ''aliases''. From the remaining equations given above, it can be seen that the other aliases for this design are <math>B\,\!</math> and <math>AC\,\!</math>, and <math>C\,\!</math> and <math>AB\,\!</math>. Therefore, the equations to calculate the effects in the present <math>{2}^{3-1}\,\!</math> design can be written as follows:

::<math>\begin{align}
& A+BC= & \frac{1}{2}(a-b-c+abc) \\
& B+AC= & \frac{1}{2}(-a+b-c+abc) \\
& C+AB= & \frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

===Calculation of Aliases===

Aliases for a fractional factorial design can be obtained using the defining relation for the design. The defining relation for the present <math>{2}^{3-1}\,\!</math> design is:

::<math>I=ABC\,\!</math>

Multiplying both sides of the previous equation by the main effect, <math>A,\,\!</math> gives the alias effect of <math>A\,\!</math> :

::<math>\begin{align}
& A\cdot I= & A\cdot ABC \\
& A= & {{A}^{2}}BC \\
& A= & BC
\end{align}\,\!</math>

Note that in calculating the alias effects, any effect multiplied by <math>I\,\!</math> remains the same (<math>A\cdot I=A\,\!</math>), while an effect multiplied by itself results in <math>I\,\!</math> (<math>{{A}^{2}}=I\,\!</math>). Other aliases can also be obtained:

::<math>\begin{align}
& B\cdot I= & B\cdot ABC \\
& B= & A{{B}^{2}}C \\
& B= & AC
\end{align}\,\!</math>

:and:

::<math>\begin{align}
& C\cdot I= & C\cdot ABC \\
& C= & AB{{C}^{2}} \\
& C= & AB
\end{align}\,\!</math>

===Fold-over Design===

If it can be assumed for this design that the two-factor interactions are unimportant, then in the absence of <math>BC\,\!</math>, <math>AC\,\!</math> and <math>AB\,\!</math>, the equations for (A+BC), (B+AC) and (C+AB) can be used to estimate the main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. However, if such an assumption is not applicable, then to uncouple the main effects from their two factor aliases, the alternate fraction that contains runs having <math>ABC\,\!</math> at the lower level should be run. The design matrix for this design is shown in the preceding figure (c). The defining relation for this design is <math>I=-ABC\,\!</math> because the four runs for this design are obtained by selecting the rows of the preceding figure (a) for which the value of the <math>ABC\,\!</math> column is <math>-1\,\!</math>. The aliases for this fraction can be obtained as explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]] as <math>A=-BC\,\!</math>, <math>B=-AC\,\!</math> and <math>C=-AB\,\!</math>. The effects for this design can be calculated as:

::<math>\begin{align}
& A-BC= & \frac{1}{2}(ab+ac-(1)-bc) \\
& B-AC= & \frac{1}{2}(ab-ac+(1)-bc) \\
& C-AB= & \frac{1}{2}(-ab+ac-(1)+bc)
\end{align}\,\!</math>

These equations can be combined with the equations for (A+BC), (B+AC) and (C+AB) to obtain the de-aliased main effects and two factor interactions. For example, adding equations (A+BC) and (A-BC) returns the main effect <math>A\,\!</math>.

::<math>\begin{align}
& 2A= & \frac{1}{2}(a-b-c+abc)+ \\
& & \frac{1}{2}(ab+ac-(1)-bc)
\end{align}\,\!</math>

The process of augmenting a fractional factorial design by a second fraction of the same size by simply reversing the signs (of all effect columns except <math>I\,\!</math>) is called ''folding over''. The combined design is referred to as a ''fold-over design''.

==Quarter and Smaller Fraction Designs==

At times, the number of runs even for a half-fraction design are very large. In these cases, smaller fractions are used. A quarter-fraction design, denoted as <math>{2}^{k-2}\,\!</math>, consists of a fourth of the runs of the full factorial design. Quarter-fraction designs require two defining relations. The first defining relation returns the half-fraction or the <math>{2}^{k-1}\,\!</math> design. The second defining relation selects half of the runs of the <math>{2}^{k-1}\,\!</math> design to give the quarter-fraction. For example, consider the <math>{2}^{4}\,\!</math> design. To obtain a <math>{2}^{4-2}\,\!</math> design from this design, first a half-fraction of this design is obtained by using a defining relation. Assume that the defining relation used is <math>I=ABCD\,\!</math>. The design matrix for the resulting <math>{2}^{4-1}\,\!</math> design is shown in figure (a) below. Now, a quarter-fraction can be obtained from the <math>{2}^{4-1}\,\!</math> design shown in figure (a) below using a second defining relation <math>I=AD\,\!</math>. The resulting <math>{2}^{4-2}\,\!</math> design obtained is shown in figure (b) below.

[[Image:doe7.37.png|center|465px|Fractions of the <math>2^4\,\!</math> design - Figure (a) shows the <math>2^{4-1}</math> design with the defining relation <math>I=ABCD\,\!</math> and (b) shows the <math>2^{4-2}\,\!</math> design with the defining relation <math>I=ABCD=AD=BC\,\!</math>.]]

The complete defining relation for this <math>{2}^{4-2}\,\!</math> design is:

::<math>I=ABCD=AD=BC\,\!</math>

Note that the effect, <math>BC,\,\!</math> in the defining relation is the generalized interaction of <math>ABCD\,\!</math> and <math>AD\,\!</math> and is obtained using <math>(ABCD)(AD)={{A}^{2}}BC{{D}^{2}}=BC\,\!</math>. In general, a <math>{2}^{k-p}\,\!</math> fractional factorial design requires <math>p\,\!</math> independent generators. The defining relation for the design consists of the <math>p\,\!</math> independent generators and their <math>{2}^{p}\,\!</math> - (<math>p\,\!</math> +1) generalized interactions.

===Calculation of Aliases===

The alias structure for the present <math>{2}^{4-2}\,\!</math> design can be obtained using the defining relation of equation (I=ABCD=AD=BC) following the procedure explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]]. For example, multiplying the defining relation by <math>A\,\!</math> returns the effects aliased with the main effect, <math>A\,\!</math>, as follows:

::<math>\begin{align}
& A\cdot I= & A\cdot ABCD=A\cdot AD=A\cdot BC \\
& A= & {{A}^{2}}BCD={{A}^{2}}D=ABC \\
& A= & BCD=D=ABC
\end{align}\,\!</math>

Therefore, in the present <math>{2}^{4-2}\,\!</math> design, it is not possible to distinguish between effects <math>A\,\!</math>, <math>D\,\!</math>, <math>BCD\,\!</math> and <math>ABC\,\!</math>. Similarly, multiplying the defining relation by <math>B\,\!</math> and <math>AB\,\!</math> returns the effects that are aliased with these effects:

::<math>\begin{align}
& B= & ACD=ABD=C \\
& AB= & CD=AD=AC
\end{align}\,\!</math>

Other aliases can be obtained in a similar way. It can be seen that each effect in this design has three aliases. In general, each effect in a <math>{2}^{k-p}\,\!</math> design has <math>{2}^{p-1}\,\!</math> aliases.
The aliases for the <math>{2}^{4-2}\,\!</math> design show that in this design the main effects are aliased with each other (<math>A\,\!</math> is aliased with <math>D\,\!</math> and <math>B\,\!</math> is aliased with <math>C\,\!</math>). Therefore, this design is not a useful design and is not available in DOE++. It is important to ensure that main effects and lower order interactions of interest are not aliased in a fractional factorial design. This is known by looking at the resolution of the fractional factorial design.

==Design Resolution==

The resolution of a fractional factorial design is defined as the number of factors in the lowest order effect in the defining relation. For example, in the defining relation <math>I=ABCD=AD=BC\,\!</math> of the previous <math>{2}^{4-2}\,\!</math> design, the lowest-order effect is either <math>AD\,\!</math> or <math>BC,\,\!</math> containing two factors. Therefore, the resolution of this design is equal to two. The resolution of a fractional factorial design is represented using Roman numerals. For example, the previously mentioned <math>{2}^{4-2}\,\!</math> design with a resolution of two can be represented as 2 <math>_{\text{II}}^{4-2}\,\!</math>. The resolution provides information about the confounding in the design as explained next:
 
 
#'''Resolution III Designs''' In these designs, the lowest order effect in the defining relation has three factors (e.g., a <math>{2}^{5-2}\,\!</math> design with the defining relation <math>I=ABDE=ABC=CDE\,\!</math>). In resolution III designs, no main effects are aliased with any other main effects, but main effects are aliased with two factor interactions. In addition, some two factor interactions are aliased with each other.
#'''Resolution IV Designs''' In these designs, the lowest order effect in the defining relation has four factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABDE\,\!</math>). In resolution IV designs, no main effects are aliased with any other main effects or two factor interactions. However, some main effects are aliased with three factor interactions and the two factor interactions are aliased with each other.
#'''Resolution V Designs''' In these designs the lowest order effect in the defining relation has five factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABCDE\,\!</math>). In resolution V designs, no main effects or two factor interactions are aliased with any other main effects or two factor interactions. However, some main effects are aliased with four factor interactions and the two factor interactions are aliased with three factor interactions.

Fractional factorial designs with the highest resolution possible should be selected because the higher the resolution of the design, the less severe the degree of confounding. In general, designs with a resolution less than III are never used because in these designs some of the main effects are aliased with each other. The table below shows fractional factorial designs with the highest available resolution for three to ten factor designs along with their defining relations.

[[Image:doet7.3.png|center|474px|Highest resolution designs available for fractional factorial designs with 3 to 10 factors.]]

All of the two level fractional factorial designs available in DOE++ are shown next.

[[Image:doe7.38.png|center|471px|Two level fractional factorial designs available in DOE++ and their resolutions.]]

===Minimum Aberration Designs===
At times, different designs with the same resolution but different aliasing may be available. The best design to select in such a case is the minimum aberration design. For example, all <math>{2}^{7-2}\,\!</math> designs in the fourth table have a resolution of four (since the generator with the minimum number of factors in each design has four factors). Design <math>1\,\!</math> has three generators of length four (<math>ABCF,\,\!</math> <math>BCDG,\,\!</math> <math>ADFG\,\!</math>). Design <math>2\,\!</math> has two generators of length four (<math>ABCF,\,\!</math> <math>ADEG\,\!</math>). Design <math>3\,\!</math> has one generator of length four (<math>CEFG\,\!</math>). Therefore, design <math>3\,\!</math> has the least number of generators with the minimum length of four. Design <math>3\,\!</math> is called the minimum aberration design. It can be seen that the alias structure for design <math>3\,\!</math> is less involved compared to the other designs. For details refer to [[DOE_References|[Wu, 2000]]].

[[Image:doet7.4.png|center|432px|Three <math>2_{IV}^{7-2}\,\!</math> designs with different defining relations.]]

====Example====

The design of an automobile fuel cone is thought to be affected by six factors in the manufacturing process: cavity temperature (factor <math>A\,\!</math>), core temperature (factor <math>B\,\!</math>), melt temperature (factor <math>C\,\!</math>), hold pressure (factor <math>D\,\!</math>), injection speed (factor <math>E\,\!</math>) and cool time (factor <math>F\,\!</math>). The manufacturer of the fuel cone is unable to run the <math>{2}^{6}=64\,\!</math> runs required to complete one replicate for a two level full factorial experiment with six factors. Instead, they decide to run a fractional factorial design. Considering that three factor and higher order interactions are likely to be inactive, the manufacturer selects a <math>{2}^{6-2}\,\!</math> design that will require only 16 runs. The manufacturer chooses the resolution IV design which will ensure that all main effects are free from aliasing (assuming three factor and higher order interactions are absent). However, in this design the two factor interactions may be aliased with each other. It is decided that, if important two factor interactions are found to be present, additional experiment trials may be conducted to separate the aliased effects. The performance of the fuel cone is measured on a scale of 1 to 15. In DOE++, the design for this experiment is set up using the properties shown in the following figure. The Fraction Generators for the design, <math>E=ABC\,\!</math> and <math>F=BCD\,\!</math>, are the same as the defaults used in DOE++. The resulting <math>{2}^{6-2}\,\!</math> design and the corresponding response values are shown in the following two figures.

[[Image:doe7_39.png|center|644px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

[[Image:doe7_40.png|center|534px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The complete alias structure for the 2 <math>_{\text{IV}}^{6-2}\,\!</math> design is shown next.

<center><math>I=ABCE=ADEF=BCDF\,\!</math></center>

 
<center><math>\begin{align}
& A= & BCE=DEF=ABCDF \\
& B= & ACE=CDF=ABDEF \\
& C= & ABE=BDF=ACDEF \\
& D= & AEF=BCF=ABCDE \\
& E= & ABC=ADF=BCDEF \\
& F= & ADE=BCD=ABCEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& AB= & CE=ACDF=BDEF \\
& AC= & BE=ABDF=CDEF \\
& AD= & EF=ABCF=BCDE \\
& AE= & BC=DF=ABCDEF \\
& AF= & DE=ABCD=BCEF \\
& BD= & CF=ABEF=ACDE \\
& BF= & CD=ABDE=ACEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& ABD= & ACF=BEF=CDE \\
& ABF= & ACD=BDE=CEF
\end{align}\,\!</math></center>

In DOE++, the alias structure is displayed in the Design Summary and as part of the Design Evaluation result, as shown next:

[[Image:doe7_41.png|center|700px|Alias structure for the experiment design in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The normal probability plot of effects for this unreplicated design shows the main effects of factors <math>C\,\!</math> and <math>D\,\!</math> and the interaction effect, <math>BF\,\!</math>, to be significant (see the following figure).

[[Image:doe7_42.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

From the alias structure, it can be seen that for the present design interaction effect, <math>BF,\,\!</math> is confounded with <math>CD\,\!</math>. Therefore, the actual source of this effect cannot be known on the basis of the present experiment. However because neither factor <math>B\,\!</math> nor <math>F\,\!</math> is found to be significant there is an indication the observed effect is likely due to interaction, <math>CD\,\!</math>. To confirm this, a follow-up <math>{2}^{2}\,\!</math> experiment is run involving only factors <math>B\,\!</math> and <math>F\,\!</math>. The interaction, <math>BF\,\!</math>, is found to be inactive, leading to the conclusion that the interaction effect in the original experiment is effect, <math>CD\,\!</math>. Given these results, the fitted regression model for the fuel cone design as per the coefficients obtained from DOE++ is shown next.

::<math>\hat{y}=7.6875+C+2\cdot D+2.1875\cdot CD\,\!</math>

[[Image:doe7_43.png|center|700px|Effect coefficients for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

==Projection==

Projection refers to the reduction of a fractional factorial design to a full factorial design by dropping out some of the factors of the design. Any fractional factorial design of resolution, <math>R,\,\!</math> can be reduced to complete factorial designs in any subset of <math>R-1\,\!</math> factors. For example, consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. The resolution of this design is four. Therefore, this design can be reduced to full factorial designs in any three (<math>4-1=3\,\!</math>) of the original seven factors (by dropping the remaining four of factors). Further, a fractional factorial design can also be reduced to a full factorial design in any <math>R\,\!</math> of the original factors, as long as these <math>R\,\!</math> factors are not part of the generator in the defining relation. Again consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. This design can be reduced to a full factorial design in four factors provided these four factors do not appear together as a generator in the defining relation. The complete defining relation for this design is:

::<math>\begin{align}
& I= & ABCE=BCDF=ACDG=ADEF=ABFG=BDEG=CEFG
\end{align}\,\!</math>

Therefore, there are seven four factor combinations out of the 35 (<math>(_{7}^{4})=35\,\!</math>) possible four-factor combinations that are used as generators in the defining relation. The designs with the remaining 28 four factor combinations would be full factorial 16-run designs. For example, factors <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math> do not occur as a generator in the defining relation of the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. If the remaining factors, <math>E\,\!</math>, <math>F\,\!</math> and <math>G\,\!</math>, are dropped, the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design will reduce to a full factorial design in <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math>.

==Resolution III Designs==

At times, the factors to be investigated in screening experiments are so large that even running a fractional factorial design is impractical. This can be partially solved by using resolution III fractional factorial designs in the cases where three factor and higher order interactions can be assumed to be unimportant. Resolution III designs, such as the 2 <math>_{\text{III}}^{3-1}\,\!</math> design, can be used to estimate <math>k\,\!</math> main effects using just <math>k+1\,\!</math> runs. In these designs, the main effects are aliased with two factor interactions. Once the results from these designs are obtained, and knowing that three factor and higher order interactions are unimportant, the experimenter can decide if there is a need to run a fold-over design to de-alias the main effects from the two factor interactions. Thus, the 2 <math>_{\text{III}}^{3-1}\,\!</math> design can be used to investigate three factors in four runs, the 2 <math>_{\text{III}}^{7-4}\,\!</math> design can be used to investigate seven factors in eight runs, the 2 <math>_{\text{III}}^{15-11}\,\!</math> design can be used to investigate fifteen factors in sixteen runs and so on.

====Example====

{{:Resolution_III_Design_Example}}

==Alias Matrix==
In [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction designs]] and [[Two_Level_Factorial_Experiments#Quarter_and_Smaller_Fraction_Designs| Quarter and Smaller Fraction Designs]], the alias structure for fractional factorial designs was obtained using the defining relation. However, this method of obtaining the alias structure is not very efficient when the alias structure is very complex or when partial aliasing is involved. One of the ways to obtain the alias structure for any design, regardless of its complexity, is to use the alias matrix. The alias matrix for a design is calculated using <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> where <math>{{X}_{1}}\,\!</math> is the portion of the design matrix, <math>X,\,\!</math> that contains the effects for which the aliases need to be calculated, and <math>{{X}_{2}}\,\!</math> contains the remaining columns of the design matrix, other than those included in <math>{{X}_{1}}\,\!</math>.

To illustrate the use of the alias matrix, consider the design matrix for the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design (using the defining relation <math>I=ABCD\,\!</math>) shown next:

[[Image:Chapter7__879.png|center|link=]]

The alias structure for this design can be obtained by defining <math>{{X}_{1}}\,\!</math> using eight columns since the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design estimates eight effects. If the first eight columns of <math>X\,\!</math> are used then <math>{{X}_{1}}\,\!</math> is:

[[Image:Chapter7__884.png|center|link=]]

<math>{{X}_{2}}\,\!</math> is obtained using the remaining columns as:

[[Image:Chapter7__886.png|center|link=]]

Then the alias matrix <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> is:

[[Image:Chapter7__888.png|center|link=]]

The alias relations can be easily obtained by observing the alias matrix as:

<center><math>\begin{align}
& I= & ABCD \\
& A= & BCD \\
& B= & ACD \\
& AB= & CD \\
& C= & ABD \\
& AC= & BD \\
& BC= & AD \\
& D= & ABC
\end{align}\,\!</math></center>

Two Level Factorial Experiments

2017-08-10T17:32:51Z

Richard House: /* Using Center Point Replicates to Test Curvature */

{{Template:Doebook|8}}
Two level factorial experiments are factorial experiments in which each factor is investigated at only two levels. The early stages of experimentation usually involve the investigation of a large number of potential factors to discover the "vital few" factors. Two level factorial experiments are used during these stages to quickly filter out unwanted effects so that attention can then be focused on the important ones.

==2''k'' Designs==
The factorial experiments, where all combination of the levels of the factors are run, are usually referred to as ''full factorial experiments''. Full factorial two level experiments are also referred to as <math>{2}^{k}\,\!</math> designs where <math>k\,\!</math> denotes the number of factors being investigated in the experiment. In Weibull++ DOE folios, these designs are referred to as 2 Level Factorial Designs as shown in the figure below.

[[Image:doe7_1.png|center|487px|Selection of full factorial experiments with two levels in Weibull++.|link=]]

A full factorial two level design with <math>k\,\!</math> factors requires <math>{{2}^{k}}\,\!</math> runs for a single replicate. For example, a two level experiment with three factors will require <math>2\times 2\times 2={{2}^{3}}=8\,\!</math> runs. The choice of the two levels of factors used in two level experiments depends on the factor; some factors naturally have two levels. For example, if gender is a factor, then male and female are the two levels. For other factors, the limits of the range of interest are usually used. For example, if temperature is a factor that varies from <math>{45}^{o}C\,\!</math> to <math>{90}^{o}C\,\!</math>, then the two levels used in the <math>{2}^{k}\,\!</math> design for this factor would be <math>{45}^{o}\,\!C\,\!</math> and <math>{90}^{o}\,\!C\,\!</math>.

The two levels of the factor in the <math>{2}^{k}\,\!</math> design are usually represented as <math>-1\,\!</math> (for the first level) and <math>1\,\!</math> (for the second level). Note that this representation is reversed from the coding used in [[General Full Factorial Designs]] for the indicator variables that represent two level factors in ANOVA models. For ANOVA models, the first level of the factor was represented using a value of <math>1\,\!</math> for the indicator variable, while the second level was represented using a value of <math>-1\,\!</math>. For details on the notation used for two level experiments refer to [[Two_Level_Factorial_Experiments#Notation| Notation]].

===The 22 Design===

The simplest of the two level factorial experiments is the <math>{2}^{2}\,\!</math> design where two factors (say factor <math>A\,\!</math> and factor <math>B\,\!</math>) are investigated at two levels. A single replicate of this design will require four runs (<math>{{2}^{2}}=2\times 2=4\,\!</math>) The effects investigated by this design are the two main effects, <math>A\,\!</math> and <math>B,\,\!</math> and the interaction effect <math>AB\,\!</math>. The treatments for this design are shown in figure (a) below. In figure (a), letters are used to represent the treatments. The presence of a letter indicates the high level of the corresponding factor and the absence indicates the low level. For example, (1) represents the treatment combination where all factors involved are at the low level or the level represented by <math>-1\,\!</math> ; <math>a\,\!</math> represents the treatment combination where factor <math>A\,\!</math> is at the high level or the level of <math>1\,\!</math>, while the remaining factors (in this case, factor <math>B\,\!</math>) are at the low level or the level of <math>-1\,\!</math>. Similarly, <math>b\,\!</math> represents the treatment combination where factor <math>B\,\!</math> is at the high level or the level of <math>1\,\!</math>, while factor <math>A\,\!</math> is at the low level and <math>ab\,\!</math> represents the treatment combination where factors <math>A\,\!</math> and <math>B\,\!</math> are at the high level or the level of the 1. Figure (b) below shows the design matrix for the <math>{2}^{2}\,\!</math> design. It can be noted that the sum of the terms resulting from the product of any two columns of the design matrix is zero. As a result the <math>{2}^{2}\,\!</math> design is an ''orthogonal design''. In fact, all <math>{2}^{k}\,\!</math> designs are orthogonal designs. This property of the <math>{2}^{k}\,\!</math> designs offers a great advantage in the analysis because of the simplifications that result from orthogonality. These simplifications are explained later on in this chapter.
The <math>{2}^{2}\,\!</math> design can also be represented geometrically using a square with the four treatment combinations lying at the four corners, as shown in figure (c) below.

[[Image:doe7.2.png|center|400px|The <math>2^2\,\!</math> design. Figure (a) displays the experiment design, (b) displays the design matrix and (c) displays the geometric representation for the design. In Figure (b), the column names I, A, B and AB are used. Column I represents the intercept term. Columns A and B represent the respective factor settings. Column AB represents the interaction and is the product of columns A and B.]]
 

===The 23 Design===

The <math>{2}^{3}\,\!</math> design is a two level factorial experiment design with three factors (say factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>). This design tests three (<math>k=3\,\!</math>) main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math> ; three (<math>(_{2}^{k})=\,\!</math> <math>(_{2}^{3})=3\,\!</math>) two factor interaction effects, <math>AB\,\!</math>, <math>BC\,\!</math>, <math>AC\,\!</math> ; and one (<math>(_{3}^{k})=\,\!</math> <math>(_{3}^{3})=1\,\!</math>) three factor interaction effect, <math>ABC\,\!</math>. The design requires eight runs per replicate. The eight treatment combinations corresponding to these runs are <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math>, <math>ab\,\!</math>, <math>c\,\!</math>, <math>ac\,\!</math>, <math>bc\,\!</math> and <math>abc\,\!</math>. Note that the treatment combinations are written in such an order that factors are introduced one by one with each new factor being combined with the preceding terms. This order of writing the treatments is called the ''standard order'' or ''Yates' order''. The <math>{2}^{3}\,\!</math> design is shown in figure (a) below. The design matrix for the <math>{2}^{3}\,\!</math> design is shown in figure (b). The design matrix can be constructed by following the standard order for the treatment combinations to obtain the columns for the main effects and then multiplying the main effects columns to obtain the interaction columns.

[[Image:doe7.3.png|center|324px|The <math>2^3\,\!</math> design. Figure (a) shows the experiment design and (b) shows the design matrix.]]

[[Image:doe7.4.png|center|290px|Geometric representation of the <math>2^3\,\!</math> design.]]

The <math>{2}^{3}\,\!</math> design can also be represented geometrically using a cube with the eight treatment combinations lying at the eight corners as shown in the figure above.

==Analysis of 2''k'' Designs==

The <math>{2}^{k}\,\!</math> designs are a special category of the factorial experiments where all the factors are at two levels. The fact that these designs contain factors at only two levels and are orthogonal greatly simplifies their analysis even when the number of factors is large. The use of <math>{2}^{k}\,\!</math> designs in investigating a large number of factors calls for a revision of the notation used previously for the ANOVA models. The case for revised notation is made stronger by the fact that the ANOVA and multiple linear regression models are identical for <math>{2}^{k}\,\!</math> designs because all factors are only at two levels. Therefore, the notation of the regression models is applied to the ANOVA models for these designs, as explained next.

===Notation===

Based on the notation used in [[General Full Factorial Designs]], the ANOVA model for a two level factorial experiment with three factors would be as follows:

::<math>\begin{align}
& Y= & \mu +{{\tau }_{1}}\cdot {{x}_{1}}+{{\delta }_{1}}\cdot {{x}_{2}}+{{(\tau \delta )}_{11}}\cdot {{x}_{1}}{{x}_{2}}+{{\gamma }_{1}}\cdot {{x}_{3}} \\
& & +{{(\tau \gamma )}_{11}}\cdot {{x}_{1}}{{x}_{3}}+{{(\delta \gamma )}_{11}}\cdot {{x}_{2}}{{x}_{3}}+{{(\tau \delta \gamma )}_{111}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where:
 
:• <math>\mu \,\!</math> represents the overall mean
:• <math>{{\tau }_{1}}\,\!</math> represents the independent effect of the first factor (factor <math>A\,\!</math>) out of the two effects <math>{{\tau }_{1}}\,\!</math> and <math>{{\tau }_{2}}\,\!</math>
:• <math>{{\delta }_{1}}\,\!</math> represents the independent effect of the second factor (factor <math>B\,\!</math>) out of the two effects <math>{{\delta }_{1}}\,\!</math> and <math>{{\delta }_{2}}\,\!</math>
:• <math>{{(\tau \delta )}_{11}}\,\!</math> represents the independent effect of the interaction <math>AB\,\!</math> out of the other interaction effects
:• <math>{{\gamma }_{1}}\,\!</math> represents the effect of the third factor (factor <math>C\,\!</math>) out of the two effects <math>{{\gamma }_{1}}\,\!</math> and <math>{{\gamma }_{2}}\,\!</math>
:• <math>{{(\tau \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>AC\,\!</math> out of the other interaction effects
:• <math>{{(\delta \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>BC\,\!</math> out of the other interaction effects
:• <math>{{(\tau \delta \gamma )}_{111}}\,\!</math> represents the effect of the interaction <math>ABC\,\!</math> out of the other interaction effects
and <math>\epsilon \,\!</math> is the random error term.

 
The notation for a linear regression model having three predictor variables with interactions is:

::<math>\begin{align}
& Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& & +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

The notation for the regression model is much more convenient, especially for the case when a large number of higher order interactions are present. In two level experiments, the ANOVA model requires only one indicator variable to represent each factor for both qualitative and quantitative factors. Therefore, the notation for the multiple linear regression model can be applied to the ANOVA model of the experiment that has all the factors at two levels. For example, for the experiment of the ANOVA model given above, <math>{{\beta }_{0}}\,\!</math> can represent the overall mean instead of <math>\mu \,\!</math>, and <math>{{\beta }_{1}}\,\!</math> can represent the independent effect, <math>{{\tau }_{1}}\,\!</math>, of factor <math>A\,\!</math>. Other main effects can be represented in a similar manner. The notation for the interaction effects is much more simplified (e.g., <math>{{\beta }_{123}}\,\!</math> can be used to represent the three factor interaction effect, <math>{{(\tau \beta \gamma )}_{111}}\,\!</math>).

As mentioned earlier, it is important to note that the coding for the indicator variables for the ANOVA models of two level factorial experiments is reversed from the coding followed in [[General Full Factorial Designs]]. Here <math>-1\,\!</math> represents the first level of the factor while <math>1\,\!</math> represents the second level. This is because for a two level factor a single variable is needed to represent the factor for both qualitative and quantitative factors. For quantitative factors, using <math>-1\,\!</math> for the first level (which is the low level) and 1 for the second level (which is the high level) keeps the coding consistent with the numerical value of the factors. The change in coding between the two coding schemes does not affect the analysis except that signs of the estimated effect coefficients will be reversed (i.e., numerical values of <math>{{\hat{\tau }}_{1}}\,\!</math>, obtained based on the coding of [[General Full Factorial Designs]], and <math>{{\hat{\beta }}_{1}}\,\!</math>, obtained based on the new coding, will be the same but their signs would be opposite).

::<math>\begin{align}
& & \text{Factor }A\text{ Coding (two level factor)} \\
& &
\end{align}\,\!</math>

::<math>\begin{matrix}
\text{Previous Coding} & {} & {} & {} & \text{Coding for }{{\text{2}}^{k}}\text{ Designs} \\
{} & {} & {} & {} & {} \\
Effect\text{ }{{\tau }_{1}}\ \ :\ \ {{x}_{1}}=1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{1}}\text{ (or }-{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=-1\text{ } \\
Effect\text{ }{{\tau }_{2}}\ \ :\ \ {{x}_{1}}=-1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{2}}\text{ (or }{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=1\text{ } \\
\end{matrix}\,\!</math>

In summary, the ANOVA model for the experiments with all factors at two levels is different from the ANOVA models for other experiments in terms of the notation in the following two ways:
 

:• The notation of the regression models is used for the effect coefficients.
:• The coding of the indicator variables is reversed.

===Special Features===

Consider the design matrix, <math>X\,\!</math>, for the <math>{2}^{3}\,\!</math> design discussed above. The (<math>{{X}^{\prime }}X\,\!</math>) <math>^{-1}\,\!</math> matrix is:

<center><math>{{({{X}^{\prime }}X)}^{-1}}=\left[ \begin{matrix}
0.125 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0.125 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0.125 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0.125 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0.125 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0.125 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0.125 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.125 \\
\end{matrix} \right]\,\!</math></center>

Notice that, due to the orthogonal design of the <math>X\,\!</math> matrix, the <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> has been simplified to a diagonal matrix which can be written as:

::<math>\begin{align}
{{({{X}^{\prime }}X)}^{-1}}= & 0.125\cdot I = & \frac{1}{8}\cdot I = & \frac{1}{{{2}^{3}}}\cdot I
\end{align}\,\!</math>

where <math>I\,\!</math> represents the identity matrix of the same order as the design matrix, <math>X\,\!</math>. Since there are eight observations per replicate of the <math>{2}^{3}\,\!</math> design, the <math>(X\,\!</math> ' <math>X{{)}^{-1}}\,\!</math> matrix for <math>m\,\!</math> replicates of this design can be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{3}}\cdot m)}\cdot I\,\!</math>

The <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> matrix for any <math>{2}^{k}\,\!</math> design can now be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{k}}\cdot m)}\cdot I\,\!</math>

Then the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is:

::<math>\begin{align}
C= & {{{\hat{\sigma }}}^{2}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & M{{S}_{E}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & \frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}\cdot I
\end{align}\,\!</math>

Note that the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is also a diagonal matrix. Therefore, the estimated effect coefficients (<math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math>, <math>{{\beta }_{12}},\,\!</math> etc.) for these designs are uncorrelated. This implies that the terms in the <math>{2}^{k}\,\!</math> design (main effects, interactions) are independent of each other. Consequently, the extra sum of squares for each of the terms in these designs is independent of the sequence of terms in the model, and also independent of the presence of other terms in the model. As a result the sequential and partial sum of squares for the terms are identical for these designs and will always add up to the model sum of squares. Multicollinearity is also not an issue for these designs.

It can also be noted from the equation given above, that in addition to the <math>C\,\!</math> matrix being diagonal, all diagonal elements of the <math>C\,\!</math> matrix are identical. This means that the variance (or its square root, the standard error) of all estimated effect coefficients are the same. The standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>, for all the coefficients is:

::<math>\begin{align}
se({{{\hat{\beta }}}_{j}})= & \sqrt{{{C}_{jj}}} = & \sqrt{\frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}}\text{ }for\text{ }all\text{ }j
\end{align}\,\!</math>

This property is used to construct the normal probability plot of effects in <math>{2}^{k}\,\!</math> designs and identify significant effects using graphical techniques. For details on the normal probability plot of effects in a Weibull++ DOE folio, refer to [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]].

====Example====
To illustrate the analysis of a full factorial <math>{2}^{k}\,\!</math> design, consider a three factor experiment to investigate the effect of honing pressure, number of strokes and cycle time on the surface finish of automobile brake drums. Each of these factors is investigated at two levels. The honing pressure is investigated at levels of 200 <math>psi\,\!</math> and 400 <math>psi\,\!</math>, the number of strokes used is 3 and 5 and the two levels of the cycle time are 3 and 5 seconds. The design for this experiment is set up in a Weibull++ DOE folio as shown in the first two following figures. It is decided to run two replicates for this experiment. The surface finish data collected from each run (using randomization) and the complete design is shown in the third following figure. The analysis of the experiment data is explained next.

[[Image:doe7_5.png|center|877px|Design properties for the experiment in the example.|link=]]

[[Image:doe7_6.png|center|859px|Design summary for the experiment in the example.|link=]]

[[Image:doe7_7.png|center|778px|Experiment design for the example to investigate the surface finish of automobile brake drums.|link=]]

The applicable model using the notation for <math>{2}^{k}\,\!</math> designs is:

::<math>\begin{align}
Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where the indicator variable, <math>{{x}_{1,}}\,\!</math> represents factor <math>A\,\!</math> (honing pressure), <math>{{x}_{1}}=-1\,\!</math> represents the low level of 200 <math>psi\,\!</math> and <math>{{x}_{1}}=1\,\!</math> represents the high level of 400 <math>psi\,\!</math>. Similarly, <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> represent factors <math>B\,\!</math> (number of strokes) and <math>C\,\!</math> (cycle time), respectively. <math>{{\beta }_{0}}\,\!</math> is the overall mean, while <math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math> and <math>{{\beta }_{3}}\,\!</math> are the effect coefficients for the main effects of factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. <math>{{\beta }_{12}}\,\!</math>, <math>{{\beta }_{13}}\,\!</math> and <math>{{\beta }_{23}}\,\!</math> are the effect coefficients for the <math>AB\,\!</math>, <math>AC\,\!</math> and <math>BC\,\!</math> interactions, while <math>{{\beta }_{123}}\,\!</math> represents the <math>ABC\,\!</math> interaction.

 
If the subscripts for the run (<math>i\,\!</math> ; <math>i=\,\!</math> 1 to 8) and replicates (<math>j\,\!</math> ; <math>j=\,\!</math> 1,2) are included, then the model can be written as:

::<math>\begin{align}
{{Y}_{ij}}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{ij1}}+{{\beta }_{2}}\cdot {{x}_{ij2}}+{{\beta }_{12}}\cdot {{x}_{ij1}}{{x}_{ij2}}+{{\beta }_{3}}\cdot {{x}_{ij3}} \\
& +{{\beta }_{13}}\cdot {{x}_{ij1}}{{x}_{ij3}}+{{\beta }_{23}}\cdot {{x}_{ij2}}{{x}_{ij3}}+{{\beta }_{123}}\cdot {{x}_{ij1}}{{x}_{ij2}}{{x}_{ij3}}+{{\epsilon }_{ij}}
\end{align}\,\!</math>

To investigate how the given factors affect the response, the following hypothesis tests need to be carried:

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{1}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{1}}\ne 0\,\!</math>

This test investigates the main effect of factor <math>A\,\!</math> (honing pressure). The statistic for this test is:

::<math>{{({{F}_{0}})}_{A}}=\frac{M{{S}_{A}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{A}}\,\!</math> is the mean square for factor <math>A\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other main effects, <math>B\,\!</math> and <math>C\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{12}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{12}}\ne 0\,\!</math>

This test investigates the two factor interaction <math>AB\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{AB}}=\frac{M{{S}_{AB}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the interaction <math>AB\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other two factor interactions, <math>AC\,\!</math> and <math>BC\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{123}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{123}}\ne 0\,\!</math>

This test investigates the three factor interaction <math>ABC\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{ABC}}=\frac{M{{S}_{ABC}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{ABC}}\,\!</math> is the mean square for the interaction <math>ABC\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square.
To calculate the test statistics, it is convenient to express the ANOVA model in the form <math>y=X\beta +\epsilon \,\!</math>.

====Expression of the ANOVA Model as <math>y=X\beta +\epsilon \,\!</math>====

In matrix notation, the ANOVA model can be expressed as:

::<math>y=X\beta +\epsilon \,\!</math>

where:

<center><math>y=\left[ \begin{matrix}
{{Y}_{11}} \\
{{Y}_{21}} \\
. \\
{{Y}_{81}} \\
{{Y}_{12}} \\
. \\
{{Y}_{82}} \\
\end{matrix} \right]=\left[ \begin{matrix}
90 \\
90 \\
. \\
90 \\
86 \\
. \\
80 \\
\end{matrix} \right]\text{ }X=\left[ \begin{matrix}
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\end{matrix} \right]\,\!</math></center>

<center><math>\beta =\left[ \begin{matrix}
{{\beta }_{0}} \\
{{\beta }_{1}} \\
{{\beta }_{2}} \\
{{\beta }_{12}} \\
{{\beta }_{3}} \\
{{\beta }_{13}} \\
{{\beta }_{23}} \\
{{\beta }_{123}} \\
\end{matrix} \right]\text{ }\epsilon =\left[ \begin{matrix}
{{\epsilon }_{11}} \\
{{\epsilon }_{21}} \\
. \\
{{\epsilon }_{81}} \\
{{\epsilon }_{12}} \\
. \\
. \\
{{\epsilon }_{82}} \\
\end{matrix} \right]\,\!</math></center>

====Calculation of the Extra Sum of Squares for the Factors====

Knowing the matrices <math>y\,\!</math>, <math>X\,\!</math> and <math>\beta \,\!</math>, the extra sum of squares for the factors can be calculated. These are used to calculate the mean squares that are used to obtain the test statistics. Since the experiment design is orthogonal, the partial and sequential extra sum of squares are identical. The extra sum of squares for each effect can be calculated as shown next. As an example, the extra sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & Model\text{ }Sum\text{ }of\text{ }Squares - Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }main\text{ }effect\text{ }of\text{ }A \\
= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }A}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }A}}={{X}_{\tilde{\ }A}}{{(X_{\tilde{\ }A}^{\prime }{{X}_{\tilde{\ }A}})}^{-1}}X_{\tilde{\ }A}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }A}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the main effect of factor <math>A\,\!</math>. Thus, the sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y \\
= & 654.4375-549.375 \\
= & 105.0625
\end{align}\,\!</math>

Similarly, the extra sum of squares for the interaction effect <math>AB\,\!</math> is:

::<math>\begin{align}
S{{S}_{AB}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }AB}}-(1/16)J]y \\
= & 654.4375-636.375 \\
= & 18.0625
\end{align}\,\!</math>

The extra sum of squares for other effects can be obtained in a similar manner.

====Calculation of the Test Statistics====

Knowing the extra sum of squares, the test statistic for the effects can be calculated. For example, the test statistic for the interaction <math>AB\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{AB}}= & \frac{M{{S}_{AB}}}{M{{S}_{E}}} \\
= & \frac{S{{S}_{AB}}/dof(S{{S}_{AB}})}{S{{S}_{E}}/dof(S{{S}_{E}})} \\
= & \frac{18.0625/1}{147.5/8} \\
= & 0.9797
\end{align}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the <math>AB\,\!</math> interaction and <math>M{{S}_{E}}\,\!</math> is the error mean square. The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{AB}}=0.9797\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{AB}}) \\
= & 1-0.6487 \\
= & 0.3513
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that the interaction between honing pressure and number of strokes does not affect the surface finish of the brake drums. Tests for other effects can be carried out in a similar manner. The results are shown in the ANOVA Table in the following figure. The values S, R-sq and R-sq(adj) in the figure indicate how well the model fits the data. The value of S represents the standard error of the model, R-sq represents the coefficient of multiple determination and R-sq(adj) represents the adjusted coefficient of multiple determination. For details on these values refer to [[Multiple_Linear_Regression_Analysis|Multiple Linear Regression Analysis]].

[[Image:doe7_8.png|center|799px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

====Calculation of Effect Coefficients====

The estimate of effect coefficients can also be obtained:

<center><math>\begin{align}
\hat{\beta }= & {{({{X}^{\prime }}X)}^{-1}}{{X}^{\prime }}y \\
= & \left[ \begin{matrix}
86.4375 \\
2.5625 \\
-4.9375 \\
1.0625 \\
-1.0625 \\
2.4375 \\
-1.3125 \\
-0.1875 \\
\end{matrix} \right]
\end{align}\,\!</math></center>

[[Image:doe7_9.png|center|800px|Regression Information table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

The coefficients and related results are shown in the Regression Information table above. In the table, the Effect column displays the effects, which are simply twice the coefficients. The Standard Error column displays the standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>. The Low CI and High CI columns display the confidence interval on the coefficients. The interval shown is the 90% interval as the significance is chosen as 0.1. The T Value column displays the <math>t\,\!</math> statistic, <math>{{t}_{0}}\,\!</math>, corresponding to the coefficients. The P Value column displays the <math>p\,\!</math> value corresponding to the <math>t\,\!</math> statistic. (For details on how these results are calculated, refer to [[General Full Factorial Designs]]). Plots of residuals can also be obtained from the DOE folio to ensure that the assumptions related to the ANOVA model are not violated.

====Model Equation====

From the analysis results in the above figure within [[Two_Level_Factorial_Experiments#Calculation_of_Effect_Coefficients|calculation of effect coefficients]] section, it is seen that effects <math>A\,\!</math>, <math>B\,\!</math> and <math>AC\,\!</math> are significant. In a DOE folio, the <math>p\,\!</math> values for the significant effects are displayed in red in the ANOVA Table for easy identification. Using the values of the estimated effect coefficients, the model for the present <math>{2}^{3}\,\!</math> design in terms of the coded values can be written as:

::<math>\begin{align}
\hat{y}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}} \\
= & 86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+2.4375{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

To make the model hierarchical, the main effect, <math>C\,\!</math>, needs to be included in the model (because the interaction <math>AC\,\!</math> is included in the model). The resulting model is:

::<math>\hat{y}=86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+1.0625{{x}_{3}}+2.4375{{x}_{1}}{{x}_{3}}\,\!</math>

This equation can be viewed in a DOE folio, as shown in the following figure, using the Show Analysis Summary icon in the Control Panel. The equation shown in the figure will match the hierarchical model once the required terms are selected using the Select Effects icon.

[[Image:doe7_10.png|center|557px|The model equation for the experiment of the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

==Replicated and Repeated Runs==

In the case of replicated experiments, it is important to note the difference between replicated runs and repeated runs. Both repeated and replicated runs are multiple response readings taken at the same factor levels. However, repeated runs are response observations taken at the same time or in succession. Replicated runs are response observations recorded in a random order. Therefore, replicated runs include more variation than repeated runs. For example, a baker, who wants to investigate the effect of two factors on the quality of cakes, will have to bake four cakes to complete one replicate of a <math>{2}^{2}\,\!</math> design. Assume that the baker bakes eight cakes in all. If, for each of the four treatments of the <math>{2}^{2}\,\!</math> design, the baker selects one treatment at random and then bakes two cakes for this treatment at the same time then this is a case of two repeated runs. If, however, the baker bakes all the eight cakes randomly, then the eight cakes represent two sets of replicated runs.
For repeated measurements, the average values of the response for each treatment should be entered into a DOE folio as shown in the following figure (a) when the two cakes for a particular treatment are baked together. For replicated measurements, when all the cakes are baked randomly, the data is entered as shown in the following figure (b).

[[Image:doe7.11.png|center|300px|Data entry for repeated and replicated runs. Figure (a) shows repeated runs and (b) shows replicated runs.]]

==Unreplicated 2''k'' Designs==

If a factorial experiment is run only for a single replicate then it is not possible to test hypotheses about the main effects and interactions as the error sum of squares cannot be obtained. This is because the number of observations in a single replicate equals the number of terms in the ANOVA model. Hence the model fits the data perfectly and no degrees of freedom are available to obtain the error sum of squares.

However, sometimes it is only possible to run a single replicate of the <math>{2}^{k}\,\!</math> design because of constraints on resources and time. In the absence of the error sum of squares, hypothesis tests to identify significant factors cannot be conducted. A number of methods of analyzing information obtained from unreplicated <math>{2}^{k}\,\!</math> designs are available. These include pooling higher order interactions, using the normal probability plot of effects or including center point replicates in the design.

===Pooling Higher Order Interactions===

One of the ways to deal with unreplicated <math>{2}^{k}\,\!</math> designs is to use the sum of squares of some of the higher order interactions as the error sum of squares provided these higher order interactions can be assumed to be insignificant. By dropping some of the higher order interactions from the model, the degrees of freedom corresponding to these interactions can be used to estimate the error mean square. Once the error mean square is known, the test statistics to conduct hypothesis tests on the factors can be calculated.

===Normal Probability Plot of Effects===
Another way to use unreplicated <math>{2}^{k}\,\!</math> designs to identify significant effects is to construct the normal probability plot of the effects. As mentioned in [[Two_Level_Factorial_Experiments#Special_Features| Special Features]], the standard error for all effect coefficients in the <math>{2}^{k}\,\!</math> designs is the same. Therefore, on a normal probability plot of effect coefficients, all non-significant effect coefficients (with <math>\beta =0\,\!</math>) will fall along the straight line representative of the normal distribution, N(<math>0,{{\sigma }^{2}}/({{2}^{k}}\cdot m)\,\!</math>). Effect coefficients that show large deviations from this line will be significant since they do not come from this normal distribution. Similarly, since effects <math>=2\times \,\!</math> effect coefficients, all non-significant effects will also follow a straight line on the normal probability plot of effects. For replicated designs, the Effects Probability plot of a DOE folio plots the normalized effect values (or the T Values) on the standard normal probability line, N(0,1). However, in the case of unreplicated <math>{2}^{k}\,\!</math> designs, <math>{{\sigma }^{2}}\,\!</math> remains unknown since <math>M{{S}_{E}}\,\!</math> cannot be obtained. Lenth's method is used in this case to estimate the variance of the effects. For details on Lenth's method, please refer to [[DOE References| Montgomery (2001)]]. The DOE folio then uses this variance value to plot effects along the N(0, Lenth's effect variance) line. The
method is illustrated in the following example.

====Example====

Vinyl panels, used as instrument panels in a certain automobile, are seen to develop defects after a certain amount of time. To investigate the issue, it is decided to carry out a two level factorial experiment. Potential factors to be investigated in the experiment are vacuum rate (factor <math>A\,\!</math>), material temperature (factor <math>B\,\!</math>), element intensity (factor <math>C\,\!</math>) and pre-stretch (factor <math>D\,\!</math>). The two levels of the factors used in the experiment are as shown in below.

[[Image:doet7.1.png|center|300px|Factors to investigate defects in vinyl panels.]]

With a <math>{2}^{4}\,\!</math> design requiring 16 runs per replicate it is only feasible for the manufacturer to run a single replicate.

The experiment design and data, collected as percent defects, are shown in the following figure. Since the present experiment design contains only a single replicate, it is not possible to obtain an estimate of the error sum of squares, <math>S{{S}_{E}}\,\!</math>. It is decided to use the normal probability plot of effects to identify the significant effects. The effect values for each term are obtained as shown in the following figure.

[[Image:doe7_13.png|center|650px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

Lenth's method uses these values to estimate the variance. As described in [[DOE_References|[Lenth, 1989]]], if all effects are arranged in ascending order, using their absolute values, then <math>{{s}_{0}}\,\!</math> is defined as 1.5 times the median value:

::<math>\begin{align}
{{s}_{0}}= & 1.5\cdot median(\left| effect \right|) \\
= & 1.5\cdot 2 \\
= & 3
\end{align}\,\!</math>

Using <math>{{s}_{0}}\,\!</math>, the "pseudo standard error" (<math>PSE\,\!</math>) is calculated as 1.5 times the median value of all effects that are less than 2.5 <math>{{s}_{0}}\,\!</math> :

::<math>\begin{align}
PSE= & 1.5\cdot median(\left| effect \right|\ \ :\ \ \left| effect \right|<2.5{{s}_{0}}) \\
= & 1.5\cdot 1.5 \\
= & 2.25
\end{align}\,\!</math>

Using <math>PSE\,\!</math> as an estimate of the effect variance, the effect variance is 2.25. Knowing the effect variance, the normal probability plot of effects for the present unreplicated experiment can be constructed as shown in the following figure. The line on this plot is the line N(0, 2.25). The plot shows that the effects <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math> do not follow the distribution represented by this line. Therefore, these effects are significant.

The significant effects can also be identified by comparing individual effect values to the margin of error or the threshold value using the pareto chart (see the third following figure). If the required significance is 0.1, then:

::<math>margin\text{ }of\text{ }error={{t}_{\alpha /2,d}}\cdot PSE\,\!</math>

The <math>t\,\!</math> statistic, <math>{{t}_{\alpha /2,d}}\,\!</math>, is calculated at a significance of <math>\alpha /2\,\!</math> (for the two-sided hypothesis) and degrees of freedom <math>d=(\,\!</math> number of effects <math>)/3\,\!</math>. Thus:

::<math>\begin{align}
margin\text{ }of\text{ }error= & {{t}_{0.05,5}}\cdot PSE \\
= & 2.015\cdot 2.25 \\
= & 4.534
\end{align}\,\!</math>

The value of 4.534 is shown as the critical value line in the third following figure. All effects with absolute values greater than the margin of error can be considered to be significant. These effects are <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math>. Therefore, the vacuum rate, the pre-stretch and their interaction have a significant effect on the defects of the vinyl panels.

[[Image:doe7_14.png|center|800px|Effect values for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_15.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_16.png|center|650px|Pareto chart for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

===Center Point Replicates===

Another method of dealing with unreplicated <math>{2}^{k}\,\!</math> designs that only have quantitative factors is to use replicated runs at the center point. The center point is the response corresponding to the treatment exactly midway between the two levels of all factors. Running multiple replicates at this point provides an estimate of pure error. Although running multiple replicates at any treatment level can provide an estimate of pure error, the other advantage of running center point replicates in the <math>{2}^{k}\,\!</math> design is in checking for the presence of curvature. The test for curvature investigates whether the model between the response and the factors is linear and is discussed in [[Two_Level_Factorial_Experiments#Using_Center_Point_Replicates_to_Test_Curvature| Center Pt. Replicates to Test Curvature]].

====Example: Use Center Point to Get Pure Error====

Consider a <math>{2}^{2}\,\!</math> experiment design to investigate the effect of two factors, <math>A\,\!</math> and <math>B\,\!</math>, on a certain response. The energy consumed when the treatments of the <math>{2}^{2}\,\!</math> design are run is considerably larger than the energy consumed for the center point run (because at the center point the factors are at their middle levels). Therefore, the analyst decides to run only a single replicate of the design and augment the design by five replicated runs at the center point as shown in the following figure. The design properties for this experiment are shown in the second following figure. The complete experiment design is shown in the third following figure. The center points can be used in the identification of significant effects as shown next.

[[Image:doe7.17.png||center|300px|<math>2^2\,\!</math> design augmented by five center point runs.]]
[[Image:doe7_18.png|center|537px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

[[Image:doe7_19.png|center|630px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

Since the present <math>{2}^{2}\,\!</math> design is unreplicated, there are no degrees of freedom available to calculate the error sum of squares. By augmenting this design with five center points, the response values at the center points, <math>y_{i}^{c}\,\!</math>, can be used to obtain an estimate of pure error, <math>S{{S}_{PE}}\,\!</math>. Let <math>{{\bar{y}}^{c}}\,\!</math> represent the average response for the five replicates at the center. Then:

::<math>S{{S}_{PE}}=Sum\text{ }of\text{ }Squares\text{ }for\text{ }center\text{ }points\,\!</math>

::<math>\begin{align}
S{{S}_{PE}}= & \underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}} \\
= & {{(25.2-25.26)}^{2}}+...+{{(25.3-25.26)}^{2}} \\
= & 0.052
\end{align}\,\!</math>

Then the corresponding mean square is:

::<math>\begin{align}
M{{S}_{PE}}= & \frac{S{{S}_{PE}}}{degrees\text{ }of\text{ }freedom} \\
= & \frac{0.052}{5-1} \\
= & 0.013
\end{align}\,\!</math>

Alternatively, <math>M{{S}_{PE}}\,\!</math> can be directly obtained by calculating the variance of the response values at the center points:

::<math>\begin{align}
M{{S}_{PE}}= & {{s}^{2}} \\
= & \frac{\underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}}}{5-1}
\end{align}\,\!</math>

Once <math>M{{S}_{PE}}\,\!</math> is known, it can be used as the error mean square, <math>M{{S}_{E}}\,\!</math>, to carry out the test of significance for each effect. For example, to test the significance of the main effect of factor <math>A,\,\!</math> the sum of squares corresponding to this effect is obtained in the usual manner by considering only the four runs of the original <math>{2}^{2}\,\!</math> design.

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/4)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/4)J]y \\
= & 0.5625
\end{align}\,\!</math>

Then, the test statistic to test the significance of the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{A}}= & \frac{M{{S}_{A}}}{M{{S}_{E}}} \\
= & \frac{0.5625/1}{0.052/4} \\
= & 43.2692
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{A}}=43.2692\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{A}}) \\
= & 1-0.9972 \\
= & 0.0028
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value < 0.1, it can be concluded that the main effect of factor <math>A\,\!</math> significantly affects the response. This result is displayed in the ANOVA table as shown in the following figure. Test for the significance of other factors can be carried out in a similar manner.

[[Image:doe7_20.png|center|649px|Results for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

===Using Center Point Replicates to Test Curvature===

Center point replicates can also be used to check for curvature in replicated or unreplicated <math>{2}^{k}\,\!</math> designs. The test for curvature investigates whether the model between the response and the factors is linear. The way DOE++ handles center point replicates is similar to its handling of blocks. The center point replicates are treated as an additional factor in the model. The factor is labeled as Curvature in the results of the DOE folio. If Curvature turns out to be a significant factor in the results, then this indicates the presence of curvature in the model.

====Example: Use Center Point to Test Curvature====

To illustrate the use of center point replicates in testing for curvature, consider again the data of the single replicate <math>{2}^{2}\,\!</math> experiment from a preceding figure(labeled "<math>2^2</math> design augmented by five center point runs"). Let <math>{{x}_{1}}\,\!</math> be the indicator variable to indicate if the run is a center point:

::<math>\begin{matrix}
{{x}_{1}}=0 & {} & \text{Center point run} \\
{{x}_{1}}=1 & {} & \text{Other run} \\
\end{matrix}\,\!</math>

If <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> are the indicator variables representing factors <math>A\,\!</math> and <math>B\,\!</math>, respectively, then the model for this experiment is:

::<math>Y={{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}\,\!</math>

To investigate the presence of curvature, the following hypotheses need to be tested:

::<math>\begin{align}
& {{H}_{0}}: & {{\beta }_{1}}=0\text{ (Curvature is absent)} \\
& {{H}_{1}}: & {{\beta }_{1}}\ne 0
\end{align}\,\!</math>

The test statistic to be used for this test is:

::<math>{{({{F}_{0}})}_{curvature}}=\frac{M{{S}_{curvature}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{curvature}}\,\!</math> is the mean square for Curvature and <math>M{{S}_{E}}\,\!</math> is the error mean square.

'''Calculation of the Sum of Squares'''

The <math>X\,\!</math> matrix and <math>y\,\!</math> vector for this experiment are:

<center><math>X=\left[ \begin{matrix}
1 & 1 & -1 & -1 & 1 \\
1 & 1 & 1 & -1 & -1 \\
1 & 1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
\end{matrix} \right]\text{ }y=\left[ \begin{matrix}
24.6 \\
25.4 \\
25.0 \\
25.7 \\
25.2 \\
25.3 \\
25.4 \\
25.1 \\
25.3 \\
\end{matrix} \right]\,\!</math></center>

The sum of squares can now be calculated. For example, the error sum of squares is:

::<math>\begin{align}
& S{{S}_{E}}= & {{y}^{\prime }}[I-H]y \\
& = & 0.052
\end{align}\,\!</math>

where <math>I\,\!</math> is the identity matrix and <math>H\,\!</math> is the hat matrix. It can be seen that this is equal to <math>S{{S}_{PE\text{ }}}\,\!</math> (the sum of squares due to pure error) because of the replicates at the center point, as obtained in the [[Two_Level_Factorial_Experiments#Example_3| example]]. The number of degrees of freedom associated with <math>S{{S}_{E}}\,\!</math>, <math>dof(S{{S}_{E}})\,\!</math> is four. The extra sum of squares corresponding to the center point replicates (or Curvature) is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & Model\text{ }Sum\text{ }of\text{ }Squares- \\
& & Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }center\text{ }point \\
& = & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Curvature}}-(1/9)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }Curvature}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }Curvature}}={{X}_{\tilde{\ }Curv}}{{(X_{\tilde{\ }Curv}^{\prime }{{X}_{\tilde{\ }Curv}})}^{-1}}X_{\tilde{\ }Curv}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }Curv}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the center point. Thus, the extra sum of squares corresponding to Curvature is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Center}}-(1/9)J]y \\
& = & 0.7036-0.6875 \\
& = & 0.0161
\end{align}\,\!</math>

This extra sum of squares can be used to test for the significance of curvature. The corresponding mean square is:

::<math>\begin{align}
& M{{S}_{Curvature}}= & \frac{Sum\text{ }of\text{ }squares\text{ }corresponding\text{ }to\text{ }Curvature}{degrees\text{ }of\text{ }freedom} \\
& = & \frac{0.0161}{1} \\
& = & 0.0161
\end{align}\,\!</math>

'''Calculation of the Test Statistic'''

Knowing the mean squares, the statistic to check the significance of curvature can be calculated.

::<math>\begin{align}
& {{({{f}_{0}})}_{Curvature}}= & \frac{M{{S}_{Curvature}}}{M{{S}_{E}}} \\
& = & \frac{0.0161/1}{0.052/4} \\
& = & 1.24
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{Curvature}}=1.24\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and four degrees of freedom in the denominator is:

::<math>\begin{align}
& p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{Curvature}}) \\
& = & 1-0.6713 \\
& = & 0.3287
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that curvature does not exist for this design. This results is shown in the ANOVA table in the figure above. The surface of the fitted model based on these results, along with the observed response values, is shown in the figure below.

[[Image:doe7.21.png|center|400px|Model surface and observed response values for the design in the [[Two_Level_Factorial_Experiments#Example_4| example]].]]

 

==Blocking in 2k Designs==

Blocking can be used in the <math>{2}^{k}\,\!</math> designs to deal with cases when replicates cannot be run under identical conditions. Randomized complete block designs that were discussed in [[Randomization and Blocking in DOE]] for factorial experiments are also applicable here. At times, even with just two levels per factor, it is not possible to run all treatment combinations for one replicate of the experiment under homogeneous conditions. For example, each replicate of the <math>{2}^{2}\,\!</math> design requires four runs. If each run requires two hours and testing facilities are available for only four hours per day, two days of testing would be required to run one complete replicate. Blocking can be used to separate the treatment runs on the two different days. Blocks that do not contain all treatments of a replicate are called incomplete blocks. In incomplete block designs, the block effect is confounded with certain effect(s) under investigation. For the <math>{2}^{2}\,\!</math> design assume that treatments <math>(1)\,\!</math> and <math>ab\,\!</math> were run on the first day and treatments <math>a\,\!</math> and <math>b\,\!</math> were run on the second day. Then, the incomplete block design for this experiment is:

::<math>\begin{matrix}
\text{Block 1} & {} & \text{Block 2} \\
\left[ \begin{matrix}
(1) \\
ab \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
b \\
\end{matrix} \right] \\
\end{matrix}\,\!</math>

For this design the block effect may be calculated as:

::<math>\begin{align}
& Block\text{ }Effect= & Average\text{ }response\text{ }for\text{ }Block\text{ }1- \\
& & Average\text{ }response\text{ }for\text{ }Block\text{ }2 \\
& = & \frac{(1)+ab}{2}-\frac{a+b}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The <math>AB\,\!</math> interaction effect is:

::<math>\begin{align}
& AB= & Average\text{ }response\text{ }at\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{low}}}- \\
& & Average\text{ }response\text{ }at\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{low}}} \\
& = & \frac{ab+(1)}{2}-\frac{b+a}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The two equations given above show that, in this design, the <math>AB\,\!</math> interaction effect cannot be distinguished from the block effect because the formulas to calculate these effects are the same. In other words, the <math>AB\,\!</math> interaction is said to be confounded with the block effect and it is not possible to say if the effect calculated based on these equations is due to the <math>AB\,\!</math> interaction effect, the block effect or both. In incomplete block designs some effects are always confounded with the blocks. Therefore, it is important to design these experiments in such a way that the important effects are not confounded with the blocks. In most cases, the experimenter can assume that higher order interactions are unimportant. In this case, it would better to use incomplete block designs that confound these effects with the blocks.
One way to design incomplete block designs is to use defining contrasts as shown next:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}+...+{{\alpha }_{k}}{{q}_{k}}\,\!</math>

where the <math>{{\alpha }_{i}}\,\!</math> s are the exponents for the factors in the effect that is to be confounded with the block effect and the <math>{{q}_{i}}\,\!</math> s are values based on the level of the <math>i\,\!</math> the factor (in a treatment that is to be allocated to a block). For <math>{2}^{k}\,\!</math> designs the <math>{{\alpha }_{i}}\,\!</math> s are either 0 or 1 and the <math>{{q}_{i}}\,\!</math> s have a value of 0 for the low level of the <math>i\,\!</math> th factor and a value of 1 for the high level of the factor in the treatment under consideration. As an example, consider the <math>{2}^{2}\,\!</math> design where the interaction effect <math>AB\,\!</math> is confounded with the block. Since there are two factors, <math>k=2\,\!</math>, with <math>i=1\,\!</math> representing factor <math>A\,\!</math> and <math>i=2\,\!</math> representing factor <math>B\,\!</math>. Therefore:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}\,\!</math>

The value of <math>{{\alpha }_{1}}\,\!</math> is one because the exponent of factor <math>A\,\!</math> in the confounded interaction <math>AB\,\!</math> is one. Similarly, the value of <math>{{\alpha }_{2}}\,\!</math> is one because the exponent of factor <math>B\,\!</math> in the confounded interaction <math>AB\,\!</math> is also one. Therefore, the defining contrast for this design can be written as:

::<math>\begin{align}
& L= & {{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}} \\
& = & 1\cdot {{q}_{1}}+1\cdot {{q}_{2}} \\
& = & {{q}_{1}}+{{q}_{2}}
\end{align}\,\!</math>

Once the defining contrast is known, it can be used to allocate treatments to the blocks. For the <math>{2}^{2}\,\!</math> design, there are four treatments <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math> and <math>ab\,\!</math>. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. In order to decide which block the treatment <math>(1)\,\!</math> belongs to, the levels of factors <math>A\,\!</math> and <math>B\,\!</math> for this run are used. Since factor <math>A\,\!</math> is at the low level in this treatment, <math>{{q}_{1}}=0\,\!</math>. Similarly, since factor <math>B\,\!</math> is also at the low level in this treatment, <math>{{q}_{2}}=0\,\!</math>. Therefore:

::<math>\begin{align}
& L= & {{q}_{1}}+{{q}_{2}} \\
& = & 0+0=0\text{ (mod 2)}
\end{align}\,\!</math>

Note that the value of <math>L\,\!</math> used to decide the block allocation is "mod 2" of the original value. This value is obtained by taking the value of 1 for odd numbers and 0 otherwise. Based on the value of <math>L\,\!</math>, treatment <math>(1)\,\!</math> is assigned to block 1. Other treatments can be assigned using the following calculations:

::<math>\begin{align}
& (1): & \text{ }L=0+0=0=0\text{ (mod 2)} \\
& a: & \text{ }L=1+0=1=1\text{ (mod 2)} \\
& b: & \text{ }L=0+1=1=1\text{ (mod 2)} \\
& ab: & \text{ }L=1+1=2=0\text{ (mod 2)}
\end{align}\,\!</math>

Therefore, to confound the interaction <math>AB\,\!</math> with the block effect in the <math>{2}^{2}\,\!</math> incomplete block design, treatments <math>(1)\,\!</math> and <math>ab\,\!</math> (with <math>L=0\,\!</math>) should be assigned to block 2 and treatment combinations <math>a\,\!</math> and <math>b\,\!</math> (with <math>L=1\,\!</math>) should be assigned to block 1.

====Example: Two Level Factorial Design with Two Blocks====

This example illustrates how treatments can be allocated to two blocks for an unreplicated <math>{2}^{k}\,\!</math> design. Consider the unreplicated <math>{2}^{4}\,\!</math> design to investigate the four factors affecting the defects in automobile vinyl panels discussed in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments required for this experiment were run by two different operators with each operator conducting 8 runs. This experiment is an example of an incomplete block design. The analyst in charge of this experiment assumed that the interaction <math>ABCD\,\!</math> was not significant and decided to allocate treatments to the two operators so that the <math>ABCD\,\!</math> interaction was confounded with the block effect (the two operators are the blocks). The allocation scheme to assign treatments to the two operators can be obtained as follows.
 
The defining contrast for the <math>{2}^{4}\,\!</math> design where the <math>ABCD\,\!</math> interaction is confounded with the blocks is:

::<math>L={{q}_{1}}+{{q}_{2}}+{{q}_{3}}+{{q}_{4}}\,\!</math>

The treatments can be allocated to the two operators using the values of the defining contrast. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. Then the value of the defining contrast for treatment <math>a\,\!</math> is:

::<math>a\ \ :\ \ \text{ }L=1+0+0+0=1=1\text{ (mod 2)}\,\!</math>

Therefore, treatment <math>a\,\!</math> should be assigned to Block 1 or the first operator. Similarly, for treatment <math>ab\,\!</math> we have:

::<math>ab\ \ :\ \ \text{ }L=1+1+0+0=2=0\text{ (mod 2)}\,\!</math>

[[Image:doe7.22.png|center|200px| Allocation of treatments to two blocks for the <math>2^4</math> design in the example by confounding interaction of <math>ABCD</math> with the blocks.]]

Therefore, <math>ab\,\!</math> should be assigned to Block 2 or the second operator. Other treatments can be allocated to the two operators in a similar manner to arrive at the allocation scheme shown in the figure below.
In DOE++, to confound the <math>ABCD\,\!</math> interaction for the <math>{2}^{4}\,\!</math> design into two blocks, the number of blocks are specified as shown in the figure below. Then the interaction <math>ABCD\,\!</math> is entered in the Block Generator window (second following figure) which is available using the Block Generator button in the following figure. The design generated by DOE++ is shown in the third of the following figures. This design matches the allocation scheme of the preceding figure.

[[Image:doe7_23.png|center|700px| Adding block properties for the experiment in the example.|link=]]

[[Image:doe7_24.png|center|700px|Specifying the interaction ABCD as the interaction to be confounded with the blocks for the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

[[Image:doe7_25.png|center|800px|Two block design for the experiment in the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

For the analysis of this design, the sum of squares for all effects are calculated assuming no blocking. Then, to account for blocking, the sum of squares corresponding to the <math>ABCD\,\!</math> interaction is considered as the sum of squares due to blocks and <math>ABCD\,\!</math>. In DOE++ this is done by displaying this sum of squares as the sum of squares due to the blocks. This is shown in the following figure where the sum of squares in question is obtained as 72.25 and is displayed against Block. The interaction ABCD, which is confounded with the blocks, is not displayed. Since the design is unreplicated, any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_26.png|center|793px|ANOVA table for the experiment of the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

===Unreplicated 2''k'' Designs in 2''p'' Blocks===

A single replicate of the <math>{2}^{k}\,\!</math> design can be run in up to <math>{2}^{p}\,\!</math> blocks where <math>p<k\,\!</math>. The number of effects confounded with the blocks equals the degrees of freedom associated with the block effect.

If two blocks are used (the block effect has two levels), then one (<math>2-1=1)\,\!</math> effect is confounded with the blocks. If four blocks are used, then three (<math>4-1=3\,\!</math>) effects are confounded with the blocks and so on. For example an unreplicated <math>{2}^{4}\,\!</math> design may be confounded in <math>{2}^{2}\,\!</math> (four) blocks using two contrasts, <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}}\,\!</math>. Let <math>AC\,\!</math> and <math>BD\,\!</math> be the effects to be confounded with the blocks. Corresponding to these two effects, the contrasts are respectively:

::<math>\begin{align}
& {{L}_{1}}= & {{q}_{1}}+{{q}_{3}} \\
& {{L}_{2}}= & {{q}_{2}}+{{q}_{4}}
\end{align}\,\!</math>

Based on the values of <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}},\,\!</math> the treatments can be assigned to the four blocks as follows:

<center><math>\begin{matrix}
\text{Block 4} & {} & \text{Block 3} & {} & \text{Block 2} & {} & \text{Block 1} \\
{{L}_{1}}=0,{{L}_{2}}=0 & {} & {{L}_{1}}=1,{{L}_{2}}=0 & {} & {{L}_{1}}=0,{{L}_{2}}=1 & {} & {{L}_{1}}=1,{{L}_{2}}=1 \\
{} & {} & {} & {} & {} & {} & {} \\
\left[ \begin{matrix}
(1) \\
ac \\
bd \\
abcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
c \\
abd \\
bcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
b \\
abc \\
d \\
acd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
ab \\
bc \\
ad \\
cd \\
\end{matrix} \right] \\
\end{matrix}\,\!</math></center>

Since the block effect has three degrees of freedom, three effects are confounded with the block effect. In addition to <math>AC\,\!</math> and <math>BD\,\!</math>, the third effect confounded with the block effect is their generalized interaction, <math>(AC)(BD)=ABCD\,\!</math>.
In general, when an unreplicated <math>{2}^{k}\,\!</math> design is confounded in <math>{2}^{p}\,\!</math> blocks, <math>p\,\!</math> contrasts are needed (<math>{{L}_{1}},{{L}_{2}}...{{L}_{p}}\,\!</math>). <math>p\,\!</math> effects are selected to define these contrasts such that none of these effects are the generalized interaction of the others. The <math>{2}^{p}\,\!</math> blocks can then be assigned the treatments using the <math>p\,\!</math> contrasts. <math>{{2}^{p}}-(p+1)\,\!</math> effects, that are also confounded with the blocks, are then obtained as the generalized interaction of the <math>p\,\!</math> effects. In the statistical analysis of these designs, the sum of squares are computed as if no blocking were used. Then the block sum of squares is obtained by adding the sum of squares for all the effects confounded with the blocks.

====Example: 2 Level Factorial Design with Four Blocks====

This example illustrates how DOE++ obtains the sum of squares when treatments for an unreplicated <math>{2}^{k}\,\!</math> design are allocated among four blocks. Consider again the unreplicated <math>{2}^{4}\,\!</math> design used to investigate the defects in automobile vinyl panels presented in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments needed to complete the experiment were run by four operators. Therefore, there are four blocks. Assume that the treatments were allocated to the blocks using the generators mentioned in the previous section, i.e., treatments were allocated among the four operators by confounding the effects, <math>AC\,\!</math> and <math>BD,\,\!</math> with the blocks. These effects can be specified as Block Generators as shown in the following figure. (The generalized interaction of these two effects, interaction <math>ABCD\,\!</math>, will also get confounded with the blocks.) The resulting design is shown in the second following figure and matches the allocation scheme obtained in the previous section.

[[Image:doe7_27.png|center|700px|Specifying the interactions AC and BD as block generators for the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

The sum of squares in this case can be obtained by calculating the sum of squares for each of the effects assuming there is no blocking. Once the individual sum of squares have been obtained, the block sum of squares can be calculated. The block sum of squares is the sum of the sum of squares of effects, <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, since these effects are confounded with the block effect. As shown in the second following figure, this sum of squares is 92.25 and is displayed against Block. The interactions <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, which are confounded with the blocks, are not displayed. Since the present design is unreplicated any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_28.png|center|800px|Design for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

[[Image:doe7_29.png|center|793px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

==Variability Analysis==

For replicated two level factorial experiments, DOE++ provides the option of conducting variability analysis (using the Variability Analysis icon under the Data menu). The analysis is used to identify the treatment that results in the least amount of variation in the product or process being investigated. Variability analysis is conducted by treating the standard deviation of the response for each treatment of the experiment as an additional response. The standard deviation for a treatment is obtained by using the replicated response values at that treatment run. As an example, consider the <math>{2}^{3}\,\!</math> design shown in the following figure where each run is replicated four times. A variability analysis can be conducted for this design. DOE++ calculates eight standard deviation values corresponding to each treatment of the design (see second following figure). Then, the design is analyzed as an unreplicated <math>{2}^{3}\,\!</math> design with the standard deviations (displayed as Y Standard Deviation. in second following figure) as the response. The normal probability plot of effects identifies <math>AC\,\!</math> as the effect that influences variability (see third figure following). Based on the effect coefficients obtained in the fourth figure following, the model for Y Std. is:

::<math>\begin{align}
& \text{Y Std}\text{.}= & 0.6779+0.2491\cdot AC \\
& = & 0.6779+0.2491{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

Based on the model, the experimenter has two choices to minimize variability (by minimizing Y Std.). The first choice is that <math>{{x}_{1}}\,\!</math> should be <math>1\,\!</math> (i.e., <math>A\,\!</math> should be set at the high level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the low level). The second choice is that <math>{{x}_{1}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>A\,\!</math> should be set at the low level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the high level). The experimenter can select the most feasible choice.

[[Image:doe7.30.png|center|391px|A <math>2^3\,\!</math> design with four replicated response values that can be used to conduct a variability analysis.]]
 

[[Image:doe7_31.png|center|727px|Variability analysis in DOE++.|link=]]
 

[[Image:doe7_32.png|center|650px|Normal probability plot of effects for the variability analysis example.|link=]]
 

[[Image:doe7_33.png|center|727px|Effect coefficients for the variability analysis example.|link=]]

 

==Two Level Fractional Factorial Designs==

As the number of factors in a two level factorial design increases, the number of runs for even a single replicate of the <math>{2}^{k}\,\!</math> design becomes very large. For example, a single replicate of an eight factor two level experiment would require 256 runs. Fractional factorial designs can be used in these cases to draw out valuable conclusions from fewer runs. The basis of fractional factorial designs is the ''sparsity of effects'' principle.[[DOE_References|[Wu, 2000]]] The principle states that, most of the time, responses are affected by a small number of main effects and lower order interactions, while higher order interactions are relatively unimportant. Fractional factorial designs are used as screening experiments during the initial stages of experimentation. At these stages, a large number of factors have to be investigated and the focus is on the main effects and two factor interactions. These designs obtain information about main effects and lower order interactions with fewer experiment runs by confounding these effects with unimportant higher order interactions. As an example, consider a <math>{2}^{8}\,\!</math> design that requires 256 runs. This design allows for the investigation of 8 main effects and 28 two factor interactions. However, 219 degrees of freedom are devoted to three factor or higher order interactions. This full factorial design can prove to be very inefficient when these higher order interactions can be assumed to be unimportant. Instead, a fractional design can be used here to identify the important factors that can then be investigated more thoroughly in subsequent experiments. In unreplicated fractional factorial designs, no degrees of freedom are available to calculate the error sum of squares and the techniques mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] should be employed for the analysis of these designs.

==Half-fraction Designs==

A half-fraction of the <math>{2}^{k}\,\!</math> design involves running only half of the treatments of the full factorial design. For example, consider a <math>{2}^{3}\,\!</math> design that requires eight runs in all. The design matrix for this design is shown in the figure (a) below. A half-fraction of this design is the design in which only four of the eight treatments are run. The fraction is denoted as <math>{2}^{3-1}\,\!</math> with the "<math>-1\,\!</math>" in the index denoting a half-fraction. Assume that the treatments chosen for the half-fraction design are the ones where the interaction <math>ABC\,\!</math> is at the high level (i.e., only those rows are chosen from the following figure (a) where the column for <math>ABC\,\!</math> has entries of 1). The resulting <math>{2}^{3-1}\,\!</math> design has a design matrix as shown in figure (b) below.

[[Image:doe7.34.png|center|330px|Half-fractions of the <math>2^3\,\!</math> design. (a) shows the full factorial <math>2^3\,\!</math> design, (b) shows the <math>2^{3-1}\,\!</math> design with the defining relation <math>I=ABC\,\!</math> and (c) shows the <math>{2}^{3-1}\,\!</math> design with the defining relation <math>I=-ABC\,\!</math>.]]

In the <math>{2}^{3-1}\,\!</math> design of figure (b), since the interaction <math>ABC\,\!</math> is always included at the same level (the high level represented by 1), it is not possible to measure this interaction effect. The effect, <math>ABC\,\!</math>, is called the ''generator'' or ''word'' for this design. It can be noted that, in the design matrix of the following figure (b), the column corresponding to the intercept, <math>I\,\!</math>, and column corresponding to the interaction <math>ABC\,\!</math>, are identical. The identical columns are written as <math>I=ABC\,\!</math> and this equation is called the ''defining relation'' for the design. In DOE++, the present <math>{2}^{3-1}\,\!</math> design can be obtained by specifying the design properties as shown in the following figure.

[[Image:doe7_35.png|center|700px|Design properties for the <math>2^{3-1}\,\!</math> design.|link=]]

The defining relation, <math>I=ABC\,\!</math>, is entered in the Fraction Generator window as shown next.

[[Image:doe7_36.png|center|700px|Specifying the defining relation for the <math>2^{3-1}\,\!</math> design.|link=]]

Note that in the figure following that, the defining relation is specified as <math>C=AB\,\!</math>. This relation is obtained by multiplying the defining relation, <math>I=ABC\,\!</math>, by the last factor, <math>C\,\!</math>, of the design.

===Calculation of Effects===

Using the four runs of the <math>{2}^{3-1}\,\!</math> design in figure (b) discussed above, the main effects can be calculated as follows:

::<math>\begin{align}
& A= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& B= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& C= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

where <math>a\,\!</math>, <math>b\,\!</math>, <math>c\,\!</math> and <math>abc\,\!</math> are the treatments included in the <math>{2}^{3-1}\,\!</math> design.

Similarly, the two factor interactions can also be obtained as:

::<math>\begin{align}
& BC= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& AC= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& AB= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

The equations for <math>A\,\!</math> and <math>BC\,\!</math> above result in the same effect values showing that effects <math>A\,\!</math> and <math>BC\,\!</math> are confounded in the present <math>{2}^{3-1}\,\!</math> design. Thus, the quantity, <math>\tfrac{1}{2}(a-b-c+abc),\,\!</math> estimates <math>A+BC\,\!</math> (i.e., both the main effect <math>A\,\!</math> and the two-factor interaction <math>BC\,\!</math>). The effects, <math>A\,\!</math> and <math>BC,\,\!</math> are called ''aliases''. From the remaining equations given above, it can be seen that the other aliases for this design are <math>B\,\!</math> and <math>AC\,\!</math>, and <math>C\,\!</math> and <math>AB\,\!</math>. Therefore, the equations to calculate the effects in the present <math>{2}^{3-1}\,\!</math> design can be written as follows:

::<math>\begin{align}
& A+BC= & \frac{1}{2}(a-b-c+abc) \\
& B+AC= & \frac{1}{2}(-a+b-c+abc) \\
& C+AB= & \frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

===Calculation of Aliases===

Aliases for a fractional factorial design can be obtained using the defining relation for the design. The defining relation for the present <math>{2}^{3-1}\,\!</math> design is:

::<math>I=ABC\,\!</math>

Multiplying both sides of the previous equation by the main effect, <math>A,\,\!</math> gives the alias effect of <math>A\,\!</math> :

::<math>\begin{align}
& A\cdot I= & A\cdot ABC \\
& A= & {{A}^{2}}BC \\
& A= & BC
\end{align}\,\!</math>

Note that in calculating the alias effects, any effect multiplied by <math>I\,\!</math> remains the same (<math>A\cdot I=A\,\!</math>), while an effect multiplied by itself results in <math>I\,\!</math> (<math>{{A}^{2}}=I\,\!</math>). Other aliases can also be obtained:

::<math>\begin{align}
& B\cdot I= & B\cdot ABC \\
& B= & A{{B}^{2}}C \\
& B= & AC
\end{align}\,\!</math>

:and:

::<math>\begin{align}
& C\cdot I= & C\cdot ABC \\
& C= & AB{{C}^{2}} \\
& C= & AB
\end{align}\,\!</math>

===Fold-over Design===

If it can be assumed for this design that the two-factor interactions are unimportant, then in the absence of <math>BC\,\!</math>, <math>AC\,\!</math> and <math>AB\,\!</math>, the equations for (A+BC), (B+AC) and (C+AB) can be used to estimate the main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. However, if such an assumption is not applicable, then to uncouple the main effects from their two factor aliases, the alternate fraction that contains runs having <math>ABC\,\!</math> at the lower level should be run. The design matrix for this design is shown in the preceding figure (c). The defining relation for this design is <math>I=-ABC\,\!</math> because the four runs for this design are obtained by selecting the rows of the preceding figure (a) for which the value of the <math>ABC\,\!</math> column is <math>-1\,\!</math>. The aliases for this fraction can be obtained as explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]] as <math>A=-BC\,\!</math>, <math>B=-AC\,\!</math> and <math>C=-AB\,\!</math>. The effects for this design can be calculated as:

::<math>\begin{align}
& A-BC= & \frac{1}{2}(ab+ac-(1)-bc) \\
& B-AC= & \frac{1}{2}(ab-ac+(1)-bc) \\
& C-AB= & \frac{1}{2}(-ab+ac-(1)+bc)
\end{align}\,\!</math>

These equations can be combined with the equations for (A+BC), (B+AC) and (C+AB) to obtain the de-aliased main effects and two factor interactions. For example, adding equations (A+BC) and (A-BC) returns the main effect <math>A\,\!</math>.

::<math>\begin{align}
& 2A= & \frac{1}{2}(a-b-c+abc)+ \\
& & \frac{1}{2}(ab+ac-(1)-bc)
\end{align}\,\!</math>

The process of augmenting a fractional factorial design by a second fraction of the same size by simply reversing the signs (of all effect columns except <math>I\,\!</math>) is called ''folding over''. The combined design is referred to as a ''fold-over design''.

==Quarter and Smaller Fraction Designs==

At times, the number of runs even for a half-fraction design are very large. In these cases, smaller fractions are used. A quarter-fraction design, denoted as <math>{2}^{k-2}\,\!</math>, consists of a fourth of the runs of the full factorial design. Quarter-fraction designs require two defining relations. The first defining relation returns the half-fraction or the <math>{2}^{k-1}\,\!</math> design. The second defining relation selects half of the runs of the <math>{2}^{k-1}\,\!</math> design to give the quarter-fraction. For example, consider the <math>{2}^{4}\,\!</math> design. To obtain a <math>{2}^{4-2}\,\!</math> design from this design, first a half-fraction of this design is obtained by using a defining relation. Assume that the defining relation used is <math>I=ABCD\,\!</math>. The design matrix for the resulting <math>{2}^{4-1}\,\!</math> design is shown in figure (a) below. Now, a quarter-fraction can be obtained from the <math>{2}^{4-1}\,\!</math> design shown in figure (a) below using a second defining relation <math>I=AD\,\!</math>. The resulting <math>{2}^{4-2}\,\!</math> design obtained is shown in figure (b) below.

[[Image:doe7.37.png|center|465px|Fractions of the <math>2^4\,\!</math> design - Figure (a) shows the <math>2^{4-1}</math> design with the defining relation <math>I=ABCD\,\!</math> and (b) shows the <math>2^{4-2}\,\!</math> design with the defining relation <math>I=ABCD=AD=BC\,\!</math>.]]

The complete defining relation for this <math>{2}^{4-2}\,\!</math> design is:

::<math>I=ABCD=AD=BC\,\!</math>

Note that the effect, <math>BC,\,\!</math> in the defining relation is the generalized interaction of <math>ABCD\,\!</math> and <math>AD\,\!</math> and is obtained using <math>(ABCD)(AD)={{A}^{2}}BC{{D}^{2}}=BC\,\!</math>. In general, a <math>{2}^{k-p}\,\!</math> fractional factorial design requires <math>p\,\!</math> independent generators. The defining relation for the design consists of the <math>p\,\!</math> independent generators and their <math>{2}^{p}\,\!</math> - (<math>p\,\!</math> +1) generalized interactions.

===Calculation of Aliases===

The alias structure for the present <math>{2}^{4-2}\,\!</math> design can be obtained using the defining relation of equation (I=ABCD=AD=BC) following the procedure explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]]. For example, multiplying the defining relation by <math>A\,\!</math> returns the effects aliased with the main effect, <math>A\,\!</math>, as follows:

::<math>\begin{align}
& A\cdot I= & A\cdot ABCD=A\cdot AD=A\cdot BC \\
& A= & {{A}^{2}}BCD={{A}^{2}}D=ABC \\
& A= & BCD=D=ABC
\end{align}\,\!</math>

Therefore, in the present <math>{2}^{4-2}\,\!</math> design, it is not possible to distinguish between effects <math>A\,\!</math>, <math>D\,\!</math>, <math>BCD\,\!</math> and <math>ABC\,\!</math>. Similarly, multiplying the defining relation by <math>B\,\!</math> and <math>AB\,\!</math> returns the effects that are aliased with these effects:

::<math>\begin{align}
& B= & ACD=ABD=C \\
& AB= & CD=AD=AC
\end{align}\,\!</math>

Other aliases can be obtained in a similar way. It can be seen that each effect in this design has three aliases. In general, each effect in a <math>{2}^{k-p}\,\!</math> design has <math>{2}^{p-1}\,\!</math> aliases.
The aliases for the <math>{2}^{4-2}\,\!</math> design show that in this design the main effects are aliased with each other (<math>A\,\!</math> is aliased with <math>D\,\!</math> and <math>B\,\!</math> is aliased with <math>C\,\!</math>). Therefore, this design is not a useful design and is not available in DOE++. It is important to ensure that main effects and lower order interactions of interest are not aliased in a fractional factorial design. This is known by looking at the resolution of the fractional factorial design.

==Design Resolution==

The resolution of a fractional factorial design is defined as the number of factors in the lowest order effect in the defining relation. For example, in the defining relation <math>I=ABCD=AD=BC\,\!</math> of the previous <math>{2}^{4-2}\,\!</math> design, the lowest-order effect is either <math>AD\,\!</math> or <math>BC,\,\!</math> containing two factors. Therefore, the resolution of this design is equal to two. The resolution of a fractional factorial design is represented using Roman numerals. For example, the previously mentioned <math>{2}^{4-2}\,\!</math> design with a resolution of two can be represented as 2 <math>_{\text{II}}^{4-2}\,\!</math>. The resolution provides information about the confounding in the design as explained next:
 
 
#'''Resolution III Designs''' In these designs, the lowest order effect in the defining relation has three factors (e.g., a <math>{2}^{5-2}\,\!</math> design with the defining relation <math>I=ABDE=ABC=CDE\,\!</math>). In resolution III designs, no main effects are aliased with any other main effects, but main effects are aliased with two factor interactions. In addition, some two factor interactions are aliased with each other.
#'''Resolution IV Designs''' In these designs, the lowest order effect in the defining relation has four factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABDE\,\!</math>). In resolution IV designs, no main effects are aliased with any other main effects or two factor interactions. However, some main effects are aliased with three factor interactions and the two factor interactions are aliased with each other.
#'''Resolution V Designs''' In these designs the lowest order effect in the defining relation has five factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABCDE\,\!</math>). In resolution V designs, no main effects or two factor interactions are aliased with any other main effects or two factor interactions. However, some main effects are aliased with four factor interactions and the two factor interactions are aliased with three factor interactions.

Fractional factorial designs with the highest resolution possible should be selected because the higher the resolution of the design, the less severe the degree of confounding. In general, designs with a resolution less than III are never used because in these designs some of the main effects are aliased with each other. The table below shows fractional factorial designs with the highest available resolution for three to ten factor designs along with their defining relations.

[[Image:doet7.3.png|center|474px|Highest resolution designs available for fractional factorial designs with 3 to 10 factors.]]

All of the two level fractional factorial designs available in DOE++ are shown next.

[[Image:doe7.38.png|center|471px|Two level fractional factorial designs available in DOE++ and their resolutions.]]

===Minimum Aberration Designs===
At times, different designs with the same resolution but different aliasing may be available. The best design to select in such a case is the minimum aberration design. For example, all <math>{2}^{7-2}\,\!</math> designs in the fourth table have a resolution of four (since the generator with the minimum number of factors in each design has four factors). Design <math>1\,\!</math> has three generators of length four (<math>ABCF,\,\!</math> <math>BCDG,\,\!</math> <math>ADFG\,\!</math>). Design <math>2\,\!</math> has two generators of length four (<math>ABCF,\,\!</math> <math>ADEG\,\!</math>). Design <math>3\,\!</math> has one generator of length four (<math>CEFG\,\!</math>). Therefore, design <math>3\,\!</math> has the least number of generators with the minimum length of four. Design <math>3\,\!</math> is called the minimum aberration design. It can be seen that the alias structure for design <math>3\,\!</math> is less involved compared to the other designs. For details refer to [[DOE_References|[Wu, 2000]]].

[[Image:doet7.4.png|center|432px|Three <math>2_{IV}^{7-2}\,\!</math> designs with different defining relations.]]

====Example====

The design of an automobile fuel cone is thought to be affected by six factors in the manufacturing process: cavity temperature (factor <math>A\,\!</math>), core temperature (factor <math>B\,\!</math>), melt temperature (factor <math>C\,\!</math>), hold pressure (factor <math>D\,\!</math>), injection speed (factor <math>E\,\!</math>) and cool time (factor <math>F\,\!</math>). The manufacturer of the fuel cone is unable to run the <math>{2}^{6}=64\,\!</math> runs required to complete one replicate for a two level full factorial experiment with six factors. Instead, they decide to run a fractional factorial design. Considering that three factor and higher order interactions are likely to be inactive, the manufacturer selects a <math>{2}^{6-2}\,\!</math> design that will require only 16 runs. The manufacturer chooses the resolution IV design which will ensure that all main effects are free from aliasing (assuming three factor and higher order interactions are absent). However, in this design the two factor interactions may be aliased with each other. It is decided that, if important two factor interactions are found to be present, additional experiment trials may be conducted to separate the aliased effects. The performance of the fuel cone is measured on a scale of 1 to 15. In DOE++, the design for this experiment is set up using the properties shown in the following figure. The Fraction Generators for the design, <math>E=ABC\,\!</math> and <math>F=BCD\,\!</math>, are the same as the defaults used in DOE++. The resulting <math>{2}^{6-2}\,\!</math> design and the corresponding response values are shown in the following two figures.

[[Image:doe7_39.png|center|644px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

[[Image:doe7_40.png|center|534px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The complete alias structure for the 2 <math>_{\text{IV}}^{6-2}\,\!</math> design is shown next.

<center><math>I=ABCE=ADEF=BCDF\,\!</math></center>

 
<center><math>\begin{align}
& A= & BCE=DEF=ABCDF \\
& B= & ACE=CDF=ABDEF \\
& C= & ABE=BDF=ACDEF \\
& D= & AEF=BCF=ABCDE \\
& E= & ABC=ADF=BCDEF \\
& F= & ADE=BCD=ABCEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& AB= & CE=ACDF=BDEF \\
& AC= & BE=ABDF=CDEF \\
& AD= & EF=ABCF=BCDE \\
& AE= & BC=DF=ABCDEF \\
& AF= & DE=ABCD=BCEF \\
& BD= & CF=ABEF=ACDE \\
& BF= & CD=ABDE=ACEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& ABD= & ACF=BEF=CDE \\
& ABF= & ACD=BDE=CEF
\end{align}\,\!</math></center>

In DOE++, the alias structure is displayed in the Design Summary and as part of the Design Evaluation result, as shown next:

[[Image:doe7_41.png|center|700px|Alias structure for the experiment design in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The normal probability plot of effects for this unreplicated design shows the main effects of factors <math>C\,\!</math> and <math>D\,\!</math> and the interaction effect, <math>BF\,\!</math>, to be significant (see the following figure).

[[Image:doe7_42.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

From the alias structure, it can be seen that for the present design interaction effect, <math>BF,\,\!</math> is confounded with <math>CD\,\!</math>. Therefore, the actual source of this effect cannot be known on the basis of the present experiment. However because neither factor <math>B\,\!</math> nor <math>F\,\!</math> is found to be significant there is an indication the observed effect is likely due to interaction, <math>CD\,\!</math>. To confirm this, a follow-up <math>{2}^{2}\,\!</math> experiment is run involving only factors <math>B\,\!</math> and <math>F\,\!</math>. The interaction, <math>BF\,\!</math>, is found to be inactive, leading to the conclusion that the interaction effect in the original experiment is effect, <math>CD\,\!</math>. Given these results, the fitted regression model for the fuel cone design as per the coefficients obtained from DOE++ is shown next.

::<math>\hat{y}=7.6875+C+2\cdot D+2.1875\cdot CD\,\!</math>

[[Image:doe7_43.png|center|700px|Effect coefficients for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

==Projection==

Projection refers to the reduction of a fractional factorial design to a full factorial design by dropping out some of the factors of the design. Any fractional factorial design of resolution, <math>R,\,\!</math> can be reduced to complete factorial designs in any subset of <math>R-1\,\!</math> factors. For example, consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. The resolution of this design is four. Therefore, this design can be reduced to full factorial designs in any three (<math>4-1=3\,\!</math>) of the original seven factors (by dropping the remaining four of factors). Further, a fractional factorial design can also be reduced to a full factorial design in any <math>R\,\!</math> of the original factors, as long as these <math>R\,\!</math> factors are not part of the generator in the defining relation. Again consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. This design can be reduced to a full factorial design in four factors provided these four factors do not appear together as a generator in the defining relation. The complete defining relation for this design is:

::<math>\begin{align}
& I= & ABCE=BCDF=ACDG=ADEF=ABFG=BDEG=CEFG
\end{align}\,\!</math>

Therefore, there are seven four factor combinations out of the 35 (<math>(_{7}^{4})=35\,\!</math>) possible four-factor combinations that are used as generators in the defining relation. The designs with the remaining 28 four factor combinations would be full factorial 16-run designs. For example, factors <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math> do not occur as a generator in the defining relation of the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. If the remaining factors, <math>E\,\!</math>, <math>F\,\!</math> and <math>G\,\!</math>, are dropped, the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design will reduce to a full factorial design in <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math>.

==Resolution III Designs==

At times, the factors to be investigated in screening experiments are so large that even running a fractional factorial design is impractical. This can be partially solved by using resolution III fractional factorial designs in the cases where three factor and higher order interactions can be assumed to be unimportant. Resolution III designs, such as the 2 <math>_{\text{III}}^{3-1}\,\!</math> design, can be used to estimate <math>k\,\!</math> main effects using just <math>k+1\,\!</math> runs. In these designs, the main effects are aliased with two factor interactions. Once the results from these designs are obtained, and knowing that three factor and higher order interactions are unimportant, the experimenter can decide if there is a need to run a fold-over design to de-alias the main effects from the two factor interactions. Thus, the 2 <math>_{\text{III}}^{3-1}\,\!</math> design can be used to investigate three factors in four runs, the 2 <math>_{\text{III}}^{7-4}\,\!</math> design can be used to investigate seven factors in eight runs, the 2 <math>_{\text{III}}^{15-11}\,\!</math> design can be used to investigate fifteen factors in sixteen runs and so on.

====Example====

{{:Resolution_III_Design_Example}}

==Alias Matrix==
In [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction designs]] and [[Two_Level_Factorial_Experiments#Quarter_and_Smaller_Fraction_Designs| Quarter and Smaller Fraction Designs]], the alias structure for fractional factorial designs was obtained using the defining relation. However, this method of obtaining the alias structure is not very efficient when the alias structure is very complex or when partial aliasing is involved. One of the ways to obtain the alias structure for any design, regardless of its complexity, is to use the alias matrix. The alias matrix for a design is calculated using <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> where <math>{{X}_{1}}\,\!</math> is the portion of the design matrix, <math>X,\,\!</math> that contains the effects for which the aliases need to be calculated, and <math>{{X}_{2}}\,\!</math> contains the remaining columns of the design matrix, other than those included in <math>{{X}_{1}}\,\!</math>.

To illustrate the use of the alias matrix, consider the design matrix for the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design (using the defining relation <math>I=ABCD\,\!</math>) shown next:

[[Image:Chapter7__879.png|center|link=]]

The alias structure for this design can be obtained by defining <math>{{X}_{1}}\,\!</math> using eight columns since the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design estimates eight effects. If the first eight columns of <math>X\,\!</math> are used then <math>{{X}_{1}}\,\!</math> is:

[[Image:Chapter7__884.png|center|link=]]

<math>{{X}_{2}}\,\!</math> is obtained using the remaining columns as:

[[Image:Chapter7__886.png|center|link=]]

Then the alias matrix <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> is:

[[Image:Chapter7__888.png|center|link=]]

The alias relations can be easily obtained by observing the alias matrix as:

<center><math>\begin{align}
& I= & ABCD \\
& A= & BCD \\
& B= & ACD \\
& AB= & CD \\
& C= & ABD \\
& AC= & BD \\
& BC= & AD \\
& D= & ABC
\end{align}\,\!</math></center>

Two Level Factorial Experiments

2017-08-10T17:31:50Z

Richard House: /* Normal Probability Plot of Effects */

{{Template:Doebook|8}}
Two level factorial experiments are factorial experiments in which each factor is investigated at only two levels. The early stages of experimentation usually involve the investigation of a large number of potential factors to discover the "vital few" factors. Two level factorial experiments are used during these stages to quickly filter out unwanted effects so that attention can then be focused on the important ones.

==2''k'' Designs==
The factorial experiments, where all combination of the levels of the factors are run, are usually referred to as ''full factorial experiments''. Full factorial two level experiments are also referred to as <math>{2}^{k}\,\!</math> designs where <math>k\,\!</math> denotes the number of factors being investigated in the experiment. In Weibull++ DOE folios, these designs are referred to as 2 Level Factorial Designs as shown in the figure below.

[[Image:doe7_1.png|center|487px|Selection of full factorial experiments with two levels in Weibull++.|link=]]

A full factorial two level design with <math>k\,\!</math> factors requires <math>{{2}^{k}}\,\!</math> runs for a single replicate. For example, a two level experiment with three factors will require <math>2\times 2\times 2={{2}^{3}}=8\,\!</math> runs. The choice of the two levels of factors used in two level experiments depends on the factor; some factors naturally have two levels. For example, if gender is a factor, then male and female are the two levels. For other factors, the limits of the range of interest are usually used. For example, if temperature is a factor that varies from <math>{45}^{o}C\,\!</math> to <math>{90}^{o}C\,\!</math>, then the two levels used in the <math>{2}^{k}\,\!</math> design for this factor would be <math>{45}^{o}\,\!C\,\!</math> and <math>{90}^{o}\,\!C\,\!</math>.

The two levels of the factor in the <math>{2}^{k}\,\!</math> design are usually represented as <math>-1\,\!</math> (for the first level) and <math>1\,\!</math> (for the second level). Note that this representation is reversed from the coding used in [[General Full Factorial Designs]] for the indicator variables that represent two level factors in ANOVA models. For ANOVA models, the first level of the factor was represented using a value of <math>1\,\!</math> for the indicator variable, while the second level was represented using a value of <math>-1\,\!</math>. For details on the notation used for two level experiments refer to [[Two_Level_Factorial_Experiments#Notation| Notation]].

===The 22 Design===

The simplest of the two level factorial experiments is the <math>{2}^{2}\,\!</math> design where two factors (say factor <math>A\,\!</math> and factor <math>B\,\!</math>) are investigated at two levels. A single replicate of this design will require four runs (<math>{{2}^{2}}=2\times 2=4\,\!</math>) The effects investigated by this design are the two main effects, <math>A\,\!</math> and <math>B,\,\!</math> and the interaction effect <math>AB\,\!</math>. The treatments for this design are shown in figure (a) below. In figure (a), letters are used to represent the treatments. The presence of a letter indicates the high level of the corresponding factor and the absence indicates the low level. For example, (1) represents the treatment combination where all factors involved are at the low level or the level represented by <math>-1\,\!</math> ; <math>a\,\!</math> represents the treatment combination where factor <math>A\,\!</math> is at the high level or the level of <math>1\,\!</math>, while the remaining factors (in this case, factor <math>B\,\!</math>) are at the low level or the level of <math>-1\,\!</math>. Similarly, <math>b\,\!</math> represents the treatment combination where factor <math>B\,\!</math> is at the high level or the level of <math>1\,\!</math>, while factor <math>A\,\!</math> is at the low level and <math>ab\,\!</math> represents the treatment combination where factors <math>A\,\!</math> and <math>B\,\!</math> are at the high level or the level of the 1. Figure (b) below shows the design matrix for the <math>{2}^{2}\,\!</math> design. It can be noted that the sum of the terms resulting from the product of any two columns of the design matrix is zero. As a result the <math>{2}^{2}\,\!</math> design is an ''orthogonal design''. In fact, all <math>{2}^{k}\,\!</math> designs are orthogonal designs. This property of the <math>{2}^{k}\,\!</math> designs offers a great advantage in the analysis because of the simplifications that result from orthogonality. These simplifications are explained later on in this chapter.
The <math>{2}^{2}\,\!</math> design can also be represented geometrically using a square with the four treatment combinations lying at the four corners, as shown in figure (c) below.

[[Image:doe7.2.png|center|400px|The <math>2^2\,\!</math> design. Figure (a) displays the experiment design, (b) displays the design matrix and (c) displays the geometric representation for the design. In Figure (b), the column names I, A, B and AB are used. Column I represents the intercept term. Columns A and B represent the respective factor settings. Column AB represents the interaction and is the product of columns A and B.]]
 

===The 23 Design===

The <math>{2}^{3}\,\!</math> design is a two level factorial experiment design with three factors (say factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>). This design tests three (<math>k=3\,\!</math>) main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math> ; three (<math>(_{2}^{k})=\,\!</math> <math>(_{2}^{3})=3\,\!</math>) two factor interaction effects, <math>AB\,\!</math>, <math>BC\,\!</math>, <math>AC\,\!</math> ; and one (<math>(_{3}^{k})=\,\!</math> <math>(_{3}^{3})=1\,\!</math>) three factor interaction effect, <math>ABC\,\!</math>. The design requires eight runs per replicate. The eight treatment combinations corresponding to these runs are <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math>, <math>ab\,\!</math>, <math>c\,\!</math>, <math>ac\,\!</math>, <math>bc\,\!</math> and <math>abc\,\!</math>. Note that the treatment combinations are written in such an order that factors are introduced one by one with each new factor being combined with the preceding terms. This order of writing the treatments is called the ''standard order'' or ''Yates' order''. The <math>{2}^{3}\,\!</math> design is shown in figure (a) below. The design matrix for the <math>{2}^{3}\,\!</math> design is shown in figure (b). The design matrix can be constructed by following the standard order for the treatment combinations to obtain the columns for the main effects and then multiplying the main effects columns to obtain the interaction columns.

[[Image:doe7.3.png|center|324px|The <math>2^3\,\!</math> design. Figure (a) shows the experiment design and (b) shows the design matrix.]]

[[Image:doe7.4.png|center|290px|Geometric representation of the <math>2^3\,\!</math> design.]]

The <math>{2}^{3}\,\!</math> design can also be represented geometrically using a cube with the eight treatment combinations lying at the eight corners as shown in the figure above.

==Analysis of 2''k'' Designs==

The <math>{2}^{k}\,\!</math> designs are a special category of the factorial experiments where all the factors are at two levels. The fact that these designs contain factors at only two levels and are orthogonal greatly simplifies their analysis even when the number of factors is large. The use of <math>{2}^{k}\,\!</math> designs in investigating a large number of factors calls for a revision of the notation used previously for the ANOVA models. The case for revised notation is made stronger by the fact that the ANOVA and multiple linear regression models are identical for <math>{2}^{k}\,\!</math> designs because all factors are only at two levels. Therefore, the notation of the regression models is applied to the ANOVA models for these designs, as explained next.

===Notation===

Based on the notation used in [[General Full Factorial Designs]], the ANOVA model for a two level factorial experiment with three factors would be as follows:

::<math>\begin{align}
& Y= & \mu +{{\tau }_{1}}\cdot {{x}_{1}}+{{\delta }_{1}}\cdot {{x}_{2}}+{{(\tau \delta )}_{11}}\cdot {{x}_{1}}{{x}_{2}}+{{\gamma }_{1}}\cdot {{x}_{3}} \\
& & +{{(\tau \gamma )}_{11}}\cdot {{x}_{1}}{{x}_{3}}+{{(\delta \gamma )}_{11}}\cdot {{x}_{2}}{{x}_{3}}+{{(\tau \delta \gamma )}_{111}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where:
 
:• <math>\mu \,\!</math> represents the overall mean
:• <math>{{\tau }_{1}}\,\!</math> represents the independent effect of the first factor (factor <math>A\,\!</math>) out of the two effects <math>{{\tau }_{1}}\,\!</math> and <math>{{\tau }_{2}}\,\!</math>
:• <math>{{\delta }_{1}}\,\!</math> represents the independent effect of the second factor (factor <math>B\,\!</math>) out of the two effects <math>{{\delta }_{1}}\,\!</math> and <math>{{\delta }_{2}}\,\!</math>
:• <math>{{(\tau \delta )}_{11}}\,\!</math> represents the independent effect of the interaction <math>AB\,\!</math> out of the other interaction effects
:• <math>{{\gamma }_{1}}\,\!</math> represents the effect of the third factor (factor <math>C\,\!</math>) out of the two effects <math>{{\gamma }_{1}}\,\!</math> and <math>{{\gamma }_{2}}\,\!</math>
:• <math>{{(\tau \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>AC\,\!</math> out of the other interaction effects
:• <math>{{(\delta \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>BC\,\!</math> out of the other interaction effects
:• <math>{{(\tau \delta \gamma )}_{111}}\,\!</math> represents the effect of the interaction <math>ABC\,\!</math> out of the other interaction effects
and <math>\epsilon \,\!</math> is the random error term.

 
The notation for a linear regression model having three predictor variables with interactions is:

::<math>\begin{align}
& Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& & +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

The notation for the regression model is much more convenient, especially for the case when a large number of higher order interactions are present. In two level experiments, the ANOVA model requires only one indicator variable to represent each factor for both qualitative and quantitative factors. Therefore, the notation for the multiple linear regression model can be applied to the ANOVA model of the experiment that has all the factors at two levels. For example, for the experiment of the ANOVA model given above, <math>{{\beta }_{0}}\,\!</math> can represent the overall mean instead of <math>\mu \,\!</math>, and <math>{{\beta }_{1}}\,\!</math> can represent the independent effect, <math>{{\tau }_{1}}\,\!</math>, of factor <math>A\,\!</math>. Other main effects can be represented in a similar manner. The notation for the interaction effects is much more simplified (e.g., <math>{{\beta }_{123}}\,\!</math> can be used to represent the three factor interaction effect, <math>{{(\tau \beta \gamma )}_{111}}\,\!</math>).

As mentioned earlier, it is important to note that the coding for the indicator variables for the ANOVA models of two level factorial experiments is reversed from the coding followed in [[General Full Factorial Designs]]. Here <math>-1\,\!</math> represents the first level of the factor while <math>1\,\!</math> represents the second level. This is because for a two level factor a single variable is needed to represent the factor for both qualitative and quantitative factors. For quantitative factors, using <math>-1\,\!</math> for the first level (which is the low level) and 1 for the second level (which is the high level) keeps the coding consistent with the numerical value of the factors. The change in coding between the two coding schemes does not affect the analysis except that signs of the estimated effect coefficients will be reversed (i.e., numerical values of <math>{{\hat{\tau }}_{1}}\,\!</math>, obtained based on the coding of [[General Full Factorial Designs]], and <math>{{\hat{\beta }}_{1}}\,\!</math>, obtained based on the new coding, will be the same but their signs would be opposite).

::<math>\begin{align}
& & \text{Factor }A\text{ Coding (two level factor)} \\
& &
\end{align}\,\!</math>

::<math>\begin{matrix}
\text{Previous Coding} & {} & {} & {} & \text{Coding for }{{\text{2}}^{k}}\text{ Designs} \\
{} & {} & {} & {} & {} \\
Effect\text{ }{{\tau }_{1}}\ \ :\ \ {{x}_{1}}=1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{1}}\text{ (or }-{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=-1\text{ } \\
Effect\text{ }{{\tau }_{2}}\ \ :\ \ {{x}_{1}}=-1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{2}}\text{ (or }{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=1\text{ } \\
\end{matrix}\,\!</math>

In summary, the ANOVA model for the experiments with all factors at two levels is different from the ANOVA models for other experiments in terms of the notation in the following two ways:
 

:• The notation of the regression models is used for the effect coefficients.
:• The coding of the indicator variables is reversed.

===Special Features===

Consider the design matrix, <math>X\,\!</math>, for the <math>{2}^{3}\,\!</math> design discussed above. The (<math>{{X}^{\prime }}X\,\!</math>) <math>^{-1}\,\!</math> matrix is:

<center><math>{{({{X}^{\prime }}X)}^{-1}}=\left[ \begin{matrix}
0.125 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0.125 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0.125 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0.125 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0.125 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0.125 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0.125 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.125 \\
\end{matrix} \right]\,\!</math></center>

Notice that, due to the orthogonal design of the <math>X\,\!</math> matrix, the <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> has been simplified to a diagonal matrix which can be written as:

::<math>\begin{align}
{{({{X}^{\prime }}X)}^{-1}}= & 0.125\cdot I = & \frac{1}{8}\cdot I = & \frac{1}{{{2}^{3}}}\cdot I
\end{align}\,\!</math>

where <math>I\,\!</math> represents the identity matrix of the same order as the design matrix, <math>X\,\!</math>. Since there are eight observations per replicate of the <math>{2}^{3}\,\!</math> design, the <math>(X\,\!</math> ' <math>X{{)}^{-1}}\,\!</math> matrix for <math>m\,\!</math> replicates of this design can be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{3}}\cdot m)}\cdot I\,\!</math>

The <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> matrix for any <math>{2}^{k}\,\!</math> design can now be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{k}}\cdot m)}\cdot I\,\!</math>

Then the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is:

::<math>\begin{align}
C= & {{{\hat{\sigma }}}^{2}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & M{{S}_{E}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & \frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}\cdot I
\end{align}\,\!</math>

Note that the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is also a diagonal matrix. Therefore, the estimated effect coefficients (<math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math>, <math>{{\beta }_{12}},\,\!</math> etc.) for these designs are uncorrelated. This implies that the terms in the <math>{2}^{k}\,\!</math> design (main effects, interactions) are independent of each other. Consequently, the extra sum of squares for each of the terms in these designs is independent of the sequence of terms in the model, and also independent of the presence of other terms in the model. As a result the sequential and partial sum of squares for the terms are identical for these designs and will always add up to the model sum of squares. Multicollinearity is also not an issue for these designs.

It can also be noted from the equation given above, that in addition to the <math>C\,\!</math> matrix being diagonal, all diagonal elements of the <math>C\,\!</math> matrix are identical. This means that the variance (or its square root, the standard error) of all estimated effect coefficients are the same. The standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>, for all the coefficients is:

::<math>\begin{align}
se({{{\hat{\beta }}}_{j}})= & \sqrt{{{C}_{jj}}} = & \sqrt{\frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}}\text{ }for\text{ }all\text{ }j
\end{align}\,\!</math>

This property is used to construct the normal probability plot of effects in <math>{2}^{k}\,\!</math> designs and identify significant effects using graphical techniques. For details on the normal probability plot of effects in a Weibull++ DOE folio, refer to [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]].

====Example====
To illustrate the analysis of a full factorial <math>{2}^{k}\,\!</math> design, consider a three factor experiment to investigate the effect of honing pressure, number of strokes and cycle time on the surface finish of automobile brake drums. Each of these factors is investigated at two levels. The honing pressure is investigated at levels of 200 <math>psi\,\!</math> and 400 <math>psi\,\!</math>, the number of strokes used is 3 and 5 and the two levels of the cycle time are 3 and 5 seconds. The design for this experiment is set up in a Weibull++ DOE folio as shown in the first two following figures. It is decided to run two replicates for this experiment. The surface finish data collected from each run (using randomization) and the complete design is shown in the third following figure. The analysis of the experiment data is explained next.

[[Image:doe7_5.png|center|877px|Design properties for the experiment in the example.|link=]]

[[Image:doe7_6.png|center|859px|Design summary for the experiment in the example.|link=]]

[[Image:doe7_7.png|center|778px|Experiment design for the example to investigate the surface finish of automobile brake drums.|link=]]

The applicable model using the notation for <math>{2}^{k}\,\!</math> designs is:

::<math>\begin{align}
Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where the indicator variable, <math>{{x}_{1,}}\,\!</math> represents factor <math>A\,\!</math> (honing pressure), <math>{{x}_{1}}=-1\,\!</math> represents the low level of 200 <math>psi\,\!</math> and <math>{{x}_{1}}=1\,\!</math> represents the high level of 400 <math>psi\,\!</math>. Similarly, <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> represent factors <math>B\,\!</math> (number of strokes) and <math>C\,\!</math> (cycle time), respectively. <math>{{\beta }_{0}}\,\!</math> is the overall mean, while <math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math> and <math>{{\beta }_{3}}\,\!</math> are the effect coefficients for the main effects of factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. <math>{{\beta }_{12}}\,\!</math>, <math>{{\beta }_{13}}\,\!</math> and <math>{{\beta }_{23}}\,\!</math> are the effect coefficients for the <math>AB\,\!</math>, <math>AC\,\!</math> and <math>BC\,\!</math> interactions, while <math>{{\beta }_{123}}\,\!</math> represents the <math>ABC\,\!</math> interaction.

 
If the subscripts for the run (<math>i\,\!</math> ; <math>i=\,\!</math> 1 to 8) and replicates (<math>j\,\!</math> ; <math>j=\,\!</math> 1,2) are included, then the model can be written as:

::<math>\begin{align}
{{Y}_{ij}}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{ij1}}+{{\beta }_{2}}\cdot {{x}_{ij2}}+{{\beta }_{12}}\cdot {{x}_{ij1}}{{x}_{ij2}}+{{\beta }_{3}}\cdot {{x}_{ij3}} \\
& +{{\beta }_{13}}\cdot {{x}_{ij1}}{{x}_{ij3}}+{{\beta }_{23}}\cdot {{x}_{ij2}}{{x}_{ij3}}+{{\beta }_{123}}\cdot {{x}_{ij1}}{{x}_{ij2}}{{x}_{ij3}}+{{\epsilon }_{ij}}
\end{align}\,\!</math>

To investigate how the given factors affect the response, the following hypothesis tests need to be carried:

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{1}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{1}}\ne 0\,\!</math>

This test investigates the main effect of factor <math>A\,\!</math> (honing pressure). The statistic for this test is:

::<math>{{({{F}_{0}})}_{A}}=\frac{M{{S}_{A}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{A}}\,\!</math> is the mean square for factor <math>A\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other main effects, <math>B\,\!</math> and <math>C\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{12}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{12}}\ne 0\,\!</math>

This test investigates the two factor interaction <math>AB\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{AB}}=\frac{M{{S}_{AB}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the interaction <math>AB\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other two factor interactions, <math>AC\,\!</math> and <math>BC\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{123}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{123}}\ne 0\,\!</math>

This test investigates the three factor interaction <math>ABC\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{ABC}}=\frac{M{{S}_{ABC}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{ABC}}\,\!</math> is the mean square for the interaction <math>ABC\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square.
To calculate the test statistics, it is convenient to express the ANOVA model in the form <math>y=X\beta +\epsilon \,\!</math>.

====Expression of the ANOVA Model as <math>y=X\beta +\epsilon \,\!</math>====

In matrix notation, the ANOVA model can be expressed as:

::<math>y=X\beta +\epsilon \,\!</math>

where:

<center><math>y=\left[ \begin{matrix}
{{Y}_{11}} \\
{{Y}_{21}} \\
. \\
{{Y}_{81}} \\
{{Y}_{12}} \\
. \\
{{Y}_{82}} \\
\end{matrix} \right]=\left[ \begin{matrix}
90 \\
90 \\
. \\
90 \\
86 \\
. \\
80 \\
\end{matrix} \right]\text{ }X=\left[ \begin{matrix}
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\end{matrix} \right]\,\!</math></center>

<center><math>\beta =\left[ \begin{matrix}
{{\beta }_{0}} \\
{{\beta }_{1}} \\
{{\beta }_{2}} \\
{{\beta }_{12}} \\
{{\beta }_{3}} \\
{{\beta }_{13}} \\
{{\beta }_{23}} \\
{{\beta }_{123}} \\
\end{matrix} \right]\text{ }\epsilon =\left[ \begin{matrix}
{{\epsilon }_{11}} \\
{{\epsilon }_{21}} \\
. \\
{{\epsilon }_{81}} \\
{{\epsilon }_{12}} \\
. \\
. \\
{{\epsilon }_{82}} \\
\end{matrix} \right]\,\!</math></center>

====Calculation of the Extra Sum of Squares for the Factors====

Knowing the matrices <math>y\,\!</math>, <math>X\,\!</math> and <math>\beta \,\!</math>, the extra sum of squares for the factors can be calculated. These are used to calculate the mean squares that are used to obtain the test statistics. Since the experiment design is orthogonal, the partial and sequential extra sum of squares are identical. The extra sum of squares for each effect can be calculated as shown next. As an example, the extra sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & Model\text{ }Sum\text{ }of\text{ }Squares - Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }main\text{ }effect\text{ }of\text{ }A \\
= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }A}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }A}}={{X}_{\tilde{\ }A}}{{(X_{\tilde{\ }A}^{\prime }{{X}_{\tilde{\ }A}})}^{-1}}X_{\tilde{\ }A}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }A}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the main effect of factor <math>A\,\!</math>. Thus, the sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y \\
= & 654.4375-549.375 \\
= & 105.0625
\end{align}\,\!</math>

Similarly, the extra sum of squares for the interaction effect <math>AB\,\!</math> is:

::<math>\begin{align}
S{{S}_{AB}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }AB}}-(1/16)J]y \\
= & 654.4375-636.375 \\
= & 18.0625
\end{align}\,\!</math>

The extra sum of squares for other effects can be obtained in a similar manner.

====Calculation of the Test Statistics====

Knowing the extra sum of squares, the test statistic for the effects can be calculated. For example, the test statistic for the interaction <math>AB\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{AB}}= & \frac{M{{S}_{AB}}}{M{{S}_{E}}} \\
= & \frac{S{{S}_{AB}}/dof(S{{S}_{AB}})}{S{{S}_{E}}/dof(S{{S}_{E}})} \\
= & \frac{18.0625/1}{147.5/8} \\
= & 0.9797
\end{align}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the <math>AB\,\!</math> interaction and <math>M{{S}_{E}}\,\!</math> is the error mean square. The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{AB}}=0.9797\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{AB}}) \\
= & 1-0.6487 \\
= & 0.3513
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that the interaction between honing pressure and number of strokes does not affect the surface finish of the brake drums. Tests for other effects can be carried out in a similar manner. The results are shown in the ANOVA Table in the following figure. The values S, R-sq and R-sq(adj) in the figure indicate how well the model fits the data. The value of S represents the standard error of the model, R-sq represents the coefficient of multiple determination and R-sq(adj) represents the adjusted coefficient of multiple determination. For details on these values refer to [[Multiple_Linear_Regression_Analysis|Multiple Linear Regression Analysis]].

[[Image:doe7_8.png|center|799px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

====Calculation of Effect Coefficients====

The estimate of effect coefficients can also be obtained:

<center><math>\begin{align}
\hat{\beta }= & {{({{X}^{\prime }}X)}^{-1}}{{X}^{\prime }}y \\
= & \left[ \begin{matrix}
86.4375 \\
2.5625 \\
-4.9375 \\
1.0625 \\
-1.0625 \\
2.4375 \\
-1.3125 \\
-0.1875 \\
\end{matrix} \right]
\end{align}\,\!</math></center>

[[Image:doe7_9.png|center|800px|Regression Information table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

The coefficients and related results are shown in the Regression Information table above. In the table, the Effect column displays the effects, which are simply twice the coefficients. The Standard Error column displays the standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>. The Low CI and High CI columns display the confidence interval on the coefficients. The interval shown is the 90% interval as the significance is chosen as 0.1. The T Value column displays the <math>t\,\!</math> statistic, <math>{{t}_{0}}\,\!</math>, corresponding to the coefficients. The P Value column displays the <math>p\,\!</math> value corresponding to the <math>t\,\!</math> statistic. (For details on how these results are calculated, refer to [[General Full Factorial Designs]]). Plots of residuals can also be obtained from the DOE folio to ensure that the assumptions related to the ANOVA model are not violated.

====Model Equation====

From the analysis results in the above figure within [[Two_Level_Factorial_Experiments#Calculation_of_Effect_Coefficients|calculation of effect coefficients]] section, it is seen that effects <math>A\,\!</math>, <math>B\,\!</math> and <math>AC\,\!</math> are significant. In a DOE folio, the <math>p\,\!</math> values for the significant effects are displayed in red in the ANOVA Table for easy identification. Using the values of the estimated effect coefficients, the model for the present <math>{2}^{3}\,\!</math> design in terms of the coded values can be written as:

::<math>\begin{align}
\hat{y}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}} \\
= & 86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+2.4375{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

To make the model hierarchical, the main effect, <math>C\,\!</math>, needs to be included in the model (because the interaction <math>AC\,\!</math> is included in the model). The resulting model is:

::<math>\hat{y}=86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+1.0625{{x}_{3}}+2.4375{{x}_{1}}{{x}_{3}}\,\!</math>

This equation can be viewed in a DOE folio, as shown in the following figure, using the Show Analysis Summary icon in the Control Panel. The equation shown in the figure will match the hierarchical model once the required terms are selected using the Select Effects icon.

[[Image:doe7_10.png|center|557px|The model equation for the experiment of the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

==Replicated and Repeated Runs==

In the case of replicated experiments, it is important to note the difference between replicated runs and repeated runs. Both repeated and replicated runs are multiple response readings taken at the same factor levels. However, repeated runs are response observations taken at the same time or in succession. Replicated runs are response observations recorded in a random order. Therefore, replicated runs include more variation than repeated runs. For example, a baker, who wants to investigate the effect of two factors on the quality of cakes, will have to bake four cakes to complete one replicate of a <math>{2}^{2}\,\!</math> design. Assume that the baker bakes eight cakes in all. If, for each of the four treatments of the <math>{2}^{2}\,\!</math> design, the baker selects one treatment at random and then bakes two cakes for this treatment at the same time then this is a case of two repeated runs. If, however, the baker bakes all the eight cakes randomly, then the eight cakes represent two sets of replicated runs.
For repeated measurements, the average values of the response for each treatment should be entered into a DOE folio as shown in the following figure (a) when the two cakes for a particular treatment are baked together. For replicated measurements, when all the cakes are baked randomly, the data is entered as shown in the following figure (b).

[[Image:doe7.11.png|center|300px|Data entry for repeated and replicated runs. Figure (a) shows repeated runs and (b) shows replicated runs.]]

==Unreplicated 2''k'' Designs==

If a factorial experiment is run only for a single replicate then it is not possible to test hypotheses about the main effects and interactions as the error sum of squares cannot be obtained. This is because the number of observations in a single replicate equals the number of terms in the ANOVA model. Hence the model fits the data perfectly and no degrees of freedom are available to obtain the error sum of squares.

However, sometimes it is only possible to run a single replicate of the <math>{2}^{k}\,\!</math> design because of constraints on resources and time. In the absence of the error sum of squares, hypothesis tests to identify significant factors cannot be conducted. A number of methods of analyzing information obtained from unreplicated <math>{2}^{k}\,\!</math> designs are available. These include pooling higher order interactions, using the normal probability plot of effects or including center point replicates in the design.

===Pooling Higher Order Interactions===

One of the ways to deal with unreplicated <math>{2}^{k}\,\!</math> designs is to use the sum of squares of some of the higher order interactions as the error sum of squares provided these higher order interactions can be assumed to be insignificant. By dropping some of the higher order interactions from the model, the degrees of freedom corresponding to these interactions can be used to estimate the error mean square. Once the error mean square is known, the test statistics to conduct hypothesis tests on the factors can be calculated.

===Normal Probability Plot of Effects===
Another way to use unreplicated <math>{2}^{k}\,\!</math> designs to identify significant effects is to construct the normal probability plot of the effects. As mentioned in [[Two_Level_Factorial_Experiments#Special_Features| Special Features]], the standard error for all effect coefficients in the <math>{2}^{k}\,\!</math> designs is the same. Therefore, on a normal probability plot of effect coefficients, all non-significant effect coefficients (with <math>\beta =0\,\!</math>) will fall along the straight line representative of the normal distribution, N(<math>0,{{\sigma }^{2}}/({{2}^{k}}\cdot m)\,\!</math>). Effect coefficients that show large deviations from this line will be significant since they do not come from this normal distribution. Similarly, since effects <math>=2\times \,\!</math> effect coefficients, all non-significant effects will also follow a straight line on the normal probability plot of effects. For replicated designs, the Effects Probability plot of a DOE folio plots the normalized effect values (or the T Values) on the standard normal probability line, N(0,1). However, in the case of unreplicated <math>{2}^{k}\,\!</math> designs, <math>{{\sigma }^{2}}\,\!</math> remains unknown since <math>M{{S}_{E}}\,\!</math> cannot be obtained. Lenth's method is used in this case to estimate the variance of the effects. For details on Lenth's method, please refer to [[DOE References| Montgomery (2001)]]. The DOE folio then uses this variance value to plot effects along the N(0, Lenth's effect variance) line. The
method is illustrated in the following example.

====Example====

Vinyl panels, used as instrument panels in a certain automobile, are seen to develop defects after a certain amount of time. To investigate the issue, it is decided to carry out a two level factorial experiment. Potential factors to be investigated in the experiment are vacuum rate (factor <math>A\,\!</math>), material temperature (factor <math>B\,\!</math>), element intensity (factor <math>C\,\!</math>) and pre-stretch (factor <math>D\,\!</math>). The two levels of the factors used in the experiment are as shown in below.

[[Image:doet7.1.png|center|300px|Factors to investigate defects in vinyl panels.]]

With a <math>{2}^{4}\,\!</math> design requiring 16 runs per replicate it is only feasible for the manufacturer to run a single replicate.

The experiment design and data, collected as percent defects, are shown in the following figure. Since the present experiment design contains only a single replicate, it is not possible to obtain an estimate of the error sum of squares, <math>S{{S}_{E}}\,\!</math>. It is decided to use the normal probability plot of effects to identify the significant effects. The effect values for each term are obtained as shown in the following figure.

[[Image:doe7_13.png|center|650px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

Lenth's method uses these values to estimate the variance. As described in [[DOE_References|[Lenth, 1989]]], if all effects are arranged in ascending order, using their absolute values, then <math>{{s}_{0}}\,\!</math> is defined as 1.5 times the median value:

::<math>\begin{align}
{{s}_{0}}= & 1.5\cdot median(\left| effect \right|) \\
= & 1.5\cdot 2 \\
= & 3
\end{align}\,\!</math>

Using <math>{{s}_{0}}\,\!</math>, the "pseudo standard error" (<math>PSE\,\!</math>) is calculated as 1.5 times the median value of all effects that are less than 2.5 <math>{{s}_{0}}\,\!</math> :

::<math>\begin{align}
PSE= & 1.5\cdot median(\left| effect \right|\ \ :\ \ \left| effect \right|<2.5{{s}_{0}}) \\
= & 1.5\cdot 1.5 \\
= & 2.25
\end{align}\,\!</math>

Using <math>PSE\,\!</math> as an estimate of the effect variance, the effect variance is 2.25. Knowing the effect variance, the normal probability plot of effects for the present unreplicated experiment can be constructed as shown in the following figure. The line on this plot is the line N(0, 2.25). The plot shows that the effects <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math> do not follow the distribution represented by this line. Therefore, these effects are significant.

The significant effects can also be identified by comparing individual effect values to the margin of error or the threshold value using the pareto chart (see the third following figure). If the required significance is 0.1, then:

::<math>margin\text{ }of\text{ }error={{t}_{\alpha /2,d}}\cdot PSE\,\!</math>

The <math>t\,\!</math> statistic, <math>{{t}_{\alpha /2,d}}\,\!</math>, is calculated at a significance of <math>\alpha /2\,\!</math> (for the two-sided hypothesis) and degrees of freedom <math>d=(\,\!</math> number of effects <math>)/3\,\!</math>. Thus:

::<math>\begin{align}
margin\text{ }of\text{ }error= & {{t}_{0.05,5}}\cdot PSE \\
= & 2.015\cdot 2.25 \\
= & 4.534
\end{align}\,\!</math>

The value of 4.534 is shown as the critical value line in the third following figure. All effects with absolute values greater than the margin of error can be considered to be significant. These effects are <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math>. Therefore, the vacuum rate, the pre-stretch and their interaction have a significant effect on the defects of the vinyl panels.

[[Image:doe7_14.png|center|800px|Effect values for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_15.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_16.png|center|650px|Pareto chart for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

===Center Point Replicates===

Another method of dealing with unreplicated <math>{2}^{k}\,\!</math> designs that only have quantitative factors is to use replicated runs at the center point. The center point is the response corresponding to the treatment exactly midway between the two levels of all factors. Running multiple replicates at this point provides an estimate of pure error. Although running multiple replicates at any treatment level can provide an estimate of pure error, the other advantage of running center point replicates in the <math>{2}^{k}\,\!</math> design is in checking for the presence of curvature. The test for curvature investigates whether the model between the response and the factors is linear and is discussed in [[Two_Level_Factorial_Experiments#Using_Center_Point_Replicates_to_Test_Curvature| Center Pt. Replicates to Test Curvature]].

====Example: Use Center Point to Get Pure Error====

Consider a <math>{2}^{2}\,\!</math> experiment design to investigate the effect of two factors, <math>A\,\!</math> and <math>B\,\!</math>, on a certain response. The energy consumed when the treatments of the <math>{2}^{2}\,\!</math> design are run is considerably larger than the energy consumed for the center point run (because at the center point the factors are at their middle levels). Therefore, the analyst decides to run only a single replicate of the design and augment the design by five replicated runs at the center point as shown in the following figure. The design properties for this experiment are shown in the second following figure. The complete experiment design is shown in the third following figure. The center points can be used in the identification of significant effects as shown next.

[[Image:doe7.17.png||center|300px|<math>2^2\,\!</math> design augmented by five center point runs.]]
[[Image:doe7_18.png|center|537px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

[[Image:doe7_19.png|center|630px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

Since the present <math>{2}^{2}\,\!</math> design is unreplicated, there are no degrees of freedom available to calculate the error sum of squares. By augmenting this design with five center points, the response values at the center points, <math>y_{i}^{c}\,\!</math>, can be used to obtain an estimate of pure error, <math>S{{S}_{PE}}\,\!</math>. Let <math>{{\bar{y}}^{c}}\,\!</math> represent the average response for the five replicates at the center. Then:

::<math>S{{S}_{PE}}=Sum\text{ }of\text{ }Squares\text{ }for\text{ }center\text{ }points\,\!</math>

::<math>\begin{align}
S{{S}_{PE}}= & \underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}} \\
= & {{(25.2-25.26)}^{2}}+...+{{(25.3-25.26)}^{2}} \\
= & 0.052
\end{align}\,\!</math>

Then the corresponding mean square is:

::<math>\begin{align}
M{{S}_{PE}}= & \frac{S{{S}_{PE}}}{degrees\text{ }of\text{ }freedom} \\
= & \frac{0.052}{5-1} \\
= & 0.013
\end{align}\,\!</math>

Alternatively, <math>M{{S}_{PE}}\,\!</math> can be directly obtained by calculating the variance of the response values at the center points:

::<math>\begin{align}
M{{S}_{PE}}= & {{s}^{2}} \\
= & \frac{\underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}}}{5-1}
\end{align}\,\!</math>

Once <math>M{{S}_{PE}}\,\!</math> is known, it can be used as the error mean square, <math>M{{S}_{E}}\,\!</math>, to carry out the test of significance for each effect. For example, to test the significance of the main effect of factor <math>A,\,\!</math> the sum of squares corresponding to this effect is obtained in the usual manner by considering only the four runs of the original <math>{2}^{2}\,\!</math> design.

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/4)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/4)J]y \\
= & 0.5625
\end{align}\,\!</math>

Then, the test statistic to test the significance of the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{A}}= & \frac{M{{S}_{A}}}{M{{S}_{E}}} \\
= & \frac{0.5625/1}{0.052/4} \\
= & 43.2692
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{A}}=43.2692\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{A}}) \\
= & 1-0.9972 \\
= & 0.0028
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value < 0.1, it can be concluded that the main effect of factor <math>A\,\!</math> significantly affects the response. This result is displayed in the ANOVA table as shown in the following figure. Test for the significance of other factors can be carried out in a similar manner.

[[Image:doe7_20.png|center|649px|Results for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

===Using Center Point Replicates to Test Curvature===

Center point replicates can also be used to check for curvature in replicated or unreplicated <math>{2}^{k}\,\!</math> designs. The test for curvature investigates whether the model between the response and the factors is linear. The way DOE++ handles center point replicates is similar to its handling of blocks. The center point replicates are treated as an additional factor in the model. The factor is labeled as Curvature in the results of DOE++. If Curvature turns out to be a significant factor in the results, then this indicates the presence of curvature in the model.

====Example: Use Center Point to Test Curvature====

To illustrate the use of center point replicates in testing for curvature, consider again the data of the single replicate <math>{2}^{2}\,\!</math> experiment from a preceding figure(labeled "<math>2^2</math> design augmented by five center point runs"). Let <math>{{x}_{1}}\,\!</math> be the indicator variable to indicate if the run is a center point:

::<math>\begin{matrix}
{{x}_{1}}=0 & {} & \text{Center point run} \\
{{x}_{1}}=1 & {} & \text{Other run} \\
\end{matrix}\,\!</math>

If <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> are the indicator variables representing factors <math>A\,\!</math> and <math>B\,\!</math>, respectively, then the model for this experiment is:

::<math>Y={{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}\,\!</math>

To investigate the presence of curvature, the following hypotheses need to be tested:

::<math>\begin{align}
& {{H}_{0}}: & {{\beta }_{1}}=0\text{ (Curvature is absent)} \\
& {{H}_{1}}: & {{\beta }_{1}}\ne 0
\end{align}\,\!</math>

The test statistic to be used for this test is:

::<math>{{({{F}_{0}})}_{curvature}}=\frac{M{{S}_{curvature}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{curvature}}\,\!</math> is the mean square for Curvature and <math>M{{S}_{E}}\,\!</math> is the error mean square.

'''Calculation of the Sum of Squares'''

The <math>X\,\!</math> matrix and <math>y\,\!</math> vector for this experiment are:

<center><math>X=\left[ \begin{matrix}
1 & 1 & -1 & -1 & 1 \\
1 & 1 & 1 & -1 & -1 \\
1 & 1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
\end{matrix} \right]\text{ }y=\left[ \begin{matrix}
24.6 \\
25.4 \\
25.0 \\
25.7 \\
25.2 \\
25.3 \\
25.4 \\
25.1 \\
25.3 \\
\end{matrix} \right]\,\!</math></center>

The sum of squares can now be calculated. For example, the error sum of squares is:

::<math>\begin{align}
& S{{S}_{E}}= & {{y}^{\prime }}[I-H]y \\
& = & 0.052
\end{align}\,\!</math>

where <math>I\,\!</math> is the identity matrix and <math>H\,\!</math> is the hat matrix. It can be seen that this is equal to <math>S{{S}_{PE\text{ }}}\,\!</math> (the sum of squares due to pure error) because of the replicates at the center point, as obtained in the [[Two_Level_Factorial_Experiments#Example_3| example]]. The number of degrees of freedom associated with <math>S{{S}_{E}}\,\!</math>, <math>dof(S{{S}_{E}})\,\!</math> is four. The extra sum of squares corresponding to the center point replicates (or Curvature) is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & Model\text{ }Sum\text{ }of\text{ }Squares- \\
& & Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }center\text{ }point \\
& = & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Curvature}}-(1/9)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }Curvature}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }Curvature}}={{X}_{\tilde{\ }Curv}}{{(X_{\tilde{\ }Curv}^{\prime }{{X}_{\tilde{\ }Curv}})}^{-1}}X_{\tilde{\ }Curv}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }Curv}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the center point. Thus, the extra sum of squares corresponding to Curvature is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Center}}-(1/9)J]y \\
& = & 0.7036-0.6875 \\
& = & 0.0161
\end{align}\,\!</math>

This extra sum of squares can be used to test for the significance of curvature. The corresponding mean square is:

::<math>\begin{align}
& M{{S}_{Curvature}}= & \frac{Sum\text{ }of\text{ }squares\text{ }corresponding\text{ }to\text{ }Curvature}{degrees\text{ }of\text{ }freedom} \\
& = & \frac{0.0161}{1} \\
& = & 0.0161
\end{align}\,\!</math>

'''Calculation of the Test Statistic'''

Knowing the mean squares, the statistic to check the significance of curvature can be calculated.

::<math>\begin{align}
& {{({{f}_{0}})}_{Curvature}}= & \frac{M{{S}_{Curvature}}}{M{{S}_{E}}} \\
& = & \frac{0.0161/1}{0.052/4} \\
& = & 1.24
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{Curvature}}=1.24\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and four degrees of freedom in the denominator is:

::<math>\begin{align}
& p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{Curvature}}) \\
& = & 1-0.6713 \\
& = & 0.3287
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that curvature does not exist for this design. This results is shown in the ANOVA table in the figure above. The surface of the fitted model based on these results, along with the observed response values, is shown in the figure below.

[[Image:doe7.21.png|center|400px|Model surface and observed response values for the design in the [[Two_Level_Factorial_Experiments#Example_4| example]].]]

 

==Blocking in 2k Designs==

Blocking can be used in the <math>{2}^{k}\,\!</math> designs to deal with cases when replicates cannot be run under identical conditions. Randomized complete block designs that were discussed in [[Randomization and Blocking in DOE]] for factorial experiments are also applicable here. At times, even with just two levels per factor, it is not possible to run all treatment combinations for one replicate of the experiment under homogeneous conditions. For example, each replicate of the <math>{2}^{2}\,\!</math> design requires four runs. If each run requires two hours and testing facilities are available for only four hours per day, two days of testing would be required to run one complete replicate. Blocking can be used to separate the treatment runs on the two different days. Blocks that do not contain all treatments of a replicate are called incomplete blocks. In incomplete block designs, the block effect is confounded with certain effect(s) under investigation. For the <math>{2}^{2}\,\!</math> design assume that treatments <math>(1)\,\!</math> and <math>ab\,\!</math> were run on the first day and treatments <math>a\,\!</math> and <math>b\,\!</math> were run on the second day. Then, the incomplete block design for this experiment is:

::<math>\begin{matrix}
\text{Block 1} & {} & \text{Block 2} \\
\left[ \begin{matrix}
(1) \\
ab \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
b \\
\end{matrix} \right] \\
\end{matrix}\,\!</math>

For this design the block effect may be calculated as:

::<math>\begin{align}
& Block\text{ }Effect= & Average\text{ }response\text{ }for\text{ }Block\text{ }1- \\
& & Average\text{ }response\text{ }for\text{ }Block\text{ }2 \\
& = & \frac{(1)+ab}{2}-\frac{a+b}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The <math>AB\,\!</math> interaction effect is:

::<math>\begin{align}
& AB= & Average\text{ }response\text{ }at\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{low}}}- \\
& & Average\text{ }response\text{ }at\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{low}}} \\
& = & \frac{ab+(1)}{2}-\frac{b+a}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The two equations given above show that, in this design, the <math>AB\,\!</math> interaction effect cannot be distinguished from the block effect because the formulas to calculate these effects are the same. In other words, the <math>AB\,\!</math> interaction is said to be confounded with the block effect and it is not possible to say if the effect calculated based on these equations is due to the <math>AB\,\!</math> interaction effect, the block effect or both. In incomplete block designs some effects are always confounded with the blocks. Therefore, it is important to design these experiments in such a way that the important effects are not confounded with the blocks. In most cases, the experimenter can assume that higher order interactions are unimportant. In this case, it would better to use incomplete block designs that confound these effects with the blocks.
One way to design incomplete block designs is to use defining contrasts as shown next:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}+...+{{\alpha }_{k}}{{q}_{k}}\,\!</math>

where the <math>{{\alpha }_{i}}\,\!</math> s are the exponents for the factors in the effect that is to be confounded with the block effect and the <math>{{q}_{i}}\,\!</math> s are values based on the level of the <math>i\,\!</math> the factor (in a treatment that is to be allocated to a block). For <math>{2}^{k}\,\!</math> designs the <math>{{\alpha }_{i}}\,\!</math> s are either 0 or 1 and the <math>{{q}_{i}}\,\!</math> s have a value of 0 for the low level of the <math>i\,\!</math> th factor and a value of 1 for the high level of the factor in the treatment under consideration. As an example, consider the <math>{2}^{2}\,\!</math> design where the interaction effect <math>AB\,\!</math> is confounded with the block. Since there are two factors, <math>k=2\,\!</math>, with <math>i=1\,\!</math> representing factor <math>A\,\!</math> and <math>i=2\,\!</math> representing factor <math>B\,\!</math>. Therefore:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}\,\!</math>

The value of <math>{{\alpha }_{1}}\,\!</math> is one because the exponent of factor <math>A\,\!</math> in the confounded interaction <math>AB\,\!</math> is one. Similarly, the value of <math>{{\alpha }_{2}}\,\!</math> is one because the exponent of factor <math>B\,\!</math> in the confounded interaction <math>AB\,\!</math> is also one. Therefore, the defining contrast for this design can be written as:

::<math>\begin{align}
& L= & {{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}} \\
& = & 1\cdot {{q}_{1}}+1\cdot {{q}_{2}} \\
& = & {{q}_{1}}+{{q}_{2}}
\end{align}\,\!</math>

Once the defining contrast is known, it can be used to allocate treatments to the blocks. For the <math>{2}^{2}\,\!</math> design, there are four treatments <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math> and <math>ab\,\!</math>. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. In order to decide which block the treatment <math>(1)\,\!</math> belongs to, the levels of factors <math>A\,\!</math> and <math>B\,\!</math> for this run are used. Since factor <math>A\,\!</math> is at the low level in this treatment, <math>{{q}_{1}}=0\,\!</math>. Similarly, since factor <math>B\,\!</math> is also at the low level in this treatment, <math>{{q}_{2}}=0\,\!</math>. Therefore:

::<math>\begin{align}
& L= & {{q}_{1}}+{{q}_{2}} \\
& = & 0+0=0\text{ (mod 2)}
\end{align}\,\!</math>

Note that the value of <math>L\,\!</math> used to decide the block allocation is "mod 2" of the original value. This value is obtained by taking the value of 1 for odd numbers and 0 otherwise. Based on the value of <math>L\,\!</math>, treatment <math>(1)\,\!</math> is assigned to block 1. Other treatments can be assigned using the following calculations:

::<math>\begin{align}
& (1): & \text{ }L=0+0=0=0\text{ (mod 2)} \\
& a: & \text{ }L=1+0=1=1\text{ (mod 2)} \\
& b: & \text{ }L=0+1=1=1\text{ (mod 2)} \\
& ab: & \text{ }L=1+1=2=0\text{ (mod 2)}
\end{align}\,\!</math>

Therefore, to confound the interaction <math>AB\,\!</math> with the block effect in the <math>{2}^{2}\,\!</math> incomplete block design, treatments <math>(1)\,\!</math> and <math>ab\,\!</math> (with <math>L=0\,\!</math>) should be assigned to block 2 and treatment combinations <math>a\,\!</math> and <math>b\,\!</math> (with <math>L=1\,\!</math>) should be assigned to block 1.

====Example: Two Level Factorial Design with Two Blocks====

This example illustrates how treatments can be allocated to two blocks for an unreplicated <math>{2}^{k}\,\!</math> design. Consider the unreplicated <math>{2}^{4}\,\!</math> design to investigate the four factors affecting the defects in automobile vinyl panels discussed in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments required for this experiment were run by two different operators with each operator conducting 8 runs. This experiment is an example of an incomplete block design. The analyst in charge of this experiment assumed that the interaction <math>ABCD\,\!</math> was not significant and decided to allocate treatments to the two operators so that the <math>ABCD\,\!</math> interaction was confounded with the block effect (the two operators are the blocks). The allocation scheme to assign treatments to the two operators can be obtained as follows.
 
The defining contrast for the <math>{2}^{4}\,\!</math> design where the <math>ABCD\,\!</math> interaction is confounded with the blocks is:

::<math>L={{q}_{1}}+{{q}_{2}}+{{q}_{3}}+{{q}_{4}}\,\!</math>

The treatments can be allocated to the two operators using the values of the defining contrast. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. Then the value of the defining contrast for treatment <math>a\,\!</math> is:

::<math>a\ \ :\ \ \text{ }L=1+0+0+0=1=1\text{ (mod 2)}\,\!</math>

Therefore, treatment <math>a\,\!</math> should be assigned to Block 1 or the first operator. Similarly, for treatment <math>ab\,\!</math> we have:

::<math>ab\ \ :\ \ \text{ }L=1+1+0+0=2=0\text{ (mod 2)}\,\!</math>

[[Image:doe7.22.png|center|200px| Allocation of treatments to two blocks for the <math>2^4</math> design in the example by confounding interaction of <math>ABCD</math> with the blocks.]]

Therefore, <math>ab\,\!</math> should be assigned to Block 2 or the second operator. Other treatments can be allocated to the two operators in a similar manner to arrive at the allocation scheme shown in the figure below.
In DOE++, to confound the <math>ABCD\,\!</math> interaction for the <math>{2}^{4}\,\!</math> design into two blocks, the number of blocks are specified as shown in the figure below. Then the interaction <math>ABCD\,\!</math> is entered in the Block Generator window (second following figure) which is available using the Block Generator button in the following figure. The design generated by DOE++ is shown in the third of the following figures. This design matches the allocation scheme of the preceding figure.

[[Image:doe7_23.png|center|700px| Adding block properties for the experiment in the example.|link=]]

[[Image:doe7_24.png|center|700px|Specifying the interaction ABCD as the interaction to be confounded with the blocks for the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

[[Image:doe7_25.png|center|800px|Two block design for the experiment in the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

For the analysis of this design, the sum of squares for all effects are calculated assuming no blocking. Then, to account for blocking, the sum of squares corresponding to the <math>ABCD\,\!</math> interaction is considered as the sum of squares due to blocks and <math>ABCD\,\!</math>. In DOE++ this is done by displaying this sum of squares as the sum of squares due to the blocks. This is shown in the following figure where the sum of squares in question is obtained as 72.25 and is displayed against Block. The interaction ABCD, which is confounded with the blocks, is not displayed. Since the design is unreplicated, any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_26.png|center|793px|ANOVA table for the experiment of the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

===Unreplicated 2''k'' Designs in 2''p'' Blocks===

A single replicate of the <math>{2}^{k}\,\!</math> design can be run in up to <math>{2}^{p}\,\!</math> blocks where <math>p<k\,\!</math>. The number of effects confounded with the blocks equals the degrees of freedom associated with the block effect.

If two blocks are used (the block effect has two levels), then one (<math>2-1=1)\,\!</math> effect is confounded with the blocks. If four blocks are used, then three (<math>4-1=3\,\!</math>) effects are confounded with the blocks and so on. For example an unreplicated <math>{2}^{4}\,\!</math> design may be confounded in <math>{2}^{2}\,\!</math> (four) blocks using two contrasts, <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}}\,\!</math>. Let <math>AC\,\!</math> and <math>BD\,\!</math> be the effects to be confounded with the blocks. Corresponding to these two effects, the contrasts are respectively:

::<math>\begin{align}
& {{L}_{1}}= & {{q}_{1}}+{{q}_{3}} \\
& {{L}_{2}}= & {{q}_{2}}+{{q}_{4}}
\end{align}\,\!</math>

Based on the values of <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}},\,\!</math> the treatments can be assigned to the four blocks as follows:

<center><math>\begin{matrix}
\text{Block 4} & {} & \text{Block 3} & {} & \text{Block 2} & {} & \text{Block 1} \\
{{L}_{1}}=0,{{L}_{2}}=0 & {} & {{L}_{1}}=1,{{L}_{2}}=0 & {} & {{L}_{1}}=0,{{L}_{2}}=1 & {} & {{L}_{1}}=1,{{L}_{2}}=1 \\
{} & {} & {} & {} & {} & {} & {} \\
\left[ \begin{matrix}
(1) \\
ac \\
bd \\
abcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
c \\
abd \\
bcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
b \\
abc \\
d \\
acd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
ab \\
bc \\
ad \\
cd \\
\end{matrix} \right] \\
\end{matrix}\,\!</math></center>

Since the block effect has three degrees of freedom, three effects are confounded with the block effect. In addition to <math>AC\,\!</math> and <math>BD\,\!</math>, the third effect confounded with the block effect is their generalized interaction, <math>(AC)(BD)=ABCD\,\!</math>.
In general, when an unreplicated <math>{2}^{k}\,\!</math> design is confounded in <math>{2}^{p}\,\!</math> blocks, <math>p\,\!</math> contrasts are needed (<math>{{L}_{1}},{{L}_{2}}...{{L}_{p}}\,\!</math>). <math>p\,\!</math> effects are selected to define these contrasts such that none of these effects are the generalized interaction of the others. The <math>{2}^{p}\,\!</math> blocks can then be assigned the treatments using the <math>p\,\!</math> contrasts. <math>{{2}^{p}}-(p+1)\,\!</math> effects, that are also confounded with the blocks, are then obtained as the generalized interaction of the <math>p\,\!</math> effects. In the statistical analysis of these designs, the sum of squares are computed as if no blocking were used. Then the block sum of squares is obtained by adding the sum of squares for all the effects confounded with the blocks.

====Example: 2 Level Factorial Design with Four Blocks====

This example illustrates how DOE++ obtains the sum of squares when treatments for an unreplicated <math>{2}^{k}\,\!</math> design are allocated among four blocks. Consider again the unreplicated <math>{2}^{4}\,\!</math> design used to investigate the defects in automobile vinyl panels presented in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments needed to complete the experiment were run by four operators. Therefore, there are four blocks. Assume that the treatments were allocated to the blocks using the generators mentioned in the previous section, i.e., treatments were allocated among the four operators by confounding the effects, <math>AC\,\!</math> and <math>BD,\,\!</math> with the blocks. These effects can be specified as Block Generators as shown in the following figure. (The generalized interaction of these two effects, interaction <math>ABCD\,\!</math>, will also get confounded with the blocks.) The resulting design is shown in the second following figure and matches the allocation scheme obtained in the previous section.

[[Image:doe7_27.png|center|700px|Specifying the interactions AC and BD as block generators for the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

The sum of squares in this case can be obtained by calculating the sum of squares for each of the effects assuming there is no blocking. Once the individual sum of squares have been obtained, the block sum of squares can be calculated. The block sum of squares is the sum of the sum of squares of effects, <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, since these effects are confounded with the block effect. As shown in the second following figure, this sum of squares is 92.25 and is displayed against Block. The interactions <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, which are confounded with the blocks, are not displayed. Since the present design is unreplicated any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_28.png|center|800px|Design for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

[[Image:doe7_29.png|center|793px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

==Variability Analysis==

For replicated two level factorial experiments, DOE++ provides the option of conducting variability analysis (using the Variability Analysis icon under the Data menu). The analysis is used to identify the treatment that results in the least amount of variation in the product or process being investigated. Variability analysis is conducted by treating the standard deviation of the response for each treatment of the experiment as an additional response. The standard deviation for a treatment is obtained by using the replicated response values at that treatment run. As an example, consider the <math>{2}^{3}\,\!</math> design shown in the following figure where each run is replicated four times. A variability analysis can be conducted for this design. DOE++ calculates eight standard deviation values corresponding to each treatment of the design (see second following figure). Then, the design is analyzed as an unreplicated <math>{2}^{3}\,\!</math> design with the standard deviations (displayed as Y Standard Deviation. in second following figure) as the response. The normal probability plot of effects identifies <math>AC\,\!</math> as the effect that influences variability (see third figure following). Based on the effect coefficients obtained in the fourth figure following, the model for Y Std. is:

::<math>\begin{align}
& \text{Y Std}\text{.}= & 0.6779+0.2491\cdot AC \\
& = & 0.6779+0.2491{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

Based on the model, the experimenter has two choices to minimize variability (by minimizing Y Std.). The first choice is that <math>{{x}_{1}}\,\!</math> should be <math>1\,\!</math> (i.e., <math>A\,\!</math> should be set at the high level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the low level). The second choice is that <math>{{x}_{1}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>A\,\!</math> should be set at the low level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the high level). The experimenter can select the most feasible choice.

[[Image:doe7.30.png|center|391px|A <math>2^3\,\!</math> design with four replicated response values that can be used to conduct a variability analysis.]]
 

[[Image:doe7_31.png|center|727px|Variability analysis in DOE++.|link=]]
 

[[Image:doe7_32.png|center|650px|Normal probability plot of effects for the variability analysis example.|link=]]
 

[[Image:doe7_33.png|center|727px|Effect coefficients for the variability analysis example.|link=]]

 

==Two Level Fractional Factorial Designs==

As the number of factors in a two level factorial design increases, the number of runs for even a single replicate of the <math>{2}^{k}\,\!</math> design becomes very large. For example, a single replicate of an eight factor two level experiment would require 256 runs. Fractional factorial designs can be used in these cases to draw out valuable conclusions from fewer runs. The basis of fractional factorial designs is the ''sparsity of effects'' principle.[[DOE_References|[Wu, 2000]]] The principle states that, most of the time, responses are affected by a small number of main effects and lower order interactions, while higher order interactions are relatively unimportant. Fractional factorial designs are used as screening experiments during the initial stages of experimentation. At these stages, a large number of factors have to be investigated and the focus is on the main effects and two factor interactions. These designs obtain information about main effects and lower order interactions with fewer experiment runs by confounding these effects with unimportant higher order interactions. As an example, consider a <math>{2}^{8}\,\!</math> design that requires 256 runs. This design allows for the investigation of 8 main effects and 28 two factor interactions. However, 219 degrees of freedom are devoted to three factor or higher order interactions. This full factorial design can prove to be very inefficient when these higher order interactions can be assumed to be unimportant. Instead, a fractional design can be used here to identify the important factors that can then be investigated more thoroughly in subsequent experiments. In unreplicated fractional factorial designs, no degrees of freedom are available to calculate the error sum of squares and the techniques mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] should be employed for the analysis of these designs.

==Half-fraction Designs==

A half-fraction of the <math>{2}^{k}\,\!</math> design involves running only half of the treatments of the full factorial design. For example, consider a <math>{2}^{3}\,\!</math> design that requires eight runs in all. The design matrix for this design is shown in the figure (a) below. A half-fraction of this design is the design in which only four of the eight treatments are run. The fraction is denoted as <math>{2}^{3-1}\,\!</math> with the "<math>-1\,\!</math>" in the index denoting a half-fraction. Assume that the treatments chosen for the half-fraction design are the ones where the interaction <math>ABC\,\!</math> is at the high level (i.e., only those rows are chosen from the following figure (a) where the column for <math>ABC\,\!</math> has entries of 1). The resulting <math>{2}^{3-1}\,\!</math> design has a design matrix as shown in figure (b) below.

[[Image:doe7.34.png|center|330px|Half-fractions of the <math>2^3\,\!</math> design. (a) shows the full factorial <math>2^3\,\!</math> design, (b) shows the <math>2^{3-1}\,\!</math> design with the defining relation <math>I=ABC\,\!</math> and (c) shows the <math>{2}^{3-1}\,\!</math> design with the defining relation <math>I=-ABC\,\!</math>.]]

In the <math>{2}^{3-1}\,\!</math> design of figure (b), since the interaction <math>ABC\,\!</math> is always included at the same level (the high level represented by 1), it is not possible to measure this interaction effect. The effect, <math>ABC\,\!</math>, is called the ''generator'' or ''word'' for this design. It can be noted that, in the design matrix of the following figure (b), the column corresponding to the intercept, <math>I\,\!</math>, and column corresponding to the interaction <math>ABC\,\!</math>, are identical. The identical columns are written as <math>I=ABC\,\!</math> and this equation is called the ''defining relation'' for the design. In DOE++, the present <math>{2}^{3-1}\,\!</math> design can be obtained by specifying the design properties as shown in the following figure.

[[Image:doe7_35.png|center|700px|Design properties for the <math>2^{3-1}\,\!</math> design.|link=]]

The defining relation, <math>I=ABC\,\!</math>, is entered in the Fraction Generator window as shown next.

[[Image:doe7_36.png|center|700px|Specifying the defining relation for the <math>2^{3-1}\,\!</math> design.|link=]]

Note that in the figure following that, the defining relation is specified as <math>C=AB\,\!</math>. This relation is obtained by multiplying the defining relation, <math>I=ABC\,\!</math>, by the last factor, <math>C\,\!</math>, of the design.

===Calculation of Effects===

Using the four runs of the <math>{2}^{3-1}\,\!</math> design in figure (b) discussed above, the main effects can be calculated as follows:

::<math>\begin{align}
& A= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& B= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& C= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

where <math>a\,\!</math>, <math>b\,\!</math>, <math>c\,\!</math> and <math>abc\,\!</math> are the treatments included in the <math>{2}^{3-1}\,\!</math> design.

Similarly, the two factor interactions can also be obtained as:

::<math>\begin{align}
& BC= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& AC= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& AB= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

The equations for <math>A\,\!</math> and <math>BC\,\!</math> above result in the same effect values showing that effects <math>A\,\!</math> and <math>BC\,\!</math> are confounded in the present <math>{2}^{3-1}\,\!</math> design. Thus, the quantity, <math>\tfrac{1}{2}(a-b-c+abc),\,\!</math> estimates <math>A+BC\,\!</math> (i.e., both the main effect <math>A\,\!</math> and the two-factor interaction <math>BC\,\!</math>). The effects, <math>A\,\!</math> and <math>BC,\,\!</math> are called ''aliases''. From the remaining equations given above, it can be seen that the other aliases for this design are <math>B\,\!</math> and <math>AC\,\!</math>, and <math>C\,\!</math> and <math>AB\,\!</math>. Therefore, the equations to calculate the effects in the present <math>{2}^{3-1}\,\!</math> design can be written as follows:

::<math>\begin{align}
& A+BC= & \frac{1}{2}(a-b-c+abc) \\
& B+AC= & \frac{1}{2}(-a+b-c+abc) \\
& C+AB= & \frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

===Calculation of Aliases===

Aliases for a fractional factorial design can be obtained using the defining relation for the design. The defining relation for the present <math>{2}^{3-1}\,\!</math> design is:

::<math>I=ABC\,\!</math>

Multiplying both sides of the previous equation by the main effect, <math>A,\,\!</math> gives the alias effect of <math>A\,\!</math> :

::<math>\begin{align}
& A\cdot I= & A\cdot ABC \\
& A= & {{A}^{2}}BC \\
& A= & BC
\end{align}\,\!</math>

Note that in calculating the alias effects, any effect multiplied by <math>I\,\!</math> remains the same (<math>A\cdot I=A\,\!</math>), while an effect multiplied by itself results in <math>I\,\!</math> (<math>{{A}^{2}}=I\,\!</math>). Other aliases can also be obtained:

::<math>\begin{align}
& B\cdot I= & B\cdot ABC \\
& B= & A{{B}^{2}}C \\
& B= & AC
\end{align}\,\!</math>

:and:

::<math>\begin{align}
& C\cdot I= & C\cdot ABC \\
& C= & AB{{C}^{2}} \\
& C= & AB
\end{align}\,\!</math>

===Fold-over Design===

If it can be assumed for this design that the two-factor interactions are unimportant, then in the absence of <math>BC\,\!</math>, <math>AC\,\!</math> and <math>AB\,\!</math>, the equations for (A+BC), (B+AC) and (C+AB) can be used to estimate the main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. However, if such an assumption is not applicable, then to uncouple the main effects from their two factor aliases, the alternate fraction that contains runs having <math>ABC\,\!</math> at the lower level should be run. The design matrix for this design is shown in the preceding figure (c). The defining relation for this design is <math>I=-ABC\,\!</math> because the four runs for this design are obtained by selecting the rows of the preceding figure (a) for which the value of the <math>ABC\,\!</math> column is <math>-1\,\!</math>. The aliases for this fraction can be obtained as explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]] as <math>A=-BC\,\!</math>, <math>B=-AC\,\!</math> and <math>C=-AB\,\!</math>. The effects for this design can be calculated as:

::<math>\begin{align}
& A-BC= & \frac{1}{2}(ab+ac-(1)-bc) \\
& B-AC= & \frac{1}{2}(ab-ac+(1)-bc) \\
& C-AB= & \frac{1}{2}(-ab+ac-(1)+bc)
\end{align}\,\!</math>

These equations can be combined with the equations for (A+BC), (B+AC) and (C+AB) to obtain the de-aliased main effects and two factor interactions. For example, adding equations (A+BC) and (A-BC) returns the main effect <math>A\,\!</math>.

::<math>\begin{align}
& 2A= & \frac{1}{2}(a-b-c+abc)+ \\
& & \frac{1}{2}(ab+ac-(1)-bc)
\end{align}\,\!</math>

The process of augmenting a fractional factorial design by a second fraction of the same size by simply reversing the signs (of all effect columns except <math>I\,\!</math>) is called ''folding over''. The combined design is referred to as a ''fold-over design''.

==Quarter and Smaller Fraction Designs==

At times, the number of runs even for a half-fraction design are very large. In these cases, smaller fractions are used. A quarter-fraction design, denoted as <math>{2}^{k-2}\,\!</math>, consists of a fourth of the runs of the full factorial design. Quarter-fraction designs require two defining relations. The first defining relation returns the half-fraction or the <math>{2}^{k-1}\,\!</math> design. The second defining relation selects half of the runs of the <math>{2}^{k-1}\,\!</math> design to give the quarter-fraction. For example, consider the <math>{2}^{4}\,\!</math> design. To obtain a <math>{2}^{4-2}\,\!</math> design from this design, first a half-fraction of this design is obtained by using a defining relation. Assume that the defining relation used is <math>I=ABCD\,\!</math>. The design matrix for the resulting <math>{2}^{4-1}\,\!</math> design is shown in figure (a) below. Now, a quarter-fraction can be obtained from the <math>{2}^{4-1}\,\!</math> design shown in figure (a) below using a second defining relation <math>I=AD\,\!</math>. The resulting <math>{2}^{4-2}\,\!</math> design obtained is shown in figure (b) below.

[[Image:doe7.37.png|center|465px|Fractions of the <math>2^4\,\!</math> design - Figure (a) shows the <math>2^{4-1}</math> design with the defining relation <math>I=ABCD\,\!</math> and (b) shows the <math>2^{4-2}\,\!</math> design with the defining relation <math>I=ABCD=AD=BC\,\!</math>.]]

The complete defining relation for this <math>{2}^{4-2}\,\!</math> design is:

::<math>I=ABCD=AD=BC\,\!</math>

Note that the effect, <math>BC,\,\!</math> in the defining relation is the generalized interaction of <math>ABCD\,\!</math> and <math>AD\,\!</math> and is obtained using <math>(ABCD)(AD)={{A}^{2}}BC{{D}^{2}}=BC\,\!</math>. In general, a <math>{2}^{k-p}\,\!</math> fractional factorial design requires <math>p\,\!</math> independent generators. The defining relation for the design consists of the <math>p\,\!</math> independent generators and their <math>{2}^{p}\,\!</math> - (<math>p\,\!</math> +1) generalized interactions.

===Calculation of Aliases===

The alias structure for the present <math>{2}^{4-2}\,\!</math> design can be obtained using the defining relation of equation (I=ABCD=AD=BC) following the procedure explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]]. For example, multiplying the defining relation by <math>A\,\!</math> returns the effects aliased with the main effect, <math>A\,\!</math>, as follows:

::<math>\begin{align}
& A\cdot I= & A\cdot ABCD=A\cdot AD=A\cdot BC \\
& A= & {{A}^{2}}BCD={{A}^{2}}D=ABC \\
& A= & BCD=D=ABC
\end{align}\,\!</math>

Therefore, in the present <math>{2}^{4-2}\,\!</math> design, it is not possible to distinguish between effects <math>A\,\!</math>, <math>D\,\!</math>, <math>BCD\,\!</math> and <math>ABC\,\!</math>. Similarly, multiplying the defining relation by <math>B\,\!</math> and <math>AB\,\!</math> returns the effects that are aliased with these effects:

::<math>\begin{align}
& B= & ACD=ABD=C \\
& AB= & CD=AD=AC
\end{align}\,\!</math>

Other aliases can be obtained in a similar way. It can be seen that each effect in this design has three aliases. In general, each effect in a <math>{2}^{k-p}\,\!</math> design has <math>{2}^{p-1}\,\!</math> aliases.
The aliases for the <math>{2}^{4-2}\,\!</math> design show that in this design the main effects are aliased with each other (<math>A\,\!</math> is aliased with <math>D\,\!</math> and <math>B\,\!</math> is aliased with <math>C\,\!</math>). Therefore, this design is not a useful design and is not available in DOE++. It is important to ensure that main effects and lower order interactions of interest are not aliased in a fractional factorial design. This is known by looking at the resolution of the fractional factorial design.

==Design Resolution==

The resolution of a fractional factorial design is defined as the number of factors in the lowest order effect in the defining relation. For example, in the defining relation <math>I=ABCD=AD=BC\,\!</math> of the previous <math>{2}^{4-2}\,\!</math> design, the lowest-order effect is either <math>AD\,\!</math> or <math>BC,\,\!</math> containing two factors. Therefore, the resolution of this design is equal to two. The resolution of a fractional factorial design is represented using Roman numerals. For example, the previously mentioned <math>{2}^{4-2}\,\!</math> design with a resolution of two can be represented as 2 <math>_{\text{II}}^{4-2}\,\!</math>. The resolution provides information about the confounding in the design as explained next:
 
 
#'''Resolution III Designs''' In these designs, the lowest order effect in the defining relation has three factors (e.g., a <math>{2}^{5-2}\,\!</math> design with the defining relation <math>I=ABDE=ABC=CDE\,\!</math>). In resolution III designs, no main effects are aliased with any other main effects, but main effects are aliased with two factor interactions. In addition, some two factor interactions are aliased with each other.
#'''Resolution IV Designs''' In these designs, the lowest order effect in the defining relation has four factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABDE\,\!</math>). In resolution IV designs, no main effects are aliased with any other main effects or two factor interactions. However, some main effects are aliased with three factor interactions and the two factor interactions are aliased with each other.
#'''Resolution V Designs''' In these designs the lowest order effect in the defining relation has five factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABCDE\,\!</math>). In resolution V designs, no main effects or two factor interactions are aliased with any other main effects or two factor interactions. However, some main effects are aliased with four factor interactions and the two factor interactions are aliased with three factor interactions.

Fractional factorial designs with the highest resolution possible should be selected because the higher the resolution of the design, the less severe the degree of confounding. In general, designs with a resolution less than III are never used because in these designs some of the main effects are aliased with each other. The table below shows fractional factorial designs with the highest available resolution for three to ten factor designs along with their defining relations.

[[Image:doet7.3.png|center|474px|Highest resolution designs available for fractional factorial designs with 3 to 10 factors.]]

All of the two level fractional factorial designs available in DOE++ are shown next.

[[Image:doe7.38.png|center|471px|Two level fractional factorial designs available in DOE++ and their resolutions.]]

===Minimum Aberration Designs===
At times, different designs with the same resolution but different aliasing may be available. The best design to select in such a case is the minimum aberration design. For example, all <math>{2}^{7-2}\,\!</math> designs in the fourth table have a resolution of four (since the generator with the minimum number of factors in each design has four factors). Design <math>1\,\!</math> has three generators of length four (<math>ABCF,\,\!</math> <math>BCDG,\,\!</math> <math>ADFG\,\!</math>). Design <math>2\,\!</math> has two generators of length four (<math>ABCF,\,\!</math> <math>ADEG\,\!</math>). Design <math>3\,\!</math> has one generator of length four (<math>CEFG\,\!</math>). Therefore, design <math>3\,\!</math> has the least number of generators with the minimum length of four. Design <math>3\,\!</math> is called the minimum aberration design. It can be seen that the alias structure for design <math>3\,\!</math> is less involved compared to the other designs. For details refer to [[DOE_References|[Wu, 2000]]].

[[Image:doet7.4.png|center|432px|Three <math>2_{IV}^{7-2}\,\!</math> designs with different defining relations.]]

====Example====

The design of an automobile fuel cone is thought to be affected by six factors in the manufacturing process: cavity temperature (factor <math>A\,\!</math>), core temperature (factor <math>B\,\!</math>), melt temperature (factor <math>C\,\!</math>), hold pressure (factor <math>D\,\!</math>), injection speed (factor <math>E\,\!</math>) and cool time (factor <math>F\,\!</math>). The manufacturer of the fuel cone is unable to run the <math>{2}^{6}=64\,\!</math> runs required to complete one replicate for a two level full factorial experiment with six factors. Instead, they decide to run a fractional factorial design. Considering that three factor and higher order interactions are likely to be inactive, the manufacturer selects a <math>{2}^{6-2}\,\!</math> design that will require only 16 runs. The manufacturer chooses the resolution IV design which will ensure that all main effects are free from aliasing (assuming three factor and higher order interactions are absent). However, in this design the two factor interactions may be aliased with each other. It is decided that, if important two factor interactions are found to be present, additional experiment trials may be conducted to separate the aliased effects. The performance of the fuel cone is measured on a scale of 1 to 15. In DOE++, the design for this experiment is set up using the properties shown in the following figure. The Fraction Generators for the design, <math>E=ABC\,\!</math> and <math>F=BCD\,\!</math>, are the same as the defaults used in DOE++. The resulting <math>{2}^{6-2}\,\!</math> design and the corresponding response values are shown in the following two figures.

[[Image:doe7_39.png|center|644px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

[[Image:doe7_40.png|center|534px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The complete alias structure for the 2 <math>_{\text{IV}}^{6-2}\,\!</math> design is shown next.

<center><math>I=ABCE=ADEF=BCDF\,\!</math></center>

 
<center><math>\begin{align}
& A= & BCE=DEF=ABCDF \\
& B= & ACE=CDF=ABDEF \\
& C= & ABE=BDF=ACDEF \\
& D= & AEF=BCF=ABCDE \\
& E= & ABC=ADF=BCDEF \\
& F= & ADE=BCD=ABCEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& AB= & CE=ACDF=BDEF \\
& AC= & BE=ABDF=CDEF \\
& AD= & EF=ABCF=BCDE \\
& AE= & BC=DF=ABCDEF \\
& AF= & DE=ABCD=BCEF \\
& BD= & CF=ABEF=ACDE \\
& BF= & CD=ABDE=ACEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& ABD= & ACF=BEF=CDE \\
& ABF= & ACD=BDE=CEF
\end{align}\,\!</math></center>

In DOE++, the alias structure is displayed in the Design Summary and as part of the Design Evaluation result, as shown next:

[[Image:doe7_41.png|center|700px|Alias structure for the experiment design in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The normal probability plot of effects for this unreplicated design shows the main effects of factors <math>C\,\!</math> and <math>D\,\!</math> and the interaction effect, <math>BF\,\!</math>, to be significant (see the following figure).

[[Image:doe7_42.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

From the alias structure, it can be seen that for the present design interaction effect, <math>BF,\,\!</math> is confounded with <math>CD\,\!</math>. Therefore, the actual source of this effect cannot be known on the basis of the present experiment. However because neither factor <math>B\,\!</math> nor <math>F\,\!</math> is found to be significant there is an indication the observed effect is likely due to interaction, <math>CD\,\!</math>. To confirm this, a follow-up <math>{2}^{2}\,\!</math> experiment is run involving only factors <math>B\,\!</math> and <math>F\,\!</math>. The interaction, <math>BF\,\!</math>, is found to be inactive, leading to the conclusion that the interaction effect in the original experiment is effect, <math>CD\,\!</math>. Given these results, the fitted regression model for the fuel cone design as per the coefficients obtained from DOE++ is shown next.

::<math>\hat{y}=7.6875+C+2\cdot D+2.1875\cdot CD\,\!</math>

[[Image:doe7_43.png|center|700px|Effect coefficients for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

==Projection==

Projection refers to the reduction of a fractional factorial design to a full factorial design by dropping out some of the factors of the design. Any fractional factorial design of resolution, <math>R,\,\!</math> can be reduced to complete factorial designs in any subset of <math>R-1\,\!</math> factors. For example, consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. The resolution of this design is four. Therefore, this design can be reduced to full factorial designs in any three (<math>4-1=3\,\!</math>) of the original seven factors (by dropping the remaining four of factors). Further, a fractional factorial design can also be reduced to a full factorial design in any <math>R\,\!</math> of the original factors, as long as these <math>R\,\!</math> factors are not part of the generator in the defining relation. Again consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. This design can be reduced to a full factorial design in four factors provided these four factors do not appear together as a generator in the defining relation. The complete defining relation for this design is:

::<math>\begin{align}
& I= & ABCE=BCDF=ACDG=ADEF=ABFG=BDEG=CEFG
\end{align}\,\!</math>

Therefore, there are seven four factor combinations out of the 35 (<math>(_{7}^{4})=35\,\!</math>) possible four-factor combinations that are used as generators in the defining relation. The designs with the remaining 28 four factor combinations would be full factorial 16-run designs. For example, factors <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math> do not occur as a generator in the defining relation of the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. If the remaining factors, <math>E\,\!</math>, <math>F\,\!</math> and <math>G\,\!</math>, are dropped, the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design will reduce to a full factorial design in <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math>.

==Resolution III Designs==

At times, the factors to be investigated in screening experiments are so large that even running a fractional factorial design is impractical. This can be partially solved by using resolution III fractional factorial designs in the cases where three factor and higher order interactions can be assumed to be unimportant. Resolution III designs, such as the 2 <math>_{\text{III}}^{3-1}\,\!</math> design, can be used to estimate <math>k\,\!</math> main effects using just <math>k+1\,\!</math> runs. In these designs, the main effects are aliased with two factor interactions. Once the results from these designs are obtained, and knowing that three factor and higher order interactions are unimportant, the experimenter can decide if there is a need to run a fold-over design to de-alias the main effects from the two factor interactions. Thus, the 2 <math>_{\text{III}}^{3-1}\,\!</math> design can be used to investigate three factors in four runs, the 2 <math>_{\text{III}}^{7-4}\,\!</math> design can be used to investigate seven factors in eight runs, the 2 <math>_{\text{III}}^{15-11}\,\!</math> design can be used to investigate fifteen factors in sixteen runs and so on.

====Example====

{{:Resolution_III_Design_Example}}

==Alias Matrix==
In [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction designs]] and [[Two_Level_Factorial_Experiments#Quarter_and_Smaller_Fraction_Designs| Quarter and Smaller Fraction Designs]], the alias structure for fractional factorial designs was obtained using the defining relation. However, this method of obtaining the alias structure is not very efficient when the alias structure is very complex or when partial aliasing is involved. One of the ways to obtain the alias structure for any design, regardless of its complexity, is to use the alias matrix. The alias matrix for a design is calculated using <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> where <math>{{X}_{1}}\,\!</math> is the portion of the design matrix, <math>X,\,\!</math> that contains the effects for which the aliases need to be calculated, and <math>{{X}_{2}}\,\!</math> contains the remaining columns of the design matrix, other than those included in <math>{{X}_{1}}\,\!</math>.

To illustrate the use of the alias matrix, consider the design matrix for the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design (using the defining relation <math>I=ABCD\,\!</math>) shown next:

[[Image:Chapter7__879.png|center|link=]]

The alias structure for this design can be obtained by defining <math>{{X}_{1}}\,\!</math> using eight columns since the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design estimates eight effects. If the first eight columns of <math>X\,\!</math> are used then <math>{{X}_{1}}\,\!</math> is:

[[Image:Chapter7__884.png|center|link=]]

<math>{{X}_{2}}\,\!</math> is obtained using the remaining columns as:

[[Image:Chapter7__886.png|center|link=]]

Then the alias matrix <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> is:

[[Image:Chapter7__888.png|center|link=]]

The alias relations can be easily obtained by observing the alias matrix as:

<center><math>\begin{align}
& I= & ABCD \\
& A= & BCD \\
& B= & ACD \\
& AB= & CD \\
& C= & ABD \\
& AC= & BD \\
& BC= & AD \\
& D= & ABC
\end{align}\,\!</math></center>

Two Level Factorial Experiments

2017-08-10T17:30:25Z

Richard House: /* Replicated and Repeated Runs */

{{Template:Doebook|8}}
Two level factorial experiments are factorial experiments in which each factor is investigated at only two levels. The early stages of experimentation usually involve the investigation of a large number of potential factors to discover the "vital few" factors. Two level factorial experiments are used during these stages to quickly filter out unwanted effects so that attention can then be focused on the important ones.

==2''k'' Designs==
The factorial experiments, where all combination of the levels of the factors are run, are usually referred to as ''full factorial experiments''. Full factorial two level experiments are also referred to as <math>{2}^{k}\,\!</math> designs where <math>k\,\!</math> denotes the number of factors being investigated in the experiment. In Weibull++ DOE folios, these designs are referred to as 2 Level Factorial Designs as shown in the figure below.

[[Image:doe7_1.png|center|487px|Selection of full factorial experiments with two levels in Weibull++.|link=]]

A full factorial two level design with <math>k\,\!</math> factors requires <math>{{2}^{k}}\,\!</math> runs for a single replicate. For example, a two level experiment with three factors will require <math>2\times 2\times 2={{2}^{3}}=8\,\!</math> runs. The choice of the two levels of factors used in two level experiments depends on the factor; some factors naturally have two levels. For example, if gender is a factor, then male and female are the two levels. For other factors, the limits of the range of interest are usually used. For example, if temperature is a factor that varies from <math>{45}^{o}C\,\!</math> to <math>{90}^{o}C\,\!</math>, then the two levels used in the <math>{2}^{k}\,\!</math> design for this factor would be <math>{45}^{o}\,\!C\,\!</math> and <math>{90}^{o}\,\!C\,\!</math>.

The two levels of the factor in the <math>{2}^{k}\,\!</math> design are usually represented as <math>-1\,\!</math> (for the first level) and <math>1\,\!</math> (for the second level). Note that this representation is reversed from the coding used in [[General Full Factorial Designs]] for the indicator variables that represent two level factors in ANOVA models. For ANOVA models, the first level of the factor was represented using a value of <math>1\,\!</math> for the indicator variable, while the second level was represented using a value of <math>-1\,\!</math>. For details on the notation used for two level experiments refer to [[Two_Level_Factorial_Experiments#Notation| Notation]].

===The 22 Design===

The simplest of the two level factorial experiments is the <math>{2}^{2}\,\!</math> design where two factors (say factor <math>A\,\!</math> and factor <math>B\,\!</math>) are investigated at two levels. A single replicate of this design will require four runs (<math>{{2}^{2}}=2\times 2=4\,\!</math>) The effects investigated by this design are the two main effects, <math>A\,\!</math> and <math>B,\,\!</math> and the interaction effect <math>AB\,\!</math>. The treatments for this design are shown in figure (a) below. In figure (a), letters are used to represent the treatments. The presence of a letter indicates the high level of the corresponding factor and the absence indicates the low level. For example, (1) represents the treatment combination where all factors involved are at the low level or the level represented by <math>-1\,\!</math> ; <math>a\,\!</math> represents the treatment combination where factor <math>A\,\!</math> is at the high level or the level of <math>1\,\!</math>, while the remaining factors (in this case, factor <math>B\,\!</math>) are at the low level or the level of <math>-1\,\!</math>. Similarly, <math>b\,\!</math> represents the treatment combination where factor <math>B\,\!</math> is at the high level or the level of <math>1\,\!</math>, while factor <math>A\,\!</math> is at the low level and <math>ab\,\!</math> represents the treatment combination where factors <math>A\,\!</math> and <math>B\,\!</math> are at the high level or the level of the 1. Figure (b) below shows the design matrix for the <math>{2}^{2}\,\!</math> design. It can be noted that the sum of the terms resulting from the product of any two columns of the design matrix is zero. As a result the <math>{2}^{2}\,\!</math> design is an ''orthogonal design''. In fact, all <math>{2}^{k}\,\!</math> designs are orthogonal designs. This property of the <math>{2}^{k}\,\!</math> designs offers a great advantage in the analysis because of the simplifications that result from orthogonality. These simplifications are explained later on in this chapter.
The <math>{2}^{2}\,\!</math> design can also be represented geometrically using a square with the four treatment combinations lying at the four corners, as shown in figure (c) below.

[[Image:doe7.2.png|center|400px|The <math>2^2\,\!</math> design. Figure (a) displays the experiment design, (b) displays the design matrix and (c) displays the geometric representation for the design. In Figure (b), the column names I, A, B and AB are used. Column I represents the intercept term. Columns A and B represent the respective factor settings. Column AB represents the interaction and is the product of columns A and B.]]
 

===The 23 Design===

The <math>{2}^{3}\,\!</math> design is a two level factorial experiment design with three factors (say factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>). This design tests three (<math>k=3\,\!</math>) main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math> ; three (<math>(_{2}^{k})=\,\!</math> <math>(_{2}^{3})=3\,\!</math>) two factor interaction effects, <math>AB\,\!</math>, <math>BC\,\!</math>, <math>AC\,\!</math> ; and one (<math>(_{3}^{k})=\,\!</math> <math>(_{3}^{3})=1\,\!</math>) three factor interaction effect, <math>ABC\,\!</math>. The design requires eight runs per replicate. The eight treatment combinations corresponding to these runs are <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math>, <math>ab\,\!</math>, <math>c\,\!</math>, <math>ac\,\!</math>, <math>bc\,\!</math> and <math>abc\,\!</math>. Note that the treatment combinations are written in such an order that factors are introduced one by one with each new factor being combined with the preceding terms. This order of writing the treatments is called the ''standard order'' or ''Yates' order''. The <math>{2}^{3}\,\!</math> design is shown in figure (a) below. The design matrix for the <math>{2}^{3}\,\!</math> design is shown in figure (b). The design matrix can be constructed by following the standard order for the treatment combinations to obtain the columns for the main effects and then multiplying the main effects columns to obtain the interaction columns.

[[Image:doe7.3.png|center|324px|The <math>2^3\,\!</math> design. Figure (a) shows the experiment design and (b) shows the design matrix.]]

[[Image:doe7.4.png|center|290px|Geometric representation of the <math>2^3\,\!</math> design.]]

The <math>{2}^{3}\,\!</math> design can also be represented geometrically using a cube with the eight treatment combinations lying at the eight corners as shown in the figure above.

==Analysis of 2''k'' Designs==

The <math>{2}^{k}\,\!</math> designs are a special category of the factorial experiments where all the factors are at two levels. The fact that these designs contain factors at only two levels and are orthogonal greatly simplifies their analysis even when the number of factors is large. The use of <math>{2}^{k}\,\!</math> designs in investigating a large number of factors calls for a revision of the notation used previously for the ANOVA models. The case for revised notation is made stronger by the fact that the ANOVA and multiple linear regression models are identical for <math>{2}^{k}\,\!</math> designs because all factors are only at two levels. Therefore, the notation of the regression models is applied to the ANOVA models for these designs, as explained next.

===Notation===

Based on the notation used in [[General Full Factorial Designs]], the ANOVA model for a two level factorial experiment with three factors would be as follows:

::<math>\begin{align}
& Y= & \mu +{{\tau }_{1}}\cdot {{x}_{1}}+{{\delta }_{1}}\cdot {{x}_{2}}+{{(\tau \delta )}_{11}}\cdot {{x}_{1}}{{x}_{2}}+{{\gamma }_{1}}\cdot {{x}_{3}} \\
& & +{{(\tau \gamma )}_{11}}\cdot {{x}_{1}}{{x}_{3}}+{{(\delta \gamma )}_{11}}\cdot {{x}_{2}}{{x}_{3}}+{{(\tau \delta \gamma )}_{111}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where:
 
:• <math>\mu \,\!</math> represents the overall mean
:• <math>{{\tau }_{1}}\,\!</math> represents the independent effect of the first factor (factor <math>A\,\!</math>) out of the two effects <math>{{\tau }_{1}}\,\!</math> and <math>{{\tau }_{2}}\,\!</math>
:• <math>{{\delta }_{1}}\,\!</math> represents the independent effect of the second factor (factor <math>B\,\!</math>) out of the two effects <math>{{\delta }_{1}}\,\!</math> and <math>{{\delta }_{2}}\,\!</math>
:• <math>{{(\tau \delta )}_{11}}\,\!</math> represents the independent effect of the interaction <math>AB\,\!</math> out of the other interaction effects
:• <math>{{\gamma }_{1}}\,\!</math> represents the effect of the third factor (factor <math>C\,\!</math>) out of the two effects <math>{{\gamma }_{1}}\,\!</math> and <math>{{\gamma }_{2}}\,\!</math>
:• <math>{{(\tau \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>AC\,\!</math> out of the other interaction effects
:• <math>{{(\delta \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>BC\,\!</math> out of the other interaction effects
:• <math>{{(\tau \delta \gamma )}_{111}}\,\!</math> represents the effect of the interaction <math>ABC\,\!</math> out of the other interaction effects
and <math>\epsilon \,\!</math> is the random error term.

 
The notation for a linear regression model having three predictor variables with interactions is:

::<math>\begin{align}
& Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& & +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

The notation for the regression model is much more convenient, especially for the case when a large number of higher order interactions are present. In two level experiments, the ANOVA model requires only one indicator variable to represent each factor for both qualitative and quantitative factors. Therefore, the notation for the multiple linear regression model can be applied to the ANOVA model of the experiment that has all the factors at two levels. For example, for the experiment of the ANOVA model given above, <math>{{\beta }_{0}}\,\!</math> can represent the overall mean instead of <math>\mu \,\!</math>, and <math>{{\beta }_{1}}\,\!</math> can represent the independent effect, <math>{{\tau }_{1}}\,\!</math>, of factor <math>A\,\!</math>. Other main effects can be represented in a similar manner. The notation for the interaction effects is much more simplified (e.g., <math>{{\beta }_{123}}\,\!</math> can be used to represent the three factor interaction effect, <math>{{(\tau \beta \gamma )}_{111}}\,\!</math>).

As mentioned earlier, it is important to note that the coding for the indicator variables for the ANOVA models of two level factorial experiments is reversed from the coding followed in [[General Full Factorial Designs]]. Here <math>-1\,\!</math> represents the first level of the factor while <math>1\,\!</math> represents the second level. This is because for a two level factor a single variable is needed to represent the factor for both qualitative and quantitative factors. For quantitative factors, using <math>-1\,\!</math> for the first level (which is the low level) and 1 for the second level (which is the high level) keeps the coding consistent with the numerical value of the factors. The change in coding between the two coding schemes does not affect the analysis except that signs of the estimated effect coefficients will be reversed (i.e., numerical values of <math>{{\hat{\tau }}_{1}}\,\!</math>, obtained based on the coding of [[General Full Factorial Designs]], and <math>{{\hat{\beta }}_{1}}\,\!</math>, obtained based on the new coding, will be the same but their signs would be opposite).

::<math>\begin{align}
& & \text{Factor }A\text{ Coding (two level factor)} \\
& &
\end{align}\,\!</math>

::<math>\begin{matrix}
\text{Previous Coding} & {} & {} & {} & \text{Coding for }{{\text{2}}^{k}}\text{ Designs} \\
{} & {} & {} & {} & {} \\
Effect\text{ }{{\tau }_{1}}\ \ :\ \ {{x}_{1}}=1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{1}}\text{ (or }-{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=-1\text{ } \\
Effect\text{ }{{\tau }_{2}}\ \ :\ \ {{x}_{1}}=-1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{2}}\text{ (or }{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=1\text{ } \\
\end{matrix}\,\!</math>

In summary, the ANOVA model for the experiments with all factors at two levels is different from the ANOVA models for other experiments in terms of the notation in the following two ways:
 

:• The notation of the regression models is used for the effect coefficients.
:• The coding of the indicator variables is reversed.

===Special Features===

Consider the design matrix, <math>X\,\!</math>, for the <math>{2}^{3}\,\!</math> design discussed above. The (<math>{{X}^{\prime }}X\,\!</math>) <math>^{-1}\,\!</math> matrix is:

<center><math>{{({{X}^{\prime }}X)}^{-1}}=\left[ \begin{matrix}
0.125 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0.125 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0.125 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0.125 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0.125 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0.125 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0.125 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.125 \\
\end{matrix} \right]\,\!</math></center>

Notice that, due to the orthogonal design of the <math>X\,\!</math> matrix, the <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> has been simplified to a diagonal matrix which can be written as:

::<math>\begin{align}
{{({{X}^{\prime }}X)}^{-1}}= & 0.125\cdot I = & \frac{1}{8}\cdot I = & \frac{1}{{{2}^{3}}}\cdot I
\end{align}\,\!</math>

where <math>I\,\!</math> represents the identity matrix of the same order as the design matrix, <math>X\,\!</math>. Since there are eight observations per replicate of the <math>{2}^{3}\,\!</math> design, the <math>(X\,\!</math> ' <math>X{{)}^{-1}}\,\!</math> matrix for <math>m\,\!</math> replicates of this design can be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{3}}\cdot m)}\cdot I\,\!</math>

The <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> matrix for any <math>{2}^{k}\,\!</math> design can now be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{k}}\cdot m)}\cdot I\,\!</math>

Then the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is:

::<math>\begin{align}
C= & {{{\hat{\sigma }}}^{2}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & M{{S}_{E}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & \frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}\cdot I
\end{align}\,\!</math>

Note that the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is also a diagonal matrix. Therefore, the estimated effect coefficients (<math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math>, <math>{{\beta }_{12}},\,\!</math> etc.) for these designs are uncorrelated. This implies that the terms in the <math>{2}^{k}\,\!</math> design (main effects, interactions) are independent of each other. Consequently, the extra sum of squares for each of the terms in these designs is independent of the sequence of terms in the model, and also independent of the presence of other terms in the model. As a result the sequential and partial sum of squares for the terms are identical for these designs and will always add up to the model sum of squares. Multicollinearity is also not an issue for these designs.

It can also be noted from the equation given above, that in addition to the <math>C\,\!</math> matrix being diagonal, all diagonal elements of the <math>C\,\!</math> matrix are identical. This means that the variance (or its square root, the standard error) of all estimated effect coefficients are the same. The standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>, for all the coefficients is:

::<math>\begin{align}
se({{{\hat{\beta }}}_{j}})= & \sqrt{{{C}_{jj}}} = & \sqrt{\frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}}\text{ }for\text{ }all\text{ }j
\end{align}\,\!</math>

This property is used to construct the normal probability plot of effects in <math>{2}^{k}\,\!</math> designs and identify significant effects using graphical techniques. For details on the normal probability plot of effects in a Weibull++ DOE folio, refer to [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]].

====Example====
To illustrate the analysis of a full factorial <math>{2}^{k}\,\!</math> design, consider a three factor experiment to investigate the effect of honing pressure, number of strokes and cycle time on the surface finish of automobile brake drums. Each of these factors is investigated at two levels. The honing pressure is investigated at levels of 200 <math>psi\,\!</math> and 400 <math>psi\,\!</math>, the number of strokes used is 3 and 5 and the two levels of the cycle time are 3 and 5 seconds. The design for this experiment is set up in a Weibull++ DOE folio as shown in the first two following figures. It is decided to run two replicates for this experiment. The surface finish data collected from each run (using randomization) and the complete design is shown in the third following figure. The analysis of the experiment data is explained next.

[[Image:doe7_5.png|center|877px|Design properties for the experiment in the example.|link=]]

[[Image:doe7_6.png|center|859px|Design summary for the experiment in the example.|link=]]

[[Image:doe7_7.png|center|778px|Experiment design for the example to investigate the surface finish of automobile brake drums.|link=]]

The applicable model using the notation for <math>{2}^{k}\,\!</math> designs is:

::<math>\begin{align}
Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where the indicator variable, <math>{{x}_{1,}}\,\!</math> represents factor <math>A\,\!</math> (honing pressure), <math>{{x}_{1}}=-1\,\!</math> represents the low level of 200 <math>psi\,\!</math> and <math>{{x}_{1}}=1\,\!</math> represents the high level of 400 <math>psi\,\!</math>. Similarly, <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> represent factors <math>B\,\!</math> (number of strokes) and <math>C\,\!</math> (cycle time), respectively. <math>{{\beta }_{0}}\,\!</math> is the overall mean, while <math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math> and <math>{{\beta }_{3}}\,\!</math> are the effect coefficients for the main effects of factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. <math>{{\beta }_{12}}\,\!</math>, <math>{{\beta }_{13}}\,\!</math> and <math>{{\beta }_{23}}\,\!</math> are the effect coefficients for the <math>AB\,\!</math>, <math>AC\,\!</math> and <math>BC\,\!</math> interactions, while <math>{{\beta }_{123}}\,\!</math> represents the <math>ABC\,\!</math> interaction.

 
If the subscripts for the run (<math>i\,\!</math> ; <math>i=\,\!</math> 1 to 8) and replicates (<math>j\,\!</math> ; <math>j=\,\!</math> 1,2) are included, then the model can be written as:

::<math>\begin{align}
{{Y}_{ij}}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{ij1}}+{{\beta }_{2}}\cdot {{x}_{ij2}}+{{\beta }_{12}}\cdot {{x}_{ij1}}{{x}_{ij2}}+{{\beta }_{3}}\cdot {{x}_{ij3}} \\
& +{{\beta }_{13}}\cdot {{x}_{ij1}}{{x}_{ij3}}+{{\beta }_{23}}\cdot {{x}_{ij2}}{{x}_{ij3}}+{{\beta }_{123}}\cdot {{x}_{ij1}}{{x}_{ij2}}{{x}_{ij3}}+{{\epsilon }_{ij}}
\end{align}\,\!</math>

To investigate how the given factors affect the response, the following hypothesis tests need to be carried:

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{1}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{1}}\ne 0\,\!</math>

This test investigates the main effect of factor <math>A\,\!</math> (honing pressure). The statistic for this test is:

::<math>{{({{F}_{0}})}_{A}}=\frac{M{{S}_{A}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{A}}\,\!</math> is the mean square for factor <math>A\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other main effects, <math>B\,\!</math> and <math>C\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{12}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{12}}\ne 0\,\!</math>

This test investigates the two factor interaction <math>AB\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{AB}}=\frac{M{{S}_{AB}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the interaction <math>AB\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other two factor interactions, <math>AC\,\!</math> and <math>BC\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{123}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{123}}\ne 0\,\!</math>

This test investigates the three factor interaction <math>ABC\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{ABC}}=\frac{M{{S}_{ABC}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{ABC}}\,\!</math> is the mean square for the interaction <math>ABC\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square.
To calculate the test statistics, it is convenient to express the ANOVA model in the form <math>y=X\beta +\epsilon \,\!</math>.

====Expression of the ANOVA Model as <math>y=X\beta +\epsilon \,\!</math>====

In matrix notation, the ANOVA model can be expressed as:

::<math>y=X\beta +\epsilon \,\!</math>

where:

<center><math>y=\left[ \begin{matrix}
{{Y}_{11}} \\
{{Y}_{21}} \\
. \\
{{Y}_{81}} \\
{{Y}_{12}} \\
. \\
{{Y}_{82}} \\
\end{matrix} \right]=\left[ \begin{matrix}
90 \\
90 \\
. \\
90 \\
86 \\
. \\
80 \\
\end{matrix} \right]\text{ }X=\left[ \begin{matrix}
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\end{matrix} \right]\,\!</math></center>

<center><math>\beta =\left[ \begin{matrix}
{{\beta }_{0}} \\
{{\beta }_{1}} \\
{{\beta }_{2}} \\
{{\beta }_{12}} \\
{{\beta }_{3}} \\
{{\beta }_{13}} \\
{{\beta }_{23}} \\
{{\beta }_{123}} \\
\end{matrix} \right]\text{ }\epsilon =\left[ \begin{matrix}
{{\epsilon }_{11}} \\
{{\epsilon }_{21}} \\
. \\
{{\epsilon }_{81}} \\
{{\epsilon }_{12}} \\
. \\
. \\
{{\epsilon }_{82}} \\
\end{matrix} \right]\,\!</math></center>

====Calculation of the Extra Sum of Squares for the Factors====

Knowing the matrices <math>y\,\!</math>, <math>X\,\!</math> and <math>\beta \,\!</math>, the extra sum of squares for the factors can be calculated. These are used to calculate the mean squares that are used to obtain the test statistics. Since the experiment design is orthogonal, the partial and sequential extra sum of squares are identical. The extra sum of squares for each effect can be calculated as shown next. As an example, the extra sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & Model\text{ }Sum\text{ }of\text{ }Squares - Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }main\text{ }effect\text{ }of\text{ }A \\
= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }A}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }A}}={{X}_{\tilde{\ }A}}{{(X_{\tilde{\ }A}^{\prime }{{X}_{\tilde{\ }A}})}^{-1}}X_{\tilde{\ }A}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }A}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the main effect of factor <math>A\,\!</math>. Thus, the sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y \\
= & 654.4375-549.375 \\
= & 105.0625
\end{align}\,\!</math>

Similarly, the extra sum of squares for the interaction effect <math>AB\,\!</math> is:

::<math>\begin{align}
S{{S}_{AB}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }AB}}-(1/16)J]y \\
= & 654.4375-636.375 \\
= & 18.0625
\end{align}\,\!</math>

The extra sum of squares for other effects can be obtained in a similar manner.

====Calculation of the Test Statistics====

Knowing the extra sum of squares, the test statistic for the effects can be calculated. For example, the test statistic for the interaction <math>AB\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{AB}}= & \frac{M{{S}_{AB}}}{M{{S}_{E}}} \\
= & \frac{S{{S}_{AB}}/dof(S{{S}_{AB}})}{S{{S}_{E}}/dof(S{{S}_{E}})} \\
= & \frac{18.0625/1}{147.5/8} \\
= & 0.9797
\end{align}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the <math>AB\,\!</math> interaction and <math>M{{S}_{E}}\,\!</math> is the error mean square. The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{AB}}=0.9797\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{AB}}) \\
= & 1-0.6487 \\
= & 0.3513
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that the interaction between honing pressure and number of strokes does not affect the surface finish of the brake drums. Tests for other effects can be carried out in a similar manner. The results are shown in the ANOVA Table in the following figure. The values S, R-sq and R-sq(adj) in the figure indicate how well the model fits the data. The value of S represents the standard error of the model, R-sq represents the coefficient of multiple determination and R-sq(adj) represents the adjusted coefficient of multiple determination. For details on these values refer to [[Multiple_Linear_Regression_Analysis|Multiple Linear Regression Analysis]].

[[Image:doe7_8.png|center|799px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

====Calculation of Effect Coefficients====

The estimate of effect coefficients can also be obtained:

<center><math>\begin{align}
\hat{\beta }= & {{({{X}^{\prime }}X)}^{-1}}{{X}^{\prime }}y \\
= & \left[ \begin{matrix}
86.4375 \\
2.5625 \\
-4.9375 \\
1.0625 \\
-1.0625 \\
2.4375 \\
-1.3125 \\
-0.1875 \\
\end{matrix} \right]
\end{align}\,\!</math></center>

[[Image:doe7_9.png|center|800px|Regression Information table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

The coefficients and related results are shown in the Regression Information table above. In the table, the Effect column displays the effects, which are simply twice the coefficients. The Standard Error column displays the standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>. The Low CI and High CI columns display the confidence interval on the coefficients. The interval shown is the 90% interval as the significance is chosen as 0.1. The T Value column displays the <math>t\,\!</math> statistic, <math>{{t}_{0}}\,\!</math>, corresponding to the coefficients. The P Value column displays the <math>p\,\!</math> value corresponding to the <math>t\,\!</math> statistic. (For details on how these results are calculated, refer to [[General Full Factorial Designs]]). Plots of residuals can also be obtained from the DOE folio to ensure that the assumptions related to the ANOVA model are not violated.

====Model Equation====

From the analysis results in the above figure within [[Two_Level_Factorial_Experiments#Calculation_of_Effect_Coefficients|calculation of effect coefficients]] section, it is seen that effects <math>A\,\!</math>, <math>B\,\!</math> and <math>AC\,\!</math> are significant. In a DOE folio, the <math>p\,\!</math> values for the significant effects are displayed in red in the ANOVA Table for easy identification. Using the values of the estimated effect coefficients, the model for the present <math>{2}^{3}\,\!</math> design in terms of the coded values can be written as:

::<math>\begin{align}
\hat{y}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}} \\
= & 86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+2.4375{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

To make the model hierarchical, the main effect, <math>C\,\!</math>, needs to be included in the model (because the interaction <math>AC\,\!</math> is included in the model). The resulting model is:

::<math>\hat{y}=86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+1.0625{{x}_{3}}+2.4375{{x}_{1}}{{x}_{3}}\,\!</math>

This equation can be viewed in a DOE folio, as shown in the following figure, using the Show Analysis Summary icon in the Control Panel. The equation shown in the figure will match the hierarchical model once the required terms are selected using the Select Effects icon.

[[Image:doe7_10.png|center|557px|The model equation for the experiment of the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

==Replicated and Repeated Runs==

In the case of replicated experiments, it is important to note the difference between replicated runs and repeated runs. Both repeated and replicated runs are multiple response readings taken at the same factor levels. However, repeated runs are response observations taken at the same time or in succession. Replicated runs are response observations recorded in a random order. Therefore, replicated runs include more variation than repeated runs. For example, a baker, who wants to investigate the effect of two factors on the quality of cakes, will have to bake four cakes to complete one replicate of a <math>{2}^{2}\,\!</math> design. Assume that the baker bakes eight cakes in all. If, for each of the four treatments of the <math>{2}^{2}\,\!</math> design, the baker selects one treatment at random and then bakes two cakes for this treatment at the same time then this is a case of two repeated runs. If, however, the baker bakes all the eight cakes randomly, then the eight cakes represent two sets of replicated runs.
For repeated measurements, the average values of the response for each treatment should be entered into a DOE folio as shown in the following figure (a) when the two cakes for a particular treatment are baked together. For replicated measurements, when all the cakes are baked randomly, the data is entered as shown in the following figure (b).

[[Image:doe7.11.png|center|300px|Data entry for repeated and replicated runs. Figure (a) shows repeated runs and (b) shows replicated runs.]]

==Unreplicated 2''k'' Designs==

If a factorial experiment is run only for a single replicate then it is not possible to test hypotheses about the main effects and interactions as the error sum of squares cannot be obtained. This is because the number of observations in a single replicate equals the number of terms in the ANOVA model. Hence the model fits the data perfectly and no degrees of freedom are available to obtain the error sum of squares.

However, sometimes it is only possible to run a single replicate of the <math>{2}^{k}\,\!</math> design because of constraints on resources and time. In the absence of the error sum of squares, hypothesis tests to identify significant factors cannot be conducted. A number of methods of analyzing information obtained from unreplicated <math>{2}^{k}\,\!</math> designs are available. These include pooling higher order interactions, using the normal probability plot of effects or including center point replicates in the design.

===Pooling Higher Order Interactions===

One of the ways to deal with unreplicated <math>{2}^{k}\,\!</math> designs is to use the sum of squares of some of the higher order interactions as the error sum of squares provided these higher order interactions can be assumed to be insignificant. By dropping some of the higher order interactions from the model, the degrees of freedom corresponding to these interactions can be used to estimate the error mean square. Once the error mean square is known, the test statistics to conduct hypothesis tests on the factors can be calculated.

===Normal Probability Plot of Effects===
Another way to use unreplicated <math>{2}^{k}\,\!</math> designs to identify significant effects is to construct the normal probability plot of the effects. As mentioned in [[Two_Level_Factorial_Experiments#Special_Features| Special Features]], the standard error for all effect coefficients in the <math>{2}^{k}\,\!</math> designs is the same. Therefore, on a normal probability plot of effect coefficients, all non-significant effect coefficients (with <math>\beta =0\,\!</math>) will fall along the straight line representative of the normal distribution, N(<math>0,{{\sigma }^{2}}/({{2}^{k}}\cdot m)\,\!</math>). Effect coefficients that show large deviations from this line will be significant since they do not come from this normal distribution. Similarly, since effects <math>=2\times \,\!</math> effect coefficients, all non-significant effects will also follow a straight line on the normal probability plot of effects. For replicated designs, the Effects Probability plot of DOE++ plots the normalized effect values (or the T Values) on the standard normal probability line, N(0,1). However, in the case of unreplicated <math>{2}^{k}\,\!</math> designs, <math>{{\sigma }^{2}}\,\!</math> remains unknown since <math>M{{S}_{E}}\,\!</math> cannot be obtained. Lenth's method is used in this case to estimate the variance of the effects. For details on Lenth's method, please refer to [[DOE References| Montgomery (2001)]]. DOE++ then uses this variance value to plot effects along the N(0, Lenth's effect variance) line. The
method is illustrated in the following example.

====Example====

Vinyl panels, used as instrument panels in a certain automobile, are seen to develop defects after a certain amount of time. To investigate the issue, it is decided to carry out a two level factorial experiment. Potential factors to be investigated in the experiment are vacuum rate (factor <math>A\,\!</math>), material temperature (factor <math>B\,\!</math>), element intensity (factor <math>C\,\!</math>) and pre-stretch (factor <math>D\,\!</math>). The two levels of the factors used in the experiment are as shown in below.

[[Image:doet7.1.png|center|300px|Factors to investigate defects in vinyl panels.]]

With a <math>{2}^{4}\,\!</math> design requiring 16 runs per replicate it is only feasible for the manufacturer to run a single replicate.

The experiment design and data, collected as percent defects, are shown in the following figure. Since the present experiment design contains only a single replicate, it is not possible to obtain an estimate of the error sum of squares, <math>S{{S}_{E}}\,\!</math>. It is decided to use the normal probability plot of effects to identify the significant effects. The effect values for each term are obtained as shown in the following figure.

[[Image:doe7_13.png|center|650px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

Lenth's method uses these values to estimate the variance. As described in [[DOE_References|[Lenth, 1989]]], if all effects are arranged in ascending order, using their absolute values, then <math>{{s}_{0}}\,\!</math> is defined as 1.5 times the median value:

::<math>\begin{align}
{{s}_{0}}= & 1.5\cdot median(\left| effect \right|) \\
= & 1.5\cdot 2 \\
= & 3
\end{align}\,\!</math>

Using <math>{{s}_{0}}\,\!</math>, the "pseudo standard error" (<math>PSE\,\!</math>) is calculated as 1.5 times the median value of all effects that are less than 2.5 <math>{{s}_{0}}\,\!</math> :

::<math>\begin{align}
PSE= & 1.5\cdot median(\left| effect \right|\ \ :\ \ \left| effect \right|<2.5{{s}_{0}}) \\
= & 1.5\cdot 1.5 \\
= & 2.25
\end{align}\,\!</math>

Using <math>PSE\,\!</math> as an estimate of the effect variance, the effect variance is 2.25. Knowing the effect variance, the normal probability plot of effects for the present unreplicated experiment can be constructed as shown in the following figure. The line on this plot is the line N(0, 2.25). The plot shows that the effects <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math> do not follow the distribution represented by this line. Therefore, these effects are significant.

The significant effects can also be identified by comparing individual effect values to the margin of error or the threshold value using the pareto chart (see the third following figure). If the required significance is 0.1, then:

::<math>margin\text{ }of\text{ }error={{t}_{\alpha /2,d}}\cdot PSE\,\!</math>

The <math>t\,\!</math> statistic, <math>{{t}_{\alpha /2,d}}\,\!</math>, is calculated at a significance of <math>\alpha /2\,\!</math> (for the two-sided hypothesis) and degrees of freedom <math>d=(\,\!</math> number of effects <math>)/3\,\!</math>. Thus:

::<math>\begin{align}
margin\text{ }of\text{ }error= & {{t}_{0.05,5}}\cdot PSE \\
= & 2.015\cdot 2.25 \\
= & 4.534
\end{align}\,\!</math>

The value of 4.534 is shown as the critical value line in the third following figure. All effects with absolute values greater than the margin of error can be considered to be significant. These effects are <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math>. Therefore, the vacuum rate, the pre-stretch and their interaction have a significant effect on the defects of the vinyl panels.

[[Image:doe7_14.png|center|800px|Effect values for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_15.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_16.png|center|650px|Pareto chart for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

===Center Point Replicates===

Another method of dealing with unreplicated <math>{2}^{k}\,\!</math> designs that only have quantitative factors is to use replicated runs at the center point. The center point is the response corresponding to the treatment exactly midway between the two levels of all factors. Running multiple replicates at this point provides an estimate of pure error. Although running multiple replicates at any treatment level can provide an estimate of pure error, the other advantage of running center point replicates in the <math>{2}^{k}\,\!</math> design is in checking for the presence of curvature. The test for curvature investigates whether the model between the response and the factors is linear and is discussed in [[Two_Level_Factorial_Experiments#Using_Center_Point_Replicates_to_Test_Curvature| Center Pt. Replicates to Test Curvature]].

====Example: Use Center Point to Get Pure Error====

Consider a <math>{2}^{2}\,\!</math> experiment design to investigate the effect of two factors, <math>A\,\!</math> and <math>B\,\!</math>, on a certain response. The energy consumed when the treatments of the <math>{2}^{2}\,\!</math> design are run is considerably larger than the energy consumed for the center point run (because at the center point the factors are at their middle levels). Therefore, the analyst decides to run only a single replicate of the design and augment the design by five replicated runs at the center point as shown in the following figure. The design properties for this experiment are shown in the second following figure. The complete experiment design is shown in the third following figure. The center points can be used in the identification of significant effects as shown next.

[[Image:doe7.17.png||center|300px|<math>2^2\,\!</math> design augmented by five center point runs.]]
[[Image:doe7_18.png|center|537px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

[[Image:doe7_19.png|center|630px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

Since the present <math>{2}^{2}\,\!</math> design is unreplicated, there are no degrees of freedom available to calculate the error sum of squares. By augmenting this design with five center points, the response values at the center points, <math>y_{i}^{c}\,\!</math>, can be used to obtain an estimate of pure error, <math>S{{S}_{PE}}\,\!</math>. Let <math>{{\bar{y}}^{c}}\,\!</math> represent the average response for the five replicates at the center. Then:

::<math>S{{S}_{PE}}=Sum\text{ }of\text{ }Squares\text{ }for\text{ }center\text{ }points\,\!</math>

::<math>\begin{align}
S{{S}_{PE}}= & \underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}} \\
= & {{(25.2-25.26)}^{2}}+...+{{(25.3-25.26)}^{2}} \\
= & 0.052
\end{align}\,\!</math>

Then the corresponding mean square is:

::<math>\begin{align}
M{{S}_{PE}}= & \frac{S{{S}_{PE}}}{degrees\text{ }of\text{ }freedom} \\
= & \frac{0.052}{5-1} \\
= & 0.013
\end{align}\,\!</math>

Alternatively, <math>M{{S}_{PE}}\,\!</math> can be directly obtained by calculating the variance of the response values at the center points:

::<math>\begin{align}
M{{S}_{PE}}= & {{s}^{2}} \\
= & \frac{\underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}}}{5-1}
\end{align}\,\!</math>

Once <math>M{{S}_{PE}}\,\!</math> is known, it can be used as the error mean square, <math>M{{S}_{E}}\,\!</math>, to carry out the test of significance for each effect. For example, to test the significance of the main effect of factor <math>A,\,\!</math> the sum of squares corresponding to this effect is obtained in the usual manner by considering only the four runs of the original <math>{2}^{2}\,\!</math> design.

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/4)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/4)J]y \\
= & 0.5625
\end{align}\,\!</math>

Then, the test statistic to test the significance of the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{A}}= & \frac{M{{S}_{A}}}{M{{S}_{E}}} \\
= & \frac{0.5625/1}{0.052/4} \\
= & 43.2692
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{A}}=43.2692\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{A}}) \\
= & 1-0.9972 \\
= & 0.0028
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value < 0.1, it can be concluded that the main effect of factor <math>A\,\!</math> significantly affects the response. This result is displayed in the ANOVA table as shown in the following figure. Test for the significance of other factors can be carried out in a similar manner.

[[Image:doe7_20.png|center|649px|Results for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

===Using Center Point Replicates to Test Curvature===

Center point replicates can also be used to check for curvature in replicated or unreplicated <math>{2}^{k}\,\!</math> designs. The test for curvature investigates whether the model between the response and the factors is linear. The way DOE++ handles center point replicates is similar to its handling of blocks. The center point replicates are treated as an additional factor in the model. The factor is labeled as Curvature in the results of DOE++. If Curvature turns out to be a significant factor in the results, then this indicates the presence of curvature in the model.

====Example: Use Center Point to Test Curvature====

To illustrate the use of center point replicates in testing for curvature, consider again the data of the single replicate <math>{2}^{2}\,\!</math> experiment from a preceding figure(labeled "<math>2^2</math> design augmented by five center point runs"). Let <math>{{x}_{1}}\,\!</math> be the indicator variable to indicate if the run is a center point:

::<math>\begin{matrix}
{{x}_{1}}=0 & {} & \text{Center point run} \\
{{x}_{1}}=1 & {} & \text{Other run} \\
\end{matrix}\,\!</math>

If <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> are the indicator variables representing factors <math>A\,\!</math> and <math>B\,\!</math>, respectively, then the model for this experiment is:

::<math>Y={{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}\,\!</math>

To investigate the presence of curvature, the following hypotheses need to be tested:

::<math>\begin{align}
& {{H}_{0}}: & {{\beta }_{1}}=0\text{ (Curvature is absent)} \\
& {{H}_{1}}: & {{\beta }_{1}}\ne 0
\end{align}\,\!</math>

The test statistic to be used for this test is:

::<math>{{({{F}_{0}})}_{curvature}}=\frac{M{{S}_{curvature}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{curvature}}\,\!</math> is the mean square for Curvature and <math>M{{S}_{E}}\,\!</math> is the error mean square.

'''Calculation of the Sum of Squares'''

The <math>X\,\!</math> matrix and <math>y\,\!</math> vector for this experiment are:

<center><math>X=\left[ \begin{matrix}
1 & 1 & -1 & -1 & 1 \\
1 & 1 & 1 & -1 & -1 \\
1 & 1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
\end{matrix} \right]\text{ }y=\left[ \begin{matrix}
24.6 \\
25.4 \\
25.0 \\
25.7 \\
25.2 \\
25.3 \\
25.4 \\
25.1 \\
25.3 \\
\end{matrix} \right]\,\!</math></center>

The sum of squares can now be calculated. For example, the error sum of squares is:

::<math>\begin{align}
& S{{S}_{E}}= & {{y}^{\prime }}[I-H]y \\
& = & 0.052
\end{align}\,\!</math>

where <math>I\,\!</math> is the identity matrix and <math>H\,\!</math> is the hat matrix. It can be seen that this is equal to <math>S{{S}_{PE\text{ }}}\,\!</math> (the sum of squares due to pure error) because of the replicates at the center point, as obtained in the [[Two_Level_Factorial_Experiments#Example_3| example]]. The number of degrees of freedom associated with <math>S{{S}_{E}}\,\!</math>, <math>dof(S{{S}_{E}})\,\!</math> is four. The extra sum of squares corresponding to the center point replicates (or Curvature) is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & Model\text{ }Sum\text{ }of\text{ }Squares- \\
& & Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }center\text{ }point \\
& = & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Curvature}}-(1/9)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }Curvature}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }Curvature}}={{X}_{\tilde{\ }Curv}}{{(X_{\tilde{\ }Curv}^{\prime }{{X}_{\tilde{\ }Curv}})}^{-1}}X_{\tilde{\ }Curv}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }Curv}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the center point. Thus, the extra sum of squares corresponding to Curvature is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Center}}-(1/9)J]y \\
& = & 0.7036-0.6875 \\
& = & 0.0161
\end{align}\,\!</math>

This extra sum of squares can be used to test for the significance of curvature. The corresponding mean square is:

::<math>\begin{align}
& M{{S}_{Curvature}}= & \frac{Sum\text{ }of\text{ }squares\text{ }corresponding\text{ }to\text{ }Curvature}{degrees\text{ }of\text{ }freedom} \\
& = & \frac{0.0161}{1} \\
& = & 0.0161
\end{align}\,\!</math>

'''Calculation of the Test Statistic'''

Knowing the mean squares, the statistic to check the significance of curvature can be calculated.

::<math>\begin{align}
& {{({{f}_{0}})}_{Curvature}}= & \frac{M{{S}_{Curvature}}}{M{{S}_{E}}} \\
& = & \frac{0.0161/1}{0.052/4} \\
& = & 1.24
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{Curvature}}=1.24\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and four degrees of freedom in the denominator is:

::<math>\begin{align}
& p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{Curvature}}) \\
& = & 1-0.6713 \\
& = & 0.3287
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that curvature does not exist for this design. This results is shown in the ANOVA table in the figure above. The surface of the fitted model based on these results, along with the observed response values, is shown in the figure below.

[[Image:doe7.21.png|center|400px|Model surface and observed response values for the design in the [[Two_Level_Factorial_Experiments#Example_4| example]].]]

 

==Blocking in 2k Designs==

Blocking can be used in the <math>{2}^{k}\,\!</math> designs to deal with cases when replicates cannot be run under identical conditions. Randomized complete block designs that were discussed in [[Randomization and Blocking in DOE]] for factorial experiments are also applicable here. At times, even with just two levels per factor, it is not possible to run all treatment combinations for one replicate of the experiment under homogeneous conditions. For example, each replicate of the <math>{2}^{2}\,\!</math> design requires four runs. If each run requires two hours and testing facilities are available for only four hours per day, two days of testing would be required to run one complete replicate. Blocking can be used to separate the treatment runs on the two different days. Blocks that do not contain all treatments of a replicate are called incomplete blocks. In incomplete block designs, the block effect is confounded with certain effect(s) under investigation. For the <math>{2}^{2}\,\!</math> design assume that treatments <math>(1)\,\!</math> and <math>ab\,\!</math> were run on the first day and treatments <math>a\,\!</math> and <math>b\,\!</math> were run on the second day. Then, the incomplete block design for this experiment is:

::<math>\begin{matrix}
\text{Block 1} & {} & \text{Block 2} \\
\left[ \begin{matrix}
(1) \\
ab \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
b \\
\end{matrix} \right] \\
\end{matrix}\,\!</math>

For this design the block effect may be calculated as:

::<math>\begin{align}
& Block\text{ }Effect= & Average\text{ }response\text{ }for\text{ }Block\text{ }1- \\
& & Average\text{ }response\text{ }for\text{ }Block\text{ }2 \\
& = & \frac{(1)+ab}{2}-\frac{a+b}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The <math>AB\,\!</math> interaction effect is:

::<math>\begin{align}
& AB= & Average\text{ }response\text{ }at\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{low}}}- \\
& & Average\text{ }response\text{ }at\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{low}}} \\
& = & \frac{ab+(1)}{2}-\frac{b+a}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The two equations given above show that, in this design, the <math>AB\,\!</math> interaction effect cannot be distinguished from the block effect because the formulas to calculate these effects are the same. In other words, the <math>AB\,\!</math> interaction is said to be confounded with the block effect and it is not possible to say if the effect calculated based on these equations is due to the <math>AB\,\!</math> interaction effect, the block effect or both. In incomplete block designs some effects are always confounded with the blocks. Therefore, it is important to design these experiments in such a way that the important effects are not confounded with the blocks. In most cases, the experimenter can assume that higher order interactions are unimportant. In this case, it would better to use incomplete block designs that confound these effects with the blocks.
One way to design incomplete block designs is to use defining contrasts as shown next:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}+...+{{\alpha }_{k}}{{q}_{k}}\,\!</math>

where the <math>{{\alpha }_{i}}\,\!</math> s are the exponents for the factors in the effect that is to be confounded with the block effect and the <math>{{q}_{i}}\,\!</math> s are values based on the level of the <math>i\,\!</math> the factor (in a treatment that is to be allocated to a block). For <math>{2}^{k}\,\!</math> designs the <math>{{\alpha }_{i}}\,\!</math> s are either 0 or 1 and the <math>{{q}_{i}}\,\!</math> s have a value of 0 for the low level of the <math>i\,\!</math> th factor and a value of 1 for the high level of the factor in the treatment under consideration. As an example, consider the <math>{2}^{2}\,\!</math> design where the interaction effect <math>AB\,\!</math> is confounded with the block. Since there are two factors, <math>k=2\,\!</math>, with <math>i=1\,\!</math> representing factor <math>A\,\!</math> and <math>i=2\,\!</math> representing factor <math>B\,\!</math>. Therefore:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}\,\!</math>

The value of <math>{{\alpha }_{1}}\,\!</math> is one because the exponent of factor <math>A\,\!</math> in the confounded interaction <math>AB\,\!</math> is one. Similarly, the value of <math>{{\alpha }_{2}}\,\!</math> is one because the exponent of factor <math>B\,\!</math> in the confounded interaction <math>AB\,\!</math> is also one. Therefore, the defining contrast for this design can be written as:

::<math>\begin{align}
& L= & {{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}} \\
& = & 1\cdot {{q}_{1}}+1\cdot {{q}_{2}} \\
& = & {{q}_{1}}+{{q}_{2}}
\end{align}\,\!</math>

Once the defining contrast is known, it can be used to allocate treatments to the blocks. For the <math>{2}^{2}\,\!</math> design, there are four treatments <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math> and <math>ab\,\!</math>. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. In order to decide which block the treatment <math>(1)\,\!</math> belongs to, the levels of factors <math>A\,\!</math> and <math>B\,\!</math> for this run are used. Since factor <math>A\,\!</math> is at the low level in this treatment, <math>{{q}_{1}}=0\,\!</math>. Similarly, since factor <math>B\,\!</math> is also at the low level in this treatment, <math>{{q}_{2}}=0\,\!</math>. Therefore:

::<math>\begin{align}
& L= & {{q}_{1}}+{{q}_{2}} \\
& = & 0+0=0\text{ (mod 2)}
\end{align}\,\!</math>

Note that the value of <math>L\,\!</math> used to decide the block allocation is "mod 2" of the original value. This value is obtained by taking the value of 1 for odd numbers and 0 otherwise. Based on the value of <math>L\,\!</math>, treatment <math>(1)\,\!</math> is assigned to block 1. Other treatments can be assigned using the following calculations:

::<math>\begin{align}
& (1): & \text{ }L=0+0=0=0\text{ (mod 2)} \\
& a: & \text{ }L=1+0=1=1\text{ (mod 2)} \\
& b: & \text{ }L=0+1=1=1\text{ (mod 2)} \\
& ab: & \text{ }L=1+1=2=0\text{ (mod 2)}
\end{align}\,\!</math>

Therefore, to confound the interaction <math>AB\,\!</math> with the block effect in the <math>{2}^{2}\,\!</math> incomplete block design, treatments <math>(1)\,\!</math> and <math>ab\,\!</math> (with <math>L=0\,\!</math>) should be assigned to block 2 and treatment combinations <math>a\,\!</math> and <math>b\,\!</math> (with <math>L=1\,\!</math>) should be assigned to block 1.

====Example: Two Level Factorial Design with Two Blocks====

This example illustrates how treatments can be allocated to two blocks for an unreplicated <math>{2}^{k}\,\!</math> design. Consider the unreplicated <math>{2}^{4}\,\!</math> design to investigate the four factors affecting the defects in automobile vinyl panels discussed in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments required for this experiment were run by two different operators with each operator conducting 8 runs. This experiment is an example of an incomplete block design. The analyst in charge of this experiment assumed that the interaction <math>ABCD\,\!</math> was not significant and decided to allocate treatments to the two operators so that the <math>ABCD\,\!</math> interaction was confounded with the block effect (the two operators are the blocks). The allocation scheme to assign treatments to the two operators can be obtained as follows.
 
The defining contrast for the <math>{2}^{4}\,\!</math> design where the <math>ABCD\,\!</math> interaction is confounded with the blocks is:

::<math>L={{q}_{1}}+{{q}_{2}}+{{q}_{3}}+{{q}_{4}}\,\!</math>

The treatments can be allocated to the two operators using the values of the defining contrast. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. Then the value of the defining contrast for treatment <math>a\,\!</math> is:

::<math>a\ \ :\ \ \text{ }L=1+0+0+0=1=1\text{ (mod 2)}\,\!</math>

Therefore, treatment <math>a\,\!</math> should be assigned to Block 1 or the first operator. Similarly, for treatment <math>ab\,\!</math> we have:

::<math>ab\ \ :\ \ \text{ }L=1+1+0+0=2=0\text{ (mod 2)}\,\!</math>

[[Image:doe7.22.png|center|200px| Allocation of treatments to two blocks for the <math>2^4</math> design in the example by confounding interaction of <math>ABCD</math> with the blocks.]]

Therefore, <math>ab\,\!</math> should be assigned to Block 2 or the second operator. Other treatments can be allocated to the two operators in a similar manner to arrive at the allocation scheme shown in the figure below.
In DOE++, to confound the <math>ABCD\,\!</math> interaction for the <math>{2}^{4}\,\!</math> design into two blocks, the number of blocks are specified as shown in the figure below. Then the interaction <math>ABCD\,\!</math> is entered in the Block Generator window (second following figure) which is available using the Block Generator button in the following figure. The design generated by DOE++ is shown in the third of the following figures. This design matches the allocation scheme of the preceding figure.

[[Image:doe7_23.png|center|700px| Adding block properties for the experiment in the example.|link=]]

[[Image:doe7_24.png|center|700px|Specifying the interaction ABCD as the interaction to be confounded with the blocks for the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

[[Image:doe7_25.png|center|800px|Two block design for the experiment in the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

For the analysis of this design, the sum of squares for all effects are calculated assuming no blocking. Then, to account for blocking, the sum of squares corresponding to the <math>ABCD\,\!</math> interaction is considered as the sum of squares due to blocks and <math>ABCD\,\!</math>. In DOE++ this is done by displaying this sum of squares as the sum of squares due to the blocks. This is shown in the following figure where the sum of squares in question is obtained as 72.25 and is displayed against Block. The interaction ABCD, which is confounded with the blocks, is not displayed. Since the design is unreplicated, any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_26.png|center|793px|ANOVA table for the experiment of the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

===Unreplicated 2''k'' Designs in 2''p'' Blocks===

A single replicate of the <math>{2}^{k}\,\!</math> design can be run in up to <math>{2}^{p}\,\!</math> blocks where <math>p<k\,\!</math>. The number of effects confounded with the blocks equals the degrees of freedom associated with the block effect.

If two blocks are used (the block effect has two levels), then one (<math>2-1=1)\,\!</math> effect is confounded with the blocks. If four blocks are used, then three (<math>4-1=3\,\!</math>) effects are confounded with the blocks and so on. For example an unreplicated <math>{2}^{4}\,\!</math> design may be confounded in <math>{2}^{2}\,\!</math> (four) blocks using two contrasts, <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}}\,\!</math>. Let <math>AC\,\!</math> and <math>BD\,\!</math> be the effects to be confounded with the blocks. Corresponding to these two effects, the contrasts are respectively:

::<math>\begin{align}
& {{L}_{1}}= & {{q}_{1}}+{{q}_{3}} \\
& {{L}_{2}}= & {{q}_{2}}+{{q}_{4}}
\end{align}\,\!</math>

Based on the values of <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}},\,\!</math> the treatments can be assigned to the four blocks as follows:

<center><math>\begin{matrix}
\text{Block 4} & {} & \text{Block 3} & {} & \text{Block 2} & {} & \text{Block 1} \\
{{L}_{1}}=0,{{L}_{2}}=0 & {} & {{L}_{1}}=1,{{L}_{2}}=0 & {} & {{L}_{1}}=0,{{L}_{2}}=1 & {} & {{L}_{1}}=1,{{L}_{2}}=1 \\
{} & {} & {} & {} & {} & {} & {} \\
\left[ \begin{matrix}
(1) \\
ac \\
bd \\
abcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
c \\
abd \\
bcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
b \\
abc \\
d \\
acd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
ab \\
bc \\
ad \\
cd \\
\end{matrix} \right] \\
\end{matrix}\,\!</math></center>

Since the block effect has three degrees of freedom, three effects are confounded with the block effect. In addition to <math>AC\,\!</math> and <math>BD\,\!</math>, the third effect confounded with the block effect is their generalized interaction, <math>(AC)(BD)=ABCD\,\!</math>.
In general, when an unreplicated <math>{2}^{k}\,\!</math> design is confounded in <math>{2}^{p}\,\!</math> blocks, <math>p\,\!</math> contrasts are needed (<math>{{L}_{1}},{{L}_{2}}...{{L}_{p}}\,\!</math>). <math>p\,\!</math> effects are selected to define these contrasts such that none of these effects are the generalized interaction of the others. The <math>{2}^{p}\,\!</math> blocks can then be assigned the treatments using the <math>p\,\!</math> contrasts. <math>{{2}^{p}}-(p+1)\,\!</math> effects, that are also confounded with the blocks, are then obtained as the generalized interaction of the <math>p\,\!</math> effects. In the statistical analysis of these designs, the sum of squares are computed as if no blocking were used. Then the block sum of squares is obtained by adding the sum of squares for all the effects confounded with the blocks.

====Example: 2 Level Factorial Design with Four Blocks====

This example illustrates how DOE++ obtains the sum of squares when treatments for an unreplicated <math>{2}^{k}\,\!</math> design are allocated among four blocks. Consider again the unreplicated <math>{2}^{4}\,\!</math> design used to investigate the defects in automobile vinyl panels presented in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments needed to complete the experiment were run by four operators. Therefore, there are four blocks. Assume that the treatments were allocated to the blocks using the generators mentioned in the previous section, i.e., treatments were allocated among the four operators by confounding the effects, <math>AC\,\!</math> and <math>BD,\,\!</math> with the blocks. These effects can be specified as Block Generators as shown in the following figure. (The generalized interaction of these two effects, interaction <math>ABCD\,\!</math>, will also get confounded with the blocks.) The resulting design is shown in the second following figure and matches the allocation scheme obtained in the previous section.

[[Image:doe7_27.png|center|700px|Specifying the interactions AC and BD as block generators for the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

The sum of squares in this case can be obtained by calculating the sum of squares for each of the effects assuming there is no blocking. Once the individual sum of squares have been obtained, the block sum of squares can be calculated. The block sum of squares is the sum of the sum of squares of effects, <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, since these effects are confounded with the block effect. As shown in the second following figure, this sum of squares is 92.25 and is displayed against Block. The interactions <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, which are confounded with the blocks, are not displayed. Since the present design is unreplicated any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_28.png|center|800px|Design for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

[[Image:doe7_29.png|center|793px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

==Variability Analysis==

For replicated two level factorial experiments, DOE++ provides the option of conducting variability analysis (using the Variability Analysis icon under the Data menu). The analysis is used to identify the treatment that results in the least amount of variation in the product or process being investigated. Variability analysis is conducted by treating the standard deviation of the response for each treatment of the experiment as an additional response. The standard deviation for a treatment is obtained by using the replicated response values at that treatment run. As an example, consider the <math>{2}^{3}\,\!</math> design shown in the following figure where each run is replicated four times. A variability analysis can be conducted for this design. DOE++ calculates eight standard deviation values corresponding to each treatment of the design (see second following figure). Then, the design is analyzed as an unreplicated <math>{2}^{3}\,\!</math> design with the standard deviations (displayed as Y Standard Deviation. in second following figure) as the response. The normal probability plot of effects identifies <math>AC\,\!</math> as the effect that influences variability (see third figure following). Based on the effect coefficients obtained in the fourth figure following, the model for Y Std. is:

::<math>\begin{align}
& \text{Y Std}\text{.}= & 0.6779+0.2491\cdot AC \\
& = & 0.6779+0.2491{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

Based on the model, the experimenter has two choices to minimize variability (by minimizing Y Std.). The first choice is that <math>{{x}_{1}}\,\!</math> should be <math>1\,\!</math> (i.e., <math>A\,\!</math> should be set at the high level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the low level). The second choice is that <math>{{x}_{1}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>A\,\!</math> should be set at the low level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the high level). The experimenter can select the most feasible choice.

[[Image:doe7.30.png|center|391px|A <math>2^3\,\!</math> design with four replicated response values that can be used to conduct a variability analysis.]]
 

[[Image:doe7_31.png|center|727px|Variability analysis in DOE++.|link=]]
 

[[Image:doe7_32.png|center|650px|Normal probability plot of effects for the variability analysis example.|link=]]
 

[[Image:doe7_33.png|center|727px|Effect coefficients for the variability analysis example.|link=]]

 

==Two Level Fractional Factorial Designs==

As the number of factors in a two level factorial design increases, the number of runs for even a single replicate of the <math>{2}^{k}\,\!</math> design becomes very large. For example, a single replicate of an eight factor two level experiment would require 256 runs. Fractional factorial designs can be used in these cases to draw out valuable conclusions from fewer runs. The basis of fractional factorial designs is the ''sparsity of effects'' principle.[[DOE_References|[Wu, 2000]]] The principle states that, most of the time, responses are affected by a small number of main effects and lower order interactions, while higher order interactions are relatively unimportant. Fractional factorial designs are used as screening experiments during the initial stages of experimentation. At these stages, a large number of factors have to be investigated and the focus is on the main effects and two factor interactions. These designs obtain information about main effects and lower order interactions with fewer experiment runs by confounding these effects with unimportant higher order interactions. As an example, consider a <math>{2}^{8}\,\!</math> design that requires 256 runs. This design allows for the investigation of 8 main effects and 28 two factor interactions. However, 219 degrees of freedom are devoted to three factor or higher order interactions. This full factorial design can prove to be very inefficient when these higher order interactions can be assumed to be unimportant. Instead, a fractional design can be used here to identify the important factors that can then be investigated more thoroughly in subsequent experiments. In unreplicated fractional factorial designs, no degrees of freedom are available to calculate the error sum of squares and the techniques mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] should be employed for the analysis of these designs.

==Half-fraction Designs==

A half-fraction of the <math>{2}^{k}\,\!</math> design involves running only half of the treatments of the full factorial design. For example, consider a <math>{2}^{3}\,\!</math> design that requires eight runs in all. The design matrix for this design is shown in the figure (a) below. A half-fraction of this design is the design in which only four of the eight treatments are run. The fraction is denoted as <math>{2}^{3-1}\,\!</math> with the "<math>-1\,\!</math>" in the index denoting a half-fraction. Assume that the treatments chosen for the half-fraction design are the ones where the interaction <math>ABC\,\!</math> is at the high level (i.e., only those rows are chosen from the following figure (a) where the column for <math>ABC\,\!</math> has entries of 1). The resulting <math>{2}^{3-1}\,\!</math> design has a design matrix as shown in figure (b) below.

[[Image:doe7.34.png|center|330px|Half-fractions of the <math>2^3\,\!</math> design. (a) shows the full factorial <math>2^3\,\!</math> design, (b) shows the <math>2^{3-1}\,\!</math> design with the defining relation <math>I=ABC\,\!</math> and (c) shows the <math>{2}^{3-1}\,\!</math> design with the defining relation <math>I=-ABC\,\!</math>.]]

In the <math>{2}^{3-1}\,\!</math> design of figure (b), since the interaction <math>ABC\,\!</math> is always included at the same level (the high level represented by 1), it is not possible to measure this interaction effect. The effect, <math>ABC\,\!</math>, is called the ''generator'' or ''word'' for this design. It can be noted that, in the design matrix of the following figure (b), the column corresponding to the intercept, <math>I\,\!</math>, and column corresponding to the interaction <math>ABC\,\!</math>, are identical. The identical columns are written as <math>I=ABC\,\!</math> and this equation is called the ''defining relation'' for the design. In DOE++, the present <math>{2}^{3-1}\,\!</math> design can be obtained by specifying the design properties as shown in the following figure.

[[Image:doe7_35.png|center|700px|Design properties for the <math>2^{3-1}\,\!</math> design.|link=]]

The defining relation, <math>I=ABC\,\!</math>, is entered in the Fraction Generator window as shown next.

[[Image:doe7_36.png|center|700px|Specifying the defining relation for the <math>2^{3-1}\,\!</math> design.|link=]]

Note that in the figure following that, the defining relation is specified as <math>C=AB\,\!</math>. This relation is obtained by multiplying the defining relation, <math>I=ABC\,\!</math>, by the last factor, <math>C\,\!</math>, of the design.

===Calculation of Effects===

Using the four runs of the <math>{2}^{3-1}\,\!</math> design in figure (b) discussed above, the main effects can be calculated as follows:

::<math>\begin{align}
& A= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& B= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& C= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

where <math>a\,\!</math>, <math>b\,\!</math>, <math>c\,\!</math> and <math>abc\,\!</math> are the treatments included in the <math>{2}^{3-1}\,\!</math> design.

Similarly, the two factor interactions can also be obtained as:

::<math>\begin{align}
& BC= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& AC= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& AB= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

The equations for <math>A\,\!</math> and <math>BC\,\!</math> above result in the same effect values showing that effects <math>A\,\!</math> and <math>BC\,\!</math> are confounded in the present <math>{2}^{3-1}\,\!</math> design. Thus, the quantity, <math>\tfrac{1}{2}(a-b-c+abc),\,\!</math> estimates <math>A+BC\,\!</math> (i.e., both the main effect <math>A\,\!</math> and the two-factor interaction <math>BC\,\!</math>). The effects, <math>A\,\!</math> and <math>BC,\,\!</math> are called ''aliases''. From the remaining equations given above, it can be seen that the other aliases for this design are <math>B\,\!</math> and <math>AC\,\!</math>, and <math>C\,\!</math> and <math>AB\,\!</math>. Therefore, the equations to calculate the effects in the present <math>{2}^{3-1}\,\!</math> design can be written as follows:

::<math>\begin{align}
& A+BC= & \frac{1}{2}(a-b-c+abc) \\
& B+AC= & \frac{1}{2}(-a+b-c+abc) \\
& C+AB= & \frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

===Calculation of Aliases===

Aliases for a fractional factorial design can be obtained using the defining relation for the design. The defining relation for the present <math>{2}^{3-1}\,\!</math> design is:

::<math>I=ABC\,\!</math>

Multiplying both sides of the previous equation by the main effect, <math>A,\,\!</math> gives the alias effect of <math>A\,\!</math> :

::<math>\begin{align}
& A\cdot I= & A\cdot ABC \\
& A= & {{A}^{2}}BC \\
& A= & BC
\end{align}\,\!</math>

Note that in calculating the alias effects, any effect multiplied by <math>I\,\!</math> remains the same (<math>A\cdot I=A\,\!</math>), while an effect multiplied by itself results in <math>I\,\!</math> (<math>{{A}^{2}}=I\,\!</math>). Other aliases can also be obtained:

::<math>\begin{align}
& B\cdot I= & B\cdot ABC \\
& B= & A{{B}^{2}}C \\
& B= & AC
\end{align}\,\!</math>

:and:

::<math>\begin{align}
& C\cdot I= & C\cdot ABC \\
& C= & AB{{C}^{2}} \\
& C= & AB
\end{align}\,\!</math>

===Fold-over Design===

If it can be assumed for this design that the two-factor interactions are unimportant, then in the absence of <math>BC\,\!</math>, <math>AC\,\!</math> and <math>AB\,\!</math>, the equations for (A+BC), (B+AC) and (C+AB) can be used to estimate the main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. However, if such an assumption is not applicable, then to uncouple the main effects from their two factor aliases, the alternate fraction that contains runs having <math>ABC\,\!</math> at the lower level should be run. The design matrix for this design is shown in the preceding figure (c). The defining relation for this design is <math>I=-ABC\,\!</math> because the four runs for this design are obtained by selecting the rows of the preceding figure (a) for which the value of the <math>ABC\,\!</math> column is <math>-1\,\!</math>. The aliases for this fraction can be obtained as explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]] as <math>A=-BC\,\!</math>, <math>B=-AC\,\!</math> and <math>C=-AB\,\!</math>. The effects for this design can be calculated as:

::<math>\begin{align}
& A-BC= & \frac{1}{2}(ab+ac-(1)-bc) \\
& B-AC= & \frac{1}{2}(ab-ac+(1)-bc) \\
& C-AB= & \frac{1}{2}(-ab+ac-(1)+bc)
\end{align}\,\!</math>

These equations can be combined with the equations for (A+BC), (B+AC) and (C+AB) to obtain the de-aliased main effects and two factor interactions. For example, adding equations (A+BC) and (A-BC) returns the main effect <math>A\,\!</math>.

::<math>\begin{align}
& 2A= & \frac{1}{2}(a-b-c+abc)+ \\
& & \frac{1}{2}(ab+ac-(1)-bc)
\end{align}\,\!</math>

The process of augmenting a fractional factorial design by a second fraction of the same size by simply reversing the signs (of all effect columns except <math>I\,\!</math>) is called ''folding over''. The combined design is referred to as a ''fold-over design''.

==Quarter and Smaller Fraction Designs==

At times, the number of runs even for a half-fraction design are very large. In these cases, smaller fractions are used. A quarter-fraction design, denoted as <math>{2}^{k-2}\,\!</math>, consists of a fourth of the runs of the full factorial design. Quarter-fraction designs require two defining relations. The first defining relation returns the half-fraction or the <math>{2}^{k-1}\,\!</math> design. The second defining relation selects half of the runs of the <math>{2}^{k-1}\,\!</math> design to give the quarter-fraction. For example, consider the <math>{2}^{4}\,\!</math> design. To obtain a <math>{2}^{4-2}\,\!</math> design from this design, first a half-fraction of this design is obtained by using a defining relation. Assume that the defining relation used is <math>I=ABCD\,\!</math>. The design matrix for the resulting <math>{2}^{4-1}\,\!</math> design is shown in figure (a) below. Now, a quarter-fraction can be obtained from the <math>{2}^{4-1}\,\!</math> design shown in figure (a) below using a second defining relation <math>I=AD\,\!</math>. The resulting <math>{2}^{4-2}\,\!</math> design obtained is shown in figure (b) below.

[[Image:doe7.37.png|center|465px|Fractions of the <math>2^4\,\!</math> design - Figure (a) shows the <math>2^{4-1}</math> design with the defining relation <math>I=ABCD\,\!</math> and (b) shows the <math>2^{4-2}\,\!</math> design with the defining relation <math>I=ABCD=AD=BC\,\!</math>.]]

The complete defining relation for this <math>{2}^{4-2}\,\!</math> design is:

::<math>I=ABCD=AD=BC\,\!</math>

Note that the effect, <math>BC,\,\!</math> in the defining relation is the generalized interaction of <math>ABCD\,\!</math> and <math>AD\,\!</math> and is obtained using <math>(ABCD)(AD)={{A}^{2}}BC{{D}^{2}}=BC\,\!</math>. In general, a <math>{2}^{k-p}\,\!</math> fractional factorial design requires <math>p\,\!</math> independent generators. The defining relation for the design consists of the <math>p\,\!</math> independent generators and their <math>{2}^{p}\,\!</math> - (<math>p\,\!</math> +1) generalized interactions.

===Calculation of Aliases===

The alias structure for the present <math>{2}^{4-2}\,\!</math> design can be obtained using the defining relation of equation (I=ABCD=AD=BC) following the procedure explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]]. For example, multiplying the defining relation by <math>A\,\!</math> returns the effects aliased with the main effect, <math>A\,\!</math>, as follows:

::<math>\begin{align}
& A\cdot I= & A\cdot ABCD=A\cdot AD=A\cdot BC \\
& A= & {{A}^{2}}BCD={{A}^{2}}D=ABC \\
& A= & BCD=D=ABC
\end{align}\,\!</math>

Therefore, in the present <math>{2}^{4-2}\,\!</math> design, it is not possible to distinguish between effects <math>A\,\!</math>, <math>D\,\!</math>, <math>BCD\,\!</math> and <math>ABC\,\!</math>. Similarly, multiplying the defining relation by <math>B\,\!</math> and <math>AB\,\!</math> returns the effects that are aliased with these effects:

::<math>\begin{align}
& B= & ACD=ABD=C \\
& AB= & CD=AD=AC
\end{align}\,\!</math>

Other aliases can be obtained in a similar way. It can be seen that each effect in this design has three aliases. In general, each effect in a <math>{2}^{k-p}\,\!</math> design has <math>{2}^{p-1}\,\!</math> aliases.
The aliases for the <math>{2}^{4-2}\,\!</math> design show that in this design the main effects are aliased with each other (<math>A\,\!</math> is aliased with <math>D\,\!</math> and <math>B\,\!</math> is aliased with <math>C\,\!</math>). Therefore, this design is not a useful design and is not available in DOE++. It is important to ensure that main effects and lower order interactions of interest are not aliased in a fractional factorial design. This is known by looking at the resolution of the fractional factorial design.

==Design Resolution==

The resolution of a fractional factorial design is defined as the number of factors in the lowest order effect in the defining relation. For example, in the defining relation <math>I=ABCD=AD=BC\,\!</math> of the previous <math>{2}^{4-2}\,\!</math> design, the lowest-order effect is either <math>AD\,\!</math> or <math>BC,\,\!</math> containing two factors. Therefore, the resolution of this design is equal to two. The resolution of a fractional factorial design is represented using Roman numerals. For example, the previously mentioned <math>{2}^{4-2}\,\!</math> design with a resolution of two can be represented as 2 <math>_{\text{II}}^{4-2}\,\!</math>. The resolution provides information about the confounding in the design as explained next:
 
 
#'''Resolution III Designs''' In these designs, the lowest order effect in the defining relation has three factors (e.g., a <math>{2}^{5-2}\,\!</math> design with the defining relation <math>I=ABDE=ABC=CDE\,\!</math>). In resolution III designs, no main effects are aliased with any other main effects, but main effects are aliased with two factor interactions. In addition, some two factor interactions are aliased with each other.
#'''Resolution IV Designs''' In these designs, the lowest order effect in the defining relation has four factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABDE\,\!</math>). In resolution IV designs, no main effects are aliased with any other main effects or two factor interactions. However, some main effects are aliased with three factor interactions and the two factor interactions are aliased with each other.
#'''Resolution V Designs''' In these designs the lowest order effect in the defining relation has five factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABCDE\,\!</math>). In resolution V designs, no main effects or two factor interactions are aliased with any other main effects or two factor interactions. However, some main effects are aliased with four factor interactions and the two factor interactions are aliased with three factor interactions.

Fractional factorial designs with the highest resolution possible should be selected because the higher the resolution of the design, the less severe the degree of confounding. In general, designs with a resolution less than III are never used because in these designs some of the main effects are aliased with each other. The table below shows fractional factorial designs with the highest available resolution for three to ten factor designs along with their defining relations.

[[Image:doet7.3.png|center|474px|Highest resolution designs available for fractional factorial designs with 3 to 10 factors.]]

All of the two level fractional factorial designs available in DOE++ are shown next.

[[Image:doe7.38.png|center|471px|Two level fractional factorial designs available in DOE++ and their resolutions.]]

===Minimum Aberration Designs===
At times, different designs with the same resolution but different aliasing may be available. The best design to select in such a case is the minimum aberration design. For example, all <math>{2}^{7-2}\,\!</math> designs in the fourth table have a resolution of four (since the generator with the minimum number of factors in each design has four factors). Design <math>1\,\!</math> has three generators of length four (<math>ABCF,\,\!</math> <math>BCDG,\,\!</math> <math>ADFG\,\!</math>). Design <math>2\,\!</math> has two generators of length four (<math>ABCF,\,\!</math> <math>ADEG\,\!</math>). Design <math>3\,\!</math> has one generator of length four (<math>CEFG\,\!</math>). Therefore, design <math>3\,\!</math> has the least number of generators with the minimum length of four. Design <math>3\,\!</math> is called the minimum aberration design. It can be seen that the alias structure for design <math>3\,\!</math> is less involved compared to the other designs. For details refer to [[DOE_References|[Wu, 2000]]].

[[Image:doet7.4.png|center|432px|Three <math>2_{IV}^{7-2}\,\!</math> designs with different defining relations.]]

====Example====

The design of an automobile fuel cone is thought to be affected by six factors in the manufacturing process: cavity temperature (factor <math>A\,\!</math>), core temperature (factor <math>B\,\!</math>), melt temperature (factor <math>C\,\!</math>), hold pressure (factor <math>D\,\!</math>), injection speed (factor <math>E\,\!</math>) and cool time (factor <math>F\,\!</math>). The manufacturer of the fuel cone is unable to run the <math>{2}^{6}=64\,\!</math> runs required to complete one replicate for a two level full factorial experiment with six factors. Instead, they decide to run a fractional factorial design. Considering that three factor and higher order interactions are likely to be inactive, the manufacturer selects a <math>{2}^{6-2}\,\!</math> design that will require only 16 runs. The manufacturer chooses the resolution IV design which will ensure that all main effects are free from aliasing (assuming three factor and higher order interactions are absent). However, in this design the two factor interactions may be aliased with each other. It is decided that, if important two factor interactions are found to be present, additional experiment trials may be conducted to separate the aliased effects. The performance of the fuel cone is measured on a scale of 1 to 15. In DOE++, the design for this experiment is set up using the properties shown in the following figure. The Fraction Generators for the design, <math>E=ABC\,\!</math> and <math>F=BCD\,\!</math>, are the same as the defaults used in DOE++. The resulting <math>{2}^{6-2}\,\!</math> design and the corresponding response values are shown in the following two figures.

[[Image:doe7_39.png|center|644px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

[[Image:doe7_40.png|center|534px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The complete alias structure for the 2 <math>_{\text{IV}}^{6-2}\,\!</math> design is shown next.

<center><math>I=ABCE=ADEF=BCDF\,\!</math></center>

 
<center><math>\begin{align}
& A= & BCE=DEF=ABCDF \\
& B= & ACE=CDF=ABDEF \\
& C= & ABE=BDF=ACDEF \\
& D= & AEF=BCF=ABCDE \\
& E= & ABC=ADF=BCDEF \\
& F= & ADE=BCD=ABCEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& AB= & CE=ACDF=BDEF \\
& AC= & BE=ABDF=CDEF \\
& AD= & EF=ABCF=BCDE \\
& AE= & BC=DF=ABCDEF \\
& AF= & DE=ABCD=BCEF \\
& BD= & CF=ABEF=ACDE \\
& BF= & CD=ABDE=ACEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& ABD= & ACF=BEF=CDE \\
& ABF= & ACD=BDE=CEF
\end{align}\,\!</math></center>

In DOE++, the alias structure is displayed in the Design Summary and as part of the Design Evaluation result, as shown next:

[[Image:doe7_41.png|center|700px|Alias structure for the experiment design in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The normal probability plot of effects for this unreplicated design shows the main effects of factors <math>C\,\!</math> and <math>D\,\!</math> and the interaction effect, <math>BF\,\!</math>, to be significant (see the following figure).

[[Image:doe7_42.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

From the alias structure, it can be seen that for the present design interaction effect, <math>BF,\,\!</math> is confounded with <math>CD\,\!</math>. Therefore, the actual source of this effect cannot be known on the basis of the present experiment. However because neither factor <math>B\,\!</math> nor <math>F\,\!</math> is found to be significant there is an indication the observed effect is likely due to interaction, <math>CD\,\!</math>. To confirm this, a follow-up <math>{2}^{2}\,\!</math> experiment is run involving only factors <math>B\,\!</math> and <math>F\,\!</math>. The interaction, <math>BF\,\!</math>, is found to be inactive, leading to the conclusion that the interaction effect in the original experiment is effect, <math>CD\,\!</math>. Given these results, the fitted regression model for the fuel cone design as per the coefficients obtained from DOE++ is shown next.

::<math>\hat{y}=7.6875+C+2\cdot D+2.1875\cdot CD\,\!</math>

[[Image:doe7_43.png|center|700px|Effect coefficients for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

==Projection==

Projection refers to the reduction of a fractional factorial design to a full factorial design by dropping out some of the factors of the design. Any fractional factorial design of resolution, <math>R,\,\!</math> can be reduced to complete factorial designs in any subset of <math>R-1\,\!</math> factors. For example, consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. The resolution of this design is four. Therefore, this design can be reduced to full factorial designs in any three (<math>4-1=3\,\!</math>) of the original seven factors (by dropping the remaining four of factors). Further, a fractional factorial design can also be reduced to a full factorial design in any <math>R\,\!</math> of the original factors, as long as these <math>R\,\!</math> factors are not part of the generator in the defining relation. Again consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. This design can be reduced to a full factorial design in four factors provided these four factors do not appear together as a generator in the defining relation. The complete defining relation for this design is:

::<math>\begin{align}
& I= & ABCE=BCDF=ACDG=ADEF=ABFG=BDEG=CEFG
\end{align}\,\!</math>

Therefore, there are seven four factor combinations out of the 35 (<math>(_{7}^{4})=35\,\!</math>) possible four-factor combinations that are used as generators in the defining relation. The designs with the remaining 28 four factor combinations would be full factorial 16-run designs. For example, factors <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math> do not occur as a generator in the defining relation of the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. If the remaining factors, <math>E\,\!</math>, <math>F\,\!</math> and <math>G\,\!</math>, are dropped, the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design will reduce to a full factorial design in <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math>.

==Resolution III Designs==

At times, the factors to be investigated in screening experiments are so large that even running a fractional factorial design is impractical. This can be partially solved by using resolution III fractional factorial designs in the cases where three factor and higher order interactions can be assumed to be unimportant. Resolution III designs, such as the 2 <math>_{\text{III}}^{3-1}\,\!</math> design, can be used to estimate <math>k\,\!</math> main effects using just <math>k+1\,\!</math> runs. In these designs, the main effects are aliased with two factor interactions. Once the results from these designs are obtained, and knowing that three factor and higher order interactions are unimportant, the experimenter can decide if there is a need to run a fold-over design to de-alias the main effects from the two factor interactions. Thus, the 2 <math>_{\text{III}}^{3-1}\,\!</math> design can be used to investigate three factors in four runs, the 2 <math>_{\text{III}}^{7-4}\,\!</math> design can be used to investigate seven factors in eight runs, the 2 <math>_{\text{III}}^{15-11}\,\!</math> design can be used to investigate fifteen factors in sixteen runs and so on.

====Example====

{{:Resolution_III_Design_Example}}

==Alias Matrix==
In [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction designs]] and [[Two_Level_Factorial_Experiments#Quarter_and_Smaller_Fraction_Designs| Quarter and Smaller Fraction Designs]], the alias structure for fractional factorial designs was obtained using the defining relation. However, this method of obtaining the alias structure is not very efficient when the alias structure is very complex or when partial aliasing is involved. One of the ways to obtain the alias structure for any design, regardless of its complexity, is to use the alias matrix. The alias matrix for a design is calculated using <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> where <math>{{X}_{1}}\,\!</math> is the portion of the design matrix, <math>X,\,\!</math> that contains the effects for which the aliases need to be calculated, and <math>{{X}_{2}}\,\!</math> contains the remaining columns of the design matrix, other than those included in <math>{{X}_{1}}\,\!</math>.

To illustrate the use of the alias matrix, consider the design matrix for the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design (using the defining relation <math>I=ABCD\,\!</math>) shown next:

[[Image:Chapter7__879.png|center|link=]]

The alias structure for this design can be obtained by defining <math>{{X}_{1}}\,\!</math> using eight columns since the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design estimates eight effects. If the first eight columns of <math>X\,\!</math> are used then <math>{{X}_{1}}\,\!</math> is:

[[Image:Chapter7__884.png|center|link=]]

<math>{{X}_{2}}\,\!</math> is obtained using the remaining columns as:

[[Image:Chapter7__886.png|center|link=]]

Then the alias matrix <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> is:

[[Image:Chapter7__888.png|center|link=]]

The alias relations can be easily obtained by observing the alias matrix as:

<center><math>\begin{align}
& I= & ABCD \\
& A= & BCD \\
& B= & ACD \\
& AB= & CD \\
& C= & ABD \\
& AC= & BD \\
& BC= & AD \\
& D= & ABC
\end{align}\,\!</math></center>

Two Level Factorial Experiments

2017-08-10T17:30:00Z

Richard House: /* Model Equation */

{{Template:Doebook|8}}
Two level factorial experiments are factorial experiments in which each factor is investigated at only two levels. The early stages of experimentation usually involve the investigation of a large number of potential factors to discover the "vital few" factors. Two level factorial experiments are used during these stages to quickly filter out unwanted effects so that attention can then be focused on the important ones.

==2''k'' Designs==
The factorial experiments, where all combination of the levels of the factors are run, are usually referred to as ''full factorial experiments''. Full factorial two level experiments are also referred to as <math>{2}^{k}\,\!</math> designs where <math>k\,\!</math> denotes the number of factors being investigated in the experiment. In Weibull++ DOE folios, these designs are referred to as 2 Level Factorial Designs as shown in the figure below.

[[Image:doe7_1.png|center|487px|Selection of full factorial experiments with two levels in Weibull++.|link=]]

A full factorial two level design with <math>k\,\!</math> factors requires <math>{{2}^{k}}\,\!</math> runs for a single replicate. For example, a two level experiment with three factors will require <math>2\times 2\times 2={{2}^{3}}=8\,\!</math> runs. The choice of the two levels of factors used in two level experiments depends on the factor; some factors naturally have two levels. For example, if gender is a factor, then male and female are the two levels. For other factors, the limits of the range of interest are usually used. For example, if temperature is a factor that varies from <math>{45}^{o}C\,\!</math> to <math>{90}^{o}C\,\!</math>, then the two levels used in the <math>{2}^{k}\,\!</math> design for this factor would be <math>{45}^{o}\,\!C\,\!</math> and <math>{90}^{o}\,\!C\,\!</math>.

The two levels of the factor in the <math>{2}^{k}\,\!</math> design are usually represented as <math>-1\,\!</math> (for the first level) and <math>1\,\!</math> (for the second level). Note that this representation is reversed from the coding used in [[General Full Factorial Designs]] for the indicator variables that represent two level factors in ANOVA models. For ANOVA models, the first level of the factor was represented using a value of <math>1\,\!</math> for the indicator variable, while the second level was represented using a value of <math>-1\,\!</math>. For details on the notation used for two level experiments refer to [[Two_Level_Factorial_Experiments#Notation| Notation]].

===The 22 Design===

The simplest of the two level factorial experiments is the <math>{2}^{2}\,\!</math> design where two factors (say factor <math>A\,\!</math> and factor <math>B\,\!</math>) are investigated at two levels. A single replicate of this design will require four runs (<math>{{2}^{2}}=2\times 2=4\,\!</math>) The effects investigated by this design are the two main effects, <math>A\,\!</math> and <math>B,\,\!</math> and the interaction effect <math>AB\,\!</math>. The treatments for this design are shown in figure (a) below. In figure (a), letters are used to represent the treatments. The presence of a letter indicates the high level of the corresponding factor and the absence indicates the low level. For example, (1) represents the treatment combination where all factors involved are at the low level or the level represented by <math>-1\,\!</math> ; <math>a\,\!</math> represents the treatment combination where factor <math>A\,\!</math> is at the high level or the level of <math>1\,\!</math>, while the remaining factors (in this case, factor <math>B\,\!</math>) are at the low level or the level of <math>-1\,\!</math>. Similarly, <math>b\,\!</math> represents the treatment combination where factor <math>B\,\!</math> is at the high level or the level of <math>1\,\!</math>, while factor <math>A\,\!</math> is at the low level and <math>ab\,\!</math> represents the treatment combination where factors <math>A\,\!</math> and <math>B\,\!</math> are at the high level or the level of the 1. Figure (b) below shows the design matrix for the <math>{2}^{2}\,\!</math> design. It can be noted that the sum of the terms resulting from the product of any two columns of the design matrix is zero. As a result the <math>{2}^{2}\,\!</math> design is an ''orthogonal design''. In fact, all <math>{2}^{k}\,\!</math> designs are orthogonal designs. This property of the <math>{2}^{k}\,\!</math> designs offers a great advantage in the analysis because of the simplifications that result from orthogonality. These simplifications are explained later on in this chapter.
The <math>{2}^{2}\,\!</math> design can also be represented geometrically using a square with the four treatment combinations lying at the four corners, as shown in figure (c) below.

[[Image:doe7.2.png|center|400px|The <math>2^2\,\!</math> design. Figure (a) displays the experiment design, (b) displays the design matrix and (c) displays the geometric representation for the design. In Figure (b), the column names I, A, B and AB are used. Column I represents the intercept term. Columns A and B represent the respective factor settings. Column AB represents the interaction and is the product of columns A and B.]]
 

===The 23 Design===

The <math>{2}^{3}\,\!</math> design is a two level factorial experiment design with three factors (say factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>). This design tests three (<math>k=3\,\!</math>) main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math> ; three (<math>(_{2}^{k})=\,\!</math> <math>(_{2}^{3})=3\,\!</math>) two factor interaction effects, <math>AB\,\!</math>, <math>BC\,\!</math>, <math>AC\,\!</math> ; and one (<math>(_{3}^{k})=\,\!</math> <math>(_{3}^{3})=1\,\!</math>) three factor interaction effect, <math>ABC\,\!</math>. The design requires eight runs per replicate. The eight treatment combinations corresponding to these runs are <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math>, <math>ab\,\!</math>, <math>c\,\!</math>, <math>ac\,\!</math>, <math>bc\,\!</math> and <math>abc\,\!</math>. Note that the treatment combinations are written in such an order that factors are introduced one by one with each new factor being combined with the preceding terms. This order of writing the treatments is called the ''standard order'' or ''Yates' order''. The <math>{2}^{3}\,\!</math> design is shown in figure (a) below. The design matrix for the <math>{2}^{3}\,\!</math> design is shown in figure (b). The design matrix can be constructed by following the standard order for the treatment combinations to obtain the columns for the main effects and then multiplying the main effects columns to obtain the interaction columns.

[[Image:doe7.3.png|center|324px|The <math>2^3\,\!</math> design. Figure (a) shows the experiment design and (b) shows the design matrix.]]

[[Image:doe7.4.png|center|290px|Geometric representation of the <math>2^3\,\!</math> design.]]

The <math>{2}^{3}\,\!</math> design can also be represented geometrically using a cube with the eight treatment combinations lying at the eight corners as shown in the figure above.

==Analysis of 2''k'' Designs==

The <math>{2}^{k}\,\!</math> designs are a special category of the factorial experiments where all the factors are at two levels. The fact that these designs contain factors at only two levels and are orthogonal greatly simplifies their analysis even when the number of factors is large. The use of <math>{2}^{k}\,\!</math> designs in investigating a large number of factors calls for a revision of the notation used previously for the ANOVA models. The case for revised notation is made stronger by the fact that the ANOVA and multiple linear regression models are identical for <math>{2}^{k}\,\!</math> designs because all factors are only at two levels. Therefore, the notation of the regression models is applied to the ANOVA models for these designs, as explained next.

===Notation===

Based on the notation used in [[General Full Factorial Designs]], the ANOVA model for a two level factorial experiment with three factors would be as follows:

::<math>\begin{align}
& Y= & \mu +{{\tau }_{1}}\cdot {{x}_{1}}+{{\delta }_{1}}\cdot {{x}_{2}}+{{(\tau \delta )}_{11}}\cdot {{x}_{1}}{{x}_{2}}+{{\gamma }_{1}}\cdot {{x}_{3}} \\
& & +{{(\tau \gamma )}_{11}}\cdot {{x}_{1}}{{x}_{3}}+{{(\delta \gamma )}_{11}}\cdot {{x}_{2}}{{x}_{3}}+{{(\tau \delta \gamma )}_{111}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where:
 
:• <math>\mu \,\!</math> represents the overall mean
:• <math>{{\tau }_{1}}\,\!</math> represents the independent effect of the first factor (factor <math>A\,\!</math>) out of the two effects <math>{{\tau }_{1}}\,\!</math> and <math>{{\tau }_{2}}\,\!</math>
:• <math>{{\delta }_{1}}\,\!</math> represents the independent effect of the second factor (factor <math>B\,\!</math>) out of the two effects <math>{{\delta }_{1}}\,\!</math> and <math>{{\delta }_{2}}\,\!</math>
:• <math>{{(\tau \delta )}_{11}}\,\!</math> represents the independent effect of the interaction <math>AB\,\!</math> out of the other interaction effects
:• <math>{{\gamma }_{1}}\,\!</math> represents the effect of the third factor (factor <math>C\,\!</math>) out of the two effects <math>{{\gamma }_{1}}\,\!</math> and <math>{{\gamma }_{2}}\,\!</math>
:• <math>{{(\tau \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>AC\,\!</math> out of the other interaction effects
:• <math>{{(\delta \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>BC\,\!</math> out of the other interaction effects
:• <math>{{(\tau \delta \gamma )}_{111}}\,\!</math> represents the effect of the interaction <math>ABC\,\!</math> out of the other interaction effects
and <math>\epsilon \,\!</math> is the random error term.

 
The notation for a linear regression model having three predictor variables with interactions is:

::<math>\begin{align}
& Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& & +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

The notation for the regression model is much more convenient, especially for the case when a large number of higher order interactions are present. In two level experiments, the ANOVA model requires only one indicator variable to represent each factor for both qualitative and quantitative factors. Therefore, the notation for the multiple linear regression model can be applied to the ANOVA model of the experiment that has all the factors at two levels. For example, for the experiment of the ANOVA model given above, <math>{{\beta }_{0}}\,\!</math> can represent the overall mean instead of <math>\mu \,\!</math>, and <math>{{\beta }_{1}}\,\!</math> can represent the independent effect, <math>{{\tau }_{1}}\,\!</math>, of factor <math>A\,\!</math>. Other main effects can be represented in a similar manner. The notation for the interaction effects is much more simplified (e.g., <math>{{\beta }_{123}}\,\!</math> can be used to represent the three factor interaction effect, <math>{{(\tau \beta \gamma )}_{111}}\,\!</math>).

As mentioned earlier, it is important to note that the coding for the indicator variables for the ANOVA models of two level factorial experiments is reversed from the coding followed in [[General Full Factorial Designs]]. Here <math>-1\,\!</math> represents the first level of the factor while <math>1\,\!</math> represents the second level. This is because for a two level factor a single variable is needed to represent the factor for both qualitative and quantitative factors. For quantitative factors, using <math>-1\,\!</math> for the first level (which is the low level) and 1 for the second level (which is the high level) keeps the coding consistent with the numerical value of the factors. The change in coding between the two coding schemes does not affect the analysis except that signs of the estimated effect coefficients will be reversed (i.e., numerical values of <math>{{\hat{\tau }}_{1}}\,\!</math>, obtained based on the coding of [[General Full Factorial Designs]], and <math>{{\hat{\beta }}_{1}}\,\!</math>, obtained based on the new coding, will be the same but their signs would be opposite).

::<math>\begin{align}
& & \text{Factor }A\text{ Coding (two level factor)} \\
& &
\end{align}\,\!</math>

::<math>\begin{matrix}
\text{Previous Coding} & {} & {} & {} & \text{Coding for }{{\text{2}}^{k}}\text{ Designs} \\
{} & {} & {} & {} & {} \\
Effect\text{ }{{\tau }_{1}}\ \ :\ \ {{x}_{1}}=1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{1}}\text{ (or }-{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=-1\text{ } \\
Effect\text{ }{{\tau }_{2}}\ \ :\ \ {{x}_{1}}=-1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{2}}\text{ (or }{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=1\text{ } \\
\end{matrix}\,\!</math>

In summary, the ANOVA model for the experiments with all factors at two levels is different from the ANOVA models for other experiments in terms of the notation in the following two ways:
 

:• The notation of the regression models is used for the effect coefficients.
:• The coding of the indicator variables is reversed.

===Special Features===

Consider the design matrix, <math>X\,\!</math>, for the <math>{2}^{3}\,\!</math> design discussed above. The (<math>{{X}^{\prime }}X\,\!</math>) <math>^{-1}\,\!</math> matrix is:

<center><math>{{({{X}^{\prime }}X)}^{-1}}=\left[ \begin{matrix}
0.125 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0.125 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0.125 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0.125 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0.125 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0.125 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0.125 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.125 \\
\end{matrix} \right]\,\!</math></center>

Notice that, due to the orthogonal design of the <math>X\,\!</math> matrix, the <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> has been simplified to a diagonal matrix which can be written as:

::<math>\begin{align}
{{({{X}^{\prime }}X)}^{-1}}= & 0.125\cdot I = & \frac{1}{8}\cdot I = & \frac{1}{{{2}^{3}}}\cdot I
\end{align}\,\!</math>

where <math>I\,\!</math> represents the identity matrix of the same order as the design matrix, <math>X\,\!</math>. Since there are eight observations per replicate of the <math>{2}^{3}\,\!</math> design, the <math>(X\,\!</math> ' <math>X{{)}^{-1}}\,\!</math> matrix for <math>m\,\!</math> replicates of this design can be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{3}}\cdot m)}\cdot I\,\!</math>

The <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> matrix for any <math>{2}^{k}\,\!</math> design can now be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{k}}\cdot m)}\cdot I\,\!</math>

Then the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is:

::<math>\begin{align}
C= & {{{\hat{\sigma }}}^{2}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & M{{S}_{E}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & \frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}\cdot I
\end{align}\,\!</math>

Note that the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is also a diagonal matrix. Therefore, the estimated effect coefficients (<math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math>, <math>{{\beta }_{12}},\,\!</math> etc.) for these designs are uncorrelated. This implies that the terms in the <math>{2}^{k}\,\!</math> design (main effects, interactions) are independent of each other. Consequently, the extra sum of squares for each of the terms in these designs is independent of the sequence of terms in the model, and also independent of the presence of other terms in the model. As a result the sequential and partial sum of squares for the terms are identical for these designs and will always add up to the model sum of squares. Multicollinearity is also not an issue for these designs.

It can also be noted from the equation given above, that in addition to the <math>C\,\!</math> matrix being diagonal, all diagonal elements of the <math>C\,\!</math> matrix are identical. This means that the variance (or its square root, the standard error) of all estimated effect coefficients are the same. The standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>, for all the coefficients is:

::<math>\begin{align}
se({{{\hat{\beta }}}_{j}})= & \sqrt{{{C}_{jj}}} = & \sqrt{\frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}}\text{ }for\text{ }all\text{ }j
\end{align}\,\!</math>

This property is used to construct the normal probability plot of effects in <math>{2}^{k}\,\!</math> designs and identify significant effects using graphical techniques. For details on the normal probability plot of effects in a Weibull++ DOE folio, refer to [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]].

====Example====
To illustrate the analysis of a full factorial <math>{2}^{k}\,\!</math> design, consider a three factor experiment to investigate the effect of honing pressure, number of strokes and cycle time on the surface finish of automobile brake drums. Each of these factors is investigated at two levels. The honing pressure is investigated at levels of 200 <math>psi\,\!</math> and 400 <math>psi\,\!</math>, the number of strokes used is 3 and 5 and the two levels of the cycle time are 3 and 5 seconds. The design for this experiment is set up in a Weibull++ DOE folio as shown in the first two following figures. It is decided to run two replicates for this experiment. The surface finish data collected from each run (using randomization) and the complete design is shown in the third following figure. The analysis of the experiment data is explained next.

[[Image:doe7_5.png|center|877px|Design properties for the experiment in the example.|link=]]

[[Image:doe7_6.png|center|859px|Design summary for the experiment in the example.|link=]]

[[Image:doe7_7.png|center|778px|Experiment design for the example to investigate the surface finish of automobile brake drums.|link=]]

The applicable model using the notation for <math>{2}^{k}\,\!</math> designs is:

::<math>\begin{align}
Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where the indicator variable, <math>{{x}_{1,}}\,\!</math> represents factor <math>A\,\!</math> (honing pressure), <math>{{x}_{1}}=-1\,\!</math> represents the low level of 200 <math>psi\,\!</math> and <math>{{x}_{1}}=1\,\!</math> represents the high level of 400 <math>psi\,\!</math>. Similarly, <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> represent factors <math>B\,\!</math> (number of strokes) and <math>C\,\!</math> (cycle time), respectively. <math>{{\beta }_{0}}\,\!</math> is the overall mean, while <math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math> and <math>{{\beta }_{3}}\,\!</math> are the effect coefficients for the main effects of factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. <math>{{\beta }_{12}}\,\!</math>, <math>{{\beta }_{13}}\,\!</math> and <math>{{\beta }_{23}}\,\!</math> are the effect coefficients for the <math>AB\,\!</math>, <math>AC\,\!</math> and <math>BC\,\!</math> interactions, while <math>{{\beta }_{123}}\,\!</math> represents the <math>ABC\,\!</math> interaction.

 
If the subscripts for the run (<math>i\,\!</math> ; <math>i=\,\!</math> 1 to 8) and replicates (<math>j\,\!</math> ; <math>j=\,\!</math> 1,2) are included, then the model can be written as:

::<math>\begin{align}
{{Y}_{ij}}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{ij1}}+{{\beta }_{2}}\cdot {{x}_{ij2}}+{{\beta }_{12}}\cdot {{x}_{ij1}}{{x}_{ij2}}+{{\beta }_{3}}\cdot {{x}_{ij3}} \\
& +{{\beta }_{13}}\cdot {{x}_{ij1}}{{x}_{ij3}}+{{\beta }_{23}}\cdot {{x}_{ij2}}{{x}_{ij3}}+{{\beta }_{123}}\cdot {{x}_{ij1}}{{x}_{ij2}}{{x}_{ij3}}+{{\epsilon }_{ij}}
\end{align}\,\!</math>

To investigate how the given factors affect the response, the following hypothesis tests need to be carried:

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{1}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{1}}\ne 0\,\!</math>

This test investigates the main effect of factor <math>A\,\!</math> (honing pressure). The statistic for this test is:

::<math>{{({{F}_{0}})}_{A}}=\frac{M{{S}_{A}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{A}}\,\!</math> is the mean square for factor <math>A\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other main effects, <math>B\,\!</math> and <math>C\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{12}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{12}}\ne 0\,\!</math>

This test investigates the two factor interaction <math>AB\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{AB}}=\frac{M{{S}_{AB}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the interaction <math>AB\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other two factor interactions, <math>AC\,\!</math> and <math>BC\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{123}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{123}}\ne 0\,\!</math>

This test investigates the three factor interaction <math>ABC\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{ABC}}=\frac{M{{S}_{ABC}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{ABC}}\,\!</math> is the mean square for the interaction <math>ABC\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square.
To calculate the test statistics, it is convenient to express the ANOVA model in the form <math>y=X\beta +\epsilon \,\!</math>.

====Expression of the ANOVA Model as <math>y=X\beta +\epsilon \,\!</math>====

In matrix notation, the ANOVA model can be expressed as:

::<math>y=X\beta +\epsilon \,\!</math>

where:

<center><math>y=\left[ \begin{matrix}
{{Y}_{11}} \\
{{Y}_{21}} \\
. \\
{{Y}_{81}} \\
{{Y}_{12}} \\
. \\
{{Y}_{82}} \\
\end{matrix} \right]=\left[ \begin{matrix}
90 \\
90 \\
. \\
90 \\
86 \\
. \\
80 \\
\end{matrix} \right]\text{ }X=\left[ \begin{matrix}
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\end{matrix} \right]\,\!</math></center>

<center><math>\beta =\left[ \begin{matrix}
{{\beta }_{0}} \\
{{\beta }_{1}} \\
{{\beta }_{2}} \\
{{\beta }_{12}} \\
{{\beta }_{3}} \\
{{\beta }_{13}} \\
{{\beta }_{23}} \\
{{\beta }_{123}} \\
\end{matrix} \right]\text{ }\epsilon =\left[ \begin{matrix}
{{\epsilon }_{11}} \\
{{\epsilon }_{21}} \\
. \\
{{\epsilon }_{81}} \\
{{\epsilon }_{12}} \\
. \\
. \\
{{\epsilon }_{82}} \\
\end{matrix} \right]\,\!</math></center>

====Calculation of the Extra Sum of Squares for the Factors====

Knowing the matrices <math>y\,\!</math>, <math>X\,\!</math> and <math>\beta \,\!</math>, the extra sum of squares for the factors can be calculated. These are used to calculate the mean squares that are used to obtain the test statistics. Since the experiment design is orthogonal, the partial and sequential extra sum of squares are identical. The extra sum of squares for each effect can be calculated as shown next. As an example, the extra sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & Model\text{ }Sum\text{ }of\text{ }Squares - Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }main\text{ }effect\text{ }of\text{ }A \\
= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }A}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }A}}={{X}_{\tilde{\ }A}}{{(X_{\tilde{\ }A}^{\prime }{{X}_{\tilde{\ }A}})}^{-1}}X_{\tilde{\ }A}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }A}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the main effect of factor <math>A\,\!</math>. Thus, the sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y \\
= & 654.4375-549.375 \\
= & 105.0625
\end{align}\,\!</math>

Similarly, the extra sum of squares for the interaction effect <math>AB\,\!</math> is:

::<math>\begin{align}
S{{S}_{AB}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }AB}}-(1/16)J]y \\
= & 654.4375-636.375 \\
= & 18.0625
\end{align}\,\!</math>

The extra sum of squares for other effects can be obtained in a similar manner.

====Calculation of the Test Statistics====

Knowing the extra sum of squares, the test statistic for the effects can be calculated. For example, the test statistic for the interaction <math>AB\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{AB}}= & \frac{M{{S}_{AB}}}{M{{S}_{E}}} \\
= & \frac{S{{S}_{AB}}/dof(S{{S}_{AB}})}{S{{S}_{E}}/dof(S{{S}_{E}})} \\
= & \frac{18.0625/1}{147.5/8} \\
= & 0.9797
\end{align}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the <math>AB\,\!</math> interaction and <math>M{{S}_{E}}\,\!</math> is the error mean square. The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{AB}}=0.9797\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{AB}}) \\
= & 1-0.6487 \\
= & 0.3513
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that the interaction between honing pressure and number of strokes does not affect the surface finish of the brake drums. Tests for other effects can be carried out in a similar manner. The results are shown in the ANOVA Table in the following figure. The values S, R-sq and R-sq(adj) in the figure indicate how well the model fits the data. The value of S represents the standard error of the model, R-sq represents the coefficient of multiple determination and R-sq(adj) represents the adjusted coefficient of multiple determination. For details on these values refer to [[Multiple_Linear_Regression_Analysis|Multiple Linear Regression Analysis]].

[[Image:doe7_8.png|center|799px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

====Calculation of Effect Coefficients====

The estimate of effect coefficients can also be obtained:

<center><math>\begin{align}
\hat{\beta }= & {{({{X}^{\prime }}X)}^{-1}}{{X}^{\prime }}y \\
= & \left[ \begin{matrix}
86.4375 \\
2.5625 \\
-4.9375 \\
1.0625 \\
-1.0625 \\
2.4375 \\
-1.3125 \\
-0.1875 \\
\end{matrix} \right]
\end{align}\,\!</math></center>

[[Image:doe7_9.png|center|800px|Regression Information table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

The coefficients and related results are shown in the Regression Information table above. In the table, the Effect column displays the effects, which are simply twice the coefficients. The Standard Error column displays the standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>. The Low CI and High CI columns display the confidence interval on the coefficients. The interval shown is the 90% interval as the significance is chosen as 0.1. The T Value column displays the <math>t\,\!</math> statistic, <math>{{t}_{0}}\,\!</math>, corresponding to the coefficients. The P Value column displays the <math>p\,\!</math> value corresponding to the <math>t\,\!</math> statistic. (For details on how these results are calculated, refer to [[General Full Factorial Designs]]). Plots of residuals can also be obtained from the DOE folio to ensure that the assumptions related to the ANOVA model are not violated.

====Model Equation====

From the analysis results in the above figure within [[Two_Level_Factorial_Experiments#Calculation_of_Effect_Coefficients|calculation of effect coefficients]] section, it is seen that effects <math>A\,\!</math>, <math>B\,\!</math> and <math>AC\,\!</math> are significant. In a DOE folio, the <math>p\,\!</math> values for the significant effects are displayed in red in the ANOVA Table for easy identification. Using the values of the estimated effect coefficients, the model for the present <math>{2}^{3}\,\!</math> design in terms of the coded values can be written as:

::<math>\begin{align}
\hat{y}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}} \\
= & 86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+2.4375{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

To make the model hierarchical, the main effect, <math>C\,\!</math>, needs to be included in the model (because the interaction <math>AC\,\!</math> is included in the model). The resulting model is:

::<math>\hat{y}=86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+1.0625{{x}_{3}}+2.4375{{x}_{1}}{{x}_{3}}\,\!</math>

This equation can be viewed in a DOE folio, as shown in the following figure, using the Show Analysis Summary icon in the Control Panel. The equation shown in the figure will match the hierarchical model once the required terms are selected using the Select Effects icon.

[[Image:doe7_10.png|center|557px|The model equation for the experiment of the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

==Replicated and Repeated Runs==

In the case of replicated experiments, it is important to note the difference between replicated runs and repeated runs. Both repeated and replicated runs are multiple response readings taken at the same factor levels. However, repeated runs are response observations taken at the same time or in succession. Replicated runs are response observations recorded in a random order. Therefore, replicated runs include more variation than repeated runs. For example, a baker, who wants to investigate the effect of two factors on the quality of cakes, will have to bake four cakes to complete one replicate of a <math>{2}^{2}\,\!</math> design. Assume that the baker bakes eight cakes in all. If, for each of the four treatments of the <math>{2}^{2}\,\!</math> design, the baker selects one treatment at random and then bakes two cakes for this treatment at the same time then this is a case of two repeated runs. If, however, the baker bakes all the eight cakes randomly, then the eight cakes represent two sets of replicated runs.
For repeated measurements, the average values of the response for each treatment should be entered into DOE++ as shown in the following figure (a) when the two cakes for a particular treatment are baked together. For replicated measurements, when all the cakes are baked randomly, the data is entered as shown in the following figure (b).

[[Image:doe7.11.png|center|300px|Data entry for repeated and replicated runs. Figure (a) shows repeated runs and (b) shows replicated runs.]]

==Unreplicated 2''k'' Designs==

If a factorial experiment is run only for a single replicate then it is not possible to test hypotheses about the main effects and interactions as the error sum of squares cannot be obtained. This is because the number of observations in a single replicate equals the number of terms in the ANOVA model. Hence the model fits the data perfectly and no degrees of freedom are available to obtain the error sum of squares.

However, sometimes it is only possible to run a single replicate of the <math>{2}^{k}\,\!</math> design because of constraints on resources and time. In the absence of the error sum of squares, hypothesis tests to identify significant factors cannot be conducted. A number of methods of analyzing information obtained from unreplicated <math>{2}^{k}\,\!</math> designs are available. These include pooling higher order interactions, using the normal probability plot of effects or including center point replicates in the design.

===Pooling Higher Order Interactions===

One of the ways to deal with unreplicated <math>{2}^{k}\,\!</math> designs is to use the sum of squares of some of the higher order interactions as the error sum of squares provided these higher order interactions can be assumed to be insignificant. By dropping some of the higher order interactions from the model, the degrees of freedom corresponding to these interactions can be used to estimate the error mean square. Once the error mean square is known, the test statistics to conduct hypothesis tests on the factors can be calculated.

===Normal Probability Plot of Effects===
Another way to use unreplicated <math>{2}^{k}\,\!</math> designs to identify significant effects is to construct the normal probability plot of the effects. As mentioned in [[Two_Level_Factorial_Experiments#Special_Features| Special Features]], the standard error for all effect coefficients in the <math>{2}^{k}\,\!</math> designs is the same. Therefore, on a normal probability plot of effect coefficients, all non-significant effect coefficients (with <math>\beta =0\,\!</math>) will fall along the straight line representative of the normal distribution, N(<math>0,{{\sigma }^{2}}/({{2}^{k}}\cdot m)\,\!</math>). Effect coefficients that show large deviations from this line will be significant since they do not come from this normal distribution. Similarly, since effects <math>=2\times \,\!</math> effect coefficients, all non-significant effects will also follow a straight line on the normal probability plot of effects. For replicated designs, the Effects Probability plot of DOE++ plots the normalized effect values (or the T Values) on the standard normal probability line, N(0,1). However, in the case of unreplicated <math>{2}^{k}\,\!</math> designs, <math>{{\sigma }^{2}}\,\!</math> remains unknown since <math>M{{S}_{E}}\,\!</math> cannot be obtained. Lenth's method is used in this case to estimate the variance of the effects. For details on Lenth's method, please refer to [[DOE References| Montgomery (2001)]]. DOE++ then uses this variance value to plot effects along the N(0, Lenth's effect variance) line. The
method is illustrated in the following example.

====Example====

Vinyl panels, used as instrument panels in a certain automobile, are seen to develop defects after a certain amount of time. To investigate the issue, it is decided to carry out a two level factorial experiment. Potential factors to be investigated in the experiment are vacuum rate (factor <math>A\,\!</math>), material temperature (factor <math>B\,\!</math>), element intensity (factor <math>C\,\!</math>) and pre-stretch (factor <math>D\,\!</math>). The two levels of the factors used in the experiment are as shown in below.

[[Image:doet7.1.png|center|300px|Factors to investigate defects in vinyl panels.]]

With a <math>{2}^{4}\,\!</math> design requiring 16 runs per replicate it is only feasible for the manufacturer to run a single replicate.

The experiment design and data, collected as percent defects, are shown in the following figure. Since the present experiment design contains only a single replicate, it is not possible to obtain an estimate of the error sum of squares, <math>S{{S}_{E}}\,\!</math>. It is decided to use the normal probability plot of effects to identify the significant effects. The effect values for each term are obtained as shown in the following figure.

[[Image:doe7_13.png|center|650px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

Lenth's method uses these values to estimate the variance. As described in [[DOE_References|[Lenth, 1989]]], if all effects are arranged in ascending order, using their absolute values, then <math>{{s}_{0}}\,\!</math> is defined as 1.5 times the median value:

::<math>\begin{align}
{{s}_{0}}= & 1.5\cdot median(\left| effect \right|) \\
= & 1.5\cdot 2 \\
= & 3
\end{align}\,\!</math>

Using <math>{{s}_{0}}\,\!</math>, the "pseudo standard error" (<math>PSE\,\!</math>) is calculated as 1.5 times the median value of all effects that are less than 2.5 <math>{{s}_{0}}\,\!</math> :

::<math>\begin{align}
PSE= & 1.5\cdot median(\left| effect \right|\ \ :\ \ \left| effect \right|<2.5{{s}_{0}}) \\
= & 1.5\cdot 1.5 \\
= & 2.25
\end{align}\,\!</math>

Using <math>PSE\,\!</math> as an estimate of the effect variance, the effect variance is 2.25. Knowing the effect variance, the normal probability plot of effects for the present unreplicated experiment can be constructed as shown in the following figure. The line on this plot is the line N(0, 2.25). The plot shows that the effects <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math> do not follow the distribution represented by this line. Therefore, these effects are significant.

The significant effects can also be identified by comparing individual effect values to the margin of error or the threshold value using the pareto chart (see the third following figure). If the required significance is 0.1, then:

::<math>margin\text{ }of\text{ }error={{t}_{\alpha /2,d}}\cdot PSE\,\!</math>

The <math>t\,\!</math> statistic, <math>{{t}_{\alpha /2,d}}\,\!</math>, is calculated at a significance of <math>\alpha /2\,\!</math> (for the two-sided hypothesis) and degrees of freedom <math>d=(\,\!</math> number of effects <math>)/3\,\!</math>. Thus:

::<math>\begin{align}
margin\text{ }of\text{ }error= & {{t}_{0.05,5}}\cdot PSE \\
= & 2.015\cdot 2.25 \\
= & 4.534
\end{align}\,\!</math>

The value of 4.534 is shown as the critical value line in the third following figure. All effects with absolute values greater than the margin of error can be considered to be significant. These effects are <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math>. Therefore, the vacuum rate, the pre-stretch and their interaction have a significant effect on the defects of the vinyl panels.

[[Image:doe7_14.png|center|800px|Effect values for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_15.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_16.png|center|650px|Pareto chart for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

===Center Point Replicates===

Another method of dealing with unreplicated <math>{2}^{k}\,\!</math> designs that only have quantitative factors is to use replicated runs at the center point. The center point is the response corresponding to the treatment exactly midway between the two levels of all factors. Running multiple replicates at this point provides an estimate of pure error. Although running multiple replicates at any treatment level can provide an estimate of pure error, the other advantage of running center point replicates in the <math>{2}^{k}\,\!</math> design is in checking for the presence of curvature. The test for curvature investigates whether the model between the response and the factors is linear and is discussed in [[Two_Level_Factorial_Experiments#Using_Center_Point_Replicates_to_Test_Curvature| Center Pt. Replicates to Test Curvature]].

====Example: Use Center Point to Get Pure Error====

Consider a <math>{2}^{2}\,\!</math> experiment design to investigate the effect of two factors, <math>A\,\!</math> and <math>B\,\!</math>, on a certain response. The energy consumed when the treatments of the <math>{2}^{2}\,\!</math> design are run is considerably larger than the energy consumed for the center point run (because at the center point the factors are at their middle levels). Therefore, the analyst decides to run only a single replicate of the design and augment the design by five replicated runs at the center point as shown in the following figure. The design properties for this experiment are shown in the second following figure. The complete experiment design is shown in the third following figure. The center points can be used in the identification of significant effects as shown next.

[[Image:doe7.17.png||center|300px|<math>2^2\,\!</math> design augmented by five center point runs.]]
[[Image:doe7_18.png|center|537px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

[[Image:doe7_19.png|center|630px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

Since the present <math>{2}^{2}\,\!</math> design is unreplicated, there are no degrees of freedom available to calculate the error sum of squares. By augmenting this design with five center points, the response values at the center points, <math>y_{i}^{c}\,\!</math>, can be used to obtain an estimate of pure error, <math>S{{S}_{PE}}\,\!</math>. Let <math>{{\bar{y}}^{c}}\,\!</math> represent the average response for the five replicates at the center. Then:

::<math>S{{S}_{PE}}=Sum\text{ }of\text{ }Squares\text{ }for\text{ }center\text{ }points\,\!</math>

::<math>\begin{align}
S{{S}_{PE}}= & \underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}} \\
= & {{(25.2-25.26)}^{2}}+...+{{(25.3-25.26)}^{2}} \\
= & 0.052
\end{align}\,\!</math>

Then the corresponding mean square is:

::<math>\begin{align}
M{{S}_{PE}}= & \frac{S{{S}_{PE}}}{degrees\text{ }of\text{ }freedom} \\
= & \frac{0.052}{5-1} \\
= & 0.013
\end{align}\,\!</math>

Alternatively, <math>M{{S}_{PE}}\,\!</math> can be directly obtained by calculating the variance of the response values at the center points:

::<math>\begin{align}
M{{S}_{PE}}= & {{s}^{2}} \\
= & \frac{\underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}}}{5-1}
\end{align}\,\!</math>

Once <math>M{{S}_{PE}}\,\!</math> is known, it can be used as the error mean square, <math>M{{S}_{E}}\,\!</math>, to carry out the test of significance for each effect. For example, to test the significance of the main effect of factor <math>A,\,\!</math> the sum of squares corresponding to this effect is obtained in the usual manner by considering only the four runs of the original <math>{2}^{2}\,\!</math> design.

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/4)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/4)J]y \\
= & 0.5625
\end{align}\,\!</math>

Then, the test statistic to test the significance of the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{A}}= & \frac{M{{S}_{A}}}{M{{S}_{E}}} \\
= & \frac{0.5625/1}{0.052/4} \\
= & 43.2692
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{A}}=43.2692\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{A}}) \\
= & 1-0.9972 \\
= & 0.0028
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value < 0.1, it can be concluded that the main effect of factor <math>A\,\!</math> significantly affects the response. This result is displayed in the ANOVA table as shown in the following figure. Test for the significance of other factors can be carried out in a similar manner.

[[Image:doe7_20.png|center|649px|Results for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

===Using Center Point Replicates to Test Curvature===

Center point replicates can also be used to check for curvature in replicated or unreplicated <math>{2}^{k}\,\!</math> designs. The test for curvature investigates whether the model between the response and the factors is linear. The way DOE++ handles center point replicates is similar to its handling of blocks. The center point replicates are treated as an additional factor in the model. The factor is labeled as Curvature in the results of DOE++. If Curvature turns out to be a significant factor in the results, then this indicates the presence of curvature in the model.

====Example: Use Center Point to Test Curvature====

To illustrate the use of center point replicates in testing for curvature, consider again the data of the single replicate <math>{2}^{2}\,\!</math> experiment from a preceding figure(labeled "<math>2^2</math> design augmented by five center point runs"). Let <math>{{x}_{1}}\,\!</math> be the indicator variable to indicate if the run is a center point:

::<math>\begin{matrix}
{{x}_{1}}=0 & {} & \text{Center point run} \\
{{x}_{1}}=1 & {} & \text{Other run} \\
\end{matrix}\,\!</math>

If <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> are the indicator variables representing factors <math>A\,\!</math> and <math>B\,\!</math>, respectively, then the model for this experiment is:

::<math>Y={{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}\,\!</math>

To investigate the presence of curvature, the following hypotheses need to be tested:

::<math>\begin{align}
& {{H}_{0}}: & {{\beta }_{1}}=0\text{ (Curvature is absent)} \\
& {{H}_{1}}: & {{\beta }_{1}}\ne 0
\end{align}\,\!</math>

The test statistic to be used for this test is:

::<math>{{({{F}_{0}})}_{curvature}}=\frac{M{{S}_{curvature}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{curvature}}\,\!</math> is the mean square for Curvature and <math>M{{S}_{E}}\,\!</math> is the error mean square.

'''Calculation of the Sum of Squares'''

The <math>X\,\!</math> matrix and <math>y\,\!</math> vector for this experiment are:

<center><math>X=\left[ \begin{matrix}
1 & 1 & -1 & -1 & 1 \\
1 & 1 & 1 & -1 & -1 \\
1 & 1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
\end{matrix} \right]\text{ }y=\left[ \begin{matrix}
24.6 \\
25.4 \\
25.0 \\
25.7 \\
25.2 \\
25.3 \\
25.4 \\
25.1 \\
25.3 \\
\end{matrix} \right]\,\!</math></center>

The sum of squares can now be calculated. For example, the error sum of squares is:

::<math>\begin{align}
& S{{S}_{E}}= & {{y}^{\prime }}[I-H]y \\
& = & 0.052
\end{align}\,\!</math>

where <math>I\,\!</math> is the identity matrix and <math>H\,\!</math> is the hat matrix. It can be seen that this is equal to <math>S{{S}_{PE\text{ }}}\,\!</math> (the sum of squares due to pure error) because of the replicates at the center point, as obtained in the [[Two_Level_Factorial_Experiments#Example_3| example]]. The number of degrees of freedom associated with <math>S{{S}_{E}}\,\!</math>, <math>dof(S{{S}_{E}})\,\!</math> is four. The extra sum of squares corresponding to the center point replicates (or Curvature) is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & Model\text{ }Sum\text{ }of\text{ }Squares- \\
& & Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }center\text{ }point \\
& = & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Curvature}}-(1/9)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }Curvature}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }Curvature}}={{X}_{\tilde{\ }Curv}}{{(X_{\tilde{\ }Curv}^{\prime }{{X}_{\tilde{\ }Curv}})}^{-1}}X_{\tilde{\ }Curv}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }Curv}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the center point. Thus, the extra sum of squares corresponding to Curvature is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Center}}-(1/9)J]y \\
& = & 0.7036-0.6875 \\
& = & 0.0161
\end{align}\,\!</math>

This extra sum of squares can be used to test for the significance of curvature. The corresponding mean square is:

::<math>\begin{align}
& M{{S}_{Curvature}}= & \frac{Sum\text{ }of\text{ }squares\text{ }corresponding\text{ }to\text{ }Curvature}{degrees\text{ }of\text{ }freedom} \\
& = & \frac{0.0161}{1} \\
& = & 0.0161
\end{align}\,\!</math>

'''Calculation of the Test Statistic'''

Knowing the mean squares, the statistic to check the significance of curvature can be calculated.

::<math>\begin{align}
& {{({{f}_{0}})}_{Curvature}}= & \frac{M{{S}_{Curvature}}}{M{{S}_{E}}} \\
& = & \frac{0.0161/1}{0.052/4} \\
& = & 1.24
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{Curvature}}=1.24\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and four degrees of freedom in the denominator is:

::<math>\begin{align}
& p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{Curvature}}) \\
& = & 1-0.6713 \\
& = & 0.3287
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that curvature does not exist for this design. This results is shown in the ANOVA table in the figure above. The surface of the fitted model based on these results, along with the observed response values, is shown in the figure below.

[[Image:doe7.21.png|center|400px|Model surface and observed response values for the design in the [[Two_Level_Factorial_Experiments#Example_4| example]].]]

 

==Blocking in 2k Designs==

Blocking can be used in the <math>{2}^{k}\,\!</math> designs to deal with cases when replicates cannot be run under identical conditions. Randomized complete block designs that were discussed in [[Randomization and Blocking in DOE]] for factorial experiments are also applicable here. At times, even with just two levels per factor, it is not possible to run all treatment combinations for one replicate of the experiment under homogeneous conditions. For example, each replicate of the <math>{2}^{2}\,\!</math> design requires four runs. If each run requires two hours and testing facilities are available for only four hours per day, two days of testing would be required to run one complete replicate. Blocking can be used to separate the treatment runs on the two different days. Blocks that do not contain all treatments of a replicate are called incomplete blocks. In incomplete block designs, the block effect is confounded with certain effect(s) under investigation. For the <math>{2}^{2}\,\!</math> design assume that treatments <math>(1)\,\!</math> and <math>ab\,\!</math> were run on the first day and treatments <math>a\,\!</math> and <math>b\,\!</math> were run on the second day. Then, the incomplete block design for this experiment is:

::<math>\begin{matrix}
\text{Block 1} & {} & \text{Block 2} \\
\left[ \begin{matrix}
(1) \\
ab \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
b \\
\end{matrix} \right] \\
\end{matrix}\,\!</math>

For this design the block effect may be calculated as:

::<math>\begin{align}
& Block\text{ }Effect= & Average\text{ }response\text{ }for\text{ }Block\text{ }1- \\
& & Average\text{ }response\text{ }for\text{ }Block\text{ }2 \\
& = & \frac{(1)+ab}{2}-\frac{a+b}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The <math>AB\,\!</math> interaction effect is:

::<math>\begin{align}
& AB= & Average\text{ }response\text{ }at\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{low}}}- \\
& & Average\text{ }response\text{ }at\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{low}}} \\
& = & \frac{ab+(1)}{2}-\frac{b+a}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The two equations given above show that, in this design, the <math>AB\,\!</math> interaction effect cannot be distinguished from the block effect because the formulas to calculate these effects are the same. In other words, the <math>AB\,\!</math> interaction is said to be confounded with the block effect and it is not possible to say if the effect calculated based on these equations is due to the <math>AB\,\!</math> interaction effect, the block effect or both. In incomplete block designs some effects are always confounded with the blocks. Therefore, it is important to design these experiments in such a way that the important effects are not confounded with the blocks. In most cases, the experimenter can assume that higher order interactions are unimportant. In this case, it would better to use incomplete block designs that confound these effects with the blocks.
One way to design incomplete block designs is to use defining contrasts as shown next:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}+...+{{\alpha }_{k}}{{q}_{k}}\,\!</math>

where the <math>{{\alpha }_{i}}\,\!</math> s are the exponents for the factors in the effect that is to be confounded with the block effect and the <math>{{q}_{i}}\,\!</math> s are values based on the level of the <math>i\,\!</math> the factor (in a treatment that is to be allocated to a block). For <math>{2}^{k}\,\!</math> designs the <math>{{\alpha }_{i}}\,\!</math> s are either 0 or 1 and the <math>{{q}_{i}}\,\!</math> s have a value of 0 for the low level of the <math>i\,\!</math> th factor and a value of 1 for the high level of the factor in the treatment under consideration. As an example, consider the <math>{2}^{2}\,\!</math> design where the interaction effect <math>AB\,\!</math> is confounded with the block. Since there are two factors, <math>k=2\,\!</math>, with <math>i=1\,\!</math> representing factor <math>A\,\!</math> and <math>i=2\,\!</math> representing factor <math>B\,\!</math>. Therefore:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}\,\!</math>

The value of <math>{{\alpha }_{1}}\,\!</math> is one because the exponent of factor <math>A\,\!</math> in the confounded interaction <math>AB\,\!</math> is one. Similarly, the value of <math>{{\alpha }_{2}}\,\!</math> is one because the exponent of factor <math>B\,\!</math> in the confounded interaction <math>AB\,\!</math> is also one. Therefore, the defining contrast for this design can be written as:

::<math>\begin{align}
& L= & {{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}} \\
& = & 1\cdot {{q}_{1}}+1\cdot {{q}_{2}} \\
& = & {{q}_{1}}+{{q}_{2}}
\end{align}\,\!</math>

Once the defining contrast is known, it can be used to allocate treatments to the blocks. For the <math>{2}^{2}\,\!</math> design, there are four treatments <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math> and <math>ab\,\!</math>. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. In order to decide which block the treatment <math>(1)\,\!</math> belongs to, the levels of factors <math>A\,\!</math> and <math>B\,\!</math> for this run are used. Since factor <math>A\,\!</math> is at the low level in this treatment, <math>{{q}_{1}}=0\,\!</math>. Similarly, since factor <math>B\,\!</math> is also at the low level in this treatment, <math>{{q}_{2}}=0\,\!</math>. Therefore:

::<math>\begin{align}
& L= & {{q}_{1}}+{{q}_{2}} \\
& = & 0+0=0\text{ (mod 2)}
\end{align}\,\!</math>

Note that the value of <math>L\,\!</math> used to decide the block allocation is "mod 2" of the original value. This value is obtained by taking the value of 1 for odd numbers and 0 otherwise. Based on the value of <math>L\,\!</math>, treatment <math>(1)\,\!</math> is assigned to block 1. Other treatments can be assigned using the following calculations:

::<math>\begin{align}
& (1): & \text{ }L=0+0=0=0\text{ (mod 2)} \\
& a: & \text{ }L=1+0=1=1\text{ (mod 2)} \\
& b: & \text{ }L=0+1=1=1\text{ (mod 2)} \\
& ab: & \text{ }L=1+1=2=0\text{ (mod 2)}
\end{align}\,\!</math>

Therefore, to confound the interaction <math>AB\,\!</math> with the block effect in the <math>{2}^{2}\,\!</math> incomplete block design, treatments <math>(1)\,\!</math> and <math>ab\,\!</math> (with <math>L=0\,\!</math>) should be assigned to block 2 and treatment combinations <math>a\,\!</math> and <math>b\,\!</math> (with <math>L=1\,\!</math>) should be assigned to block 1.

====Example: Two Level Factorial Design with Two Blocks====

This example illustrates how treatments can be allocated to two blocks for an unreplicated <math>{2}^{k}\,\!</math> design. Consider the unreplicated <math>{2}^{4}\,\!</math> design to investigate the four factors affecting the defects in automobile vinyl panels discussed in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments required for this experiment were run by two different operators with each operator conducting 8 runs. This experiment is an example of an incomplete block design. The analyst in charge of this experiment assumed that the interaction <math>ABCD\,\!</math> was not significant and decided to allocate treatments to the two operators so that the <math>ABCD\,\!</math> interaction was confounded with the block effect (the two operators are the blocks). The allocation scheme to assign treatments to the two operators can be obtained as follows.
 
The defining contrast for the <math>{2}^{4}\,\!</math> design where the <math>ABCD\,\!</math> interaction is confounded with the blocks is:

::<math>L={{q}_{1}}+{{q}_{2}}+{{q}_{3}}+{{q}_{4}}\,\!</math>

The treatments can be allocated to the two operators using the values of the defining contrast. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. Then the value of the defining contrast for treatment <math>a\,\!</math> is:

::<math>a\ \ :\ \ \text{ }L=1+0+0+0=1=1\text{ (mod 2)}\,\!</math>

Therefore, treatment <math>a\,\!</math> should be assigned to Block 1 or the first operator. Similarly, for treatment <math>ab\,\!</math> we have:

::<math>ab\ \ :\ \ \text{ }L=1+1+0+0=2=0\text{ (mod 2)}\,\!</math>

[[Image:doe7.22.png|center|200px| Allocation of treatments to two blocks for the <math>2^4</math> design in the example by confounding interaction of <math>ABCD</math> with the blocks.]]

Therefore, <math>ab\,\!</math> should be assigned to Block 2 or the second operator. Other treatments can be allocated to the two operators in a similar manner to arrive at the allocation scheme shown in the figure below.
In DOE++, to confound the <math>ABCD\,\!</math> interaction for the <math>{2}^{4}\,\!</math> design into two blocks, the number of blocks are specified as shown in the figure below. Then the interaction <math>ABCD\,\!</math> is entered in the Block Generator window (second following figure) which is available using the Block Generator button in the following figure. The design generated by DOE++ is shown in the third of the following figures. This design matches the allocation scheme of the preceding figure.

[[Image:doe7_23.png|center|700px| Adding block properties for the experiment in the example.|link=]]

[[Image:doe7_24.png|center|700px|Specifying the interaction ABCD as the interaction to be confounded with the blocks for the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

[[Image:doe7_25.png|center|800px|Two block design for the experiment in the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

For the analysis of this design, the sum of squares for all effects are calculated assuming no blocking. Then, to account for blocking, the sum of squares corresponding to the <math>ABCD\,\!</math> interaction is considered as the sum of squares due to blocks and <math>ABCD\,\!</math>. In DOE++ this is done by displaying this sum of squares as the sum of squares due to the blocks. This is shown in the following figure where the sum of squares in question is obtained as 72.25 and is displayed against Block. The interaction ABCD, which is confounded with the blocks, is not displayed. Since the design is unreplicated, any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_26.png|center|793px|ANOVA table for the experiment of the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

===Unreplicated 2''k'' Designs in 2''p'' Blocks===

A single replicate of the <math>{2}^{k}\,\!</math> design can be run in up to <math>{2}^{p}\,\!</math> blocks where <math>p<k\,\!</math>. The number of effects confounded with the blocks equals the degrees of freedom associated with the block effect.

If two blocks are used (the block effect has two levels), then one (<math>2-1=1)\,\!</math> effect is confounded with the blocks. If four blocks are used, then three (<math>4-1=3\,\!</math>) effects are confounded with the blocks and so on. For example an unreplicated <math>{2}^{4}\,\!</math> design may be confounded in <math>{2}^{2}\,\!</math> (four) blocks using two contrasts, <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}}\,\!</math>. Let <math>AC\,\!</math> and <math>BD\,\!</math> be the effects to be confounded with the blocks. Corresponding to these two effects, the contrasts are respectively:

::<math>\begin{align}
& {{L}_{1}}= & {{q}_{1}}+{{q}_{3}} \\
& {{L}_{2}}= & {{q}_{2}}+{{q}_{4}}
\end{align}\,\!</math>

Based on the values of <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}},\,\!</math> the treatments can be assigned to the four blocks as follows:

<center><math>\begin{matrix}
\text{Block 4} & {} & \text{Block 3} & {} & \text{Block 2} & {} & \text{Block 1} \\
{{L}_{1}}=0,{{L}_{2}}=0 & {} & {{L}_{1}}=1,{{L}_{2}}=0 & {} & {{L}_{1}}=0,{{L}_{2}}=1 & {} & {{L}_{1}}=1,{{L}_{2}}=1 \\
{} & {} & {} & {} & {} & {} & {} \\
\left[ \begin{matrix}
(1) \\
ac \\
bd \\
abcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
c \\
abd \\
bcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
b \\
abc \\
d \\
acd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
ab \\
bc \\
ad \\
cd \\
\end{matrix} \right] \\
\end{matrix}\,\!</math></center>

Since the block effect has three degrees of freedom, three effects are confounded with the block effect. In addition to <math>AC\,\!</math> and <math>BD\,\!</math>, the third effect confounded with the block effect is their generalized interaction, <math>(AC)(BD)=ABCD\,\!</math>.
In general, when an unreplicated <math>{2}^{k}\,\!</math> design is confounded in <math>{2}^{p}\,\!</math> blocks, <math>p\,\!</math> contrasts are needed (<math>{{L}_{1}},{{L}_{2}}...{{L}_{p}}\,\!</math>). <math>p\,\!</math> effects are selected to define these contrasts such that none of these effects are the generalized interaction of the others. The <math>{2}^{p}\,\!</math> blocks can then be assigned the treatments using the <math>p\,\!</math> contrasts. <math>{{2}^{p}}-(p+1)\,\!</math> effects, that are also confounded with the blocks, are then obtained as the generalized interaction of the <math>p\,\!</math> effects. In the statistical analysis of these designs, the sum of squares are computed as if no blocking were used. Then the block sum of squares is obtained by adding the sum of squares for all the effects confounded with the blocks.

====Example: 2 Level Factorial Design with Four Blocks====

This example illustrates how DOE++ obtains the sum of squares when treatments for an unreplicated <math>{2}^{k}\,\!</math> design are allocated among four blocks. Consider again the unreplicated <math>{2}^{4}\,\!</math> design used to investigate the defects in automobile vinyl panels presented in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments needed to complete the experiment were run by four operators. Therefore, there are four blocks. Assume that the treatments were allocated to the blocks using the generators mentioned in the previous section, i.e., treatments were allocated among the four operators by confounding the effects, <math>AC\,\!</math> and <math>BD,\,\!</math> with the blocks. These effects can be specified as Block Generators as shown in the following figure. (The generalized interaction of these two effects, interaction <math>ABCD\,\!</math>, will also get confounded with the blocks.) The resulting design is shown in the second following figure and matches the allocation scheme obtained in the previous section.

[[Image:doe7_27.png|center|700px|Specifying the interactions AC and BD as block generators for the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

The sum of squares in this case can be obtained by calculating the sum of squares for each of the effects assuming there is no blocking. Once the individual sum of squares have been obtained, the block sum of squares can be calculated. The block sum of squares is the sum of the sum of squares of effects, <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, since these effects are confounded with the block effect. As shown in the second following figure, this sum of squares is 92.25 and is displayed against Block. The interactions <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, which are confounded with the blocks, are not displayed. Since the present design is unreplicated any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_28.png|center|800px|Design for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

[[Image:doe7_29.png|center|793px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

==Variability Analysis==

For replicated two level factorial experiments, DOE++ provides the option of conducting variability analysis (using the Variability Analysis icon under the Data menu). The analysis is used to identify the treatment that results in the least amount of variation in the product or process being investigated. Variability analysis is conducted by treating the standard deviation of the response for each treatment of the experiment as an additional response. The standard deviation for a treatment is obtained by using the replicated response values at that treatment run. As an example, consider the <math>{2}^{3}\,\!</math> design shown in the following figure where each run is replicated four times. A variability analysis can be conducted for this design. DOE++ calculates eight standard deviation values corresponding to each treatment of the design (see second following figure). Then, the design is analyzed as an unreplicated <math>{2}^{3}\,\!</math> design with the standard deviations (displayed as Y Standard Deviation. in second following figure) as the response. The normal probability plot of effects identifies <math>AC\,\!</math> as the effect that influences variability (see third figure following). Based on the effect coefficients obtained in the fourth figure following, the model for Y Std. is:

::<math>\begin{align}
& \text{Y Std}\text{.}= & 0.6779+0.2491\cdot AC \\
& = & 0.6779+0.2491{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

Based on the model, the experimenter has two choices to minimize variability (by minimizing Y Std.). The first choice is that <math>{{x}_{1}}\,\!</math> should be <math>1\,\!</math> (i.e., <math>A\,\!</math> should be set at the high level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the low level). The second choice is that <math>{{x}_{1}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>A\,\!</math> should be set at the low level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the high level). The experimenter can select the most feasible choice.

[[Image:doe7.30.png|center|391px|A <math>2^3\,\!</math> design with four replicated response values that can be used to conduct a variability analysis.]]
 

[[Image:doe7_31.png|center|727px|Variability analysis in DOE++.|link=]]
 

[[Image:doe7_32.png|center|650px|Normal probability plot of effects for the variability analysis example.|link=]]
 

[[Image:doe7_33.png|center|727px|Effect coefficients for the variability analysis example.|link=]]

 

==Two Level Fractional Factorial Designs==

As the number of factors in a two level factorial design increases, the number of runs for even a single replicate of the <math>{2}^{k}\,\!</math> design becomes very large. For example, a single replicate of an eight factor two level experiment would require 256 runs. Fractional factorial designs can be used in these cases to draw out valuable conclusions from fewer runs. The basis of fractional factorial designs is the ''sparsity of effects'' principle.[[DOE_References|[Wu, 2000]]] The principle states that, most of the time, responses are affected by a small number of main effects and lower order interactions, while higher order interactions are relatively unimportant. Fractional factorial designs are used as screening experiments during the initial stages of experimentation. At these stages, a large number of factors have to be investigated and the focus is on the main effects and two factor interactions. These designs obtain information about main effects and lower order interactions with fewer experiment runs by confounding these effects with unimportant higher order interactions. As an example, consider a <math>{2}^{8}\,\!</math> design that requires 256 runs. This design allows for the investigation of 8 main effects and 28 two factor interactions. However, 219 degrees of freedom are devoted to three factor or higher order interactions. This full factorial design can prove to be very inefficient when these higher order interactions can be assumed to be unimportant. Instead, a fractional design can be used here to identify the important factors that can then be investigated more thoroughly in subsequent experiments. In unreplicated fractional factorial designs, no degrees of freedom are available to calculate the error sum of squares and the techniques mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] should be employed for the analysis of these designs.

==Half-fraction Designs==

A half-fraction of the <math>{2}^{k}\,\!</math> design involves running only half of the treatments of the full factorial design. For example, consider a <math>{2}^{3}\,\!</math> design that requires eight runs in all. The design matrix for this design is shown in the figure (a) below. A half-fraction of this design is the design in which only four of the eight treatments are run. The fraction is denoted as <math>{2}^{3-1}\,\!</math> with the "<math>-1\,\!</math>" in the index denoting a half-fraction. Assume that the treatments chosen for the half-fraction design are the ones where the interaction <math>ABC\,\!</math> is at the high level (i.e., only those rows are chosen from the following figure (a) where the column for <math>ABC\,\!</math> has entries of 1). The resulting <math>{2}^{3-1}\,\!</math> design has a design matrix as shown in figure (b) below.

[[Image:doe7.34.png|center|330px|Half-fractions of the <math>2^3\,\!</math> design. (a) shows the full factorial <math>2^3\,\!</math> design, (b) shows the <math>2^{3-1}\,\!</math> design with the defining relation <math>I=ABC\,\!</math> and (c) shows the <math>{2}^{3-1}\,\!</math> design with the defining relation <math>I=-ABC\,\!</math>.]]

In the <math>{2}^{3-1}\,\!</math> design of figure (b), since the interaction <math>ABC\,\!</math> is always included at the same level (the high level represented by 1), it is not possible to measure this interaction effect. The effect, <math>ABC\,\!</math>, is called the ''generator'' or ''word'' for this design. It can be noted that, in the design matrix of the following figure (b), the column corresponding to the intercept, <math>I\,\!</math>, and column corresponding to the interaction <math>ABC\,\!</math>, are identical. The identical columns are written as <math>I=ABC\,\!</math> and this equation is called the ''defining relation'' for the design. In DOE++, the present <math>{2}^{3-1}\,\!</math> design can be obtained by specifying the design properties as shown in the following figure.

[[Image:doe7_35.png|center|700px|Design properties for the <math>2^{3-1}\,\!</math> design.|link=]]

The defining relation, <math>I=ABC\,\!</math>, is entered in the Fraction Generator window as shown next.

[[Image:doe7_36.png|center|700px|Specifying the defining relation for the <math>2^{3-1}\,\!</math> design.|link=]]

Note that in the figure following that, the defining relation is specified as <math>C=AB\,\!</math>. This relation is obtained by multiplying the defining relation, <math>I=ABC\,\!</math>, by the last factor, <math>C\,\!</math>, of the design.

===Calculation of Effects===

Using the four runs of the <math>{2}^{3-1}\,\!</math> design in figure (b) discussed above, the main effects can be calculated as follows:

::<math>\begin{align}
& A= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& B= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& C= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

where <math>a\,\!</math>, <math>b\,\!</math>, <math>c\,\!</math> and <math>abc\,\!</math> are the treatments included in the <math>{2}^{3-1}\,\!</math> design.

Similarly, the two factor interactions can also be obtained as:

::<math>\begin{align}
& BC= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& AC= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& AB= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

The equations for <math>A\,\!</math> and <math>BC\,\!</math> above result in the same effect values showing that effects <math>A\,\!</math> and <math>BC\,\!</math> are confounded in the present <math>{2}^{3-1}\,\!</math> design. Thus, the quantity, <math>\tfrac{1}{2}(a-b-c+abc),\,\!</math> estimates <math>A+BC\,\!</math> (i.e., both the main effect <math>A\,\!</math> and the two-factor interaction <math>BC\,\!</math>). The effects, <math>A\,\!</math> and <math>BC,\,\!</math> are called ''aliases''. From the remaining equations given above, it can be seen that the other aliases for this design are <math>B\,\!</math> and <math>AC\,\!</math>, and <math>C\,\!</math> and <math>AB\,\!</math>. Therefore, the equations to calculate the effects in the present <math>{2}^{3-1}\,\!</math> design can be written as follows:

::<math>\begin{align}
& A+BC= & \frac{1}{2}(a-b-c+abc) \\
& B+AC= & \frac{1}{2}(-a+b-c+abc) \\
& C+AB= & \frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

===Calculation of Aliases===

Aliases for a fractional factorial design can be obtained using the defining relation for the design. The defining relation for the present <math>{2}^{3-1}\,\!</math> design is:

::<math>I=ABC\,\!</math>

Multiplying both sides of the previous equation by the main effect, <math>A,\,\!</math> gives the alias effect of <math>A\,\!</math> :

::<math>\begin{align}
& A\cdot I= & A\cdot ABC \\
& A= & {{A}^{2}}BC \\
& A= & BC
\end{align}\,\!</math>

Note that in calculating the alias effects, any effect multiplied by <math>I\,\!</math> remains the same (<math>A\cdot I=A\,\!</math>), while an effect multiplied by itself results in <math>I\,\!</math> (<math>{{A}^{2}}=I\,\!</math>). Other aliases can also be obtained:

::<math>\begin{align}
& B\cdot I= & B\cdot ABC \\
& B= & A{{B}^{2}}C \\
& B= & AC
\end{align}\,\!</math>

:and:

::<math>\begin{align}
& C\cdot I= & C\cdot ABC \\
& C= & AB{{C}^{2}} \\
& C= & AB
\end{align}\,\!</math>

===Fold-over Design===

If it can be assumed for this design that the two-factor interactions are unimportant, then in the absence of <math>BC\,\!</math>, <math>AC\,\!</math> and <math>AB\,\!</math>, the equations for (A+BC), (B+AC) and (C+AB) can be used to estimate the main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. However, if such an assumption is not applicable, then to uncouple the main effects from their two factor aliases, the alternate fraction that contains runs having <math>ABC\,\!</math> at the lower level should be run. The design matrix for this design is shown in the preceding figure (c). The defining relation for this design is <math>I=-ABC\,\!</math> because the four runs for this design are obtained by selecting the rows of the preceding figure (a) for which the value of the <math>ABC\,\!</math> column is <math>-1\,\!</math>. The aliases for this fraction can be obtained as explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]] as <math>A=-BC\,\!</math>, <math>B=-AC\,\!</math> and <math>C=-AB\,\!</math>. The effects for this design can be calculated as:

::<math>\begin{align}
& A-BC= & \frac{1}{2}(ab+ac-(1)-bc) \\
& B-AC= & \frac{1}{2}(ab-ac+(1)-bc) \\
& C-AB= & \frac{1}{2}(-ab+ac-(1)+bc)
\end{align}\,\!</math>

These equations can be combined with the equations for (A+BC), (B+AC) and (C+AB) to obtain the de-aliased main effects and two factor interactions. For example, adding equations (A+BC) and (A-BC) returns the main effect <math>A\,\!</math>.

::<math>\begin{align}
& 2A= & \frac{1}{2}(a-b-c+abc)+ \\
& & \frac{1}{2}(ab+ac-(1)-bc)
\end{align}\,\!</math>

The process of augmenting a fractional factorial design by a second fraction of the same size by simply reversing the signs (of all effect columns except <math>I\,\!</math>) is called ''folding over''. The combined design is referred to as a ''fold-over design''.

==Quarter and Smaller Fraction Designs==

At times, the number of runs even for a half-fraction design are very large. In these cases, smaller fractions are used. A quarter-fraction design, denoted as <math>{2}^{k-2}\,\!</math>, consists of a fourth of the runs of the full factorial design. Quarter-fraction designs require two defining relations. The first defining relation returns the half-fraction or the <math>{2}^{k-1}\,\!</math> design. The second defining relation selects half of the runs of the <math>{2}^{k-1}\,\!</math> design to give the quarter-fraction. For example, consider the <math>{2}^{4}\,\!</math> design. To obtain a <math>{2}^{4-2}\,\!</math> design from this design, first a half-fraction of this design is obtained by using a defining relation. Assume that the defining relation used is <math>I=ABCD\,\!</math>. The design matrix for the resulting <math>{2}^{4-1}\,\!</math> design is shown in figure (a) below. Now, a quarter-fraction can be obtained from the <math>{2}^{4-1}\,\!</math> design shown in figure (a) below using a second defining relation <math>I=AD\,\!</math>. The resulting <math>{2}^{4-2}\,\!</math> design obtained is shown in figure (b) below.

[[Image:doe7.37.png|center|465px|Fractions of the <math>2^4\,\!</math> design - Figure (a) shows the <math>2^{4-1}</math> design with the defining relation <math>I=ABCD\,\!</math> and (b) shows the <math>2^{4-2}\,\!</math> design with the defining relation <math>I=ABCD=AD=BC\,\!</math>.]]

The complete defining relation for this <math>{2}^{4-2}\,\!</math> design is:

::<math>I=ABCD=AD=BC\,\!</math>

Note that the effect, <math>BC,\,\!</math> in the defining relation is the generalized interaction of <math>ABCD\,\!</math> and <math>AD\,\!</math> and is obtained using <math>(ABCD)(AD)={{A}^{2}}BC{{D}^{2}}=BC\,\!</math>. In general, a <math>{2}^{k-p}\,\!</math> fractional factorial design requires <math>p\,\!</math> independent generators. The defining relation for the design consists of the <math>p\,\!</math> independent generators and their <math>{2}^{p}\,\!</math> - (<math>p\,\!</math> +1) generalized interactions.

===Calculation of Aliases===

The alias structure for the present <math>{2}^{4-2}\,\!</math> design can be obtained using the defining relation of equation (I=ABCD=AD=BC) following the procedure explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]]. For example, multiplying the defining relation by <math>A\,\!</math> returns the effects aliased with the main effect, <math>A\,\!</math>, as follows:

::<math>\begin{align}
& A\cdot I= & A\cdot ABCD=A\cdot AD=A\cdot BC \\
& A= & {{A}^{2}}BCD={{A}^{2}}D=ABC \\
& A= & BCD=D=ABC
\end{align}\,\!</math>

Therefore, in the present <math>{2}^{4-2}\,\!</math> design, it is not possible to distinguish between effects <math>A\,\!</math>, <math>D\,\!</math>, <math>BCD\,\!</math> and <math>ABC\,\!</math>. Similarly, multiplying the defining relation by <math>B\,\!</math> and <math>AB\,\!</math> returns the effects that are aliased with these effects:

::<math>\begin{align}
& B= & ACD=ABD=C \\
& AB= & CD=AD=AC
\end{align}\,\!</math>

Other aliases can be obtained in a similar way. It can be seen that each effect in this design has three aliases. In general, each effect in a <math>{2}^{k-p}\,\!</math> design has <math>{2}^{p-1}\,\!</math> aliases.
The aliases for the <math>{2}^{4-2}\,\!</math> design show that in this design the main effects are aliased with each other (<math>A\,\!</math> is aliased with <math>D\,\!</math> and <math>B\,\!</math> is aliased with <math>C\,\!</math>). Therefore, this design is not a useful design and is not available in DOE++. It is important to ensure that main effects and lower order interactions of interest are not aliased in a fractional factorial design. This is known by looking at the resolution of the fractional factorial design.

==Design Resolution==

The resolution of a fractional factorial design is defined as the number of factors in the lowest order effect in the defining relation. For example, in the defining relation <math>I=ABCD=AD=BC\,\!</math> of the previous <math>{2}^{4-2}\,\!</math> design, the lowest-order effect is either <math>AD\,\!</math> or <math>BC,\,\!</math> containing two factors. Therefore, the resolution of this design is equal to two. The resolution of a fractional factorial design is represented using Roman numerals. For example, the previously mentioned <math>{2}^{4-2}\,\!</math> design with a resolution of two can be represented as 2 <math>_{\text{II}}^{4-2}\,\!</math>. The resolution provides information about the confounding in the design as explained next:
 
 
#'''Resolution III Designs''' In these designs, the lowest order effect in the defining relation has three factors (e.g., a <math>{2}^{5-2}\,\!</math> design with the defining relation <math>I=ABDE=ABC=CDE\,\!</math>). In resolution III designs, no main effects are aliased with any other main effects, but main effects are aliased with two factor interactions. In addition, some two factor interactions are aliased with each other.
#'''Resolution IV Designs''' In these designs, the lowest order effect in the defining relation has four factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABDE\,\!</math>). In resolution IV designs, no main effects are aliased with any other main effects or two factor interactions. However, some main effects are aliased with three factor interactions and the two factor interactions are aliased with each other.
#'''Resolution V Designs''' In these designs the lowest order effect in the defining relation has five factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABCDE\,\!</math>). In resolution V designs, no main effects or two factor interactions are aliased with any other main effects or two factor interactions. However, some main effects are aliased with four factor interactions and the two factor interactions are aliased with three factor interactions.

Fractional factorial designs with the highest resolution possible should be selected because the higher the resolution of the design, the less severe the degree of confounding. In general, designs with a resolution less than III are never used because in these designs some of the main effects are aliased with each other. The table below shows fractional factorial designs with the highest available resolution for three to ten factor designs along with their defining relations.

[[Image:doet7.3.png|center|474px|Highest resolution designs available for fractional factorial designs with 3 to 10 factors.]]

All of the two level fractional factorial designs available in DOE++ are shown next.

[[Image:doe7.38.png|center|471px|Two level fractional factorial designs available in DOE++ and their resolutions.]]

===Minimum Aberration Designs===
At times, different designs with the same resolution but different aliasing may be available. The best design to select in such a case is the minimum aberration design. For example, all <math>{2}^{7-2}\,\!</math> designs in the fourth table have a resolution of four (since the generator with the minimum number of factors in each design has four factors). Design <math>1\,\!</math> has three generators of length four (<math>ABCF,\,\!</math> <math>BCDG,\,\!</math> <math>ADFG\,\!</math>). Design <math>2\,\!</math> has two generators of length four (<math>ABCF,\,\!</math> <math>ADEG\,\!</math>). Design <math>3\,\!</math> has one generator of length four (<math>CEFG\,\!</math>). Therefore, design <math>3\,\!</math> has the least number of generators with the minimum length of four. Design <math>3\,\!</math> is called the minimum aberration design. It can be seen that the alias structure for design <math>3\,\!</math> is less involved compared to the other designs. For details refer to [[DOE_References|[Wu, 2000]]].

[[Image:doet7.4.png|center|432px|Three <math>2_{IV}^{7-2}\,\!</math> designs with different defining relations.]]

====Example====

The design of an automobile fuel cone is thought to be affected by six factors in the manufacturing process: cavity temperature (factor <math>A\,\!</math>), core temperature (factor <math>B\,\!</math>), melt temperature (factor <math>C\,\!</math>), hold pressure (factor <math>D\,\!</math>), injection speed (factor <math>E\,\!</math>) and cool time (factor <math>F\,\!</math>). The manufacturer of the fuel cone is unable to run the <math>{2}^{6}=64\,\!</math> runs required to complete one replicate for a two level full factorial experiment with six factors. Instead, they decide to run a fractional factorial design. Considering that three factor and higher order interactions are likely to be inactive, the manufacturer selects a <math>{2}^{6-2}\,\!</math> design that will require only 16 runs. The manufacturer chooses the resolution IV design which will ensure that all main effects are free from aliasing (assuming three factor and higher order interactions are absent). However, in this design the two factor interactions may be aliased with each other. It is decided that, if important two factor interactions are found to be present, additional experiment trials may be conducted to separate the aliased effects. The performance of the fuel cone is measured on a scale of 1 to 15. In DOE++, the design for this experiment is set up using the properties shown in the following figure. The Fraction Generators for the design, <math>E=ABC\,\!</math> and <math>F=BCD\,\!</math>, are the same as the defaults used in DOE++. The resulting <math>{2}^{6-2}\,\!</math> design and the corresponding response values are shown in the following two figures.

[[Image:doe7_39.png|center|644px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

[[Image:doe7_40.png|center|534px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The complete alias structure for the 2 <math>_{\text{IV}}^{6-2}\,\!</math> design is shown next.

<center><math>I=ABCE=ADEF=BCDF\,\!</math></center>

 
<center><math>\begin{align}
& A= & BCE=DEF=ABCDF \\
& B= & ACE=CDF=ABDEF \\
& C= & ABE=BDF=ACDEF \\
& D= & AEF=BCF=ABCDE \\
& E= & ABC=ADF=BCDEF \\
& F= & ADE=BCD=ABCEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& AB= & CE=ACDF=BDEF \\
& AC= & BE=ABDF=CDEF \\
& AD= & EF=ABCF=BCDE \\
& AE= & BC=DF=ABCDEF \\
& AF= & DE=ABCD=BCEF \\
& BD= & CF=ABEF=ACDE \\
& BF= & CD=ABDE=ACEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& ABD= & ACF=BEF=CDE \\
& ABF= & ACD=BDE=CEF
\end{align}\,\!</math></center>

In DOE++, the alias structure is displayed in the Design Summary and as part of the Design Evaluation result, as shown next:

[[Image:doe7_41.png|center|700px|Alias structure for the experiment design in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The normal probability plot of effects for this unreplicated design shows the main effects of factors <math>C\,\!</math> and <math>D\,\!</math> and the interaction effect, <math>BF\,\!</math>, to be significant (see the following figure).

[[Image:doe7_42.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

From the alias structure, it can be seen that for the present design interaction effect, <math>BF,\,\!</math> is confounded with <math>CD\,\!</math>. Therefore, the actual source of this effect cannot be known on the basis of the present experiment. However because neither factor <math>B\,\!</math> nor <math>F\,\!</math> is found to be significant there is an indication the observed effect is likely due to interaction, <math>CD\,\!</math>. To confirm this, a follow-up <math>{2}^{2}\,\!</math> experiment is run involving only factors <math>B\,\!</math> and <math>F\,\!</math>. The interaction, <math>BF\,\!</math>, is found to be inactive, leading to the conclusion that the interaction effect in the original experiment is effect, <math>CD\,\!</math>. Given these results, the fitted regression model for the fuel cone design as per the coefficients obtained from DOE++ is shown next.

::<math>\hat{y}=7.6875+C+2\cdot D+2.1875\cdot CD\,\!</math>

[[Image:doe7_43.png|center|700px|Effect coefficients for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

==Projection==

Projection refers to the reduction of a fractional factorial design to a full factorial design by dropping out some of the factors of the design. Any fractional factorial design of resolution, <math>R,\,\!</math> can be reduced to complete factorial designs in any subset of <math>R-1\,\!</math> factors. For example, consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. The resolution of this design is four. Therefore, this design can be reduced to full factorial designs in any three (<math>4-1=3\,\!</math>) of the original seven factors (by dropping the remaining four of factors). Further, a fractional factorial design can also be reduced to a full factorial design in any <math>R\,\!</math> of the original factors, as long as these <math>R\,\!</math> factors are not part of the generator in the defining relation. Again consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. This design can be reduced to a full factorial design in four factors provided these four factors do not appear together as a generator in the defining relation. The complete defining relation for this design is:

::<math>\begin{align}
& I= & ABCE=BCDF=ACDG=ADEF=ABFG=BDEG=CEFG
\end{align}\,\!</math>

Therefore, there are seven four factor combinations out of the 35 (<math>(_{7}^{4})=35\,\!</math>) possible four-factor combinations that are used as generators in the defining relation. The designs with the remaining 28 four factor combinations would be full factorial 16-run designs. For example, factors <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math> do not occur as a generator in the defining relation of the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. If the remaining factors, <math>E\,\!</math>, <math>F\,\!</math> and <math>G\,\!</math>, are dropped, the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design will reduce to a full factorial design in <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math>.

==Resolution III Designs==

At times, the factors to be investigated in screening experiments are so large that even running a fractional factorial design is impractical. This can be partially solved by using resolution III fractional factorial designs in the cases where three factor and higher order interactions can be assumed to be unimportant. Resolution III designs, such as the 2 <math>_{\text{III}}^{3-1}\,\!</math> design, can be used to estimate <math>k\,\!</math> main effects using just <math>k+1\,\!</math> runs. In these designs, the main effects are aliased with two factor interactions. Once the results from these designs are obtained, and knowing that three factor and higher order interactions are unimportant, the experimenter can decide if there is a need to run a fold-over design to de-alias the main effects from the two factor interactions. Thus, the 2 <math>_{\text{III}}^{3-1}\,\!</math> design can be used to investigate three factors in four runs, the 2 <math>_{\text{III}}^{7-4}\,\!</math> design can be used to investigate seven factors in eight runs, the 2 <math>_{\text{III}}^{15-11}\,\!</math> design can be used to investigate fifteen factors in sixteen runs and so on.

====Example====

{{:Resolution_III_Design_Example}}

==Alias Matrix==
In [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction designs]] and [[Two_Level_Factorial_Experiments#Quarter_and_Smaller_Fraction_Designs| Quarter and Smaller Fraction Designs]], the alias structure for fractional factorial designs was obtained using the defining relation. However, this method of obtaining the alias structure is not very efficient when the alias structure is very complex or when partial aliasing is involved. One of the ways to obtain the alias structure for any design, regardless of its complexity, is to use the alias matrix. The alias matrix for a design is calculated using <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> where <math>{{X}_{1}}\,\!</math> is the portion of the design matrix, <math>X,\,\!</math> that contains the effects for which the aliases need to be calculated, and <math>{{X}_{2}}\,\!</math> contains the remaining columns of the design matrix, other than those included in <math>{{X}_{1}}\,\!</math>.

To illustrate the use of the alias matrix, consider the design matrix for the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design (using the defining relation <math>I=ABCD\,\!</math>) shown next:

[[Image:Chapter7__879.png|center|link=]]

The alias structure for this design can be obtained by defining <math>{{X}_{1}}\,\!</math> using eight columns since the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design estimates eight effects. If the first eight columns of <math>X\,\!</math> are used then <math>{{X}_{1}}\,\!</math> is:

[[Image:Chapter7__884.png|center|link=]]

<math>{{X}_{2}}\,\!</math> is obtained using the remaining columns as:

[[Image:Chapter7__886.png|center|link=]]

Then the alias matrix <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> is:

[[Image:Chapter7__888.png|center|link=]]

The alias relations can be easily obtained by observing the alias matrix as:

<center><math>\begin{align}
& I= & ABCD \\
& A= & BCD \\
& B= & ACD \\
& AB= & CD \\
& C= & ABD \\
& AC= & BD \\
& BC= & AD \\
& D= & ABC
\end{align}\,\!</math></center>

Two Level Factorial Experiments

2017-08-10T17:29:12Z

Richard House: /* Calculation of Effect Coefficients */

{{Template:Doebook|8}}
Two level factorial experiments are factorial experiments in which each factor is investigated at only two levels. The early stages of experimentation usually involve the investigation of a large number of potential factors to discover the "vital few" factors. Two level factorial experiments are used during these stages to quickly filter out unwanted effects so that attention can then be focused on the important ones.

==2''k'' Designs==
The factorial experiments, where all combination of the levels of the factors are run, are usually referred to as ''full factorial experiments''. Full factorial two level experiments are also referred to as <math>{2}^{k}\,\!</math> designs where <math>k\,\!</math> denotes the number of factors being investigated in the experiment. In Weibull++ DOE folios, these designs are referred to as 2 Level Factorial Designs as shown in the figure below.

[[Image:doe7_1.png|center|487px|Selection of full factorial experiments with two levels in Weibull++.|link=]]

A full factorial two level design with <math>k\,\!</math> factors requires <math>{{2}^{k}}\,\!</math> runs for a single replicate. For example, a two level experiment with three factors will require <math>2\times 2\times 2={{2}^{3}}=8\,\!</math> runs. The choice of the two levels of factors used in two level experiments depends on the factor; some factors naturally have two levels. For example, if gender is a factor, then male and female are the two levels. For other factors, the limits of the range of interest are usually used. For example, if temperature is a factor that varies from <math>{45}^{o}C\,\!</math> to <math>{90}^{o}C\,\!</math>, then the two levels used in the <math>{2}^{k}\,\!</math> design for this factor would be <math>{45}^{o}\,\!C\,\!</math> and <math>{90}^{o}\,\!C\,\!</math>.

The two levels of the factor in the <math>{2}^{k}\,\!</math> design are usually represented as <math>-1\,\!</math> (for the first level) and <math>1\,\!</math> (for the second level). Note that this representation is reversed from the coding used in [[General Full Factorial Designs]] for the indicator variables that represent two level factors in ANOVA models. For ANOVA models, the first level of the factor was represented using a value of <math>1\,\!</math> for the indicator variable, while the second level was represented using a value of <math>-1\,\!</math>. For details on the notation used for two level experiments refer to [[Two_Level_Factorial_Experiments#Notation| Notation]].

===The 22 Design===

The simplest of the two level factorial experiments is the <math>{2}^{2}\,\!</math> design where two factors (say factor <math>A\,\!</math> and factor <math>B\,\!</math>) are investigated at two levels. A single replicate of this design will require four runs (<math>{{2}^{2}}=2\times 2=4\,\!</math>) The effects investigated by this design are the two main effects, <math>A\,\!</math> and <math>B,\,\!</math> and the interaction effect <math>AB\,\!</math>. The treatments for this design are shown in figure (a) below. In figure (a), letters are used to represent the treatments. The presence of a letter indicates the high level of the corresponding factor and the absence indicates the low level. For example, (1) represents the treatment combination where all factors involved are at the low level or the level represented by <math>-1\,\!</math> ; <math>a\,\!</math> represents the treatment combination where factor <math>A\,\!</math> is at the high level or the level of <math>1\,\!</math>, while the remaining factors (in this case, factor <math>B\,\!</math>) are at the low level or the level of <math>-1\,\!</math>. Similarly, <math>b\,\!</math> represents the treatment combination where factor <math>B\,\!</math> is at the high level or the level of <math>1\,\!</math>, while factor <math>A\,\!</math> is at the low level and <math>ab\,\!</math> represents the treatment combination where factors <math>A\,\!</math> and <math>B\,\!</math> are at the high level or the level of the 1. Figure (b) below shows the design matrix for the <math>{2}^{2}\,\!</math> design. It can be noted that the sum of the terms resulting from the product of any two columns of the design matrix is zero. As a result the <math>{2}^{2}\,\!</math> design is an ''orthogonal design''. In fact, all <math>{2}^{k}\,\!</math> designs are orthogonal designs. This property of the <math>{2}^{k}\,\!</math> designs offers a great advantage in the analysis because of the simplifications that result from orthogonality. These simplifications are explained later on in this chapter.
The <math>{2}^{2}\,\!</math> design can also be represented geometrically using a square with the four treatment combinations lying at the four corners, as shown in figure (c) below.

[[Image:doe7.2.png|center|400px|The <math>2^2\,\!</math> design. Figure (a) displays the experiment design, (b) displays the design matrix and (c) displays the geometric representation for the design. In Figure (b), the column names I, A, B and AB are used. Column I represents the intercept term. Columns A and B represent the respective factor settings. Column AB represents the interaction and is the product of columns A and B.]]
 

===The 23 Design===

The <math>{2}^{3}\,\!</math> design is a two level factorial experiment design with three factors (say factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>). This design tests three (<math>k=3\,\!</math>) main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math> ; three (<math>(_{2}^{k})=\,\!</math> <math>(_{2}^{3})=3\,\!</math>) two factor interaction effects, <math>AB\,\!</math>, <math>BC\,\!</math>, <math>AC\,\!</math> ; and one (<math>(_{3}^{k})=\,\!</math> <math>(_{3}^{3})=1\,\!</math>) three factor interaction effect, <math>ABC\,\!</math>. The design requires eight runs per replicate. The eight treatment combinations corresponding to these runs are <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math>, <math>ab\,\!</math>, <math>c\,\!</math>, <math>ac\,\!</math>, <math>bc\,\!</math> and <math>abc\,\!</math>. Note that the treatment combinations are written in such an order that factors are introduced one by one with each new factor being combined with the preceding terms. This order of writing the treatments is called the ''standard order'' or ''Yates' order''. The <math>{2}^{3}\,\!</math> design is shown in figure (a) below. The design matrix for the <math>{2}^{3}\,\!</math> design is shown in figure (b). The design matrix can be constructed by following the standard order for the treatment combinations to obtain the columns for the main effects and then multiplying the main effects columns to obtain the interaction columns.

[[Image:doe7.3.png|center|324px|The <math>2^3\,\!</math> design. Figure (a) shows the experiment design and (b) shows the design matrix.]]

[[Image:doe7.4.png|center|290px|Geometric representation of the <math>2^3\,\!</math> design.]]

The <math>{2}^{3}\,\!</math> design can also be represented geometrically using a cube with the eight treatment combinations lying at the eight corners as shown in the figure above.

==Analysis of 2''k'' Designs==

The <math>{2}^{k}\,\!</math> designs are a special category of the factorial experiments where all the factors are at two levels. The fact that these designs contain factors at only two levels and are orthogonal greatly simplifies their analysis even when the number of factors is large. The use of <math>{2}^{k}\,\!</math> designs in investigating a large number of factors calls for a revision of the notation used previously for the ANOVA models. The case for revised notation is made stronger by the fact that the ANOVA and multiple linear regression models are identical for <math>{2}^{k}\,\!</math> designs because all factors are only at two levels. Therefore, the notation of the regression models is applied to the ANOVA models for these designs, as explained next.

===Notation===

Based on the notation used in [[General Full Factorial Designs]], the ANOVA model for a two level factorial experiment with three factors would be as follows:

::<math>\begin{align}
& Y= & \mu +{{\tau }_{1}}\cdot {{x}_{1}}+{{\delta }_{1}}\cdot {{x}_{2}}+{{(\tau \delta )}_{11}}\cdot {{x}_{1}}{{x}_{2}}+{{\gamma }_{1}}\cdot {{x}_{3}} \\
& & +{{(\tau \gamma )}_{11}}\cdot {{x}_{1}}{{x}_{3}}+{{(\delta \gamma )}_{11}}\cdot {{x}_{2}}{{x}_{3}}+{{(\tau \delta \gamma )}_{111}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where:
 
:• <math>\mu \,\!</math> represents the overall mean
:• <math>{{\tau }_{1}}\,\!</math> represents the independent effect of the first factor (factor <math>A\,\!</math>) out of the two effects <math>{{\tau }_{1}}\,\!</math> and <math>{{\tau }_{2}}\,\!</math>
:• <math>{{\delta }_{1}}\,\!</math> represents the independent effect of the second factor (factor <math>B\,\!</math>) out of the two effects <math>{{\delta }_{1}}\,\!</math> and <math>{{\delta }_{2}}\,\!</math>
:• <math>{{(\tau \delta )}_{11}}\,\!</math> represents the independent effect of the interaction <math>AB\,\!</math> out of the other interaction effects
:• <math>{{\gamma }_{1}}\,\!</math> represents the effect of the third factor (factor <math>C\,\!</math>) out of the two effects <math>{{\gamma }_{1}}\,\!</math> and <math>{{\gamma }_{2}}\,\!</math>
:• <math>{{(\tau \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>AC\,\!</math> out of the other interaction effects
:• <math>{{(\delta \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>BC\,\!</math> out of the other interaction effects
:• <math>{{(\tau \delta \gamma )}_{111}}\,\!</math> represents the effect of the interaction <math>ABC\,\!</math> out of the other interaction effects
and <math>\epsilon \,\!</math> is the random error term.

 
The notation for a linear regression model having three predictor variables with interactions is:

::<math>\begin{align}
& Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& & +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

The notation for the regression model is much more convenient, especially for the case when a large number of higher order interactions are present. In two level experiments, the ANOVA model requires only one indicator variable to represent each factor for both qualitative and quantitative factors. Therefore, the notation for the multiple linear regression model can be applied to the ANOVA model of the experiment that has all the factors at two levels. For example, for the experiment of the ANOVA model given above, <math>{{\beta }_{0}}\,\!</math> can represent the overall mean instead of <math>\mu \,\!</math>, and <math>{{\beta }_{1}}\,\!</math> can represent the independent effect, <math>{{\tau }_{1}}\,\!</math>, of factor <math>A\,\!</math>. Other main effects can be represented in a similar manner. The notation for the interaction effects is much more simplified (e.g., <math>{{\beta }_{123}}\,\!</math> can be used to represent the three factor interaction effect, <math>{{(\tau \beta \gamma )}_{111}}\,\!</math>).

As mentioned earlier, it is important to note that the coding for the indicator variables for the ANOVA models of two level factorial experiments is reversed from the coding followed in [[General Full Factorial Designs]]. Here <math>-1\,\!</math> represents the first level of the factor while <math>1\,\!</math> represents the second level. This is because for a two level factor a single variable is needed to represent the factor for both qualitative and quantitative factors. For quantitative factors, using <math>-1\,\!</math> for the first level (which is the low level) and 1 for the second level (which is the high level) keeps the coding consistent with the numerical value of the factors. The change in coding between the two coding schemes does not affect the analysis except that signs of the estimated effect coefficients will be reversed (i.e., numerical values of <math>{{\hat{\tau }}_{1}}\,\!</math>, obtained based on the coding of [[General Full Factorial Designs]], and <math>{{\hat{\beta }}_{1}}\,\!</math>, obtained based on the new coding, will be the same but their signs would be opposite).

::<math>\begin{align}
& & \text{Factor }A\text{ Coding (two level factor)} \\
& &
\end{align}\,\!</math>

::<math>\begin{matrix}
\text{Previous Coding} & {} & {} & {} & \text{Coding for }{{\text{2}}^{k}}\text{ Designs} \\
{} & {} & {} & {} & {} \\
Effect\text{ }{{\tau }_{1}}\ \ :\ \ {{x}_{1}}=1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{1}}\text{ (or }-{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=-1\text{ } \\
Effect\text{ }{{\tau }_{2}}\ \ :\ \ {{x}_{1}}=-1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{2}}\text{ (or }{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=1\text{ } \\
\end{matrix}\,\!</math>

In summary, the ANOVA model for the experiments with all factors at two levels is different from the ANOVA models for other experiments in terms of the notation in the following two ways:
 

:• The notation of the regression models is used for the effect coefficients.
:• The coding of the indicator variables is reversed.

===Special Features===

Consider the design matrix, <math>X\,\!</math>, for the <math>{2}^{3}\,\!</math> design discussed above. The (<math>{{X}^{\prime }}X\,\!</math>) <math>^{-1}\,\!</math> matrix is:

<center><math>{{({{X}^{\prime }}X)}^{-1}}=\left[ \begin{matrix}
0.125 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0.125 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0.125 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0.125 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0.125 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0.125 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0.125 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.125 \\
\end{matrix} \right]\,\!</math></center>

Notice that, due to the orthogonal design of the <math>X\,\!</math> matrix, the <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> has been simplified to a diagonal matrix which can be written as:

::<math>\begin{align}
{{({{X}^{\prime }}X)}^{-1}}= & 0.125\cdot I = & \frac{1}{8}\cdot I = & \frac{1}{{{2}^{3}}}\cdot I
\end{align}\,\!</math>

where <math>I\,\!</math> represents the identity matrix of the same order as the design matrix, <math>X\,\!</math>. Since there are eight observations per replicate of the <math>{2}^{3}\,\!</math> design, the <math>(X\,\!</math> ' <math>X{{)}^{-1}}\,\!</math> matrix for <math>m\,\!</math> replicates of this design can be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{3}}\cdot m)}\cdot I\,\!</math>

The <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> matrix for any <math>{2}^{k}\,\!</math> design can now be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{k}}\cdot m)}\cdot I\,\!</math>

Then the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is:

::<math>\begin{align}
C= & {{{\hat{\sigma }}}^{2}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & M{{S}_{E}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & \frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}\cdot I
\end{align}\,\!</math>

Note that the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is also a diagonal matrix. Therefore, the estimated effect coefficients (<math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math>, <math>{{\beta }_{12}},\,\!</math> etc.) for these designs are uncorrelated. This implies that the terms in the <math>{2}^{k}\,\!</math> design (main effects, interactions) are independent of each other. Consequently, the extra sum of squares for each of the terms in these designs is independent of the sequence of terms in the model, and also independent of the presence of other terms in the model. As a result the sequential and partial sum of squares for the terms are identical for these designs and will always add up to the model sum of squares. Multicollinearity is also not an issue for these designs.

It can also be noted from the equation given above, that in addition to the <math>C\,\!</math> matrix being diagonal, all diagonal elements of the <math>C\,\!</math> matrix are identical. This means that the variance (or its square root, the standard error) of all estimated effect coefficients are the same. The standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>, for all the coefficients is:

::<math>\begin{align}
se({{{\hat{\beta }}}_{j}})= & \sqrt{{{C}_{jj}}} = & \sqrt{\frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}}\text{ }for\text{ }all\text{ }j
\end{align}\,\!</math>

This property is used to construct the normal probability plot of effects in <math>{2}^{k}\,\!</math> designs and identify significant effects using graphical techniques. For details on the normal probability plot of effects in a Weibull++ DOE folio, refer to [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]].

====Example====
To illustrate the analysis of a full factorial <math>{2}^{k}\,\!</math> design, consider a three factor experiment to investigate the effect of honing pressure, number of strokes and cycle time on the surface finish of automobile brake drums. Each of these factors is investigated at two levels. The honing pressure is investigated at levels of 200 <math>psi\,\!</math> and 400 <math>psi\,\!</math>, the number of strokes used is 3 and 5 and the two levels of the cycle time are 3 and 5 seconds. The design for this experiment is set up in a Weibull++ DOE folio as shown in the first two following figures. It is decided to run two replicates for this experiment. The surface finish data collected from each run (using randomization) and the complete design is shown in the third following figure. The analysis of the experiment data is explained next.

[[Image:doe7_5.png|center|877px|Design properties for the experiment in the example.|link=]]

[[Image:doe7_6.png|center|859px|Design summary for the experiment in the example.|link=]]

[[Image:doe7_7.png|center|778px|Experiment design for the example to investigate the surface finish of automobile brake drums.|link=]]

The applicable model using the notation for <math>{2}^{k}\,\!</math> designs is:

::<math>\begin{align}
Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where the indicator variable, <math>{{x}_{1,}}\,\!</math> represents factor <math>A\,\!</math> (honing pressure), <math>{{x}_{1}}=-1\,\!</math> represents the low level of 200 <math>psi\,\!</math> and <math>{{x}_{1}}=1\,\!</math> represents the high level of 400 <math>psi\,\!</math>. Similarly, <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> represent factors <math>B\,\!</math> (number of strokes) and <math>C\,\!</math> (cycle time), respectively. <math>{{\beta }_{0}}\,\!</math> is the overall mean, while <math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math> and <math>{{\beta }_{3}}\,\!</math> are the effect coefficients for the main effects of factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. <math>{{\beta }_{12}}\,\!</math>, <math>{{\beta }_{13}}\,\!</math> and <math>{{\beta }_{23}}\,\!</math> are the effect coefficients for the <math>AB\,\!</math>, <math>AC\,\!</math> and <math>BC\,\!</math> interactions, while <math>{{\beta }_{123}}\,\!</math> represents the <math>ABC\,\!</math> interaction.

 
If the subscripts for the run (<math>i\,\!</math> ; <math>i=\,\!</math> 1 to 8) and replicates (<math>j\,\!</math> ; <math>j=\,\!</math> 1,2) are included, then the model can be written as:

::<math>\begin{align}
{{Y}_{ij}}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{ij1}}+{{\beta }_{2}}\cdot {{x}_{ij2}}+{{\beta }_{12}}\cdot {{x}_{ij1}}{{x}_{ij2}}+{{\beta }_{3}}\cdot {{x}_{ij3}} \\
& +{{\beta }_{13}}\cdot {{x}_{ij1}}{{x}_{ij3}}+{{\beta }_{23}}\cdot {{x}_{ij2}}{{x}_{ij3}}+{{\beta }_{123}}\cdot {{x}_{ij1}}{{x}_{ij2}}{{x}_{ij3}}+{{\epsilon }_{ij}}
\end{align}\,\!</math>

To investigate how the given factors affect the response, the following hypothesis tests need to be carried:

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{1}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{1}}\ne 0\,\!</math>

This test investigates the main effect of factor <math>A\,\!</math> (honing pressure). The statistic for this test is:

::<math>{{({{F}_{0}})}_{A}}=\frac{M{{S}_{A}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{A}}\,\!</math> is the mean square for factor <math>A\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other main effects, <math>B\,\!</math> and <math>C\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{12}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{12}}\ne 0\,\!</math>

This test investigates the two factor interaction <math>AB\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{AB}}=\frac{M{{S}_{AB}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the interaction <math>AB\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other two factor interactions, <math>AC\,\!</math> and <math>BC\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{123}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{123}}\ne 0\,\!</math>

This test investigates the three factor interaction <math>ABC\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{ABC}}=\frac{M{{S}_{ABC}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{ABC}}\,\!</math> is the mean square for the interaction <math>ABC\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square.
To calculate the test statistics, it is convenient to express the ANOVA model in the form <math>y=X\beta +\epsilon \,\!</math>.

====Expression of the ANOVA Model as <math>y=X\beta +\epsilon \,\!</math>====

In matrix notation, the ANOVA model can be expressed as:

::<math>y=X\beta +\epsilon \,\!</math>

where:

<center><math>y=\left[ \begin{matrix}
{{Y}_{11}} \\
{{Y}_{21}} \\
. \\
{{Y}_{81}} \\
{{Y}_{12}} \\
. \\
{{Y}_{82}} \\
\end{matrix} \right]=\left[ \begin{matrix}
90 \\
90 \\
. \\
90 \\
86 \\
. \\
80 \\
\end{matrix} \right]\text{ }X=\left[ \begin{matrix}
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\end{matrix} \right]\,\!</math></center>

<center><math>\beta =\left[ \begin{matrix}
{{\beta }_{0}} \\
{{\beta }_{1}} \\
{{\beta }_{2}} \\
{{\beta }_{12}} \\
{{\beta }_{3}} \\
{{\beta }_{13}} \\
{{\beta }_{23}} \\
{{\beta }_{123}} \\
\end{matrix} \right]\text{ }\epsilon =\left[ \begin{matrix}
{{\epsilon }_{11}} \\
{{\epsilon }_{21}} \\
. \\
{{\epsilon }_{81}} \\
{{\epsilon }_{12}} \\
. \\
. \\
{{\epsilon }_{82}} \\
\end{matrix} \right]\,\!</math></center>

====Calculation of the Extra Sum of Squares for the Factors====

Knowing the matrices <math>y\,\!</math>, <math>X\,\!</math> and <math>\beta \,\!</math>, the extra sum of squares for the factors can be calculated. These are used to calculate the mean squares that are used to obtain the test statistics. Since the experiment design is orthogonal, the partial and sequential extra sum of squares are identical. The extra sum of squares for each effect can be calculated as shown next. As an example, the extra sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & Model\text{ }Sum\text{ }of\text{ }Squares - Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }main\text{ }effect\text{ }of\text{ }A \\
= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }A}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }A}}={{X}_{\tilde{\ }A}}{{(X_{\tilde{\ }A}^{\prime }{{X}_{\tilde{\ }A}})}^{-1}}X_{\tilde{\ }A}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }A}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the main effect of factor <math>A\,\!</math>. Thus, the sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y \\
= & 654.4375-549.375 \\
= & 105.0625
\end{align}\,\!</math>

Similarly, the extra sum of squares for the interaction effect <math>AB\,\!</math> is:

::<math>\begin{align}
S{{S}_{AB}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }AB}}-(1/16)J]y \\
= & 654.4375-636.375 \\
= & 18.0625
\end{align}\,\!</math>

The extra sum of squares for other effects can be obtained in a similar manner.

====Calculation of the Test Statistics====

Knowing the extra sum of squares, the test statistic for the effects can be calculated. For example, the test statistic for the interaction <math>AB\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{AB}}= & \frac{M{{S}_{AB}}}{M{{S}_{E}}} \\
= & \frac{S{{S}_{AB}}/dof(S{{S}_{AB}})}{S{{S}_{E}}/dof(S{{S}_{E}})} \\
= & \frac{18.0625/1}{147.5/8} \\
= & 0.9797
\end{align}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the <math>AB\,\!</math> interaction and <math>M{{S}_{E}}\,\!</math> is the error mean square. The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{AB}}=0.9797\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{AB}}) \\
= & 1-0.6487 \\
= & 0.3513
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that the interaction between honing pressure and number of strokes does not affect the surface finish of the brake drums. Tests for other effects can be carried out in a similar manner. The results are shown in the ANOVA Table in the following figure. The values S, R-sq and R-sq(adj) in the figure indicate how well the model fits the data. The value of S represents the standard error of the model, R-sq represents the coefficient of multiple determination and R-sq(adj) represents the adjusted coefficient of multiple determination. For details on these values refer to [[Multiple_Linear_Regression_Analysis|Multiple Linear Regression Analysis]].

[[Image:doe7_8.png|center|799px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

====Calculation of Effect Coefficients====

The estimate of effect coefficients can also be obtained:

<center><math>\begin{align}
\hat{\beta }= & {{({{X}^{\prime }}X)}^{-1}}{{X}^{\prime }}y \\
= & \left[ \begin{matrix}
86.4375 \\
2.5625 \\
-4.9375 \\
1.0625 \\
-1.0625 \\
2.4375 \\
-1.3125 \\
-0.1875 \\
\end{matrix} \right]
\end{align}\,\!</math></center>

[[Image:doe7_9.png|center|800px|Regression Information table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

The coefficients and related results are shown in the Regression Information table above. In the table, the Effect column displays the effects, which are simply twice the coefficients. The Standard Error column displays the standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>. The Low CI and High CI columns display the confidence interval on the coefficients. The interval shown is the 90% interval as the significance is chosen as 0.1. The T Value column displays the <math>t\,\!</math> statistic, <math>{{t}_{0}}\,\!</math>, corresponding to the coefficients. The P Value column displays the <math>p\,\!</math> value corresponding to the <math>t\,\!</math> statistic. (For details on how these results are calculated, refer to [[General Full Factorial Designs]]). Plots of residuals can also be obtained from the DOE folio to ensure that the assumptions related to the ANOVA model are not violated.

====Model Equation====

From the analysis results in the above figure within [[Two_Level_Factorial_Experiments#Calculation_of_Effect_Coefficients|calculation of effect coefficients]] section, it is seen that effects <math>A\,\!</math>, <math>B\,\!</math> and <math>AC\,\!</math> are significant. In DOE++, the <math>p\,\!</math> values for the significant effects are displayed in red in the ANOVA Table for easy identification. Using the values of the estimated effect coefficients, the model for the present <math>{2}^{3}\,\!</math> design in terms of the coded values can be written as:

::<math>\begin{align}
\hat{y}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}} \\
= & 86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+2.4375{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

To make the model hierarchical, the main effect, <math>C\,\!</math>, needs to be included in the model (because the interaction <math>AC\,\!</math> is included in the model). The resulting model is:

::<math>\hat{y}=86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+1.0625{{x}_{3}}+2.4375{{x}_{1}}{{x}_{3}}\,\!</math>

This equation can be viewed in DOE++, as shown in the following figure, using the Show Analysis Summary icon in the Control Panel. The equation shown in the figure will match the hierarchical model once the required terms are selected using the Select Effects icon.

[[Image:doe7_10.png|center|557px|The model equation for the experiment of the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

==Replicated and Repeated Runs==

In the case of replicated experiments, it is important to note the difference between replicated runs and repeated runs. Both repeated and replicated runs are multiple response readings taken at the same factor levels. However, repeated runs are response observations taken at the same time or in succession. Replicated runs are response observations recorded in a random order. Therefore, replicated runs include more variation than repeated runs. For example, a baker, who wants to investigate the effect of two factors on the quality of cakes, will have to bake four cakes to complete one replicate of a <math>{2}^{2}\,\!</math> design. Assume that the baker bakes eight cakes in all. If, for each of the four treatments of the <math>{2}^{2}\,\!</math> design, the baker selects one treatment at random and then bakes two cakes for this treatment at the same time then this is a case of two repeated runs. If, however, the baker bakes all the eight cakes randomly, then the eight cakes represent two sets of replicated runs.
For repeated measurements, the average values of the response for each treatment should be entered into DOE++ as shown in the following figure (a) when the two cakes for a particular treatment are baked together. For replicated measurements, when all the cakes are baked randomly, the data is entered as shown in the following figure (b).

[[Image:doe7.11.png|center|300px|Data entry for repeated and replicated runs. Figure (a) shows repeated runs and (b) shows replicated runs.]]

==Unreplicated 2''k'' Designs==

If a factorial experiment is run only for a single replicate then it is not possible to test hypotheses about the main effects and interactions as the error sum of squares cannot be obtained. This is because the number of observations in a single replicate equals the number of terms in the ANOVA model. Hence the model fits the data perfectly and no degrees of freedom are available to obtain the error sum of squares.

However, sometimes it is only possible to run a single replicate of the <math>{2}^{k}\,\!</math> design because of constraints on resources and time. In the absence of the error sum of squares, hypothesis tests to identify significant factors cannot be conducted. A number of methods of analyzing information obtained from unreplicated <math>{2}^{k}\,\!</math> designs are available. These include pooling higher order interactions, using the normal probability plot of effects or including center point replicates in the design.

===Pooling Higher Order Interactions===

One of the ways to deal with unreplicated <math>{2}^{k}\,\!</math> designs is to use the sum of squares of some of the higher order interactions as the error sum of squares provided these higher order interactions can be assumed to be insignificant. By dropping some of the higher order interactions from the model, the degrees of freedom corresponding to these interactions can be used to estimate the error mean square. Once the error mean square is known, the test statistics to conduct hypothesis tests on the factors can be calculated.

===Normal Probability Plot of Effects===
Another way to use unreplicated <math>{2}^{k}\,\!</math> designs to identify significant effects is to construct the normal probability plot of the effects. As mentioned in [[Two_Level_Factorial_Experiments#Special_Features| Special Features]], the standard error for all effect coefficients in the <math>{2}^{k}\,\!</math> designs is the same. Therefore, on a normal probability plot of effect coefficients, all non-significant effect coefficients (with <math>\beta =0\,\!</math>) will fall along the straight line representative of the normal distribution, N(<math>0,{{\sigma }^{2}}/({{2}^{k}}\cdot m)\,\!</math>). Effect coefficients that show large deviations from this line will be significant since they do not come from this normal distribution. Similarly, since effects <math>=2\times \,\!</math> effect coefficients, all non-significant effects will also follow a straight line on the normal probability plot of effects. For replicated designs, the Effects Probability plot of DOE++ plots the normalized effect values (or the T Values) on the standard normal probability line, N(0,1). However, in the case of unreplicated <math>{2}^{k}\,\!</math> designs, <math>{{\sigma }^{2}}\,\!</math> remains unknown since <math>M{{S}_{E}}\,\!</math> cannot be obtained. Lenth's method is used in this case to estimate the variance of the effects. For details on Lenth's method, please refer to [[DOE References| Montgomery (2001)]]. DOE++ then uses this variance value to plot effects along the N(0, Lenth's effect variance) line. The
method is illustrated in the following example.

====Example====

Vinyl panels, used as instrument panels in a certain automobile, are seen to develop defects after a certain amount of time. To investigate the issue, it is decided to carry out a two level factorial experiment. Potential factors to be investigated in the experiment are vacuum rate (factor <math>A\,\!</math>), material temperature (factor <math>B\,\!</math>), element intensity (factor <math>C\,\!</math>) and pre-stretch (factor <math>D\,\!</math>). The two levels of the factors used in the experiment are as shown in below.

[[Image:doet7.1.png|center|300px|Factors to investigate defects in vinyl panels.]]

With a <math>{2}^{4}\,\!</math> design requiring 16 runs per replicate it is only feasible for the manufacturer to run a single replicate.

The experiment design and data, collected as percent defects, are shown in the following figure. Since the present experiment design contains only a single replicate, it is not possible to obtain an estimate of the error sum of squares, <math>S{{S}_{E}}\,\!</math>. It is decided to use the normal probability plot of effects to identify the significant effects. The effect values for each term are obtained as shown in the following figure.

[[Image:doe7_13.png|center|650px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

Lenth's method uses these values to estimate the variance. As described in [[DOE_References|[Lenth, 1989]]], if all effects are arranged in ascending order, using their absolute values, then <math>{{s}_{0}}\,\!</math> is defined as 1.5 times the median value:

::<math>\begin{align}
{{s}_{0}}= & 1.5\cdot median(\left| effect \right|) \\
= & 1.5\cdot 2 \\
= & 3
\end{align}\,\!</math>

Using <math>{{s}_{0}}\,\!</math>, the "pseudo standard error" (<math>PSE\,\!</math>) is calculated as 1.5 times the median value of all effects that are less than 2.5 <math>{{s}_{0}}\,\!</math> :

::<math>\begin{align}
PSE= & 1.5\cdot median(\left| effect \right|\ \ :\ \ \left| effect \right|<2.5{{s}_{0}}) \\
= & 1.5\cdot 1.5 \\
= & 2.25
\end{align}\,\!</math>

Using <math>PSE\,\!</math> as an estimate of the effect variance, the effect variance is 2.25. Knowing the effect variance, the normal probability plot of effects for the present unreplicated experiment can be constructed as shown in the following figure. The line on this plot is the line N(0, 2.25). The plot shows that the effects <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math> do not follow the distribution represented by this line. Therefore, these effects are significant.

The significant effects can also be identified by comparing individual effect values to the margin of error or the threshold value using the pareto chart (see the third following figure). If the required significance is 0.1, then:

::<math>margin\text{ }of\text{ }error={{t}_{\alpha /2,d}}\cdot PSE\,\!</math>

The <math>t\,\!</math> statistic, <math>{{t}_{\alpha /2,d}}\,\!</math>, is calculated at a significance of <math>\alpha /2\,\!</math> (for the two-sided hypothesis) and degrees of freedom <math>d=(\,\!</math> number of effects <math>)/3\,\!</math>. Thus:

::<math>\begin{align}
margin\text{ }of\text{ }error= & {{t}_{0.05,5}}\cdot PSE \\
= & 2.015\cdot 2.25 \\
= & 4.534
\end{align}\,\!</math>

The value of 4.534 is shown as the critical value line in the third following figure. All effects with absolute values greater than the margin of error can be considered to be significant. These effects are <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math>. Therefore, the vacuum rate, the pre-stretch and their interaction have a significant effect on the defects of the vinyl panels.

[[Image:doe7_14.png|center|800px|Effect values for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_15.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_16.png|center|650px|Pareto chart for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

===Center Point Replicates===

Another method of dealing with unreplicated <math>{2}^{k}\,\!</math> designs that only have quantitative factors is to use replicated runs at the center point. The center point is the response corresponding to the treatment exactly midway between the two levels of all factors. Running multiple replicates at this point provides an estimate of pure error. Although running multiple replicates at any treatment level can provide an estimate of pure error, the other advantage of running center point replicates in the <math>{2}^{k}\,\!</math> design is in checking for the presence of curvature. The test for curvature investigates whether the model between the response and the factors is linear and is discussed in [[Two_Level_Factorial_Experiments#Using_Center_Point_Replicates_to_Test_Curvature| Center Pt. Replicates to Test Curvature]].

====Example: Use Center Point to Get Pure Error====

Consider a <math>{2}^{2}\,\!</math> experiment design to investigate the effect of two factors, <math>A\,\!</math> and <math>B\,\!</math>, on a certain response. The energy consumed when the treatments of the <math>{2}^{2}\,\!</math> design are run is considerably larger than the energy consumed for the center point run (because at the center point the factors are at their middle levels). Therefore, the analyst decides to run only a single replicate of the design and augment the design by five replicated runs at the center point as shown in the following figure. The design properties for this experiment are shown in the second following figure. The complete experiment design is shown in the third following figure. The center points can be used in the identification of significant effects as shown next.

[[Image:doe7.17.png||center|300px|<math>2^2\,\!</math> design augmented by five center point runs.]]
[[Image:doe7_18.png|center|537px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

[[Image:doe7_19.png|center|630px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

Since the present <math>{2}^{2}\,\!</math> design is unreplicated, there are no degrees of freedom available to calculate the error sum of squares. By augmenting this design with five center points, the response values at the center points, <math>y_{i}^{c}\,\!</math>, can be used to obtain an estimate of pure error, <math>S{{S}_{PE}}\,\!</math>. Let <math>{{\bar{y}}^{c}}\,\!</math> represent the average response for the five replicates at the center. Then:

::<math>S{{S}_{PE}}=Sum\text{ }of\text{ }Squares\text{ }for\text{ }center\text{ }points\,\!</math>

::<math>\begin{align}
S{{S}_{PE}}= & \underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}} \\
= & {{(25.2-25.26)}^{2}}+...+{{(25.3-25.26)}^{2}} \\
= & 0.052
\end{align}\,\!</math>

Then the corresponding mean square is:

::<math>\begin{align}
M{{S}_{PE}}= & \frac{S{{S}_{PE}}}{degrees\text{ }of\text{ }freedom} \\
= & \frac{0.052}{5-1} \\
= & 0.013
\end{align}\,\!</math>

Alternatively, <math>M{{S}_{PE}}\,\!</math> can be directly obtained by calculating the variance of the response values at the center points:

::<math>\begin{align}
M{{S}_{PE}}= & {{s}^{2}} \\
= & \frac{\underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}}}{5-1}
\end{align}\,\!</math>

Once <math>M{{S}_{PE}}\,\!</math> is known, it can be used as the error mean square, <math>M{{S}_{E}}\,\!</math>, to carry out the test of significance for each effect. For example, to test the significance of the main effect of factor <math>A,\,\!</math> the sum of squares corresponding to this effect is obtained in the usual manner by considering only the four runs of the original <math>{2}^{2}\,\!</math> design.

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/4)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/4)J]y \\
= & 0.5625
\end{align}\,\!</math>

Then, the test statistic to test the significance of the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{A}}= & \frac{M{{S}_{A}}}{M{{S}_{E}}} \\
= & \frac{0.5625/1}{0.052/4} \\
= & 43.2692
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{A}}=43.2692\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{A}}) \\
= & 1-0.9972 \\
= & 0.0028
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value < 0.1, it can be concluded that the main effect of factor <math>A\,\!</math> significantly affects the response. This result is displayed in the ANOVA table as shown in the following figure. Test for the significance of other factors can be carried out in a similar manner.

[[Image:doe7_20.png|center|649px|Results for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

===Using Center Point Replicates to Test Curvature===

Center point replicates can also be used to check for curvature in replicated or unreplicated <math>{2}^{k}\,\!</math> designs. The test for curvature investigates whether the model between the response and the factors is linear. The way DOE++ handles center point replicates is similar to its handling of blocks. The center point replicates are treated as an additional factor in the model. The factor is labeled as Curvature in the results of DOE++. If Curvature turns out to be a significant factor in the results, then this indicates the presence of curvature in the model.

====Example: Use Center Point to Test Curvature====

To illustrate the use of center point replicates in testing for curvature, consider again the data of the single replicate <math>{2}^{2}\,\!</math> experiment from a preceding figure(labeled "<math>2^2</math> design augmented by five center point runs"). Let <math>{{x}_{1}}\,\!</math> be the indicator variable to indicate if the run is a center point:

::<math>\begin{matrix}
{{x}_{1}}=0 & {} & \text{Center point run} \\
{{x}_{1}}=1 & {} & \text{Other run} \\
\end{matrix}\,\!</math>

If <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> are the indicator variables representing factors <math>A\,\!</math> and <math>B\,\!</math>, respectively, then the model for this experiment is:

::<math>Y={{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}\,\!</math>

To investigate the presence of curvature, the following hypotheses need to be tested:

::<math>\begin{align}
& {{H}_{0}}: & {{\beta }_{1}}=0\text{ (Curvature is absent)} \\
& {{H}_{1}}: & {{\beta }_{1}}\ne 0
\end{align}\,\!</math>

The test statistic to be used for this test is:

::<math>{{({{F}_{0}})}_{curvature}}=\frac{M{{S}_{curvature}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{curvature}}\,\!</math> is the mean square for Curvature and <math>M{{S}_{E}}\,\!</math> is the error mean square.

'''Calculation of the Sum of Squares'''

The <math>X\,\!</math> matrix and <math>y\,\!</math> vector for this experiment are:

<center><math>X=\left[ \begin{matrix}
1 & 1 & -1 & -1 & 1 \\
1 & 1 & 1 & -1 & -1 \\
1 & 1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
\end{matrix} \right]\text{ }y=\left[ \begin{matrix}
24.6 \\
25.4 \\
25.0 \\
25.7 \\
25.2 \\
25.3 \\
25.4 \\
25.1 \\
25.3 \\
\end{matrix} \right]\,\!</math></center>

The sum of squares can now be calculated. For example, the error sum of squares is:

::<math>\begin{align}
& S{{S}_{E}}= & {{y}^{\prime }}[I-H]y \\
& = & 0.052
\end{align}\,\!</math>

where <math>I\,\!</math> is the identity matrix and <math>H\,\!</math> is the hat matrix. It can be seen that this is equal to <math>S{{S}_{PE\text{ }}}\,\!</math> (the sum of squares due to pure error) because of the replicates at the center point, as obtained in the [[Two_Level_Factorial_Experiments#Example_3| example]]. The number of degrees of freedom associated with <math>S{{S}_{E}}\,\!</math>, <math>dof(S{{S}_{E}})\,\!</math> is four. The extra sum of squares corresponding to the center point replicates (or Curvature) is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & Model\text{ }Sum\text{ }of\text{ }Squares- \\
& & Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }center\text{ }point \\
& = & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Curvature}}-(1/9)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }Curvature}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }Curvature}}={{X}_{\tilde{\ }Curv}}{{(X_{\tilde{\ }Curv}^{\prime }{{X}_{\tilde{\ }Curv}})}^{-1}}X_{\tilde{\ }Curv}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }Curv}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the center point. Thus, the extra sum of squares corresponding to Curvature is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Center}}-(1/9)J]y \\
& = & 0.7036-0.6875 \\
& = & 0.0161
\end{align}\,\!</math>

This extra sum of squares can be used to test for the significance of curvature. The corresponding mean square is:

::<math>\begin{align}
& M{{S}_{Curvature}}= & \frac{Sum\text{ }of\text{ }squares\text{ }corresponding\text{ }to\text{ }Curvature}{degrees\text{ }of\text{ }freedom} \\
& = & \frac{0.0161}{1} \\
& = & 0.0161
\end{align}\,\!</math>

'''Calculation of the Test Statistic'''

Knowing the mean squares, the statistic to check the significance of curvature can be calculated.

::<math>\begin{align}
& {{({{f}_{0}})}_{Curvature}}= & \frac{M{{S}_{Curvature}}}{M{{S}_{E}}} \\
& = & \frac{0.0161/1}{0.052/4} \\
& = & 1.24
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{Curvature}}=1.24\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and four degrees of freedom in the denominator is:

::<math>\begin{align}
& p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{Curvature}}) \\
& = & 1-0.6713 \\
& = & 0.3287
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that curvature does not exist for this design. This results is shown in the ANOVA table in the figure above. The surface of the fitted model based on these results, along with the observed response values, is shown in the figure below.

[[Image:doe7.21.png|center|400px|Model surface and observed response values for the design in the [[Two_Level_Factorial_Experiments#Example_4| example]].]]

 

==Blocking in 2k Designs==

Blocking can be used in the <math>{2}^{k}\,\!</math> designs to deal with cases when replicates cannot be run under identical conditions. Randomized complete block designs that were discussed in [[Randomization and Blocking in DOE]] for factorial experiments are also applicable here. At times, even with just two levels per factor, it is not possible to run all treatment combinations for one replicate of the experiment under homogeneous conditions. For example, each replicate of the <math>{2}^{2}\,\!</math> design requires four runs. If each run requires two hours and testing facilities are available for only four hours per day, two days of testing would be required to run one complete replicate. Blocking can be used to separate the treatment runs on the two different days. Blocks that do not contain all treatments of a replicate are called incomplete blocks. In incomplete block designs, the block effect is confounded with certain effect(s) under investigation. For the <math>{2}^{2}\,\!</math> design assume that treatments <math>(1)\,\!</math> and <math>ab\,\!</math> were run on the first day and treatments <math>a\,\!</math> and <math>b\,\!</math> were run on the second day. Then, the incomplete block design for this experiment is:

::<math>\begin{matrix}
\text{Block 1} & {} & \text{Block 2} \\
\left[ \begin{matrix}
(1) \\
ab \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
b \\
\end{matrix} \right] \\
\end{matrix}\,\!</math>

For this design the block effect may be calculated as:

::<math>\begin{align}
& Block\text{ }Effect= & Average\text{ }response\text{ }for\text{ }Block\text{ }1- \\
& & Average\text{ }response\text{ }for\text{ }Block\text{ }2 \\
& = & \frac{(1)+ab}{2}-\frac{a+b}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The <math>AB\,\!</math> interaction effect is:

::<math>\begin{align}
& AB= & Average\text{ }response\text{ }at\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{low}}}- \\
& & Average\text{ }response\text{ }at\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{low}}} \\
& = & \frac{ab+(1)}{2}-\frac{b+a}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The two equations given above show that, in this design, the <math>AB\,\!</math> interaction effect cannot be distinguished from the block effect because the formulas to calculate these effects are the same. In other words, the <math>AB\,\!</math> interaction is said to be confounded with the block effect and it is not possible to say if the effect calculated based on these equations is due to the <math>AB\,\!</math> interaction effect, the block effect or both. In incomplete block designs some effects are always confounded with the blocks. Therefore, it is important to design these experiments in such a way that the important effects are not confounded with the blocks. In most cases, the experimenter can assume that higher order interactions are unimportant. In this case, it would better to use incomplete block designs that confound these effects with the blocks.
One way to design incomplete block designs is to use defining contrasts as shown next:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}+...+{{\alpha }_{k}}{{q}_{k}}\,\!</math>

where the <math>{{\alpha }_{i}}\,\!</math> s are the exponents for the factors in the effect that is to be confounded with the block effect and the <math>{{q}_{i}}\,\!</math> s are values based on the level of the <math>i\,\!</math> the factor (in a treatment that is to be allocated to a block). For <math>{2}^{k}\,\!</math> designs the <math>{{\alpha }_{i}}\,\!</math> s are either 0 or 1 and the <math>{{q}_{i}}\,\!</math> s have a value of 0 for the low level of the <math>i\,\!</math> th factor and a value of 1 for the high level of the factor in the treatment under consideration. As an example, consider the <math>{2}^{2}\,\!</math> design where the interaction effect <math>AB\,\!</math> is confounded with the block. Since there are two factors, <math>k=2\,\!</math>, with <math>i=1\,\!</math> representing factor <math>A\,\!</math> and <math>i=2\,\!</math> representing factor <math>B\,\!</math>. Therefore:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}\,\!</math>

The value of <math>{{\alpha }_{1}}\,\!</math> is one because the exponent of factor <math>A\,\!</math> in the confounded interaction <math>AB\,\!</math> is one. Similarly, the value of <math>{{\alpha }_{2}}\,\!</math> is one because the exponent of factor <math>B\,\!</math> in the confounded interaction <math>AB\,\!</math> is also one. Therefore, the defining contrast for this design can be written as:

::<math>\begin{align}
& L= & {{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}} \\
& = & 1\cdot {{q}_{1}}+1\cdot {{q}_{2}} \\
& = & {{q}_{1}}+{{q}_{2}}
\end{align}\,\!</math>

Once the defining contrast is known, it can be used to allocate treatments to the blocks. For the <math>{2}^{2}\,\!</math> design, there are four treatments <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math> and <math>ab\,\!</math>. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. In order to decide which block the treatment <math>(1)\,\!</math> belongs to, the levels of factors <math>A\,\!</math> and <math>B\,\!</math> for this run are used. Since factor <math>A\,\!</math> is at the low level in this treatment, <math>{{q}_{1}}=0\,\!</math>. Similarly, since factor <math>B\,\!</math> is also at the low level in this treatment, <math>{{q}_{2}}=0\,\!</math>. Therefore:

::<math>\begin{align}
& L= & {{q}_{1}}+{{q}_{2}} \\
& = & 0+0=0\text{ (mod 2)}
\end{align}\,\!</math>

Note that the value of <math>L\,\!</math> used to decide the block allocation is "mod 2" of the original value. This value is obtained by taking the value of 1 for odd numbers and 0 otherwise. Based on the value of <math>L\,\!</math>, treatment <math>(1)\,\!</math> is assigned to block 1. Other treatments can be assigned using the following calculations:

::<math>\begin{align}
& (1): & \text{ }L=0+0=0=0\text{ (mod 2)} \\
& a: & \text{ }L=1+0=1=1\text{ (mod 2)} \\
& b: & \text{ }L=0+1=1=1\text{ (mod 2)} \\
& ab: & \text{ }L=1+1=2=0\text{ (mod 2)}
\end{align}\,\!</math>

Therefore, to confound the interaction <math>AB\,\!</math> with the block effect in the <math>{2}^{2}\,\!</math> incomplete block design, treatments <math>(1)\,\!</math> and <math>ab\,\!</math> (with <math>L=0\,\!</math>) should be assigned to block 2 and treatment combinations <math>a\,\!</math> and <math>b\,\!</math> (with <math>L=1\,\!</math>) should be assigned to block 1.

====Example: Two Level Factorial Design with Two Blocks====

This example illustrates how treatments can be allocated to two blocks for an unreplicated <math>{2}^{k}\,\!</math> design. Consider the unreplicated <math>{2}^{4}\,\!</math> design to investigate the four factors affecting the defects in automobile vinyl panels discussed in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments required for this experiment were run by two different operators with each operator conducting 8 runs. This experiment is an example of an incomplete block design. The analyst in charge of this experiment assumed that the interaction <math>ABCD\,\!</math> was not significant and decided to allocate treatments to the two operators so that the <math>ABCD\,\!</math> interaction was confounded with the block effect (the two operators are the blocks). The allocation scheme to assign treatments to the two operators can be obtained as follows.
 
The defining contrast for the <math>{2}^{4}\,\!</math> design where the <math>ABCD\,\!</math> interaction is confounded with the blocks is:

::<math>L={{q}_{1}}+{{q}_{2}}+{{q}_{3}}+{{q}_{4}}\,\!</math>

The treatments can be allocated to the two operators using the values of the defining contrast. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. Then the value of the defining contrast for treatment <math>a\,\!</math> is:

::<math>a\ \ :\ \ \text{ }L=1+0+0+0=1=1\text{ (mod 2)}\,\!</math>

Therefore, treatment <math>a\,\!</math> should be assigned to Block 1 or the first operator. Similarly, for treatment <math>ab\,\!</math> we have:

::<math>ab\ \ :\ \ \text{ }L=1+1+0+0=2=0\text{ (mod 2)}\,\!</math>

[[Image:doe7.22.png|center|200px| Allocation of treatments to two blocks for the <math>2^4</math> design in the example by confounding interaction of <math>ABCD</math> with the blocks.]]

Therefore, <math>ab\,\!</math> should be assigned to Block 2 or the second operator. Other treatments can be allocated to the two operators in a similar manner to arrive at the allocation scheme shown in the figure below.
In DOE++, to confound the <math>ABCD\,\!</math> interaction for the <math>{2}^{4}\,\!</math> design into two blocks, the number of blocks are specified as shown in the figure below. Then the interaction <math>ABCD\,\!</math> is entered in the Block Generator window (second following figure) which is available using the Block Generator button in the following figure. The design generated by DOE++ is shown in the third of the following figures. This design matches the allocation scheme of the preceding figure.

[[Image:doe7_23.png|center|700px| Adding block properties for the experiment in the example.|link=]]

[[Image:doe7_24.png|center|700px|Specifying the interaction ABCD as the interaction to be confounded with the blocks for the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

[[Image:doe7_25.png|center|800px|Two block design for the experiment in the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

For the analysis of this design, the sum of squares for all effects are calculated assuming no blocking. Then, to account for blocking, the sum of squares corresponding to the <math>ABCD\,\!</math> interaction is considered as the sum of squares due to blocks and <math>ABCD\,\!</math>. In DOE++ this is done by displaying this sum of squares as the sum of squares due to the blocks. This is shown in the following figure where the sum of squares in question is obtained as 72.25 and is displayed against Block. The interaction ABCD, which is confounded with the blocks, is not displayed. Since the design is unreplicated, any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_26.png|center|793px|ANOVA table for the experiment of the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

===Unreplicated 2''k'' Designs in 2''p'' Blocks===

A single replicate of the <math>{2}^{k}\,\!</math> design can be run in up to <math>{2}^{p}\,\!</math> blocks where <math>p<k\,\!</math>. The number of effects confounded with the blocks equals the degrees of freedom associated with the block effect.

If two blocks are used (the block effect has two levels), then one (<math>2-1=1)\,\!</math> effect is confounded with the blocks. If four blocks are used, then three (<math>4-1=3\,\!</math>) effects are confounded with the blocks and so on. For example an unreplicated <math>{2}^{4}\,\!</math> design may be confounded in <math>{2}^{2}\,\!</math> (four) blocks using two contrasts, <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}}\,\!</math>. Let <math>AC\,\!</math> and <math>BD\,\!</math> be the effects to be confounded with the blocks. Corresponding to these two effects, the contrasts are respectively:

::<math>\begin{align}
& {{L}_{1}}= & {{q}_{1}}+{{q}_{3}} \\
& {{L}_{2}}= & {{q}_{2}}+{{q}_{4}}
\end{align}\,\!</math>

Based on the values of <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}},\,\!</math> the treatments can be assigned to the four blocks as follows:

<center><math>\begin{matrix}
\text{Block 4} & {} & \text{Block 3} & {} & \text{Block 2} & {} & \text{Block 1} \\
{{L}_{1}}=0,{{L}_{2}}=0 & {} & {{L}_{1}}=1,{{L}_{2}}=0 & {} & {{L}_{1}}=0,{{L}_{2}}=1 & {} & {{L}_{1}}=1,{{L}_{2}}=1 \\
{} & {} & {} & {} & {} & {} & {} \\
\left[ \begin{matrix}
(1) \\
ac \\
bd \\
abcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
c \\
abd \\
bcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
b \\
abc \\
d \\
acd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
ab \\
bc \\
ad \\
cd \\
\end{matrix} \right] \\
\end{matrix}\,\!</math></center>

Since the block effect has three degrees of freedom, three effects are confounded with the block effect. In addition to <math>AC\,\!</math> and <math>BD\,\!</math>, the third effect confounded with the block effect is their generalized interaction, <math>(AC)(BD)=ABCD\,\!</math>.
In general, when an unreplicated <math>{2}^{k}\,\!</math> design is confounded in <math>{2}^{p}\,\!</math> blocks, <math>p\,\!</math> contrasts are needed (<math>{{L}_{1}},{{L}_{2}}...{{L}_{p}}\,\!</math>). <math>p\,\!</math> effects are selected to define these contrasts such that none of these effects are the generalized interaction of the others. The <math>{2}^{p}\,\!</math> blocks can then be assigned the treatments using the <math>p\,\!</math> contrasts. <math>{{2}^{p}}-(p+1)\,\!</math> effects, that are also confounded with the blocks, are then obtained as the generalized interaction of the <math>p\,\!</math> effects. In the statistical analysis of these designs, the sum of squares are computed as if no blocking were used. Then the block sum of squares is obtained by adding the sum of squares for all the effects confounded with the blocks.

====Example: 2 Level Factorial Design with Four Blocks====

This example illustrates how DOE++ obtains the sum of squares when treatments for an unreplicated <math>{2}^{k}\,\!</math> design are allocated among four blocks. Consider again the unreplicated <math>{2}^{4}\,\!</math> design used to investigate the defects in automobile vinyl panels presented in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments needed to complete the experiment were run by four operators. Therefore, there are four blocks. Assume that the treatments were allocated to the blocks using the generators mentioned in the previous section, i.e., treatments were allocated among the four operators by confounding the effects, <math>AC\,\!</math> and <math>BD,\,\!</math> with the blocks. These effects can be specified as Block Generators as shown in the following figure. (The generalized interaction of these two effects, interaction <math>ABCD\,\!</math>, will also get confounded with the blocks.) The resulting design is shown in the second following figure and matches the allocation scheme obtained in the previous section.

[[Image:doe7_27.png|center|700px|Specifying the interactions AC and BD as block generators for the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

The sum of squares in this case can be obtained by calculating the sum of squares for each of the effects assuming there is no blocking. Once the individual sum of squares have been obtained, the block sum of squares can be calculated. The block sum of squares is the sum of the sum of squares of effects, <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, since these effects are confounded with the block effect. As shown in the second following figure, this sum of squares is 92.25 and is displayed against Block. The interactions <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, which are confounded with the blocks, are not displayed. Since the present design is unreplicated any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_28.png|center|800px|Design for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

[[Image:doe7_29.png|center|793px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

==Variability Analysis==

For replicated two level factorial experiments, DOE++ provides the option of conducting variability analysis (using the Variability Analysis icon under the Data menu). The analysis is used to identify the treatment that results in the least amount of variation in the product or process being investigated. Variability analysis is conducted by treating the standard deviation of the response for each treatment of the experiment as an additional response. The standard deviation for a treatment is obtained by using the replicated response values at that treatment run. As an example, consider the <math>{2}^{3}\,\!</math> design shown in the following figure where each run is replicated four times. A variability analysis can be conducted for this design. DOE++ calculates eight standard deviation values corresponding to each treatment of the design (see second following figure). Then, the design is analyzed as an unreplicated <math>{2}^{3}\,\!</math> design with the standard deviations (displayed as Y Standard Deviation. in second following figure) as the response. The normal probability plot of effects identifies <math>AC\,\!</math> as the effect that influences variability (see third figure following). Based on the effect coefficients obtained in the fourth figure following, the model for Y Std. is:

::<math>\begin{align}
& \text{Y Std}\text{.}= & 0.6779+0.2491\cdot AC \\
& = & 0.6779+0.2491{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

Based on the model, the experimenter has two choices to minimize variability (by minimizing Y Std.). The first choice is that <math>{{x}_{1}}\,\!</math> should be <math>1\,\!</math> (i.e., <math>A\,\!</math> should be set at the high level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the low level). The second choice is that <math>{{x}_{1}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>A\,\!</math> should be set at the low level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the high level). The experimenter can select the most feasible choice.

[[Image:doe7.30.png|center|391px|A <math>2^3\,\!</math> design with four replicated response values that can be used to conduct a variability analysis.]]
 

[[Image:doe7_31.png|center|727px|Variability analysis in DOE++.|link=]]
 

[[Image:doe7_32.png|center|650px|Normal probability plot of effects for the variability analysis example.|link=]]
 

[[Image:doe7_33.png|center|727px|Effect coefficients for the variability analysis example.|link=]]

 

==Two Level Fractional Factorial Designs==

As the number of factors in a two level factorial design increases, the number of runs for even a single replicate of the <math>{2}^{k}\,\!</math> design becomes very large. For example, a single replicate of an eight factor two level experiment would require 256 runs. Fractional factorial designs can be used in these cases to draw out valuable conclusions from fewer runs. The basis of fractional factorial designs is the ''sparsity of effects'' principle.[[DOE_References|[Wu, 2000]]] The principle states that, most of the time, responses are affected by a small number of main effects and lower order interactions, while higher order interactions are relatively unimportant. Fractional factorial designs are used as screening experiments during the initial stages of experimentation. At these stages, a large number of factors have to be investigated and the focus is on the main effects and two factor interactions. These designs obtain information about main effects and lower order interactions with fewer experiment runs by confounding these effects with unimportant higher order interactions. As an example, consider a <math>{2}^{8}\,\!</math> design that requires 256 runs. This design allows for the investigation of 8 main effects and 28 two factor interactions. However, 219 degrees of freedom are devoted to three factor or higher order interactions. This full factorial design can prove to be very inefficient when these higher order interactions can be assumed to be unimportant. Instead, a fractional design can be used here to identify the important factors that can then be investigated more thoroughly in subsequent experiments. In unreplicated fractional factorial designs, no degrees of freedom are available to calculate the error sum of squares and the techniques mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] should be employed for the analysis of these designs.

==Half-fraction Designs==

A half-fraction of the <math>{2}^{k}\,\!</math> design involves running only half of the treatments of the full factorial design. For example, consider a <math>{2}^{3}\,\!</math> design that requires eight runs in all. The design matrix for this design is shown in the figure (a) below. A half-fraction of this design is the design in which only four of the eight treatments are run. The fraction is denoted as <math>{2}^{3-1}\,\!</math> with the "<math>-1\,\!</math>" in the index denoting a half-fraction. Assume that the treatments chosen for the half-fraction design are the ones where the interaction <math>ABC\,\!</math> is at the high level (i.e., only those rows are chosen from the following figure (a) where the column for <math>ABC\,\!</math> has entries of 1). The resulting <math>{2}^{3-1}\,\!</math> design has a design matrix as shown in figure (b) below.

[[Image:doe7.34.png|center|330px|Half-fractions of the <math>2^3\,\!</math> design. (a) shows the full factorial <math>2^3\,\!</math> design, (b) shows the <math>2^{3-1}\,\!</math> design with the defining relation <math>I=ABC\,\!</math> and (c) shows the <math>{2}^{3-1}\,\!</math> design with the defining relation <math>I=-ABC\,\!</math>.]]

In the <math>{2}^{3-1}\,\!</math> design of figure (b), since the interaction <math>ABC\,\!</math> is always included at the same level (the high level represented by 1), it is not possible to measure this interaction effect. The effect, <math>ABC\,\!</math>, is called the ''generator'' or ''word'' for this design. It can be noted that, in the design matrix of the following figure (b), the column corresponding to the intercept, <math>I\,\!</math>, and column corresponding to the interaction <math>ABC\,\!</math>, are identical. The identical columns are written as <math>I=ABC\,\!</math> and this equation is called the ''defining relation'' for the design. In DOE++, the present <math>{2}^{3-1}\,\!</math> design can be obtained by specifying the design properties as shown in the following figure.

[[Image:doe7_35.png|center|700px|Design properties for the <math>2^{3-1}\,\!</math> design.|link=]]

The defining relation, <math>I=ABC\,\!</math>, is entered in the Fraction Generator window as shown next.

[[Image:doe7_36.png|center|700px|Specifying the defining relation for the <math>2^{3-1}\,\!</math> design.|link=]]

Note that in the figure following that, the defining relation is specified as <math>C=AB\,\!</math>. This relation is obtained by multiplying the defining relation, <math>I=ABC\,\!</math>, by the last factor, <math>C\,\!</math>, of the design.

===Calculation of Effects===

Using the four runs of the <math>{2}^{3-1}\,\!</math> design in figure (b) discussed above, the main effects can be calculated as follows:

::<math>\begin{align}
& A= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& B= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& C= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

where <math>a\,\!</math>, <math>b\,\!</math>, <math>c\,\!</math> and <math>abc\,\!</math> are the treatments included in the <math>{2}^{3-1}\,\!</math> design.

Similarly, the two factor interactions can also be obtained as:

::<math>\begin{align}
& BC= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& AC= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& AB= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

The equations for <math>A\,\!</math> and <math>BC\,\!</math> above result in the same effect values showing that effects <math>A\,\!</math> and <math>BC\,\!</math> are confounded in the present <math>{2}^{3-1}\,\!</math> design. Thus, the quantity, <math>\tfrac{1}{2}(a-b-c+abc),\,\!</math> estimates <math>A+BC\,\!</math> (i.e., both the main effect <math>A\,\!</math> and the two-factor interaction <math>BC\,\!</math>). The effects, <math>A\,\!</math> and <math>BC,\,\!</math> are called ''aliases''. From the remaining equations given above, it can be seen that the other aliases for this design are <math>B\,\!</math> and <math>AC\,\!</math>, and <math>C\,\!</math> and <math>AB\,\!</math>. Therefore, the equations to calculate the effects in the present <math>{2}^{3-1}\,\!</math> design can be written as follows:

::<math>\begin{align}
& A+BC= & \frac{1}{2}(a-b-c+abc) \\
& B+AC= & \frac{1}{2}(-a+b-c+abc) \\
& C+AB= & \frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

===Calculation of Aliases===

Aliases for a fractional factorial design can be obtained using the defining relation for the design. The defining relation for the present <math>{2}^{3-1}\,\!</math> design is:

::<math>I=ABC\,\!</math>

Multiplying both sides of the previous equation by the main effect, <math>A,\,\!</math> gives the alias effect of <math>A\,\!</math> :

::<math>\begin{align}
& A\cdot I= & A\cdot ABC \\
& A= & {{A}^{2}}BC \\
& A= & BC
\end{align}\,\!</math>

Note that in calculating the alias effects, any effect multiplied by <math>I\,\!</math> remains the same (<math>A\cdot I=A\,\!</math>), while an effect multiplied by itself results in <math>I\,\!</math> (<math>{{A}^{2}}=I\,\!</math>). Other aliases can also be obtained:

::<math>\begin{align}
& B\cdot I= & B\cdot ABC \\
& B= & A{{B}^{2}}C \\
& B= & AC
\end{align}\,\!</math>

:and:

::<math>\begin{align}
& C\cdot I= & C\cdot ABC \\
& C= & AB{{C}^{2}} \\
& C= & AB
\end{align}\,\!</math>

===Fold-over Design===

If it can be assumed for this design that the two-factor interactions are unimportant, then in the absence of <math>BC\,\!</math>, <math>AC\,\!</math> and <math>AB\,\!</math>, the equations for (A+BC), (B+AC) and (C+AB) can be used to estimate the main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. However, if such an assumption is not applicable, then to uncouple the main effects from their two factor aliases, the alternate fraction that contains runs having <math>ABC\,\!</math> at the lower level should be run. The design matrix for this design is shown in the preceding figure (c). The defining relation for this design is <math>I=-ABC\,\!</math> because the four runs for this design are obtained by selecting the rows of the preceding figure (a) for which the value of the <math>ABC\,\!</math> column is <math>-1\,\!</math>. The aliases for this fraction can be obtained as explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]] as <math>A=-BC\,\!</math>, <math>B=-AC\,\!</math> and <math>C=-AB\,\!</math>. The effects for this design can be calculated as:

::<math>\begin{align}
& A-BC= & \frac{1}{2}(ab+ac-(1)-bc) \\
& B-AC= & \frac{1}{2}(ab-ac+(1)-bc) \\
& C-AB= & \frac{1}{2}(-ab+ac-(1)+bc)
\end{align}\,\!</math>

These equations can be combined with the equations for (A+BC), (B+AC) and (C+AB) to obtain the de-aliased main effects and two factor interactions. For example, adding equations (A+BC) and (A-BC) returns the main effect <math>A\,\!</math>.

::<math>\begin{align}
& 2A= & \frac{1}{2}(a-b-c+abc)+ \\
& & \frac{1}{2}(ab+ac-(1)-bc)
\end{align}\,\!</math>

The process of augmenting a fractional factorial design by a second fraction of the same size by simply reversing the signs (of all effect columns except <math>I\,\!</math>) is called ''folding over''. The combined design is referred to as a ''fold-over design''.

==Quarter and Smaller Fraction Designs==

At times, the number of runs even for a half-fraction design are very large. In these cases, smaller fractions are used. A quarter-fraction design, denoted as <math>{2}^{k-2}\,\!</math>, consists of a fourth of the runs of the full factorial design. Quarter-fraction designs require two defining relations. The first defining relation returns the half-fraction or the <math>{2}^{k-1}\,\!</math> design. The second defining relation selects half of the runs of the <math>{2}^{k-1}\,\!</math> design to give the quarter-fraction. For example, consider the <math>{2}^{4}\,\!</math> design. To obtain a <math>{2}^{4-2}\,\!</math> design from this design, first a half-fraction of this design is obtained by using a defining relation. Assume that the defining relation used is <math>I=ABCD\,\!</math>. The design matrix for the resulting <math>{2}^{4-1}\,\!</math> design is shown in figure (a) below. Now, a quarter-fraction can be obtained from the <math>{2}^{4-1}\,\!</math> design shown in figure (a) below using a second defining relation <math>I=AD\,\!</math>. The resulting <math>{2}^{4-2}\,\!</math> design obtained is shown in figure (b) below.

[[Image:doe7.37.png|center|465px|Fractions of the <math>2^4\,\!</math> design - Figure (a) shows the <math>2^{4-1}</math> design with the defining relation <math>I=ABCD\,\!</math> and (b) shows the <math>2^{4-2}\,\!</math> design with the defining relation <math>I=ABCD=AD=BC\,\!</math>.]]

The complete defining relation for this <math>{2}^{4-2}\,\!</math> design is:

::<math>I=ABCD=AD=BC\,\!</math>

Note that the effect, <math>BC,\,\!</math> in the defining relation is the generalized interaction of <math>ABCD\,\!</math> and <math>AD\,\!</math> and is obtained using <math>(ABCD)(AD)={{A}^{2}}BC{{D}^{2}}=BC\,\!</math>. In general, a <math>{2}^{k-p}\,\!</math> fractional factorial design requires <math>p\,\!</math> independent generators. The defining relation for the design consists of the <math>p\,\!</math> independent generators and their <math>{2}^{p}\,\!</math> - (<math>p\,\!</math> +1) generalized interactions.

===Calculation of Aliases===

The alias structure for the present <math>{2}^{4-2}\,\!</math> design can be obtained using the defining relation of equation (I=ABCD=AD=BC) following the procedure explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]]. For example, multiplying the defining relation by <math>A\,\!</math> returns the effects aliased with the main effect, <math>A\,\!</math>, as follows:

::<math>\begin{align}
& A\cdot I= & A\cdot ABCD=A\cdot AD=A\cdot BC \\
& A= & {{A}^{2}}BCD={{A}^{2}}D=ABC \\
& A= & BCD=D=ABC
\end{align}\,\!</math>

Therefore, in the present <math>{2}^{4-2}\,\!</math> design, it is not possible to distinguish between effects <math>A\,\!</math>, <math>D\,\!</math>, <math>BCD\,\!</math> and <math>ABC\,\!</math>. Similarly, multiplying the defining relation by <math>B\,\!</math> and <math>AB\,\!</math> returns the effects that are aliased with these effects:

::<math>\begin{align}
& B= & ACD=ABD=C \\
& AB= & CD=AD=AC
\end{align}\,\!</math>

Other aliases can be obtained in a similar way. It can be seen that each effect in this design has three aliases. In general, each effect in a <math>{2}^{k-p}\,\!</math> design has <math>{2}^{p-1}\,\!</math> aliases.
The aliases for the <math>{2}^{4-2}\,\!</math> design show that in this design the main effects are aliased with each other (<math>A\,\!</math> is aliased with <math>D\,\!</math> and <math>B\,\!</math> is aliased with <math>C\,\!</math>). Therefore, this design is not a useful design and is not available in DOE++. It is important to ensure that main effects and lower order interactions of interest are not aliased in a fractional factorial design. This is known by looking at the resolution of the fractional factorial design.

==Design Resolution==

The resolution of a fractional factorial design is defined as the number of factors in the lowest order effect in the defining relation. For example, in the defining relation <math>I=ABCD=AD=BC\,\!</math> of the previous <math>{2}^{4-2}\,\!</math> design, the lowest-order effect is either <math>AD\,\!</math> or <math>BC,\,\!</math> containing two factors. Therefore, the resolution of this design is equal to two. The resolution of a fractional factorial design is represented using Roman numerals. For example, the previously mentioned <math>{2}^{4-2}\,\!</math> design with a resolution of two can be represented as 2 <math>_{\text{II}}^{4-2}\,\!</math>. The resolution provides information about the confounding in the design as explained next:
 
 
#'''Resolution III Designs''' In these designs, the lowest order effect in the defining relation has three factors (e.g., a <math>{2}^{5-2}\,\!</math> design with the defining relation <math>I=ABDE=ABC=CDE\,\!</math>). In resolution III designs, no main effects are aliased with any other main effects, but main effects are aliased with two factor interactions. In addition, some two factor interactions are aliased with each other.
#'''Resolution IV Designs''' In these designs, the lowest order effect in the defining relation has four factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABDE\,\!</math>). In resolution IV designs, no main effects are aliased with any other main effects or two factor interactions. However, some main effects are aliased with three factor interactions and the two factor interactions are aliased with each other.
#'''Resolution V Designs''' In these designs the lowest order effect in the defining relation has five factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABCDE\,\!</math>). In resolution V designs, no main effects or two factor interactions are aliased with any other main effects or two factor interactions. However, some main effects are aliased with four factor interactions and the two factor interactions are aliased with three factor interactions.

Fractional factorial designs with the highest resolution possible should be selected because the higher the resolution of the design, the less severe the degree of confounding. In general, designs with a resolution less than III are never used because in these designs some of the main effects are aliased with each other. The table below shows fractional factorial designs with the highest available resolution for three to ten factor designs along with their defining relations.

[[Image:doet7.3.png|center|474px|Highest resolution designs available for fractional factorial designs with 3 to 10 factors.]]

All of the two level fractional factorial designs available in DOE++ are shown next.

[[Image:doe7.38.png|center|471px|Two level fractional factorial designs available in DOE++ and their resolutions.]]

===Minimum Aberration Designs===
At times, different designs with the same resolution but different aliasing may be available. The best design to select in such a case is the minimum aberration design. For example, all <math>{2}^{7-2}\,\!</math> designs in the fourth table have a resolution of four (since the generator with the minimum number of factors in each design has four factors). Design <math>1\,\!</math> has three generators of length four (<math>ABCF,\,\!</math> <math>BCDG,\,\!</math> <math>ADFG\,\!</math>). Design <math>2\,\!</math> has two generators of length four (<math>ABCF,\,\!</math> <math>ADEG\,\!</math>). Design <math>3\,\!</math> has one generator of length four (<math>CEFG\,\!</math>). Therefore, design <math>3\,\!</math> has the least number of generators with the minimum length of four. Design <math>3\,\!</math> is called the minimum aberration design. It can be seen that the alias structure for design <math>3\,\!</math> is less involved compared to the other designs. For details refer to [[DOE_References|[Wu, 2000]]].

[[Image:doet7.4.png|center|432px|Three <math>2_{IV}^{7-2}\,\!</math> designs with different defining relations.]]

====Example====

The design of an automobile fuel cone is thought to be affected by six factors in the manufacturing process: cavity temperature (factor <math>A\,\!</math>), core temperature (factor <math>B\,\!</math>), melt temperature (factor <math>C\,\!</math>), hold pressure (factor <math>D\,\!</math>), injection speed (factor <math>E\,\!</math>) and cool time (factor <math>F\,\!</math>). The manufacturer of the fuel cone is unable to run the <math>{2}^{6}=64\,\!</math> runs required to complete one replicate for a two level full factorial experiment with six factors. Instead, they decide to run a fractional factorial design. Considering that three factor and higher order interactions are likely to be inactive, the manufacturer selects a <math>{2}^{6-2}\,\!</math> design that will require only 16 runs. The manufacturer chooses the resolution IV design which will ensure that all main effects are free from aliasing (assuming three factor and higher order interactions are absent). However, in this design the two factor interactions may be aliased with each other. It is decided that, if important two factor interactions are found to be present, additional experiment trials may be conducted to separate the aliased effects. The performance of the fuel cone is measured on a scale of 1 to 15. In DOE++, the design for this experiment is set up using the properties shown in the following figure. The Fraction Generators for the design, <math>E=ABC\,\!</math> and <math>F=BCD\,\!</math>, are the same as the defaults used in DOE++. The resulting <math>{2}^{6-2}\,\!</math> design and the corresponding response values are shown in the following two figures.

[[Image:doe7_39.png|center|644px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

[[Image:doe7_40.png|center|534px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The complete alias structure for the 2 <math>_{\text{IV}}^{6-2}\,\!</math> design is shown next.

<center><math>I=ABCE=ADEF=BCDF\,\!</math></center>

 
<center><math>\begin{align}
& A= & BCE=DEF=ABCDF \\
& B= & ACE=CDF=ABDEF \\
& C= & ABE=BDF=ACDEF \\
& D= & AEF=BCF=ABCDE \\
& E= & ABC=ADF=BCDEF \\
& F= & ADE=BCD=ABCEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& AB= & CE=ACDF=BDEF \\
& AC= & BE=ABDF=CDEF \\
& AD= & EF=ABCF=BCDE \\
& AE= & BC=DF=ABCDEF \\
& AF= & DE=ABCD=BCEF \\
& BD= & CF=ABEF=ACDE \\
& BF= & CD=ABDE=ACEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& ABD= & ACF=BEF=CDE \\
& ABF= & ACD=BDE=CEF
\end{align}\,\!</math></center>

In DOE++, the alias structure is displayed in the Design Summary and as part of the Design Evaluation result, as shown next:

[[Image:doe7_41.png|center|700px|Alias structure for the experiment design in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The normal probability plot of effects for this unreplicated design shows the main effects of factors <math>C\,\!</math> and <math>D\,\!</math> and the interaction effect, <math>BF\,\!</math>, to be significant (see the following figure).

[[Image:doe7_42.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

From the alias structure, it can be seen that for the present design interaction effect, <math>BF,\,\!</math> is confounded with <math>CD\,\!</math>. Therefore, the actual source of this effect cannot be known on the basis of the present experiment. However because neither factor <math>B\,\!</math> nor <math>F\,\!</math> is found to be significant there is an indication the observed effect is likely due to interaction, <math>CD\,\!</math>. To confirm this, a follow-up <math>{2}^{2}\,\!</math> experiment is run involving only factors <math>B\,\!</math> and <math>F\,\!</math>. The interaction, <math>BF\,\!</math>, is found to be inactive, leading to the conclusion that the interaction effect in the original experiment is effect, <math>CD\,\!</math>. Given these results, the fitted regression model for the fuel cone design as per the coefficients obtained from DOE++ is shown next.

::<math>\hat{y}=7.6875+C+2\cdot D+2.1875\cdot CD\,\!</math>

[[Image:doe7_43.png|center|700px|Effect coefficients for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

==Projection==

Projection refers to the reduction of a fractional factorial design to a full factorial design by dropping out some of the factors of the design. Any fractional factorial design of resolution, <math>R,\,\!</math> can be reduced to complete factorial designs in any subset of <math>R-1\,\!</math> factors. For example, consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. The resolution of this design is four. Therefore, this design can be reduced to full factorial designs in any three (<math>4-1=3\,\!</math>) of the original seven factors (by dropping the remaining four of factors). Further, a fractional factorial design can also be reduced to a full factorial design in any <math>R\,\!</math> of the original factors, as long as these <math>R\,\!</math> factors are not part of the generator in the defining relation. Again consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. This design can be reduced to a full factorial design in four factors provided these four factors do not appear together as a generator in the defining relation. The complete defining relation for this design is:

::<math>\begin{align}
& I= & ABCE=BCDF=ACDG=ADEF=ABFG=BDEG=CEFG
\end{align}\,\!</math>

Therefore, there are seven four factor combinations out of the 35 (<math>(_{7}^{4})=35\,\!</math>) possible four-factor combinations that are used as generators in the defining relation. The designs with the remaining 28 four factor combinations would be full factorial 16-run designs. For example, factors <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math> do not occur as a generator in the defining relation of the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. If the remaining factors, <math>E\,\!</math>, <math>F\,\!</math> and <math>G\,\!</math>, are dropped, the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design will reduce to a full factorial design in <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math>.

==Resolution III Designs==

At times, the factors to be investigated in screening experiments are so large that even running a fractional factorial design is impractical. This can be partially solved by using resolution III fractional factorial designs in the cases where three factor and higher order interactions can be assumed to be unimportant. Resolution III designs, such as the 2 <math>_{\text{III}}^{3-1}\,\!</math> design, can be used to estimate <math>k\,\!</math> main effects using just <math>k+1\,\!</math> runs. In these designs, the main effects are aliased with two factor interactions. Once the results from these designs are obtained, and knowing that three factor and higher order interactions are unimportant, the experimenter can decide if there is a need to run a fold-over design to de-alias the main effects from the two factor interactions. Thus, the 2 <math>_{\text{III}}^{3-1}\,\!</math> design can be used to investigate three factors in four runs, the 2 <math>_{\text{III}}^{7-4}\,\!</math> design can be used to investigate seven factors in eight runs, the 2 <math>_{\text{III}}^{15-11}\,\!</math> design can be used to investigate fifteen factors in sixteen runs and so on.

====Example====

{{:Resolution_III_Design_Example}}

==Alias Matrix==
In [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction designs]] and [[Two_Level_Factorial_Experiments#Quarter_and_Smaller_Fraction_Designs| Quarter and Smaller Fraction Designs]], the alias structure for fractional factorial designs was obtained using the defining relation. However, this method of obtaining the alias structure is not very efficient when the alias structure is very complex or when partial aliasing is involved. One of the ways to obtain the alias structure for any design, regardless of its complexity, is to use the alias matrix. The alias matrix for a design is calculated using <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> where <math>{{X}_{1}}\,\!</math> is the portion of the design matrix, <math>X,\,\!</math> that contains the effects for which the aliases need to be calculated, and <math>{{X}_{2}}\,\!</math> contains the remaining columns of the design matrix, other than those included in <math>{{X}_{1}}\,\!</math>.

To illustrate the use of the alias matrix, consider the design matrix for the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design (using the defining relation <math>I=ABCD\,\!</math>) shown next:

[[Image:Chapter7__879.png|center|link=]]

The alias structure for this design can be obtained by defining <math>{{X}_{1}}\,\!</math> using eight columns since the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design estimates eight effects. If the first eight columns of <math>X\,\!</math> are used then <math>{{X}_{1}}\,\!</math> is:

[[Image:Chapter7__884.png|center|link=]]

<math>{{X}_{2}}\,\!</math> is obtained using the remaining columns as:

[[Image:Chapter7__886.png|center|link=]]

Then the alias matrix <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> is:

[[Image:Chapter7__888.png|center|link=]]

The alias relations can be easily obtained by observing the alias matrix as:

<center><math>\begin{align}
& I= & ABCD \\
& A= & BCD \\
& B= & ACD \\
& AB= & CD \\
& C= & ABD \\
& AC= & BD \\
& BC= & AD \\
& D= & ABC
\end{align}\,\!</math></center>

Two Level Factorial Experiments

2017-08-10T17:28:52Z

Richard House: /* Special Features */

{{Template:Doebook|8}}
Two level factorial experiments are factorial experiments in which each factor is investigated at only two levels. The early stages of experimentation usually involve the investigation of a large number of potential factors to discover the "vital few" factors. Two level factorial experiments are used during these stages to quickly filter out unwanted effects so that attention can then be focused on the important ones.

==2''k'' Designs==
The factorial experiments, where all combination of the levels of the factors are run, are usually referred to as ''full factorial experiments''. Full factorial two level experiments are also referred to as <math>{2}^{k}\,\!</math> designs where <math>k\,\!</math> denotes the number of factors being investigated in the experiment. In Weibull++ DOE folios, these designs are referred to as 2 Level Factorial Designs as shown in the figure below.

[[Image:doe7_1.png|center|487px|Selection of full factorial experiments with two levels in Weibull++.|link=]]

A full factorial two level design with <math>k\,\!</math> factors requires <math>{{2}^{k}}\,\!</math> runs for a single replicate. For example, a two level experiment with three factors will require <math>2\times 2\times 2={{2}^{3}}=8\,\!</math> runs. The choice of the two levels of factors used in two level experiments depends on the factor; some factors naturally have two levels. For example, if gender is a factor, then male and female are the two levels. For other factors, the limits of the range of interest are usually used. For example, if temperature is a factor that varies from <math>{45}^{o}C\,\!</math> to <math>{90}^{o}C\,\!</math>, then the two levels used in the <math>{2}^{k}\,\!</math> design for this factor would be <math>{45}^{o}\,\!C\,\!</math> and <math>{90}^{o}\,\!C\,\!</math>.

The two levels of the factor in the <math>{2}^{k}\,\!</math> design are usually represented as <math>-1\,\!</math> (for the first level) and <math>1\,\!</math> (for the second level). Note that this representation is reversed from the coding used in [[General Full Factorial Designs]] for the indicator variables that represent two level factors in ANOVA models. For ANOVA models, the first level of the factor was represented using a value of <math>1\,\!</math> for the indicator variable, while the second level was represented using a value of <math>-1\,\!</math>. For details on the notation used for two level experiments refer to [[Two_Level_Factorial_Experiments#Notation| Notation]].

===The 22 Design===

The simplest of the two level factorial experiments is the <math>{2}^{2}\,\!</math> design where two factors (say factor <math>A\,\!</math> and factor <math>B\,\!</math>) are investigated at two levels. A single replicate of this design will require four runs (<math>{{2}^{2}}=2\times 2=4\,\!</math>) The effects investigated by this design are the two main effects, <math>A\,\!</math> and <math>B,\,\!</math> and the interaction effect <math>AB\,\!</math>. The treatments for this design are shown in figure (a) below. In figure (a), letters are used to represent the treatments. The presence of a letter indicates the high level of the corresponding factor and the absence indicates the low level. For example, (1) represents the treatment combination where all factors involved are at the low level or the level represented by <math>-1\,\!</math> ; <math>a\,\!</math> represents the treatment combination where factor <math>A\,\!</math> is at the high level or the level of <math>1\,\!</math>, while the remaining factors (in this case, factor <math>B\,\!</math>) are at the low level or the level of <math>-1\,\!</math>. Similarly, <math>b\,\!</math> represents the treatment combination where factor <math>B\,\!</math> is at the high level or the level of <math>1\,\!</math>, while factor <math>A\,\!</math> is at the low level and <math>ab\,\!</math> represents the treatment combination where factors <math>A\,\!</math> and <math>B\,\!</math> are at the high level or the level of the 1. Figure (b) below shows the design matrix for the <math>{2}^{2}\,\!</math> design. It can be noted that the sum of the terms resulting from the product of any two columns of the design matrix is zero. As a result the <math>{2}^{2}\,\!</math> design is an ''orthogonal design''. In fact, all <math>{2}^{k}\,\!</math> designs are orthogonal designs. This property of the <math>{2}^{k}\,\!</math> designs offers a great advantage in the analysis because of the simplifications that result from orthogonality. These simplifications are explained later on in this chapter.
The <math>{2}^{2}\,\!</math> design can also be represented geometrically using a square with the four treatment combinations lying at the four corners, as shown in figure (c) below.

[[Image:doe7.2.png|center|400px|The <math>2^2\,\!</math> design. Figure (a) displays the experiment design, (b) displays the design matrix and (c) displays the geometric representation for the design. In Figure (b), the column names I, A, B and AB are used. Column I represents the intercept term. Columns A and B represent the respective factor settings. Column AB represents the interaction and is the product of columns A and B.]]
 

===The 23 Design===

The <math>{2}^{3}\,\!</math> design is a two level factorial experiment design with three factors (say factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>). This design tests three (<math>k=3\,\!</math>) main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math> ; three (<math>(_{2}^{k})=\,\!</math> <math>(_{2}^{3})=3\,\!</math>) two factor interaction effects, <math>AB\,\!</math>, <math>BC\,\!</math>, <math>AC\,\!</math> ; and one (<math>(_{3}^{k})=\,\!</math> <math>(_{3}^{3})=1\,\!</math>) three factor interaction effect, <math>ABC\,\!</math>. The design requires eight runs per replicate. The eight treatment combinations corresponding to these runs are <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math>, <math>ab\,\!</math>, <math>c\,\!</math>, <math>ac\,\!</math>, <math>bc\,\!</math> and <math>abc\,\!</math>. Note that the treatment combinations are written in such an order that factors are introduced one by one with each new factor being combined with the preceding terms. This order of writing the treatments is called the ''standard order'' or ''Yates' order''. The <math>{2}^{3}\,\!</math> design is shown in figure (a) below. The design matrix for the <math>{2}^{3}\,\!</math> design is shown in figure (b). The design matrix can be constructed by following the standard order for the treatment combinations to obtain the columns for the main effects and then multiplying the main effects columns to obtain the interaction columns.

[[Image:doe7.3.png|center|324px|The <math>2^3\,\!</math> design. Figure (a) shows the experiment design and (b) shows the design matrix.]]

[[Image:doe7.4.png|center|290px|Geometric representation of the <math>2^3\,\!</math> design.]]

The <math>{2}^{3}\,\!</math> design can also be represented geometrically using a cube with the eight treatment combinations lying at the eight corners as shown in the figure above.

==Analysis of 2''k'' Designs==

The <math>{2}^{k}\,\!</math> designs are a special category of the factorial experiments where all the factors are at two levels. The fact that these designs contain factors at only two levels and are orthogonal greatly simplifies their analysis even when the number of factors is large. The use of <math>{2}^{k}\,\!</math> designs in investigating a large number of factors calls for a revision of the notation used previously for the ANOVA models. The case for revised notation is made stronger by the fact that the ANOVA and multiple linear regression models are identical for <math>{2}^{k}\,\!</math> designs because all factors are only at two levels. Therefore, the notation of the regression models is applied to the ANOVA models for these designs, as explained next.

===Notation===

Based on the notation used in [[General Full Factorial Designs]], the ANOVA model for a two level factorial experiment with three factors would be as follows:

::<math>\begin{align}
& Y= & \mu +{{\tau }_{1}}\cdot {{x}_{1}}+{{\delta }_{1}}\cdot {{x}_{2}}+{{(\tau \delta )}_{11}}\cdot {{x}_{1}}{{x}_{2}}+{{\gamma }_{1}}\cdot {{x}_{3}} \\
& & +{{(\tau \gamma )}_{11}}\cdot {{x}_{1}}{{x}_{3}}+{{(\delta \gamma )}_{11}}\cdot {{x}_{2}}{{x}_{3}}+{{(\tau \delta \gamma )}_{111}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where:
 
:• <math>\mu \,\!</math> represents the overall mean
:• <math>{{\tau }_{1}}\,\!</math> represents the independent effect of the first factor (factor <math>A\,\!</math>) out of the two effects <math>{{\tau }_{1}}\,\!</math> and <math>{{\tau }_{2}}\,\!</math>
:• <math>{{\delta }_{1}}\,\!</math> represents the independent effect of the second factor (factor <math>B\,\!</math>) out of the two effects <math>{{\delta }_{1}}\,\!</math> and <math>{{\delta }_{2}}\,\!</math>
:• <math>{{(\tau \delta )}_{11}}\,\!</math> represents the independent effect of the interaction <math>AB\,\!</math> out of the other interaction effects
:• <math>{{\gamma }_{1}}\,\!</math> represents the effect of the third factor (factor <math>C\,\!</math>) out of the two effects <math>{{\gamma }_{1}}\,\!</math> and <math>{{\gamma }_{2}}\,\!</math>
:• <math>{{(\tau \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>AC\,\!</math> out of the other interaction effects
:• <math>{{(\delta \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>BC\,\!</math> out of the other interaction effects
:• <math>{{(\tau \delta \gamma )}_{111}}\,\!</math> represents the effect of the interaction <math>ABC\,\!</math> out of the other interaction effects
and <math>\epsilon \,\!</math> is the random error term.

 
The notation for a linear regression model having three predictor variables with interactions is:

::<math>\begin{align}
& Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& & +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

The notation for the regression model is much more convenient, especially for the case when a large number of higher order interactions are present. In two level experiments, the ANOVA model requires only one indicator variable to represent each factor for both qualitative and quantitative factors. Therefore, the notation for the multiple linear regression model can be applied to the ANOVA model of the experiment that has all the factors at two levels. For example, for the experiment of the ANOVA model given above, <math>{{\beta }_{0}}\,\!</math> can represent the overall mean instead of <math>\mu \,\!</math>, and <math>{{\beta }_{1}}\,\!</math> can represent the independent effect, <math>{{\tau }_{1}}\,\!</math>, of factor <math>A\,\!</math>. Other main effects can be represented in a similar manner. The notation for the interaction effects is much more simplified (e.g., <math>{{\beta }_{123}}\,\!</math> can be used to represent the three factor interaction effect, <math>{{(\tau \beta \gamma )}_{111}}\,\!</math>).

As mentioned earlier, it is important to note that the coding for the indicator variables for the ANOVA models of two level factorial experiments is reversed from the coding followed in [[General Full Factorial Designs]]. Here <math>-1\,\!</math> represents the first level of the factor while <math>1\,\!</math> represents the second level. This is because for a two level factor a single variable is needed to represent the factor for both qualitative and quantitative factors. For quantitative factors, using <math>-1\,\!</math> for the first level (which is the low level) and 1 for the second level (which is the high level) keeps the coding consistent with the numerical value of the factors. The change in coding between the two coding schemes does not affect the analysis except that signs of the estimated effect coefficients will be reversed (i.e., numerical values of <math>{{\hat{\tau }}_{1}}\,\!</math>, obtained based on the coding of [[General Full Factorial Designs]], and <math>{{\hat{\beta }}_{1}}\,\!</math>, obtained based on the new coding, will be the same but their signs would be opposite).

::<math>\begin{align}
& & \text{Factor }A\text{ Coding (two level factor)} \\
& &
\end{align}\,\!</math>

::<math>\begin{matrix}
\text{Previous Coding} & {} & {} & {} & \text{Coding for }{{\text{2}}^{k}}\text{ Designs} \\
{} & {} & {} & {} & {} \\
Effect\text{ }{{\tau }_{1}}\ \ :\ \ {{x}_{1}}=1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{1}}\text{ (or }-{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=-1\text{ } \\
Effect\text{ }{{\tau }_{2}}\ \ :\ \ {{x}_{1}}=-1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{2}}\text{ (or }{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=1\text{ } \\
\end{matrix}\,\!</math>

In summary, the ANOVA model for the experiments with all factors at two levels is different from the ANOVA models for other experiments in terms of the notation in the following two ways:
 

:• The notation of the regression models is used for the effect coefficients.
:• The coding of the indicator variables is reversed.

===Special Features===

Consider the design matrix, <math>X\,\!</math>, for the <math>{2}^{3}\,\!</math> design discussed above. The (<math>{{X}^{\prime }}X\,\!</math>) <math>^{-1}\,\!</math> matrix is:

<center><math>{{({{X}^{\prime }}X)}^{-1}}=\left[ \begin{matrix}
0.125 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0.125 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0.125 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0.125 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0.125 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0.125 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0.125 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.125 \\
\end{matrix} \right]\,\!</math></center>

Notice that, due to the orthogonal design of the <math>X\,\!</math> matrix, the <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> has been simplified to a diagonal matrix which can be written as:

::<math>\begin{align}
{{({{X}^{\prime }}X)}^{-1}}= & 0.125\cdot I = & \frac{1}{8}\cdot I = & \frac{1}{{{2}^{3}}}\cdot I
\end{align}\,\!</math>

where <math>I\,\!</math> represents the identity matrix of the same order as the design matrix, <math>X\,\!</math>. Since there are eight observations per replicate of the <math>{2}^{3}\,\!</math> design, the <math>(X\,\!</math> ' <math>X{{)}^{-1}}\,\!</math> matrix for <math>m\,\!</math> replicates of this design can be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{3}}\cdot m)}\cdot I\,\!</math>

The <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> matrix for any <math>{2}^{k}\,\!</math> design can now be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{k}}\cdot m)}\cdot I\,\!</math>

Then the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is:

::<math>\begin{align}
C= & {{{\hat{\sigma }}}^{2}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & M{{S}_{E}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & \frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}\cdot I
\end{align}\,\!</math>

Note that the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is also a diagonal matrix. Therefore, the estimated effect coefficients (<math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math>, <math>{{\beta }_{12}},\,\!</math> etc.) for these designs are uncorrelated. This implies that the terms in the <math>{2}^{k}\,\!</math> design (main effects, interactions) are independent of each other. Consequently, the extra sum of squares for each of the terms in these designs is independent of the sequence of terms in the model, and also independent of the presence of other terms in the model. As a result the sequential and partial sum of squares for the terms are identical for these designs and will always add up to the model sum of squares. Multicollinearity is also not an issue for these designs.

It can also be noted from the equation given above, that in addition to the <math>C\,\!</math> matrix being diagonal, all diagonal elements of the <math>C\,\!</math> matrix are identical. This means that the variance (or its square root, the standard error) of all estimated effect coefficients are the same. The standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>, for all the coefficients is:

::<math>\begin{align}
se({{{\hat{\beta }}}_{j}})= & \sqrt{{{C}_{jj}}} = & \sqrt{\frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}}\text{ }for\text{ }all\text{ }j
\end{align}\,\!</math>

This property is used to construct the normal probability plot of effects in <math>{2}^{k}\,\!</math> designs and identify significant effects using graphical techniques. For details on the normal probability plot of effects in a Weibull++ DOE folio, refer to [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]].

====Example====
To illustrate the analysis of a full factorial <math>{2}^{k}\,\!</math> design, consider a three factor experiment to investigate the effect of honing pressure, number of strokes and cycle time on the surface finish of automobile brake drums. Each of these factors is investigated at two levels. The honing pressure is investigated at levels of 200 <math>psi\,\!</math> and 400 <math>psi\,\!</math>, the number of strokes used is 3 and 5 and the two levels of the cycle time are 3 and 5 seconds. The design for this experiment is set up in a Weibull++ DOE folio as shown in the first two following figures. It is decided to run two replicates for this experiment. The surface finish data collected from each run (using randomization) and the complete design is shown in the third following figure. The analysis of the experiment data is explained next.

[[Image:doe7_5.png|center|877px|Design properties for the experiment in the example.|link=]]

[[Image:doe7_6.png|center|859px|Design summary for the experiment in the example.|link=]]

[[Image:doe7_7.png|center|778px|Experiment design for the example to investigate the surface finish of automobile brake drums.|link=]]

The applicable model using the notation for <math>{2}^{k}\,\!</math> designs is:

::<math>\begin{align}
Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where the indicator variable, <math>{{x}_{1,}}\,\!</math> represents factor <math>A\,\!</math> (honing pressure), <math>{{x}_{1}}=-1\,\!</math> represents the low level of 200 <math>psi\,\!</math> and <math>{{x}_{1}}=1\,\!</math> represents the high level of 400 <math>psi\,\!</math>. Similarly, <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> represent factors <math>B\,\!</math> (number of strokes) and <math>C\,\!</math> (cycle time), respectively. <math>{{\beta }_{0}}\,\!</math> is the overall mean, while <math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math> and <math>{{\beta }_{3}}\,\!</math> are the effect coefficients for the main effects of factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. <math>{{\beta }_{12}}\,\!</math>, <math>{{\beta }_{13}}\,\!</math> and <math>{{\beta }_{23}}\,\!</math> are the effect coefficients for the <math>AB\,\!</math>, <math>AC\,\!</math> and <math>BC\,\!</math> interactions, while <math>{{\beta }_{123}}\,\!</math> represents the <math>ABC\,\!</math> interaction.

 
If the subscripts for the run (<math>i\,\!</math> ; <math>i=\,\!</math> 1 to 8) and replicates (<math>j\,\!</math> ; <math>j=\,\!</math> 1,2) are included, then the model can be written as:

::<math>\begin{align}
{{Y}_{ij}}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{ij1}}+{{\beta }_{2}}\cdot {{x}_{ij2}}+{{\beta }_{12}}\cdot {{x}_{ij1}}{{x}_{ij2}}+{{\beta }_{3}}\cdot {{x}_{ij3}} \\
& +{{\beta }_{13}}\cdot {{x}_{ij1}}{{x}_{ij3}}+{{\beta }_{23}}\cdot {{x}_{ij2}}{{x}_{ij3}}+{{\beta }_{123}}\cdot {{x}_{ij1}}{{x}_{ij2}}{{x}_{ij3}}+{{\epsilon }_{ij}}
\end{align}\,\!</math>

To investigate how the given factors affect the response, the following hypothesis tests need to be carried:

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{1}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{1}}\ne 0\,\!</math>

This test investigates the main effect of factor <math>A\,\!</math> (honing pressure). The statistic for this test is:

::<math>{{({{F}_{0}})}_{A}}=\frac{M{{S}_{A}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{A}}\,\!</math> is the mean square for factor <math>A\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other main effects, <math>B\,\!</math> and <math>C\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{12}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{12}}\ne 0\,\!</math>

This test investigates the two factor interaction <math>AB\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{AB}}=\frac{M{{S}_{AB}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the interaction <math>AB\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other two factor interactions, <math>AC\,\!</math> and <math>BC\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{123}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{123}}\ne 0\,\!</math>

This test investigates the three factor interaction <math>ABC\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{ABC}}=\frac{M{{S}_{ABC}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{ABC}}\,\!</math> is the mean square for the interaction <math>ABC\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square.
To calculate the test statistics, it is convenient to express the ANOVA model in the form <math>y=X\beta +\epsilon \,\!</math>.

====Expression of the ANOVA Model as <math>y=X\beta +\epsilon \,\!</math>====

In matrix notation, the ANOVA model can be expressed as:

::<math>y=X\beta +\epsilon \,\!</math>

where:

<center><math>y=\left[ \begin{matrix}
{{Y}_{11}} \\
{{Y}_{21}} \\
. \\
{{Y}_{81}} \\
{{Y}_{12}} \\
. \\
{{Y}_{82}} \\
\end{matrix} \right]=\left[ \begin{matrix}
90 \\
90 \\
. \\
90 \\
86 \\
. \\
80 \\
\end{matrix} \right]\text{ }X=\left[ \begin{matrix}
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\end{matrix} \right]\,\!</math></center>

<center><math>\beta =\left[ \begin{matrix}
{{\beta }_{0}} \\
{{\beta }_{1}} \\
{{\beta }_{2}} \\
{{\beta }_{12}} \\
{{\beta }_{3}} \\
{{\beta }_{13}} \\
{{\beta }_{23}} \\
{{\beta }_{123}} \\
\end{matrix} \right]\text{ }\epsilon =\left[ \begin{matrix}
{{\epsilon }_{11}} \\
{{\epsilon }_{21}} \\
. \\
{{\epsilon }_{81}} \\
{{\epsilon }_{12}} \\
. \\
. \\
{{\epsilon }_{82}} \\
\end{matrix} \right]\,\!</math></center>

====Calculation of the Extra Sum of Squares for the Factors====

Knowing the matrices <math>y\,\!</math>, <math>X\,\!</math> and <math>\beta \,\!</math>, the extra sum of squares for the factors can be calculated. These are used to calculate the mean squares that are used to obtain the test statistics. Since the experiment design is orthogonal, the partial and sequential extra sum of squares are identical. The extra sum of squares for each effect can be calculated as shown next. As an example, the extra sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & Model\text{ }Sum\text{ }of\text{ }Squares - Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }main\text{ }effect\text{ }of\text{ }A \\
= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }A}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }A}}={{X}_{\tilde{\ }A}}{{(X_{\tilde{\ }A}^{\prime }{{X}_{\tilde{\ }A}})}^{-1}}X_{\tilde{\ }A}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }A}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the main effect of factor <math>A\,\!</math>. Thus, the sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y \\
= & 654.4375-549.375 \\
= & 105.0625
\end{align}\,\!</math>

Similarly, the extra sum of squares for the interaction effect <math>AB\,\!</math> is:

::<math>\begin{align}
S{{S}_{AB}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }AB}}-(1/16)J]y \\
= & 654.4375-636.375 \\
= & 18.0625
\end{align}\,\!</math>

The extra sum of squares for other effects can be obtained in a similar manner.

====Calculation of the Test Statistics====

Knowing the extra sum of squares, the test statistic for the effects can be calculated. For example, the test statistic for the interaction <math>AB\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{AB}}= & \frac{M{{S}_{AB}}}{M{{S}_{E}}} \\
= & \frac{S{{S}_{AB}}/dof(S{{S}_{AB}})}{S{{S}_{E}}/dof(S{{S}_{E}})} \\
= & \frac{18.0625/1}{147.5/8} \\
= & 0.9797
\end{align}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the <math>AB\,\!</math> interaction and <math>M{{S}_{E}}\,\!</math> is the error mean square. The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{AB}}=0.9797\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{AB}}) \\
= & 1-0.6487 \\
= & 0.3513
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that the interaction between honing pressure and number of strokes does not affect the surface finish of the brake drums. Tests for other effects can be carried out in a similar manner. The results are shown in the ANOVA Table in the following figure. The values S, R-sq and R-sq(adj) in the figure indicate how well the model fits the data. The value of S represents the standard error of the model, R-sq represents the coefficient of multiple determination and R-sq(adj) represents the adjusted coefficient of multiple determination. For details on these values refer to [[Multiple_Linear_Regression_Analysis|Multiple Linear Regression Analysis]].

[[Image:doe7_8.png|center|799px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

====Calculation of Effect Coefficients====

The estimate of effect coefficients can also be obtained:

<center><math>\begin{align}
\hat{\beta }= & {{({{X}^{\prime }}X)}^{-1}}{{X}^{\prime }}y \\
= & \left[ \begin{matrix}
86.4375 \\
2.5625 \\
-4.9375 \\
1.0625 \\
-1.0625 \\
2.4375 \\
-1.3125 \\
-0.1875 \\
\end{matrix} \right]
\end{align}\,\!</math></center>

[[Image:doe7_9.png|center|800px|Regression Information table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

The coefficients and related results are shown in the Regression Information table above. In the table, the Effect column displays the effects, which are simply twice the coefficients. The Standard Error column displays the standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>. The Low CI and High CI columns display the confidence interval on the coefficients. The interval shown is the 90% interval as the significance is chosen as 0.1. The T Value column displays the <math>t\,\!</math> statistic, <math>{{t}_{0}}\,\!</math>, corresponding to the coefficients. The P Value column displays the <math>p\,\!</math> value corresponding to the <math>t\,\!</math> statistic. (For details on how these results are calculated, refer to [[General Full Factorial Designs]]). Plots of residuals can also be obtained from DOE++ to ensure that the assumptions related to the ANOVA model are not violated.

====Model Equation====

From the analysis results in the above figure within [[Two_Level_Factorial_Experiments#Calculation_of_Effect_Coefficients|calculation of effect coefficients]] section, it is seen that effects <math>A\,\!</math>, <math>B\,\!</math> and <math>AC\,\!</math> are significant. In DOE++, the <math>p\,\!</math> values for the significant effects are displayed in red in the ANOVA Table for easy identification. Using the values of the estimated effect coefficients, the model for the present <math>{2}^{3}\,\!</math> design in terms of the coded values can be written as:

::<math>\begin{align}
\hat{y}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}} \\
= & 86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+2.4375{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

To make the model hierarchical, the main effect, <math>C\,\!</math>, needs to be included in the model (because the interaction <math>AC\,\!</math> is included in the model). The resulting model is:

::<math>\hat{y}=86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+1.0625{{x}_{3}}+2.4375{{x}_{1}}{{x}_{3}}\,\!</math>

This equation can be viewed in DOE++, as shown in the following figure, using the Show Analysis Summary icon in the Control Panel. The equation shown in the figure will match the hierarchical model once the required terms are selected using the Select Effects icon.

[[Image:doe7_10.png|center|557px|The model equation for the experiment of the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

==Replicated and Repeated Runs==

In the case of replicated experiments, it is important to note the difference between replicated runs and repeated runs. Both repeated and replicated runs are multiple response readings taken at the same factor levels. However, repeated runs are response observations taken at the same time or in succession. Replicated runs are response observations recorded in a random order. Therefore, replicated runs include more variation than repeated runs. For example, a baker, who wants to investigate the effect of two factors on the quality of cakes, will have to bake four cakes to complete one replicate of a <math>{2}^{2}\,\!</math> design. Assume that the baker bakes eight cakes in all. If, for each of the four treatments of the <math>{2}^{2}\,\!</math> design, the baker selects one treatment at random and then bakes two cakes for this treatment at the same time then this is a case of two repeated runs. If, however, the baker bakes all the eight cakes randomly, then the eight cakes represent two sets of replicated runs.
For repeated measurements, the average values of the response for each treatment should be entered into DOE++ as shown in the following figure (a) when the two cakes for a particular treatment are baked together. For replicated measurements, when all the cakes are baked randomly, the data is entered as shown in the following figure (b).

[[Image:doe7.11.png|center|300px|Data entry for repeated and replicated runs. Figure (a) shows repeated runs and (b) shows replicated runs.]]

==Unreplicated 2''k'' Designs==

If a factorial experiment is run only for a single replicate then it is not possible to test hypotheses about the main effects and interactions as the error sum of squares cannot be obtained. This is because the number of observations in a single replicate equals the number of terms in the ANOVA model. Hence the model fits the data perfectly and no degrees of freedom are available to obtain the error sum of squares.

However, sometimes it is only possible to run a single replicate of the <math>{2}^{k}\,\!</math> design because of constraints on resources and time. In the absence of the error sum of squares, hypothesis tests to identify significant factors cannot be conducted. A number of methods of analyzing information obtained from unreplicated <math>{2}^{k}\,\!</math> designs are available. These include pooling higher order interactions, using the normal probability plot of effects or including center point replicates in the design.

===Pooling Higher Order Interactions===

One of the ways to deal with unreplicated <math>{2}^{k}\,\!</math> designs is to use the sum of squares of some of the higher order interactions as the error sum of squares provided these higher order interactions can be assumed to be insignificant. By dropping some of the higher order interactions from the model, the degrees of freedom corresponding to these interactions can be used to estimate the error mean square. Once the error mean square is known, the test statistics to conduct hypothesis tests on the factors can be calculated.

===Normal Probability Plot of Effects===
Another way to use unreplicated <math>{2}^{k}\,\!</math> designs to identify significant effects is to construct the normal probability plot of the effects. As mentioned in [[Two_Level_Factorial_Experiments#Special_Features| Special Features]], the standard error for all effect coefficients in the <math>{2}^{k}\,\!</math> designs is the same. Therefore, on a normal probability plot of effect coefficients, all non-significant effect coefficients (with <math>\beta =0\,\!</math>) will fall along the straight line representative of the normal distribution, N(<math>0,{{\sigma }^{2}}/({{2}^{k}}\cdot m)\,\!</math>). Effect coefficients that show large deviations from this line will be significant since they do not come from this normal distribution. Similarly, since effects <math>=2\times \,\!</math> effect coefficients, all non-significant effects will also follow a straight line on the normal probability plot of effects. For replicated designs, the Effects Probability plot of DOE++ plots the normalized effect values (or the T Values) on the standard normal probability line, N(0,1). However, in the case of unreplicated <math>{2}^{k}\,\!</math> designs, <math>{{\sigma }^{2}}\,\!</math> remains unknown since <math>M{{S}_{E}}\,\!</math> cannot be obtained. Lenth's method is used in this case to estimate the variance of the effects. For details on Lenth's method, please refer to [[DOE References| Montgomery (2001)]]. DOE++ then uses this variance value to plot effects along the N(0, Lenth's effect variance) line. The
method is illustrated in the following example.

====Example====

Vinyl panels, used as instrument panels in a certain automobile, are seen to develop defects after a certain amount of time. To investigate the issue, it is decided to carry out a two level factorial experiment. Potential factors to be investigated in the experiment are vacuum rate (factor <math>A\,\!</math>), material temperature (factor <math>B\,\!</math>), element intensity (factor <math>C\,\!</math>) and pre-stretch (factor <math>D\,\!</math>). The two levels of the factors used in the experiment are as shown in below.

[[Image:doet7.1.png|center|300px|Factors to investigate defects in vinyl panels.]]

With a <math>{2}^{4}\,\!</math> design requiring 16 runs per replicate it is only feasible for the manufacturer to run a single replicate.

The experiment design and data, collected as percent defects, are shown in the following figure. Since the present experiment design contains only a single replicate, it is not possible to obtain an estimate of the error sum of squares, <math>S{{S}_{E}}\,\!</math>. It is decided to use the normal probability plot of effects to identify the significant effects. The effect values for each term are obtained as shown in the following figure.

[[Image:doe7_13.png|center|650px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

Lenth's method uses these values to estimate the variance. As described in [[DOE_References|[Lenth, 1989]]], if all effects are arranged in ascending order, using their absolute values, then <math>{{s}_{0}}\,\!</math> is defined as 1.5 times the median value:

::<math>\begin{align}
{{s}_{0}}= & 1.5\cdot median(\left| effect \right|) \\
= & 1.5\cdot 2 \\
= & 3
\end{align}\,\!</math>

Using <math>{{s}_{0}}\,\!</math>, the "pseudo standard error" (<math>PSE\,\!</math>) is calculated as 1.5 times the median value of all effects that are less than 2.5 <math>{{s}_{0}}\,\!</math> :

::<math>\begin{align}
PSE= & 1.5\cdot median(\left| effect \right|\ \ :\ \ \left| effect \right|<2.5{{s}_{0}}) \\
= & 1.5\cdot 1.5 \\
= & 2.25
\end{align}\,\!</math>

Using <math>PSE\,\!</math> as an estimate of the effect variance, the effect variance is 2.25. Knowing the effect variance, the normal probability plot of effects for the present unreplicated experiment can be constructed as shown in the following figure. The line on this plot is the line N(0, 2.25). The plot shows that the effects <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math> do not follow the distribution represented by this line. Therefore, these effects are significant.

The significant effects can also be identified by comparing individual effect values to the margin of error or the threshold value using the pareto chart (see the third following figure). If the required significance is 0.1, then:

::<math>margin\text{ }of\text{ }error={{t}_{\alpha /2,d}}\cdot PSE\,\!</math>

The <math>t\,\!</math> statistic, <math>{{t}_{\alpha /2,d}}\,\!</math>, is calculated at a significance of <math>\alpha /2\,\!</math> (for the two-sided hypothesis) and degrees of freedom <math>d=(\,\!</math> number of effects <math>)/3\,\!</math>. Thus:

::<math>\begin{align}
margin\text{ }of\text{ }error= & {{t}_{0.05,5}}\cdot PSE \\
= & 2.015\cdot 2.25 \\
= & 4.534
\end{align}\,\!</math>

The value of 4.534 is shown as the critical value line in the third following figure. All effects with absolute values greater than the margin of error can be considered to be significant. These effects are <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math>. Therefore, the vacuum rate, the pre-stretch and their interaction have a significant effect on the defects of the vinyl panels.

[[Image:doe7_14.png|center|800px|Effect values for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_15.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_16.png|center|650px|Pareto chart for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

===Center Point Replicates===

Another method of dealing with unreplicated <math>{2}^{k}\,\!</math> designs that only have quantitative factors is to use replicated runs at the center point. The center point is the response corresponding to the treatment exactly midway between the two levels of all factors. Running multiple replicates at this point provides an estimate of pure error. Although running multiple replicates at any treatment level can provide an estimate of pure error, the other advantage of running center point replicates in the <math>{2}^{k}\,\!</math> design is in checking for the presence of curvature. The test for curvature investigates whether the model between the response and the factors is linear and is discussed in [[Two_Level_Factorial_Experiments#Using_Center_Point_Replicates_to_Test_Curvature| Center Pt. Replicates to Test Curvature]].

====Example: Use Center Point to Get Pure Error====

Consider a <math>{2}^{2}\,\!</math> experiment design to investigate the effect of two factors, <math>A\,\!</math> and <math>B\,\!</math>, on a certain response. The energy consumed when the treatments of the <math>{2}^{2}\,\!</math> design are run is considerably larger than the energy consumed for the center point run (because at the center point the factors are at their middle levels). Therefore, the analyst decides to run only a single replicate of the design and augment the design by five replicated runs at the center point as shown in the following figure. The design properties for this experiment are shown in the second following figure. The complete experiment design is shown in the third following figure. The center points can be used in the identification of significant effects as shown next.

[[Image:doe7.17.png||center|300px|<math>2^2\,\!</math> design augmented by five center point runs.]]
[[Image:doe7_18.png|center|537px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

[[Image:doe7_19.png|center|630px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

Since the present <math>{2}^{2}\,\!</math> design is unreplicated, there are no degrees of freedom available to calculate the error sum of squares. By augmenting this design with five center points, the response values at the center points, <math>y_{i}^{c}\,\!</math>, can be used to obtain an estimate of pure error, <math>S{{S}_{PE}}\,\!</math>. Let <math>{{\bar{y}}^{c}}\,\!</math> represent the average response for the five replicates at the center. Then:

::<math>S{{S}_{PE}}=Sum\text{ }of\text{ }Squares\text{ }for\text{ }center\text{ }points\,\!</math>

::<math>\begin{align}
S{{S}_{PE}}= & \underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}} \\
= & {{(25.2-25.26)}^{2}}+...+{{(25.3-25.26)}^{2}} \\
= & 0.052
\end{align}\,\!</math>

Then the corresponding mean square is:

::<math>\begin{align}
M{{S}_{PE}}= & \frac{S{{S}_{PE}}}{degrees\text{ }of\text{ }freedom} \\
= & \frac{0.052}{5-1} \\
= & 0.013
\end{align}\,\!</math>

Alternatively, <math>M{{S}_{PE}}\,\!</math> can be directly obtained by calculating the variance of the response values at the center points:

::<math>\begin{align}
M{{S}_{PE}}= & {{s}^{2}} \\
= & \frac{\underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}}}{5-1}
\end{align}\,\!</math>

Once <math>M{{S}_{PE}}\,\!</math> is known, it can be used as the error mean square, <math>M{{S}_{E}}\,\!</math>, to carry out the test of significance for each effect. For example, to test the significance of the main effect of factor <math>A,\,\!</math> the sum of squares corresponding to this effect is obtained in the usual manner by considering only the four runs of the original <math>{2}^{2}\,\!</math> design.

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/4)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/4)J]y \\
= & 0.5625
\end{align}\,\!</math>

Then, the test statistic to test the significance of the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{A}}= & \frac{M{{S}_{A}}}{M{{S}_{E}}} \\
= & \frac{0.5625/1}{0.052/4} \\
= & 43.2692
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{A}}=43.2692\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{A}}) \\
= & 1-0.9972 \\
= & 0.0028
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value < 0.1, it can be concluded that the main effect of factor <math>A\,\!</math> significantly affects the response. This result is displayed in the ANOVA table as shown in the following figure. Test for the significance of other factors can be carried out in a similar manner.

[[Image:doe7_20.png|center|649px|Results for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

===Using Center Point Replicates to Test Curvature===

Center point replicates can also be used to check for curvature in replicated or unreplicated <math>{2}^{k}\,\!</math> designs. The test for curvature investigates whether the model between the response and the factors is linear. The way DOE++ handles center point replicates is similar to its handling of blocks. The center point replicates are treated as an additional factor in the model. The factor is labeled as Curvature in the results of DOE++. If Curvature turns out to be a significant factor in the results, then this indicates the presence of curvature in the model.

====Example: Use Center Point to Test Curvature====

To illustrate the use of center point replicates in testing for curvature, consider again the data of the single replicate <math>{2}^{2}\,\!</math> experiment from a preceding figure(labeled "<math>2^2</math> design augmented by five center point runs"). Let <math>{{x}_{1}}\,\!</math> be the indicator variable to indicate if the run is a center point:

::<math>\begin{matrix}
{{x}_{1}}=0 & {} & \text{Center point run} \\
{{x}_{1}}=1 & {} & \text{Other run} \\
\end{matrix}\,\!</math>

If <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> are the indicator variables representing factors <math>A\,\!</math> and <math>B\,\!</math>, respectively, then the model for this experiment is:

::<math>Y={{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}\,\!</math>

To investigate the presence of curvature, the following hypotheses need to be tested:

::<math>\begin{align}
& {{H}_{0}}: & {{\beta }_{1}}=0\text{ (Curvature is absent)} \\
& {{H}_{1}}: & {{\beta }_{1}}\ne 0
\end{align}\,\!</math>

The test statistic to be used for this test is:

::<math>{{({{F}_{0}})}_{curvature}}=\frac{M{{S}_{curvature}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{curvature}}\,\!</math> is the mean square for Curvature and <math>M{{S}_{E}}\,\!</math> is the error mean square.

'''Calculation of the Sum of Squares'''

The <math>X\,\!</math> matrix and <math>y\,\!</math> vector for this experiment are:

<center><math>X=\left[ \begin{matrix}
1 & 1 & -1 & -1 & 1 \\
1 & 1 & 1 & -1 & -1 \\
1 & 1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
\end{matrix} \right]\text{ }y=\left[ \begin{matrix}
24.6 \\
25.4 \\
25.0 \\
25.7 \\
25.2 \\
25.3 \\
25.4 \\
25.1 \\
25.3 \\
\end{matrix} \right]\,\!</math></center>

The sum of squares can now be calculated. For example, the error sum of squares is:

::<math>\begin{align}
& S{{S}_{E}}= & {{y}^{\prime }}[I-H]y \\
& = & 0.052
\end{align}\,\!</math>

where <math>I\,\!</math> is the identity matrix and <math>H\,\!</math> is the hat matrix. It can be seen that this is equal to <math>S{{S}_{PE\text{ }}}\,\!</math> (the sum of squares due to pure error) because of the replicates at the center point, as obtained in the [[Two_Level_Factorial_Experiments#Example_3| example]]. The number of degrees of freedom associated with <math>S{{S}_{E}}\,\!</math>, <math>dof(S{{S}_{E}})\,\!</math> is four. The extra sum of squares corresponding to the center point replicates (or Curvature) is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & Model\text{ }Sum\text{ }of\text{ }Squares- \\
& & Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }center\text{ }point \\
& = & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Curvature}}-(1/9)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }Curvature}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }Curvature}}={{X}_{\tilde{\ }Curv}}{{(X_{\tilde{\ }Curv}^{\prime }{{X}_{\tilde{\ }Curv}})}^{-1}}X_{\tilde{\ }Curv}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }Curv}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the center point. Thus, the extra sum of squares corresponding to Curvature is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Center}}-(1/9)J]y \\
& = & 0.7036-0.6875 \\
& = & 0.0161
\end{align}\,\!</math>

This extra sum of squares can be used to test for the significance of curvature. The corresponding mean square is:

::<math>\begin{align}
& M{{S}_{Curvature}}= & \frac{Sum\text{ }of\text{ }squares\text{ }corresponding\text{ }to\text{ }Curvature}{degrees\text{ }of\text{ }freedom} \\
& = & \frac{0.0161}{1} \\
& = & 0.0161
\end{align}\,\!</math>

'''Calculation of the Test Statistic'''

Knowing the mean squares, the statistic to check the significance of curvature can be calculated.

::<math>\begin{align}
& {{({{f}_{0}})}_{Curvature}}= & \frac{M{{S}_{Curvature}}}{M{{S}_{E}}} \\
& = & \frac{0.0161/1}{0.052/4} \\
& = & 1.24
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{Curvature}}=1.24\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and four degrees of freedom in the denominator is:

::<math>\begin{align}
& p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{Curvature}}) \\
& = & 1-0.6713 \\
& = & 0.3287
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that curvature does not exist for this design. This results is shown in the ANOVA table in the figure above. The surface of the fitted model based on these results, along with the observed response values, is shown in the figure below.

[[Image:doe7.21.png|center|400px|Model surface and observed response values for the design in the [[Two_Level_Factorial_Experiments#Example_4| example]].]]

 

==Blocking in 2k Designs==

Blocking can be used in the <math>{2}^{k}\,\!</math> designs to deal with cases when replicates cannot be run under identical conditions. Randomized complete block designs that were discussed in [[Randomization and Blocking in DOE]] for factorial experiments are also applicable here. At times, even with just two levels per factor, it is not possible to run all treatment combinations for one replicate of the experiment under homogeneous conditions. For example, each replicate of the <math>{2}^{2}\,\!</math> design requires four runs. If each run requires two hours and testing facilities are available for only four hours per day, two days of testing would be required to run one complete replicate. Blocking can be used to separate the treatment runs on the two different days. Blocks that do not contain all treatments of a replicate are called incomplete blocks. In incomplete block designs, the block effect is confounded with certain effect(s) under investigation. For the <math>{2}^{2}\,\!</math> design assume that treatments <math>(1)\,\!</math> and <math>ab\,\!</math> were run on the first day and treatments <math>a\,\!</math> and <math>b\,\!</math> were run on the second day. Then, the incomplete block design for this experiment is:

::<math>\begin{matrix}
\text{Block 1} & {} & \text{Block 2} \\
\left[ \begin{matrix}
(1) \\
ab \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
b \\
\end{matrix} \right] \\
\end{matrix}\,\!</math>

For this design the block effect may be calculated as:

::<math>\begin{align}
& Block\text{ }Effect= & Average\text{ }response\text{ }for\text{ }Block\text{ }1- \\
& & Average\text{ }response\text{ }for\text{ }Block\text{ }2 \\
& = & \frac{(1)+ab}{2}-\frac{a+b}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The <math>AB\,\!</math> interaction effect is:

::<math>\begin{align}
& AB= & Average\text{ }response\text{ }at\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{low}}}- \\
& & Average\text{ }response\text{ }at\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{low}}} \\
& = & \frac{ab+(1)}{2}-\frac{b+a}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The two equations given above show that, in this design, the <math>AB\,\!</math> interaction effect cannot be distinguished from the block effect because the formulas to calculate these effects are the same. In other words, the <math>AB\,\!</math> interaction is said to be confounded with the block effect and it is not possible to say if the effect calculated based on these equations is due to the <math>AB\,\!</math> interaction effect, the block effect or both. In incomplete block designs some effects are always confounded with the blocks. Therefore, it is important to design these experiments in such a way that the important effects are not confounded with the blocks. In most cases, the experimenter can assume that higher order interactions are unimportant. In this case, it would better to use incomplete block designs that confound these effects with the blocks.
One way to design incomplete block designs is to use defining contrasts as shown next:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}+...+{{\alpha }_{k}}{{q}_{k}}\,\!</math>

where the <math>{{\alpha }_{i}}\,\!</math> s are the exponents for the factors in the effect that is to be confounded with the block effect and the <math>{{q}_{i}}\,\!</math> s are values based on the level of the <math>i\,\!</math> the factor (in a treatment that is to be allocated to a block). For <math>{2}^{k}\,\!</math> designs the <math>{{\alpha }_{i}}\,\!</math> s are either 0 or 1 and the <math>{{q}_{i}}\,\!</math> s have a value of 0 for the low level of the <math>i\,\!</math> th factor and a value of 1 for the high level of the factor in the treatment under consideration. As an example, consider the <math>{2}^{2}\,\!</math> design where the interaction effect <math>AB\,\!</math> is confounded with the block. Since there are two factors, <math>k=2\,\!</math>, with <math>i=1\,\!</math> representing factor <math>A\,\!</math> and <math>i=2\,\!</math> representing factor <math>B\,\!</math>. Therefore:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}\,\!</math>

The value of <math>{{\alpha }_{1}}\,\!</math> is one because the exponent of factor <math>A\,\!</math> in the confounded interaction <math>AB\,\!</math> is one. Similarly, the value of <math>{{\alpha }_{2}}\,\!</math> is one because the exponent of factor <math>B\,\!</math> in the confounded interaction <math>AB\,\!</math> is also one. Therefore, the defining contrast for this design can be written as:

::<math>\begin{align}
& L= & {{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}} \\
& = & 1\cdot {{q}_{1}}+1\cdot {{q}_{2}} \\
& = & {{q}_{1}}+{{q}_{2}}
\end{align}\,\!</math>

Once the defining contrast is known, it can be used to allocate treatments to the blocks. For the <math>{2}^{2}\,\!</math> design, there are four treatments <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math> and <math>ab\,\!</math>. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. In order to decide which block the treatment <math>(1)\,\!</math> belongs to, the levels of factors <math>A\,\!</math> and <math>B\,\!</math> for this run are used. Since factor <math>A\,\!</math> is at the low level in this treatment, <math>{{q}_{1}}=0\,\!</math>. Similarly, since factor <math>B\,\!</math> is also at the low level in this treatment, <math>{{q}_{2}}=0\,\!</math>. Therefore:

::<math>\begin{align}
& L= & {{q}_{1}}+{{q}_{2}} \\
& = & 0+0=0\text{ (mod 2)}
\end{align}\,\!</math>

Note that the value of <math>L\,\!</math> used to decide the block allocation is "mod 2" of the original value. This value is obtained by taking the value of 1 for odd numbers and 0 otherwise. Based on the value of <math>L\,\!</math>, treatment <math>(1)\,\!</math> is assigned to block 1. Other treatments can be assigned using the following calculations:

::<math>\begin{align}
& (1): & \text{ }L=0+0=0=0\text{ (mod 2)} \\
& a: & \text{ }L=1+0=1=1\text{ (mod 2)} \\
& b: & \text{ }L=0+1=1=1\text{ (mod 2)} \\
& ab: & \text{ }L=1+1=2=0\text{ (mod 2)}
\end{align}\,\!</math>

Therefore, to confound the interaction <math>AB\,\!</math> with the block effect in the <math>{2}^{2}\,\!</math> incomplete block design, treatments <math>(1)\,\!</math> and <math>ab\,\!</math> (with <math>L=0\,\!</math>) should be assigned to block 2 and treatment combinations <math>a\,\!</math> and <math>b\,\!</math> (with <math>L=1\,\!</math>) should be assigned to block 1.

====Example: Two Level Factorial Design with Two Blocks====

This example illustrates how treatments can be allocated to two blocks for an unreplicated <math>{2}^{k}\,\!</math> design. Consider the unreplicated <math>{2}^{4}\,\!</math> design to investigate the four factors affecting the defects in automobile vinyl panels discussed in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments required for this experiment were run by two different operators with each operator conducting 8 runs. This experiment is an example of an incomplete block design. The analyst in charge of this experiment assumed that the interaction <math>ABCD\,\!</math> was not significant and decided to allocate treatments to the two operators so that the <math>ABCD\,\!</math> interaction was confounded with the block effect (the two operators are the blocks). The allocation scheme to assign treatments to the two operators can be obtained as follows.
 
The defining contrast for the <math>{2}^{4}\,\!</math> design where the <math>ABCD\,\!</math> interaction is confounded with the blocks is:

::<math>L={{q}_{1}}+{{q}_{2}}+{{q}_{3}}+{{q}_{4}}\,\!</math>

The treatments can be allocated to the two operators using the values of the defining contrast. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. Then the value of the defining contrast for treatment <math>a\,\!</math> is:

::<math>a\ \ :\ \ \text{ }L=1+0+0+0=1=1\text{ (mod 2)}\,\!</math>

Therefore, treatment <math>a\,\!</math> should be assigned to Block 1 or the first operator. Similarly, for treatment <math>ab\,\!</math> we have:

::<math>ab\ \ :\ \ \text{ }L=1+1+0+0=2=0\text{ (mod 2)}\,\!</math>

[[Image:doe7.22.png|center|200px| Allocation of treatments to two blocks for the <math>2^4</math> design in the example by confounding interaction of <math>ABCD</math> with the blocks.]]

Therefore, <math>ab\,\!</math> should be assigned to Block 2 or the second operator. Other treatments can be allocated to the two operators in a similar manner to arrive at the allocation scheme shown in the figure below.
In DOE++, to confound the <math>ABCD\,\!</math> interaction for the <math>{2}^{4}\,\!</math> design into two blocks, the number of blocks are specified as shown in the figure below. Then the interaction <math>ABCD\,\!</math> is entered in the Block Generator window (second following figure) which is available using the Block Generator button in the following figure. The design generated by DOE++ is shown in the third of the following figures. This design matches the allocation scheme of the preceding figure.

[[Image:doe7_23.png|center|700px| Adding block properties for the experiment in the example.|link=]]

[[Image:doe7_24.png|center|700px|Specifying the interaction ABCD as the interaction to be confounded with the blocks for the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

[[Image:doe7_25.png|center|800px|Two block design for the experiment in the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

For the analysis of this design, the sum of squares for all effects are calculated assuming no blocking. Then, to account for blocking, the sum of squares corresponding to the <math>ABCD\,\!</math> interaction is considered as the sum of squares due to blocks and <math>ABCD\,\!</math>. In DOE++ this is done by displaying this sum of squares as the sum of squares due to the blocks. This is shown in the following figure where the sum of squares in question is obtained as 72.25 and is displayed against Block. The interaction ABCD, which is confounded with the blocks, is not displayed. Since the design is unreplicated, any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_26.png|center|793px|ANOVA table for the experiment of the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

===Unreplicated 2''k'' Designs in 2''p'' Blocks===

A single replicate of the <math>{2}^{k}\,\!</math> design can be run in up to <math>{2}^{p}\,\!</math> blocks where <math>p<k\,\!</math>. The number of effects confounded with the blocks equals the degrees of freedom associated with the block effect.

If two blocks are used (the block effect has two levels), then one (<math>2-1=1)\,\!</math> effect is confounded with the blocks. If four blocks are used, then three (<math>4-1=3\,\!</math>) effects are confounded with the blocks and so on. For example an unreplicated <math>{2}^{4}\,\!</math> design may be confounded in <math>{2}^{2}\,\!</math> (four) blocks using two contrasts, <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}}\,\!</math>. Let <math>AC\,\!</math> and <math>BD\,\!</math> be the effects to be confounded with the blocks. Corresponding to these two effects, the contrasts are respectively:

::<math>\begin{align}
& {{L}_{1}}= & {{q}_{1}}+{{q}_{3}} \\
& {{L}_{2}}= & {{q}_{2}}+{{q}_{4}}
\end{align}\,\!</math>

Based on the values of <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}},\,\!</math> the treatments can be assigned to the four blocks as follows:

<center><math>\begin{matrix}
\text{Block 4} & {} & \text{Block 3} & {} & \text{Block 2} & {} & \text{Block 1} \\
{{L}_{1}}=0,{{L}_{2}}=0 & {} & {{L}_{1}}=1,{{L}_{2}}=0 & {} & {{L}_{1}}=0,{{L}_{2}}=1 & {} & {{L}_{1}}=1,{{L}_{2}}=1 \\
{} & {} & {} & {} & {} & {} & {} \\
\left[ \begin{matrix}
(1) \\
ac \\
bd \\
abcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
c \\
abd \\
bcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
b \\
abc \\
d \\
acd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
ab \\
bc \\
ad \\
cd \\
\end{matrix} \right] \\
\end{matrix}\,\!</math></center>

Since the block effect has three degrees of freedom, three effects are confounded with the block effect. In addition to <math>AC\,\!</math> and <math>BD\,\!</math>, the third effect confounded with the block effect is their generalized interaction, <math>(AC)(BD)=ABCD\,\!</math>.
In general, when an unreplicated <math>{2}^{k}\,\!</math> design is confounded in <math>{2}^{p}\,\!</math> blocks, <math>p\,\!</math> contrasts are needed (<math>{{L}_{1}},{{L}_{2}}...{{L}_{p}}\,\!</math>). <math>p\,\!</math> effects are selected to define these contrasts such that none of these effects are the generalized interaction of the others. The <math>{2}^{p}\,\!</math> blocks can then be assigned the treatments using the <math>p\,\!</math> contrasts. <math>{{2}^{p}}-(p+1)\,\!</math> effects, that are also confounded with the blocks, are then obtained as the generalized interaction of the <math>p\,\!</math> effects. In the statistical analysis of these designs, the sum of squares are computed as if no blocking were used. Then the block sum of squares is obtained by adding the sum of squares for all the effects confounded with the blocks.

====Example: 2 Level Factorial Design with Four Blocks====

This example illustrates how DOE++ obtains the sum of squares when treatments for an unreplicated <math>{2}^{k}\,\!</math> design are allocated among four blocks. Consider again the unreplicated <math>{2}^{4}\,\!</math> design used to investigate the defects in automobile vinyl panels presented in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments needed to complete the experiment were run by four operators. Therefore, there are four blocks. Assume that the treatments were allocated to the blocks using the generators mentioned in the previous section, i.e., treatments were allocated among the four operators by confounding the effects, <math>AC\,\!</math> and <math>BD,\,\!</math> with the blocks. These effects can be specified as Block Generators as shown in the following figure. (The generalized interaction of these two effects, interaction <math>ABCD\,\!</math>, will also get confounded with the blocks.) The resulting design is shown in the second following figure and matches the allocation scheme obtained in the previous section.

[[Image:doe7_27.png|center|700px|Specifying the interactions AC and BD as block generators for the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

The sum of squares in this case can be obtained by calculating the sum of squares for each of the effects assuming there is no blocking. Once the individual sum of squares have been obtained, the block sum of squares can be calculated. The block sum of squares is the sum of the sum of squares of effects, <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, since these effects are confounded with the block effect. As shown in the second following figure, this sum of squares is 92.25 and is displayed against Block. The interactions <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, which are confounded with the blocks, are not displayed. Since the present design is unreplicated any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_28.png|center|800px|Design for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

[[Image:doe7_29.png|center|793px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

==Variability Analysis==

For replicated two level factorial experiments, DOE++ provides the option of conducting variability analysis (using the Variability Analysis icon under the Data menu). The analysis is used to identify the treatment that results in the least amount of variation in the product or process being investigated. Variability analysis is conducted by treating the standard deviation of the response for each treatment of the experiment as an additional response. The standard deviation for a treatment is obtained by using the replicated response values at that treatment run. As an example, consider the <math>{2}^{3}\,\!</math> design shown in the following figure where each run is replicated four times. A variability analysis can be conducted for this design. DOE++ calculates eight standard deviation values corresponding to each treatment of the design (see second following figure). Then, the design is analyzed as an unreplicated <math>{2}^{3}\,\!</math> design with the standard deviations (displayed as Y Standard Deviation. in second following figure) as the response. The normal probability plot of effects identifies <math>AC\,\!</math> as the effect that influences variability (see third figure following). Based on the effect coefficients obtained in the fourth figure following, the model for Y Std. is:

::<math>\begin{align}
& \text{Y Std}\text{.}= & 0.6779+0.2491\cdot AC \\
& = & 0.6779+0.2491{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

Based on the model, the experimenter has two choices to minimize variability (by minimizing Y Std.). The first choice is that <math>{{x}_{1}}\,\!</math> should be <math>1\,\!</math> (i.e., <math>A\,\!</math> should be set at the high level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the low level). The second choice is that <math>{{x}_{1}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>A\,\!</math> should be set at the low level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the high level). The experimenter can select the most feasible choice.

[[Image:doe7.30.png|center|391px|A <math>2^3\,\!</math> design with four replicated response values that can be used to conduct a variability analysis.]]
 

[[Image:doe7_31.png|center|727px|Variability analysis in DOE++.|link=]]
 

[[Image:doe7_32.png|center|650px|Normal probability plot of effects for the variability analysis example.|link=]]
 

[[Image:doe7_33.png|center|727px|Effect coefficients for the variability analysis example.|link=]]

 

==Two Level Fractional Factorial Designs==

As the number of factors in a two level factorial design increases, the number of runs for even a single replicate of the <math>{2}^{k}\,\!</math> design becomes very large. For example, a single replicate of an eight factor two level experiment would require 256 runs. Fractional factorial designs can be used in these cases to draw out valuable conclusions from fewer runs. The basis of fractional factorial designs is the ''sparsity of effects'' principle.[[DOE_References|[Wu, 2000]]] The principle states that, most of the time, responses are affected by a small number of main effects and lower order interactions, while higher order interactions are relatively unimportant. Fractional factorial designs are used as screening experiments during the initial stages of experimentation. At these stages, a large number of factors have to be investigated and the focus is on the main effects and two factor interactions. These designs obtain information about main effects and lower order interactions with fewer experiment runs by confounding these effects with unimportant higher order interactions. As an example, consider a <math>{2}^{8}\,\!</math> design that requires 256 runs. This design allows for the investigation of 8 main effects and 28 two factor interactions. However, 219 degrees of freedom are devoted to three factor or higher order interactions. This full factorial design can prove to be very inefficient when these higher order interactions can be assumed to be unimportant. Instead, a fractional design can be used here to identify the important factors that can then be investigated more thoroughly in subsequent experiments. In unreplicated fractional factorial designs, no degrees of freedom are available to calculate the error sum of squares and the techniques mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] should be employed for the analysis of these designs.

==Half-fraction Designs==

A half-fraction of the <math>{2}^{k}\,\!</math> design involves running only half of the treatments of the full factorial design. For example, consider a <math>{2}^{3}\,\!</math> design that requires eight runs in all. The design matrix for this design is shown in the figure (a) below. A half-fraction of this design is the design in which only four of the eight treatments are run. The fraction is denoted as <math>{2}^{3-1}\,\!</math> with the "<math>-1\,\!</math>" in the index denoting a half-fraction. Assume that the treatments chosen for the half-fraction design are the ones where the interaction <math>ABC\,\!</math> is at the high level (i.e., only those rows are chosen from the following figure (a) where the column for <math>ABC\,\!</math> has entries of 1). The resulting <math>{2}^{3-1}\,\!</math> design has a design matrix as shown in figure (b) below.

[[Image:doe7.34.png|center|330px|Half-fractions of the <math>2^3\,\!</math> design. (a) shows the full factorial <math>2^3\,\!</math> design, (b) shows the <math>2^{3-1}\,\!</math> design with the defining relation <math>I=ABC\,\!</math> and (c) shows the <math>{2}^{3-1}\,\!</math> design with the defining relation <math>I=-ABC\,\!</math>.]]

In the <math>{2}^{3-1}\,\!</math> design of figure (b), since the interaction <math>ABC\,\!</math> is always included at the same level (the high level represented by 1), it is not possible to measure this interaction effect. The effect, <math>ABC\,\!</math>, is called the ''generator'' or ''word'' for this design. It can be noted that, in the design matrix of the following figure (b), the column corresponding to the intercept, <math>I\,\!</math>, and column corresponding to the interaction <math>ABC\,\!</math>, are identical. The identical columns are written as <math>I=ABC\,\!</math> and this equation is called the ''defining relation'' for the design. In DOE++, the present <math>{2}^{3-1}\,\!</math> design can be obtained by specifying the design properties as shown in the following figure.

[[Image:doe7_35.png|center|700px|Design properties for the <math>2^{3-1}\,\!</math> design.|link=]]

The defining relation, <math>I=ABC\,\!</math>, is entered in the Fraction Generator window as shown next.

[[Image:doe7_36.png|center|700px|Specifying the defining relation for the <math>2^{3-1}\,\!</math> design.|link=]]

Note that in the figure following that, the defining relation is specified as <math>C=AB\,\!</math>. This relation is obtained by multiplying the defining relation, <math>I=ABC\,\!</math>, by the last factor, <math>C\,\!</math>, of the design.

===Calculation of Effects===

Using the four runs of the <math>{2}^{3-1}\,\!</math> design in figure (b) discussed above, the main effects can be calculated as follows:

::<math>\begin{align}
& A= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& B= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& C= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

where <math>a\,\!</math>, <math>b\,\!</math>, <math>c\,\!</math> and <math>abc\,\!</math> are the treatments included in the <math>{2}^{3-1}\,\!</math> design.

Similarly, the two factor interactions can also be obtained as:

::<math>\begin{align}
& BC= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& AC= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& AB= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

The equations for <math>A\,\!</math> and <math>BC\,\!</math> above result in the same effect values showing that effects <math>A\,\!</math> and <math>BC\,\!</math> are confounded in the present <math>{2}^{3-1}\,\!</math> design. Thus, the quantity, <math>\tfrac{1}{2}(a-b-c+abc),\,\!</math> estimates <math>A+BC\,\!</math> (i.e., both the main effect <math>A\,\!</math> and the two-factor interaction <math>BC\,\!</math>). The effects, <math>A\,\!</math> and <math>BC,\,\!</math> are called ''aliases''. From the remaining equations given above, it can be seen that the other aliases for this design are <math>B\,\!</math> and <math>AC\,\!</math>, and <math>C\,\!</math> and <math>AB\,\!</math>. Therefore, the equations to calculate the effects in the present <math>{2}^{3-1}\,\!</math> design can be written as follows:

::<math>\begin{align}
& A+BC= & \frac{1}{2}(a-b-c+abc) \\
& B+AC= & \frac{1}{2}(-a+b-c+abc) \\
& C+AB= & \frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

===Calculation of Aliases===

Aliases for a fractional factorial design can be obtained using the defining relation for the design. The defining relation for the present <math>{2}^{3-1}\,\!</math> design is:

::<math>I=ABC\,\!</math>

Multiplying both sides of the previous equation by the main effect, <math>A,\,\!</math> gives the alias effect of <math>A\,\!</math> :

::<math>\begin{align}
& A\cdot I= & A\cdot ABC \\
& A= & {{A}^{2}}BC \\
& A= & BC
\end{align}\,\!</math>

Note that in calculating the alias effects, any effect multiplied by <math>I\,\!</math> remains the same (<math>A\cdot I=A\,\!</math>), while an effect multiplied by itself results in <math>I\,\!</math> (<math>{{A}^{2}}=I\,\!</math>). Other aliases can also be obtained:

::<math>\begin{align}
& B\cdot I= & B\cdot ABC \\
& B= & A{{B}^{2}}C \\
& B= & AC
\end{align}\,\!</math>

:and:

::<math>\begin{align}
& C\cdot I= & C\cdot ABC \\
& C= & AB{{C}^{2}} \\
& C= & AB
\end{align}\,\!</math>

===Fold-over Design===

If it can be assumed for this design that the two-factor interactions are unimportant, then in the absence of <math>BC\,\!</math>, <math>AC\,\!</math> and <math>AB\,\!</math>, the equations for (A+BC), (B+AC) and (C+AB) can be used to estimate the main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. However, if such an assumption is not applicable, then to uncouple the main effects from their two factor aliases, the alternate fraction that contains runs having <math>ABC\,\!</math> at the lower level should be run. The design matrix for this design is shown in the preceding figure (c). The defining relation for this design is <math>I=-ABC\,\!</math> because the four runs for this design are obtained by selecting the rows of the preceding figure (a) for which the value of the <math>ABC\,\!</math> column is <math>-1\,\!</math>. The aliases for this fraction can be obtained as explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]] as <math>A=-BC\,\!</math>, <math>B=-AC\,\!</math> and <math>C=-AB\,\!</math>. The effects for this design can be calculated as:

::<math>\begin{align}
& A-BC= & \frac{1}{2}(ab+ac-(1)-bc) \\
& B-AC= & \frac{1}{2}(ab-ac+(1)-bc) \\
& C-AB= & \frac{1}{2}(-ab+ac-(1)+bc)
\end{align}\,\!</math>

These equations can be combined with the equations for (A+BC), (B+AC) and (C+AB) to obtain the de-aliased main effects and two factor interactions. For example, adding equations (A+BC) and (A-BC) returns the main effect <math>A\,\!</math>.

::<math>\begin{align}
& 2A= & \frac{1}{2}(a-b-c+abc)+ \\
& & \frac{1}{2}(ab+ac-(1)-bc)
\end{align}\,\!</math>

The process of augmenting a fractional factorial design by a second fraction of the same size by simply reversing the signs (of all effect columns except <math>I\,\!</math>) is called ''folding over''. The combined design is referred to as a ''fold-over design''.

==Quarter and Smaller Fraction Designs==

At times, the number of runs even for a half-fraction design are very large. In these cases, smaller fractions are used. A quarter-fraction design, denoted as <math>{2}^{k-2}\,\!</math>, consists of a fourth of the runs of the full factorial design. Quarter-fraction designs require two defining relations. The first defining relation returns the half-fraction or the <math>{2}^{k-1}\,\!</math> design. The second defining relation selects half of the runs of the <math>{2}^{k-1}\,\!</math> design to give the quarter-fraction. For example, consider the <math>{2}^{4}\,\!</math> design. To obtain a <math>{2}^{4-2}\,\!</math> design from this design, first a half-fraction of this design is obtained by using a defining relation. Assume that the defining relation used is <math>I=ABCD\,\!</math>. The design matrix for the resulting <math>{2}^{4-1}\,\!</math> design is shown in figure (a) below. Now, a quarter-fraction can be obtained from the <math>{2}^{4-1}\,\!</math> design shown in figure (a) below using a second defining relation <math>I=AD\,\!</math>. The resulting <math>{2}^{4-2}\,\!</math> design obtained is shown in figure (b) below.

[[Image:doe7.37.png|center|465px|Fractions of the <math>2^4\,\!</math> design - Figure (a) shows the <math>2^{4-1}</math> design with the defining relation <math>I=ABCD\,\!</math> and (b) shows the <math>2^{4-2}\,\!</math> design with the defining relation <math>I=ABCD=AD=BC\,\!</math>.]]

The complete defining relation for this <math>{2}^{4-2}\,\!</math> design is:

::<math>I=ABCD=AD=BC\,\!</math>

Note that the effect, <math>BC,\,\!</math> in the defining relation is the generalized interaction of <math>ABCD\,\!</math> and <math>AD\,\!</math> and is obtained using <math>(ABCD)(AD)={{A}^{2}}BC{{D}^{2}}=BC\,\!</math>. In general, a <math>{2}^{k-p}\,\!</math> fractional factorial design requires <math>p\,\!</math> independent generators. The defining relation for the design consists of the <math>p\,\!</math> independent generators and their <math>{2}^{p}\,\!</math> - (<math>p\,\!</math> +1) generalized interactions.

===Calculation of Aliases===

The alias structure for the present <math>{2}^{4-2}\,\!</math> design can be obtained using the defining relation of equation (I=ABCD=AD=BC) following the procedure explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]]. For example, multiplying the defining relation by <math>A\,\!</math> returns the effects aliased with the main effect, <math>A\,\!</math>, as follows:

::<math>\begin{align}
& A\cdot I= & A\cdot ABCD=A\cdot AD=A\cdot BC \\
& A= & {{A}^{2}}BCD={{A}^{2}}D=ABC \\
& A= & BCD=D=ABC
\end{align}\,\!</math>

Therefore, in the present <math>{2}^{4-2}\,\!</math> design, it is not possible to distinguish between effects <math>A\,\!</math>, <math>D\,\!</math>, <math>BCD\,\!</math> and <math>ABC\,\!</math>. Similarly, multiplying the defining relation by <math>B\,\!</math> and <math>AB\,\!</math> returns the effects that are aliased with these effects:

::<math>\begin{align}
& B= & ACD=ABD=C \\
& AB= & CD=AD=AC
\end{align}\,\!</math>

Other aliases can be obtained in a similar way. It can be seen that each effect in this design has three aliases. In general, each effect in a <math>{2}^{k-p}\,\!</math> design has <math>{2}^{p-1}\,\!</math> aliases.
The aliases for the <math>{2}^{4-2}\,\!</math> design show that in this design the main effects are aliased with each other (<math>A\,\!</math> is aliased with <math>D\,\!</math> and <math>B\,\!</math> is aliased with <math>C\,\!</math>). Therefore, this design is not a useful design and is not available in DOE++. It is important to ensure that main effects and lower order interactions of interest are not aliased in a fractional factorial design. This is known by looking at the resolution of the fractional factorial design.

==Design Resolution==

The resolution of a fractional factorial design is defined as the number of factors in the lowest order effect in the defining relation. For example, in the defining relation <math>I=ABCD=AD=BC\,\!</math> of the previous <math>{2}^{4-2}\,\!</math> design, the lowest-order effect is either <math>AD\,\!</math> or <math>BC,\,\!</math> containing two factors. Therefore, the resolution of this design is equal to two. The resolution of a fractional factorial design is represented using Roman numerals. For example, the previously mentioned <math>{2}^{4-2}\,\!</math> design with a resolution of two can be represented as 2 <math>_{\text{II}}^{4-2}\,\!</math>. The resolution provides information about the confounding in the design as explained next:
 
 
#'''Resolution III Designs''' In these designs, the lowest order effect in the defining relation has three factors (e.g., a <math>{2}^{5-2}\,\!</math> design with the defining relation <math>I=ABDE=ABC=CDE\,\!</math>). In resolution III designs, no main effects are aliased with any other main effects, but main effects are aliased with two factor interactions. In addition, some two factor interactions are aliased with each other.
#'''Resolution IV Designs''' In these designs, the lowest order effect in the defining relation has four factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABDE\,\!</math>). In resolution IV designs, no main effects are aliased with any other main effects or two factor interactions. However, some main effects are aliased with three factor interactions and the two factor interactions are aliased with each other.
#'''Resolution V Designs''' In these designs the lowest order effect in the defining relation has five factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABCDE\,\!</math>). In resolution V designs, no main effects or two factor interactions are aliased with any other main effects or two factor interactions. However, some main effects are aliased with four factor interactions and the two factor interactions are aliased with three factor interactions.

Fractional factorial designs with the highest resolution possible should be selected because the higher the resolution of the design, the less severe the degree of confounding. In general, designs with a resolution less than III are never used because in these designs some of the main effects are aliased with each other. The table below shows fractional factorial designs with the highest available resolution for three to ten factor designs along with their defining relations.

[[Image:doet7.3.png|center|474px|Highest resolution designs available for fractional factorial designs with 3 to 10 factors.]]

All of the two level fractional factorial designs available in DOE++ are shown next.

[[Image:doe7.38.png|center|471px|Two level fractional factorial designs available in DOE++ and their resolutions.]]

===Minimum Aberration Designs===
At times, different designs with the same resolution but different aliasing may be available. The best design to select in such a case is the minimum aberration design. For example, all <math>{2}^{7-2}\,\!</math> designs in the fourth table have a resolution of four (since the generator with the minimum number of factors in each design has four factors). Design <math>1\,\!</math> has three generators of length four (<math>ABCF,\,\!</math> <math>BCDG,\,\!</math> <math>ADFG\,\!</math>). Design <math>2\,\!</math> has two generators of length four (<math>ABCF,\,\!</math> <math>ADEG\,\!</math>). Design <math>3\,\!</math> has one generator of length four (<math>CEFG\,\!</math>). Therefore, design <math>3\,\!</math> has the least number of generators with the minimum length of four. Design <math>3\,\!</math> is called the minimum aberration design. It can be seen that the alias structure for design <math>3\,\!</math> is less involved compared to the other designs. For details refer to [[DOE_References|[Wu, 2000]]].

[[Image:doet7.4.png|center|432px|Three <math>2_{IV}^{7-2}\,\!</math> designs with different defining relations.]]

====Example====

The design of an automobile fuel cone is thought to be affected by six factors in the manufacturing process: cavity temperature (factor <math>A\,\!</math>), core temperature (factor <math>B\,\!</math>), melt temperature (factor <math>C\,\!</math>), hold pressure (factor <math>D\,\!</math>), injection speed (factor <math>E\,\!</math>) and cool time (factor <math>F\,\!</math>). The manufacturer of the fuel cone is unable to run the <math>{2}^{6}=64\,\!</math> runs required to complete one replicate for a two level full factorial experiment with six factors. Instead, they decide to run a fractional factorial design. Considering that three factor and higher order interactions are likely to be inactive, the manufacturer selects a <math>{2}^{6-2}\,\!</math> design that will require only 16 runs. The manufacturer chooses the resolution IV design which will ensure that all main effects are free from aliasing (assuming three factor and higher order interactions are absent). However, in this design the two factor interactions may be aliased with each other. It is decided that, if important two factor interactions are found to be present, additional experiment trials may be conducted to separate the aliased effects. The performance of the fuel cone is measured on a scale of 1 to 15. In DOE++, the design for this experiment is set up using the properties shown in the following figure. The Fraction Generators for the design, <math>E=ABC\,\!</math> and <math>F=BCD\,\!</math>, are the same as the defaults used in DOE++. The resulting <math>{2}^{6-2}\,\!</math> design and the corresponding response values are shown in the following two figures.

[[Image:doe7_39.png|center|644px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

[[Image:doe7_40.png|center|534px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The complete alias structure for the 2 <math>_{\text{IV}}^{6-2}\,\!</math> design is shown next.

<center><math>I=ABCE=ADEF=BCDF\,\!</math></center>

 
<center><math>\begin{align}
& A= & BCE=DEF=ABCDF \\
& B= & ACE=CDF=ABDEF \\
& C= & ABE=BDF=ACDEF \\
& D= & AEF=BCF=ABCDE \\
& E= & ABC=ADF=BCDEF \\
& F= & ADE=BCD=ABCEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& AB= & CE=ACDF=BDEF \\
& AC= & BE=ABDF=CDEF \\
& AD= & EF=ABCF=BCDE \\
& AE= & BC=DF=ABCDEF \\
& AF= & DE=ABCD=BCEF \\
& BD= & CF=ABEF=ACDE \\
& BF= & CD=ABDE=ACEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& ABD= & ACF=BEF=CDE \\
& ABF= & ACD=BDE=CEF
\end{align}\,\!</math></center>

In DOE++, the alias structure is displayed in the Design Summary and as part of the Design Evaluation result, as shown next:

[[Image:doe7_41.png|center|700px|Alias structure for the experiment design in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The normal probability plot of effects for this unreplicated design shows the main effects of factors <math>C\,\!</math> and <math>D\,\!</math> and the interaction effect, <math>BF\,\!</math>, to be significant (see the following figure).

[[Image:doe7_42.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

From the alias structure, it can be seen that for the present design interaction effect, <math>BF,\,\!</math> is confounded with <math>CD\,\!</math>. Therefore, the actual source of this effect cannot be known on the basis of the present experiment. However because neither factor <math>B\,\!</math> nor <math>F\,\!</math> is found to be significant there is an indication the observed effect is likely due to interaction, <math>CD\,\!</math>. To confirm this, a follow-up <math>{2}^{2}\,\!</math> experiment is run involving only factors <math>B\,\!</math> and <math>F\,\!</math>. The interaction, <math>BF\,\!</math>, is found to be inactive, leading to the conclusion that the interaction effect in the original experiment is effect, <math>CD\,\!</math>. Given these results, the fitted regression model for the fuel cone design as per the coefficients obtained from DOE++ is shown next.

::<math>\hat{y}=7.6875+C+2\cdot D+2.1875\cdot CD\,\!</math>

[[Image:doe7_43.png|center|700px|Effect coefficients for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

==Projection==

Projection refers to the reduction of a fractional factorial design to a full factorial design by dropping out some of the factors of the design. Any fractional factorial design of resolution, <math>R,\,\!</math> can be reduced to complete factorial designs in any subset of <math>R-1\,\!</math> factors. For example, consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. The resolution of this design is four. Therefore, this design can be reduced to full factorial designs in any three (<math>4-1=3\,\!</math>) of the original seven factors (by dropping the remaining four of factors). Further, a fractional factorial design can also be reduced to a full factorial design in any <math>R\,\!</math> of the original factors, as long as these <math>R\,\!</math> factors are not part of the generator in the defining relation. Again consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. This design can be reduced to a full factorial design in four factors provided these four factors do not appear together as a generator in the defining relation. The complete defining relation for this design is:

::<math>\begin{align}
& I= & ABCE=BCDF=ACDG=ADEF=ABFG=BDEG=CEFG
\end{align}\,\!</math>

Therefore, there are seven four factor combinations out of the 35 (<math>(_{7}^{4})=35\,\!</math>) possible four-factor combinations that are used as generators in the defining relation. The designs with the remaining 28 four factor combinations would be full factorial 16-run designs. For example, factors <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math> do not occur as a generator in the defining relation of the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. If the remaining factors, <math>E\,\!</math>, <math>F\,\!</math> and <math>G\,\!</math>, are dropped, the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design will reduce to a full factorial design in <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math>.

==Resolution III Designs==

At times, the factors to be investigated in screening experiments are so large that even running a fractional factorial design is impractical. This can be partially solved by using resolution III fractional factorial designs in the cases where three factor and higher order interactions can be assumed to be unimportant. Resolution III designs, such as the 2 <math>_{\text{III}}^{3-1}\,\!</math> design, can be used to estimate <math>k\,\!</math> main effects using just <math>k+1\,\!</math> runs. In these designs, the main effects are aliased with two factor interactions. Once the results from these designs are obtained, and knowing that three factor and higher order interactions are unimportant, the experimenter can decide if there is a need to run a fold-over design to de-alias the main effects from the two factor interactions. Thus, the 2 <math>_{\text{III}}^{3-1}\,\!</math> design can be used to investigate three factors in four runs, the 2 <math>_{\text{III}}^{7-4}\,\!</math> design can be used to investigate seven factors in eight runs, the 2 <math>_{\text{III}}^{15-11}\,\!</math> design can be used to investigate fifteen factors in sixteen runs and so on.

====Example====

{{:Resolution_III_Design_Example}}

==Alias Matrix==
In [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction designs]] and [[Two_Level_Factorial_Experiments#Quarter_and_Smaller_Fraction_Designs| Quarter and Smaller Fraction Designs]], the alias structure for fractional factorial designs was obtained using the defining relation. However, this method of obtaining the alias structure is not very efficient when the alias structure is very complex or when partial aliasing is involved. One of the ways to obtain the alias structure for any design, regardless of its complexity, is to use the alias matrix. The alias matrix for a design is calculated using <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> where <math>{{X}_{1}}\,\!</math> is the portion of the design matrix, <math>X,\,\!</math> that contains the effects for which the aliases need to be calculated, and <math>{{X}_{2}}\,\!</math> contains the remaining columns of the design matrix, other than those included in <math>{{X}_{1}}\,\!</math>.

To illustrate the use of the alias matrix, consider the design matrix for the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design (using the defining relation <math>I=ABCD\,\!</math>) shown next:

[[Image:Chapter7__879.png|center|link=]]

The alias structure for this design can be obtained by defining <math>{{X}_{1}}\,\!</math> using eight columns since the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design estimates eight effects. If the first eight columns of <math>X\,\!</math> are used then <math>{{X}_{1}}\,\!</math> is:

[[Image:Chapter7__884.png|center|link=]]

<math>{{X}_{2}}\,\!</math> is obtained using the remaining columns as:

[[Image:Chapter7__886.png|center|link=]]

Then the alias matrix <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> is:

[[Image:Chapter7__888.png|center|link=]]

The alias relations can be easily obtained by observing the alias matrix as:

<center><math>\begin{align}
& I= & ABCD \\
& A= & BCD \\
& B= & ACD \\
& AB= & CD \\
& C= & ABD \\
& AC= & BD \\
& BC= & AD \\
& D= & ABC
\end{align}\,\!</math></center>

Two Level Factorial Experiments

2017-08-10T17:26:42Z

Richard House: /* 2k Designs */

{{Template:Doebook|8}}
Two level factorial experiments are factorial experiments in which each factor is investigated at only two levels. The early stages of experimentation usually involve the investigation of a large number of potential factors to discover the "vital few" factors. Two level factorial experiments are used during these stages to quickly filter out unwanted effects so that attention can then be focused on the important ones.

==2''k'' Designs==
The factorial experiments, where all combination of the levels of the factors are run, are usually referred to as ''full factorial experiments''. Full factorial two level experiments are also referred to as <math>{2}^{k}\,\!</math> designs where <math>k\,\!</math> denotes the number of factors being investigated in the experiment. In Weibull++ DOE folios, these designs are referred to as 2 Level Factorial Designs as shown in the figure below.

[[Image:doe7_1.png|center|487px|Selection of full factorial experiments with two levels in Weibull++.|link=]]

A full factorial two level design with <math>k\,\!</math> factors requires <math>{{2}^{k}}\,\!</math> runs for a single replicate. For example, a two level experiment with three factors will require <math>2\times 2\times 2={{2}^{3}}=8\,\!</math> runs. The choice of the two levels of factors used in two level experiments depends on the factor; some factors naturally have two levels. For example, if gender is a factor, then male and female are the two levels. For other factors, the limits of the range of interest are usually used. For example, if temperature is a factor that varies from <math>{45}^{o}C\,\!</math> to <math>{90}^{o}C\,\!</math>, then the two levels used in the <math>{2}^{k}\,\!</math> design for this factor would be <math>{45}^{o}\,\!C\,\!</math> and <math>{90}^{o}\,\!C\,\!</math>.

The two levels of the factor in the <math>{2}^{k}\,\!</math> design are usually represented as <math>-1\,\!</math> (for the first level) and <math>1\,\!</math> (for the second level). Note that this representation is reversed from the coding used in [[General Full Factorial Designs]] for the indicator variables that represent two level factors in ANOVA models. For ANOVA models, the first level of the factor was represented using a value of <math>1\,\!</math> for the indicator variable, while the second level was represented using a value of <math>-1\,\!</math>. For details on the notation used for two level experiments refer to [[Two_Level_Factorial_Experiments#Notation| Notation]].

===The 22 Design===

The simplest of the two level factorial experiments is the <math>{2}^{2}\,\!</math> design where two factors (say factor <math>A\,\!</math> and factor <math>B\,\!</math>) are investigated at two levels. A single replicate of this design will require four runs (<math>{{2}^{2}}=2\times 2=4\,\!</math>) The effects investigated by this design are the two main effects, <math>A\,\!</math> and <math>B,\,\!</math> and the interaction effect <math>AB\,\!</math>. The treatments for this design are shown in figure (a) below. In figure (a), letters are used to represent the treatments. The presence of a letter indicates the high level of the corresponding factor and the absence indicates the low level. For example, (1) represents the treatment combination where all factors involved are at the low level or the level represented by <math>-1\,\!</math> ; <math>a\,\!</math> represents the treatment combination where factor <math>A\,\!</math> is at the high level or the level of <math>1\,\!</math>, while the remaining factors (in this case, factor <math>B\,\!</math>) are at the low level or the level of <math>-1\,\!</math>. Similarly, <math>b\,\!</math> represents the treatment combination where factor <math>B\,\!</math> is at the high level or the level of <math>1\,\!</math>, while factor <math>A\,\!</math> is at the low level and <math>ab\,\!</math> represents the treatment combination where factors <math>A\,\!</math> and <math>B\,\!</math> are at the high level or the level of the 1. Figure (b) below shows the design matrix for the <math>{2}^{2}\,\!</math> design. It can be noted that the sum of the terms resulting from the product of any two columns of the design matrix is zero. As a result the <math>{2}^{2}\,\!</math> design is an ''orthogonal design''. In fact, all <math>{2}^{k}\,\!</math> designs are orthogonal designs. This property of the <math>{2}^{k}\,\!</math> designs offers a great advantage in the analysis because of the simplifications that result from orthogonality. These simplifications are explained later on in this chapter.
The <math>{2}^{2}\,\!</math> design can also be represented geometrically using a square with the four treatment combinations lying at the four corners, as shown in figure (c) below.

[[Image:doe7.2.png|center|400px|The <math>2^2\,\!</math> design. Figure (a) displays the experiment design, (b) displays the design matrix and (c) displays the geometric representation for the design. In Figure (b), the column names I, A, B and AB are used. Column I represents the intercept term. Columns A and B represent the respective factor settings. Column AB represents the interaction and is the product of columns A and B.]]
 

===The 23 Design===

The <math>{2}^{3}\,\!</math> design is a two level factorial experiment design with three factors (say factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>). This design tests three (<math>k=3\,\!</math>) main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math> ; three (<math>(_{2}^{k})=\,\!</math> <math>(_{2}^{3})=3\,\!</math>) two factor interaction effects, <math>AB\,\!</math>, <math>BC\,\!</math>, <math>AC\,\!</math> ; and one (<math>(_{3}^{k})=\,\!</math> <math>(_{3}^{3})=1\,\!</math>) three factor interaction effect, <math>ABC\,\!</math>. The design requires eight runs per replicate. The eight treatment combinations corresponding to these runs are <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math>, <math>ab\,\!</math>, <math>c\,\!</math>, <math>ac\,\!</math>, <math>bc\,\!</math> and <math>abc\,\!</math>. Note that the treatment combinations are written in such an order that factors are introduced one by one with each new factor being combined with the preceding terms. This order of writing the treatments is called the ''standard order'' or ''Yates' order''. The <math>{2}^{3}\,\!</math> design is shown in figure (a) below. The design matrix for the <math>{2}^{3}\,\!</math> design is shown in figure (b). The design matrix can be constructed by following the standard order for the treatment combinations to obtain the columns for the main effects and then multiplying the main effects columns to obtain the interaction columns.

[[Image:doe7.3.png|center|324px|The <math>2^3\,\!</math> design. Figure (a) shows the experiment design and (b) shows the design matrix.]]

[[Image:doe7.4.png|center|290px|Geometric representation of the <math>2^3\,\!</math> design.]]

The <math>{2}^{3}\,\!</math> design can also be represented geometrically using a cube with the eight treatment combinations lying at the eight corners as shown in the figure above.

==Analysis of 2''k'' Designs==

The <math>{2}^{k}\,\!</math> designs are a special category of the factorial experiments where all the factors are at two levels. The fact that these designs contain factors at only two levels and are orthogonal greatly simplifies their analysis even when the number of factors is large. The use of <math>{2}^{k}\,\!</math> designs in investigating a large number of factors calls for a revision of the notation used previously for the ANOVA models. The case for revised notation is made stronger by the fact that the ANOVA and multiple linear regression models are identical for <math>{2}^{k}\,\!</math> designs because all factors are only at two levels. Therefore, the notation of the regression models is applied to the ANOVA models for these designs, as explained next.

===Notation===

Based on the notation used in [[General Full Factorial Designs]], the ANOVA model for a two level factorial experiment with three factors would be as follows:

::<math>\begin{align}
& Y= & \mu +{{\tau }_{1}}\cdot {{x}_{1}}+{{\delta }_{1}}\cdot {{x}_{2}}+{{(\tau \delta )}_{11}}\cdot {{x}_{1}}{{x}_{2}}+{{\gamma }_{1}}\cdot {{x}_{3}} \\
& & +{{(\tau \gamma )}_{11}}\cdot {{x}_{1}}{{x}_{3}}+{{(\delta \gamma )}_{11}}\cdot {{x}_{2}}{{x}_{3}}+{{(\tau \delta \gamma )}_{111}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where:
 
:• <math>\mu \,\!</math> represents the overall mean
:• <math>{{\tau }_{1}}\,\!</math> represents the independent effect of the first factor (factor <math>A\,\!</math>) out of the two effects <math>{{\tau }_{1}}\,\!</math> and <math>{{\tau }_{2}}\,\!</math>
:• <math>{{\delta }_{1}}\,\!</math> represents the independent effect of the second factor (factor <math>B\,\!</math>) out of the two effects <math>{{\delta }_{1}}\,\!</math> and <math>{{\delta }_{2}}\,\!</math>
:• <math>{{(\tau \delta )}_{11}}\,\!</math> represents the independent effect of the interaction <math>AB\,\!</math> out of the other interaction effects
:• <math>{{\gamma }_{1}}\,\!</math> represents the effect of the third factor (factor <math>C\,\!</math>) out of the two effects <math>{{\gamma }_{1}}\,\!</math> and <math>{{\gamma }_{2}}\,\!</math>
:• <math>{{(\tau \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>AC\,\!</math> out of the other interaction effects
:• <math>{{(\delta \gamma )}_{11}}\,\!</math> represents the effect of the interaction <math>BC\,\!</math> out of the other interaction effects
:• <math>{{(\tau \delta \gamma )}_{111}}\,\!</math> represents the effect of the interaction <math>ABC\,\!</math> out of the other interaction effects
and <math>\epsilon \,\!</math> is the random error term.

 
The notation for a linear regression model having three predictor variables with interactions is:

::<math>\begin{align}
& Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& & +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

The notation for the regression model is much more convenient, especially for the case when a large number of higher order interactions are present. In two level experiments, the ANOVA model requires only one indicator variable to represent each factor for both qualitative and quantitative factors. Therefore, the notation for the multiple linear regression model can be applied to the ANOVA model of the experiment that has all the factors at two levels. For example, for the experiment of the ANOVA model given above, <math>{{\beta }_{0}}\,\!</math> can represent the overall mean instead of <math>\mu \,\!</math>, and <math>{{\beta }_{1}}\,\!</math> can represent the independent effect, <math>{{\tau }_{1}}\,\!</math>, of factor <math>A\,\!</math>. Other main effects can be represented in a similar manner. The notation for the interaction effects is much more simplified (e.g., <math>{{\beta }_{123}}\,\!</math> can be used to represent the three factor interaction effect, <math>{{(\tau \beta \gamma )}_{111}}\,\!</math>).

As mentioned earlier, it is important to note that the coding for the indicator variables for the ANOVA models of two level factorial experiments is reversed from the coding followed in [[General Full Factorial Designs]]. Here <math>-1\,\!</math> represents the first level of the factor while <math>1\,\!</math> represents the second level. This is because for a two level factor a single variable is needed to represent the factor for both qualitative and quantitative factors. For quantitative factors, using <math>-1\,\!</math> for the first level (which is the low level) and 1 for the second level (which is the high level) keeps the coding consistent with the numerical value of the factors. The change in coding between the two coding schemes does not affect the analysis except that signs of the estimated effect coefficients will be reversed (i.e., numerical values of <math>{{\hat{\tau }}_{1}}\,\!</math>, obtained based on the coding of [[General Full Factorial Designs]], and <math>{{\hat{\beta }}_{1}}\,\!</math>, obtained based on the new coding, will be the same but their signs would be opposite).

::<math>\begin{align}
& & \text{Factor }A\text{ Coding (two level factor)} \\
& &
\end{align}\,\!</math>

::<math>\begin{matrix}
\text{Previous Coding} & {} & {} & {} & \text{Coding for }{{\text{2}}^{k}}\text{ Designs} \\
{} & {} & {} & {} & {} \\
Effect\text{ }{{\tau }_{1}}\ \ :\ \ {{x}_{1}}=1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{1}}\text{ (or }-{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=-1\text{ } \\
Effect\text{ }{{\tau }_{2}}\ \ :\ \ {{x}_{1}}=-1\text{ } & {} & {} & {} & Effect\text{ }{{\tau }_{2}}\text{ (or }{{\beta }_{1}}\text{)}\ \ :\ \ {{x}_{1}}=1\text{ } \\
\end{matrix}\,\!</math>

In summary, the ANOVA model for the experiments with all factors at two levels is different from the ANOVA models for other experiments in terms of the notation in the following two ways:
 

:• The notation of the regression models is used for the effect coefficients.
:• The coding of the indicator variables is reversed.

===Special Features===

Consider the design matrix, <math>X\,\!</math>, for the <math>{2}^{3}\,\!</math> design discussed above. The (<math>{{X}^{\prime }}X\,\!</math>) <math>^{-1}\,\!</math> matrix is:

<center><math>{{({{X}^{\prime }}X)}^{-1}}=\left[ \begin{matrix}
0.125 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0.125 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0.125 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0.125 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0.125 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0.125 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0.125 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.125 \\
\end{matrix} \right]\,\!</math></center>

Notice that, due to the orthogonal design of the <math>X\,\!</math> matrix, the <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> has been simplified to a diagonal matrix which can be written as:

::<math>\begin{align}
{{({{X}^{\prime }}X)}^{-1}}= & 0.125\cdot I = & \frac{1}{8}\cdot I = & \frac{1}{{{2}^{3}}}\cdot I
\end{align}\,\!</math>

where <math>I\,\!</math> represents the identity matrix of the same order as the design matrix, <math>X\,\!</math>. Since there are eight observations per replicate of the <math>{2}^{3}\,\!</math> design, the <math>(X\,\!</math> ' <math>X{{)}^{-1}}\,\!</math> matrix for <math>m\,\!</math> replicates of this design can be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{3}}\cdot m)}\cdot I\,\!</math>

The <math>{{({{X}^{\prime }}X)}^{-1}}\,\!</math> matrix for any <math>{2}^{k}\,\!</math> design can now be written as:

::<math>{{({{X}^{\prime }}X)}^{-1}}=\frac{1}{({{2}^{k}}\cdot m)}\cdot I\,\!</math>

Then the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is:

::<math>\begin{align}
C= & {{{\hat{\sigma }}}^{2}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & M{{S}_{E}}\cdot {{({{X}^{\prime }}X)}^{-1}} = & \frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}\cdot I
\end{align}\,\!</math>

Note that the variance-covariance matrix for the <math>{2}^{k}\,\!</math> design is also a diagonal matrix. Therefore, the estimated effect coefficients (<math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math>, <math>{{\beta }_{12}},\,\!</math> etc.) for these designs are uncorrelated. This implies that the terms in the <math>{2}^{k}\,\!</math> design (main effects, interactions) are independent of each other. Consequently, the extra sum of squares for each of the terms in these designs is independent of the sequence of terms in the model, and also independent of the presence of other terms in the model. As a result the sequential and partial sum of squares for the terms are identical for these designs and will always add up to the model sum of squares. Multicollinearity is also not an issue for these designs.

It can also be noted from the equation given above, that in addition to the <math>C\,\!</math> matrix being diagonal, all diagonal elements of the <math>C\,\!</math> matrix are identical. This means that the variance (or its square root, the standard error) of all estimated effect coefficients are the same. The standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>, for all the coefficients is:

::<math>\begin{align}
se({{{\hat{\beta }}}_{j}})= & \sqrt{{{C}_{jj}}} = & \sqrt{\frac{M{{S}_{E}}}{({{2}^{k}}\cdot m)}}\text{ }for\text{ }all\text{ }j
\end{align}\,\!</math>

This property is used to construct the normal probability plot of effects in <math>{2}^{k}\,\!</math> designs and identify significant effects using graphical techniques. For details on the normal probability plot of effects in DOE++, refer to [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]].

====Example====
To illustrate the analysis of a full factorial <math>{2}^{k}\,\!</math> design, consider a three factor experiment to investigate the effect of honing pressure, number of strokes and cycle time on the surface finish of automobile brake drums. Each of these factors is investigated at two levels. The honing pressure is investigated at levels of 200 <math>psi\,\!</math> and 400 <math>psi\,\!</math>, the number of strokes used is 3 and 5 and the two levels of the cycle time are 3 and 5 seconds. The design for this experiment is set up in DOE++ as shown in the first two following figures. It is decided to run two replicates for this experiment. The surface finish data collected from each run (using randomization) and the complete design is shown in the third following figure. The analysis of the experiment data is explained next.

[[Image:doe7_5.png|center|877px|Design properties for the experiment in the example.|link=]]

[[Image:doe7_6.png|center|859px|Design summary for the experiment in the example.|link=]]

[[Image:doe7_7.png|center|778px|Experiment design for the example to investigate the surface finish of automobile brake drums.|link=]]

The applicable model using the notation for <math>{2}^{k}\,\!</math> designs is:

::<math>\begin{align}
Y= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{12}}\cdot {{x}_{1}}{{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}} \\
& +{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}+{{\beta }_{123}}\cdot {{x}_{1}}{{x}_{2}}{{x}_{3}}+\epsilon
\end{align}\,\!</math>

where the indicator variable, <math>{{x}_{1,}}\,\!</math> represents factor <math>A\,\!</math> (honing pressure), <math>{{x}_{1}}=-1\,\!</math> represents the low level of 200 <math>psi\,\!</math> and <math>{{x}_{1}}=1\,\!</math> represents the high level of 400 <math>psi\,\!</math>. Similarly, <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> represent factors <math>B\,\!</math> (number of strokes) and <math>C\,\!</math> (cycle time), respectively. <math>{{\beta }_{0}}\,\!</math> is the overall mean, while <math>{{\beta }_{1}}\,\!</math>, <math>{{\beta }_{2}}\,\!</math> and <math>{{\beta }_{3}}\,\!</math> are the effect coefficients for the main effects of factors <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. <math>{{\beta }_{12}}\,\!</math>, <math>{{\beta }_{13}}\,\!</math> and <math>{{\beta }_{23}}\,\!</math> are the effect coefficients for the <math>AB\,\!</math>, <math>AC\,\!</math> and <math>BC\,\!</math> interactions, while <math>{{\beta }_{123}}\,\!</math> represents the <math>ABC\,\!</math> interaction.

 
If the subscripts for the run (<math>i\,\!</math> ; <math>i=\,\!</math> 1 to 8) and replicates (<math>j\,\!</math> ; <math>j=\,\!</math> 1,2) are included, then the model can be written as:

::<math>\begin{align}
{{Y}_{ij}}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{ij1}}+{{\beta }_{2}}\cdot {{x}_{ij2}}+{{\beta }_{12}}\cdot {{x}_{ij1}}{{x}_{ij2}}+{{\beta }_{3}}\cdot {{x}_{ij3}} \\
& +{{\beta }_{13}}\cdot {{x}_{ij1}}{{x}_{ij3}}+{{\beta }_{23}}\cdot {{x}_{ij2}}{{x}_{ij3}}+{{\beta }_{123}}\cdot {{x}_{ij1}}{{x}_{ij2}}{{x}_{ij3}}+{{\epsilon }_{ij}}
\end{align}\,\!</math>

To investigate how the given factors affect the response, the following hypothesis tests need to be carried:

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{1}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{1}}\ne 0\,\!</math>

This test investigates the main effect of factor <math>A\,\!</math> (honing pressure). The statistic for this test is:

::<math>{{({{F}_{0}})}_{A}}=\frac{M{{S}_{A}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{A}}\,\!</math> is the mean square for factor <math>A\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other main effects, <math>B\,\!</math> and <math>C\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{12}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{12}}\ne 0\,\!</math>

This test investigates the two factor interaction <math>AB\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{AB}}=\frac{M{{S}_{AB}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the interaction <math>AB\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square. Hypotheses for the other two factor interactions, <math>AC\,\!</math> and <math>BC\,\!</math>, can be written in a similar manner.

:<math>{{H}_{0}}\ \ :\ \ {{\beta }_{123}}=0\,\!</math>
:<math>{{H}_{1}}\ \ :\ \ {{\beta }_{123}}\ne 0\,\!</math>

This test investigates the three factor interaction <math>ABC\,\!</math>. The statistic for this test is:

::<math>{{({{F}_{0}})}_{ABC}}=\frac{M{{S}_{ABC}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{ABC}}\,\!</math> is the mean square for the interaction <math>ABC\,\!</math> and <math>M{{S}_{E}}\,\!</math> is the error mean square.
To calculate the test statistics, it is convenient to express the ANOVA model in the form <math>y=X\beta +\epsilon \,\!</math>.

====Expression of the ANOVA Model as <math>y=X\beta +\epsilon \,\!</math>====

In matrix notation, the ANOVA model can be expressed as:

::<math>y=X\beta +\epsilon \,\!</math>

where:

<center><math>y=\left[ \begin{matrix}
{{Y}_{11}} \\
{{Y}_{21}} \\
. \\
{{Y}_{81}} \\
{{Y}_{12}} \\
. \\
{{Y}_{82}} \\
\end{matrix} \right]=\left[ \begin{matrix}
90 \\
90 \\
. \\
90 \\
86 \\
. \\
80 \\
\end{matrix} \right]\text{ }X=\left[ \begin{matrix}
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\
. & . & . & . & . & . & . & . \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\end{matrix} \right]\,\!</math></center>

<center><math>\beta =\left[ \begin{matrix}
{{\beta }_{0}} \\
{{\beta }_{1}} \\
{{\beta }_{2}} \\
{{\beta }_{12}} \\
{{\beta }_{3}} \\
{{\beta }_{13}} \\
{{\beta }_{23}} \\
{{\beta }_{123}} \\
\end{matrix} \right]\text{ }\epsilon =\left[ \begin{matrix}
{{\epsilon }_{11}} \\
{{\epsilon }_{21}} \\
. \\
{{\epsilon }_{81}} \\
{{\epsilon }_{12}} \\
. \\
. \\
{{\epsilon }_{82}} \\
\end{matrix} \right]\,\!</math></center>

====Calculation of the Extra Sum of Squares for the Factors====

Knowing the matrices <math>y\,\!</math>, <math>X\,\!</math> and <math>\beta \,\!</math>, the extra sum of squares for the factors can be calculated. These are used to calculate the mean squares that are used to obtain the test statistics. Since the experiment design is orthogonal, the partial and sequential extra sum of squares are identical. The extra sum of squares for each effect can be calculated as shown next. As an example, the extra sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & Model\text{ }Sum\text{ }of\text{ }Squares - Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }main\text{ }effect\text{ }of\text{ }A \\
= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }A}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }A}}={{X}_{\tilde{\ }A}}{{(X_{\tilde{\ }A}^{\prime }{{X}_{\tilde{\ }A}})}^{-1}}X_{\tilde{\ }A}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }A}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the main effect of factor <math>A\,\!</math>. Thus, the sum of squares for the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/16)J]y \\
= & 654.4375-549.375 \\
= & 105.0625
\end{align}\,\!</math>

Similarly, the extra sum of squares for the interaction effect <math>AB\,\!</math> is:

::<math>\begin{align}
S{{S}_{AB}}= & {{y}^{\prime }}[H-(1/16)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }AB}}-(1/16)J]y \\
= & 654.4375-636.375 \\
= & 18.0625
\end{align}\,\!</math>

The extra sum of squares for other effects can be obtained in a similar manner.

====Calculation of the Test Statistics====

Knowing the extra sum of squares, the test statistic for the effects can be calculated. For example, the test statistic for the interaction <math>AB\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{AB}}= & \frac{M{{S}_{AB}}}{M{{S}_{E}}} \\
= & \frac{S{{S}_{AB}}/dof(S{{S}_{AB}})}{S{{S}_{E}}/dof(S{{S}_{E}})} \\
= & \frac{18.0625/1}{147.5/8} \\
= & 0.9797
\end{align}\,\!</math>

where <math>M{{S}_{AB}}\,\!</math> is the mean square for the <math>AB\,\!</math> interaction and <math>M{{S}_{E}}\,\!</math> is the error mean square. The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{AB}}=0.9797\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{AB}}) \\
= & 1-0.6487 \\
= & 0.3513
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that the interaction between honing pressure and number of strokes does not affect the surface finish of the brake drums. Tests for other effects can be carried out in a similar manner. The results are shown in the ANOVA Table in the following figure. The values S, R-sq and R-sq(adj) in the figure indicate how well the model fits the data. The value of S represents the standard error of the model, R-sq represents the coefficient of multiple determination and R-sq(adj) represents the adjusted coefficient of multiple determination. For details on these values refer to [[Multiple_Linear_Regression_Analysis|Multiple Linear Regression Analysis]].

[[Image:doe7_8.png|center|799px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

====Calculation of Effect Coefficients====

The estimate of effect coefficients can also be obtained:

<center><math>\begin{align}
\hat{\beta }= & {{({{X}^{\prime }}X)}^{-1}}{{X}^{\prime }}y \\
= & \left[ \begin{matrix}
86.4375 \\
2.5625 \\
-4.9375 \\
1.0625 \\
-1.0625 \\
2.4375 \\
-1.3125 \\
-0.1875 \\
\end{matrix} \right]
\end{align}\,\!</math></center>

[[Image:doe7_9.png|center|800px|Regression Information table for the experiment in the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

The coefficients and related results are shown in the Regression Information table above. In the table, the Effect column displays the effects, which are simply twice the coefficients. The Standard Error column displays the standard error, <math>se({{\hat{\beta }}_{j}})\,\!</math>. The Low CI and High CI columns display the confidence interval on the coefficients. The interval shown is the 90% interval as the significance is chosen as 0.1. The T Value column displays the <math>t\,\!</math> statistic, <math>{{t}_{0}}\,\!</math>, corresponding to the coefficients. The P Value column displays the <math>p\,\!</math> value corresponding to the <math>t\,\!</math> statistic. (For details on how these results are calculated, refer to [[General Full Factorial Designs]]). Plots of residuals can also be obtained from DOE++ to ensure that the assumptions related to the ANOVA model are not violated.

====Model Equation====

From the analysis results in the above figure within [[Two_Level_Factorial_Experiments#Calculation_of_Effect_Coefficients|calculation of effect coefficients]] section, it is seen that effects <math>A\,\!</math>, <math>B\,\!</math> and <math>AC\,\!</math> are significant. In DOE++, the <math>p\,\!</math> values for the significant effects are displayed in red in the ANOVA Table for easy identification. Using the values of the estimated effect coefficients, the model for the present <math>{2}^{3}\,\!</math> design in terms of the coded values can be written as:

::<math>\begin{align}
\hat{y}= & {{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{13}}\cdot {{x}_{1}}{{x}_{3}} \\
= & 86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+2.4375{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

To make the model hierarchical, the main effect, <math>C\,\!</math>, needs to be included in the model (because the interaction <math>AC\,\!</math> is included in the model). The resulting model is:

::<math>\hat{y}=86.4375+2.5625{{x}_{1}}-4.9375{{x}_{2}}+1.0625{{x}_{3}}+2.4375{{x}_{1}}{{x}_{3}}\,\!</math>

This equation can be viewed in DOE++, as shown in the following figure, using the Show Analysis Summary icon in the Control Panel. The equation shown in the figure will match the hierarchical model once the required terms are selected using the Select Effects icon.

[[Image:doe7_10.png|center|557px|The model equation for the experiment of the [[Two_Level_Factorial_Experiments#Example| example]].|link=]]

==Replicated and Repeated Runs==

In the case of replicated experiments, it is important to note the difference between replicated runs and repeated runs. Both repeated and replicated runs are multiple response readings taken at the same factor levels. However, repeated runs are response observations taken at the same time or in succession. Replicated runs are response observations recorded in a random order. Therefore, replicated runs include more variation than repeated runs. For example, a baker, who wants to investigate the effect of two factors on the quality of cakes, will have to bake four cakes to complete one replicate of a <math>{2}^{2}\,\!</math> design. Assume that the baker bakes eight cakes in all. If, for each of the four treatments of the <math>{2}^{2}\,\!</math> design, the baker selects one treatment at random and then bakes two cakes for this treatment at the same time then this is a case of two repeated runs. If, however, the baker bakes all the eight cakes randomly, then the eight cakes represent two sets of replicated runs.
For repeated measurements, the average values of the response for each treatment should be entered into DOE++ as shown in the following figure (a) when the two cakes for a particular treatment are baked together. For replicated measurements, when all the cakes are baked randomly, the data is entered as shown in the following figure (b).

[[Image:doe7.11.png|center|300px|Data entry for repeated and replicated runs. Figure (a) shows repeated runs and (b) shows replicated runs.]]

==Unreplicated 2''k'' Designs==

If a factorial experiment is run only for a single replicate then it is not possible to test hypotheses about the main effects and interactions as the error sum of squares cannot be obtained. This is because the number of observations in a single replicate equals the number of terms in the ANOVA model. Hence the model fits the data perfectly and no degrees of freedom are available to obtain the error sum of squares.

However, sometimes it is only possible to run a single replicate of the <math>{2}^{k}\,\!</math> design because of constraints on resources and time. In the absence of the error sum of squares, hypothesis tests to identify significant factors cannot be conducted. A number of methods of analyzing information obtained from unreplicated <math>{2}^{k}\,\!</math> designs are available. These include pooling higher order interactions, using the normal probability plot of effects or including center point replicates in the design.

===Pooling Higher Order Interactions===

One of the ways to deal with unreplicated <math>{2}^{k}\,\!</math> designs is to use the sum of squares of some of the higher order interactions as the error sum of squares provided these higher order interactions can be assumed to be insignificant. By dropping some of the higher order interactions from the model, the degrees of freedom corresponding to these interactions can be used to estimate the error mean square. Once the error mean square is known, the test statistics to conduct hypothesis tests on the factors can be calculated.

===Normal Probability Plot of Effects===
Another way to use unreplicated <math>{2}^{k}\,\!</math> designs to identify significant effects is to construct the normal probability plot of the effects. As mentioned in [[Two_Level_Factorial_Experiments#Special_Features| Special Features]], the standard error for all effect coefficients in the <math>{2}^{k}\,\!</math> designs is the same. Therefore, on a normal probability plot of effect coefficients, all non-significant effect coefficients (with <math>\beta =0\,\!</math>) will fall along the straight line representative of the normal distribution, N(<math>0,{{\sigma }^{2}}/({{2}^{k}}\cdot m)\,\!</math>). Effect coefficients that show large deviations from this line will be significant since they do not come from this normal distribution. Similarly, since effects <math>=2\times \,\!</math> effect coefficients, all non-significant effects will also follow a straight line on the normal probability plot of effects. For replicated designs, the Effects Probability plot of DOE++ plots the normalized effect values (or the T Values) on the standard normal probability line, N(0,1). However, in the case of unreplicated <math>{2}^{k}\,\!</math> designs, <math>{{\sigma }^{2}}\,\!</math> remains unknown since <math>M{{S}_{E}}\,\!</math> cannot be obtained. Lenth's method is used in this case to estimate the variance of the effects. For details on Lenth's method, please refer to [[DOE References| Montgomery (2001)]]. DOE++ then uses this variance value to plot effects along the N(0, Lenth's effect variance) line. The
method is illustrated in the following example.

====Example====

Vinyl panels, used as instrument panels in a certain automobile, are seen to develop defects after a certain amount of time. To investigate the issue, it is decided to carry out a two level factorial experiment. Potential factors to be investigated in the experiment are vacuum rate (factor <math>A\,\!</math>), material temperature (factor <math>B\,\!</math>), element intensity (factor <math>C\,\!</math>) and pre-stretch (factor <math>D\,\!</math>). The two levels of the factors used in the experiment are as shown in below.

[[Image:doet7.1.png|center|300px|Factors to investigate defects in vinyl panels.]]

With a <math>{2}^{4}\,\!</math> design requiring 16 runs per replicate it is only feasible for the manufacturer to run a single replicate.

The experiment design and data, collected as percent defects, are shown in the following figure. Since the present experiment design contains only a single replicate, it is not possible to obtain an estimate of the error sum of squares, <math>S{{S}_{E}}\,\!</math>. It is decided to use the normal probability plot of effects to identify the significant effects. The effect values for each term are obtained as shown in the following figure.

[[Image:doe7_13.png|center|650px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

Lenth's method uses these values to estimate the variance. As described in [[DOE_References|[Lenth, 1989]]], if all effects are arranged in ascending order, using their absolute values, then <math>{{s}_{0}}\,\!</math> is defined as 1.5 times the median value:

::<math>\begin{align}
{{s}_{0}}= & 1.5\cdot median(\left| effect \right|) \\
= & 1.5\cdot 2 \\
= & 3
\end{align}\,\!</math>

Using <math>{{s}_{0}}\,\!</math>, the "pseudo standard error" (<math>PSE\,\!</math>) is calculated as 1.5 times the median value of all effects that are less than 2.5 <math>{{s}_{0}}\,\!</math> :

::<math>\begin{align}
PSE= & 1.5\cdot median(\left| effect \right|\ \ :\ \ \left| effect \right|<2.5{{s}_{0}}) \\
= & 1.5\cdot 1.5 \\
= & 2.25
\end{align}\,\!</math>

Using <math>PSE\,\!</math> as an estimate of the effect variance, the effect variance is 2.25. Knowing the effect variance, the normal probability plot of effects for the present unreplicated experiment can be constructed as shown in the following figure. The line on this plot is the line N(0, 2.25). The plot shows that the effects <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math> do not follow the distribution represented by this line. Therefore, these effects are significant.

The significant effects can also be identified by comparing individual effect values to the margin of error or the threshold value using the pareto chart (see the third following figure). If the required significance is 0.1, then:

::<math>margin\text{ }of\text{ }error={{t}_{\alpha /2,d}}\cdot PSE\,\!</math>

The <math>t\,\!</math> statistic, <math>{{t}_{\alpha /2,d}}\,\!</math>, is calculated at a significance of <math>\alpha /2\,\!</math> (for the two-sided hypothesis) and degrees of freedom <math>d=(\,\!</math> number of effects <math>)/3\,\!</math>. Thus:

::<math>\begin{align}
margin\text{ }of\text{ }error= & {{t}_{0.05,5}}\cdot PSE \\
= & 2.015\cdot 2.25 \\
= & 4.534
\end{align}\,\!</math>

The value of 4.534 is shown as the critical value line in the third following figure. All effects with absolute values greater than the margin of error can be considered to be significant. These effects are <math>A\,\!</math>, <math>D\,\!</math> and the interaction <math>AD\,\!</math>. Therefore, the vacuum rate, the pre-stretch and their interaction have a significant effect on the defects of the vinyl panels.

[[Image:doe7_14.png|center|800px|Effect values for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_15.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

[[Image:doe7_16.png|center|650px|Pareto chart for the experiment in the [[Two_Level_Factorial_Experiments#Example_2| example]].|link=]]

===Center Point Replicates===

Another method of dealing with unreplicated <math>{2}^{k}\,\!</math> designs that only have quantitative factors is to use replicated runs at the center point. The center point is the response corresponding to the treatment exactly midway between the two levels of all factors. Running multiple replicates at this point provides an estimate of pure error. Although running multiple replicates at any treatment level can provide an estimate of pure error, the other advantage of running center point replicates in the <math>{2}^{k}\,\!</math> design is in checking for the presence of curvature. The test for curvature investigates whether the model between the response and the factors is linear and is discussed in [[Two_Level_Factorial_Experiments#Using_Center_Point_Replicates_to_Test_Curvature| Center Pt. Replicates to Test Curvature]].

====Example: Use Center Point to Get Pure Error====

Consider a <math>{2}^{2}\,\!</math> experiment design to investigate the effect of two factors, <math>A\,\!</math> and <math>B\,\!</math>, on a certain response. The energy consumed when the treatments of the <math>{2}^{2}\,\!</math> design are run is considerably larger than the energy consumed for the center point run (because at the center point the factors are at their middle levels). Therefore, the analyst decides to run only a single replicate of the design and augment the design by five replicated runs at the center point as shown in the following figure. The design properties for this experiment are shown in the second following figure. The complete experiment design is shown in the third following figure. The center points can be used in the identification of significant effects as shown next.

[[Image:doe7.17.png||center|300px|<math>2^2\,\!</math> design augmented by five center point runs.]]
[[Image:doe7_18.png|center|537px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

[[Image:doe7_19.png|center|630px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

Since the present <math>{2}^{2}\,\!</math> design is unreplicated, there are no degrees of freedom available to calculate the error sum of squares. By augmenting this design with five center points, the response values at the center points, <math>y_{i}^{c}\,\!</math>, can be used to obtain an estimate of pure error, <math>S{{S}_{PE}}\,\!</math>. Let <math>{{\bar{y}}^{c}}\,\!</math> represent the average response for the five replicates at the center. Then:

::<math>S{{S}_{PE}}=Sum\text{ }of\text{ }Squares\text{ }for\text{ }center\text{ }points\,\!</math>

::<math>\begin{align}
S{{S}_{PE}}= & \underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}} \\
= & {{(25.2-25.26)}^{2}}+...+{{(25.3-25.26)}^{2}} \\
= & 0.052
\end{align}\,\!</math>

Then the corresponding mean square is:

::<math>\begin{align}
M{{S}_{PE}}= & \frac{S{{S}_{PE}}}{degrees\text{ }of\text{ }freedom} \\
= & \frac{0.052}{5-1} \\
= & 0.013
\end{align}\,\!</math>

Alternatively, <math>M{{S}_{PE}}\,\!</math> can be directly obtained by calculating the variance of the response values at the center points:

::<math>\begin{align}
M{{S}_{PE}}= & {{s}^{2}} \\
= & \frac{\underset{i=1}{\overset{5}{\mathop{\sum }}}\,{{(y_{i}^{c}-{{{\bar{y}}}^{c}})}^{2}}}{5-1}
\end{align}\,\!</math>

Once <math>M{{S}_{PE}}\,\!</math> is known, it can be used as the error mean square, <math>M{{S}_{E}}\,\!</math>, to carry out the test of significance for each effect. For example, to test the significance of the main effect of factor <math>A,\,\!</math> the sum of squares corresponding to this effect is obtained in the usual manner by considering only the four runs of the original <math>{2}^{2}\,\!</math> design.

::<math>\begin{align}
S{{S}_{A}}= & {{y}^{\prime }}[H-(1/4)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }A}}-(1/4)J]y \\
= & 0.5625
\end{align}\,\!</math>

Then, the test statistic to test the significance of the main effect of factor <math>A\,\!</math> is:

::<math>\begin{align}
{{({{f}_{0}})}_{A}}= & \frac{M{{S}_{A}}}{M{{S}_{E}}} \\
= & \frac{0.5625/1}{0.052/4} \\
= & 43.2692
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{A}}=43.2692\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and eight degrees of freedom in the denominator is:

::<math>\begin{align}
p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{A}}) \\
= & 1-0.9972 \\
= & 0.0028
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value < 0.1, it can be concluded that the main effect of factor <math>A\,\!</math> significantly affects the response. This result is displayed in the ANOVA table as shown in the following figure. Test for the significance of other factors can be carried out in a similar manner.

[[Image:doe7_20.png|center|649px|Results for the experiment in the [[Two_Level_Factorial_Experiments#Example_3| example]].]]

===Using Center Point Replicates to Test Curvature===

Center point replicates can also be used to check for curvature in replicated or unreplicated <math>{2}^{k}\,\!</math> designs. The test for curvature investigates whether the model between the response and the factors is linear. The way DOE++ handles center point replicates is similar to its handling of blocks. The center point replicates are treated as an additional factor in the model. The factor is labeled as Curvature in the results of DOE++. If Curvature turns out to be a significant factor in the results, then this indicates the presence of curvature in the model.

====Example: Use Center Point to Test Curvature====

To illustrate the use of center point replicates in testing for curvature, consider again the data of the single replicate <math>{2}^{2}\,\!</math> experiment from a preceding figure(labeled "<math>2^2</math> design augmented by five center point runs"). Let <math>{{x}_{1}}\,\!</math> be the indicator variable to indicate if the run is a center point:

::<math>\begin{matrix}
{{x}_{1}}=0 & {} & \text{Center point run} \\
{{x}_{1}}=1 & {} & \text{Other run} \\
\end{matrix}\,\!</math>

If <math>{{x}_{2}}\,\!</math> and <math>{{x}_{3}}\,\!</math> are the indicator variables representing factors <math>A\,\!</math> and <math>B\,\!</math>, respectively, then the model for this experiment is:

::<math>Y={{\beta }_{0}}+{{\beta }_{1}}\cdot {{x}_{1}}+{{\beta }_{2}}\cdot {{x}_{2}}+{{\beta }_{3}}\cdot {{x}_{3}}+{{\beta }_{23}}\cdot {{x}_{2}}{{x}_{3}}\,\!</math>

To investigate the presence of curvature, the following hypotheses need to be tested:

::<math>\begin{align}
& {{H}_{0}}: & {{\beta }_{1}}=0\text{ (Curvature is absent)} \\
& {{H}_{1}}: & {{\beta }_{1}}\ne 0
\end{align}\,\!</math>

The test statistic to be used for this test is:

::<math>{{({{F}_{0}})}_{curvature}}=\frac{M{{S}_{curvature}}}{M{{S}_{E}}}\,\!</math>

where <math>M{{S}_{curvature}}\,\!</math> is the mean square for Curvature and <math>M{{S}_{E}}\,\!</math> is the error mean square.

'''Calculation of the Sum of Squares'''

The <math>X\,\!</math> matrix and <math>y\,\!</math> vector for this experiment are:

<center><math>X=\left[ \begin{matrix}
1 & 1 & -1 & -1 & 1 \\
1 & 1 & 1 & -1 & -1 \\
1 & 1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
\end{matrix} \right]\text{ }y=\left[ \begin{matrix}
24.6 \\
25.4 \\
25.0 \\
25.7 \\
25.2 \\
25.3 \\
25.4 \\
25.1 \\
25.3 \\
\end{matrix} \right]\,\!</math></center>

The sum of squares can now be calculated. For example, the error sum of squares is:

::<math>\begin{align}
& S{{S}_{E}}= & {{y}^{\prime }}[I-H]y \\
& = & 0.052
\end{align}\,\!</math>

where <math>I\,\!</math> is the identity matrix and <math>H\,\!</math> is the hat matrix. It can be seen that this is equal to <math>S{{S}_{PE\text{ }}}\,\!</math> (the sum of squares due to pure error) because of the replicates at the center point, as obtained in the [[Two_Level_Factorial_Experiments#Example_3| example]]. The number of degrees of freedom associated with <math>S{{S}_{E}}\,\!</math>, <math>dof(S{{S}_{E}})\,\!</math> is four. The extra sum of squares corresponding to the center point replicates (or Curvature) is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & Model\text{ }Sum\text{ }of\text{ }Squares- \\
& & Sum\text{ }of\text{ }Squares\text{ }of\text{ }model\text{ }excluding\text{ }the\text{ }center\text{ }point \\
& = & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Curvature}}-(1/9)J]y
\end{align}\,\!</math>

where <math>H\,\!</math> is the hat matrix and <math>J\,\!</math> is the matrix of ones. The matrix <math>{{H}_{\tilde{\ }Curvature}}\,\!</math> can be calculated using <math>{{H}_{\tilde{\ }Curvature}}={{X}_{\tilde{\ }Curv}}{{(X_{\tilde{\ }Curv}^{\prime }{{X}_{\tilde{\ }Curv}})}^{-1}}X_{\tilde{\ }Curv}^{\prime }\,\!</math> where <math>{{X}_{\tilde{\ }Curv}}\,\!</math> is the design matrix, <math>X\,\!</math>, excluding the second column that represents the center point. Thus, the extra sum of squares corresponding to Curvature is:

::<math>\begin{align}
& S{{S}_{Curvature}}= & {{y}^{\prime }}[H-(1/9)J]y-{{y}^{\prime }}[{{H}_{\tilde{\ }Center}}-(1/9)J]y \\
& = & 0.7036-0.6875 \\
& = & 0.0161
\end{align}\,\!</math>

This extra sum of squares can be used to test for the significance of curvature. The corresponding mean square is:

::<math>\begin{align}
& M{{S}_{Curvature}}= & \frac{Sum\text{ }of\text{ }squares\text{ }corresponding\text{ }to\text{ }Curvature}{degrees\text{ }of\text{ }freedom} \\
& = & \frac{0.0161}{1} \\
& = & 0.0161
\end{align}\,\!</math>

'''Calculation of the Test Statistic'''

Knowing the mean squares, the statistic to check the significance of curvature can be calculated.

::<math>\begin{align}
& {{({{f}_{0}})}_{Curvature}}= & \frac{M{{S}_{Curvature}}}{M{{S}_{E}}} \\
& = & \frac{0.0161/1}{0.052/4} \\
& = & 1.24
\end{align}\,\!</math>

The <math>p\,\!</math> value corresponding to the statistic, <math>{{({{f}_{0}})}_{Curvature}}=1.24\,\!</math>, based on the <math>F\,\!</math> distribution with one degree of freedom in the numerator and four degrees of freedom in the denominator is:

::<math>\begin{align}
& p\text{ }value= & 1-P(F\le {{({{f}_{0}})}_{Curvature}}) \\
& = & 1-0.6713 \\
& = & 0.3287
\end{align}\,\!</math>

Assuming that the desired significance is 0.1, since <math>p\,\!</math> value > 0.1, it can be concluded that curvature does not exist for this design. This results is shown in the ANOVA table in the figure above. The surface of the fitted model based on these results, along with the observed response values, is shown in the figure below.

[[Image:doe7.21.png|center|400px|Model surface and observed response values for the design in the [[Two_Level_Factorial_Experiments#Example_4| example]].]]

 

==Blocking in 2k Designs==

Blocking can be used in the <math>{2}^{k}\,\!</math> designs to deal with cases when replicates cannot be run under identical conditions. Randomized complete block designs that were discussed in [[Randomization and Blocking in DOE]] for factorial experiments are also applicable here. At times, even with just two levels per factor, it is not possible to run all treatment combinations for one replicate of the experiment under homogeneous conditions. For example, each replicate of the <math>{2}^{2}\,\!</math> design requires four runs. If each run requires two hours and testing facilities are available for only four hours per day, two days of testing would be required to run one complete replicate. Blocking can be used to separate the treatment runs on the two different days. Blocks that do not contain all treatments of a replicate are called incomplete blocks. In incomplete block designs, the block effect is confounded with certain effect(s) under investigation. For the <math>{2}^{2}\,\!</math> design assume that treatments <math>(1)\,\!</math> and <math>ab\,\!</math> were run on the first day and treatments <math>a\,\!</math> and <math>b\,\!</math> were run on the second day. Then, the incomplete block design for this experiment is:

::<math>\begin{matrix}
\text{Block 1} & {} & \text{Block 2} \\
\left[ \begin{matrix}
(1) \\
ab \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
b \\
\end{matrix} \right] \\
\end{matrix}\,\!</math>

For this design the block effect may be calculated as:

::<math>\begin{align}
& Block\text{ }Effect= & Average\text{ }response\text{ }for\text{ }Block\text{ }1- \\
& & Average\text{ }response\text{ }for\text{ }Block\text{ }2 \\
& = & \frac{(1)+ab}{2}-\frac{a+b}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The <math>AB\,\!</math> interaction effect is:

::<math>\begin{align}
& AB= & Average\text{ }response\text{ }at\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{low}}}- \\
& & Average\text{ }response\text{ }at\text{ }{{A}_{\text{low}}}\text{-}{{B}_{\text{high}}}\text{ }and\text{ }{{A}_{\text{high}}}\text{-}{{B}_{\text{low}}} \\
& = & \frac{ab+(1)}{2}-\frac{b+a}{2} \\
& = & \frac{1}{2}[(1)+ab-a-b]
\end{align}\,\!</math>

The two equations given above show that, in this design, the <math>AB\,\!</math> interaction effect cannot be distinguished from the block effect because the formulas to calculate these effects are the same. In other words, the <math>AB\,\!</math> interaction is said to be confounded with the block effect and it is not possible to say if the effect calculated based on these equations is due to the <math>AB\,\!</math> interaction effect, the block effect or both. In incomplete block designs some effects are always confounded with the blocks. Therefore, it is important to design these experiments in such a way that the important effects are not confounded with the blocks. In most cases, the experimenter can assume that higher order interactions are unimportant. In this case, it would better to use incomplete block designs that confound these effects with the blocks.
One way to design incomplete block designs is to use defining contrasts as shown next:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}+...+{{\alpha }_{k}}{{q}_{k}}\,\!</math>

where the <math>{{\alpha }_{i}}\,\!</math> s are the exponents for the factors in the effect that is to be confounded with the block effect and the <math>{{q}_{i}}\,\!</math> s are values based on the level of the <math>i\,\!</math> the factor (in a treatment that is to be allocated to a block). For <math>{2}^{k}\,\!</math> designs the <math>{{\alpha }_{i}}\,\!</math> s are either 0 or 1 and the <math>{{q}_{i}}\,\!</math> s have a value of 0 for the low level of the <math>i\,\!</math> th factor and a value of 1 for the high level of the factor in the treatment under consideration. As an example, consider the <math>{2}^{2}\,\!</math> design where the interaction effect <math>AB\,\!</math> is confounded with the block. Since there are two factors, <math>k=2\,\!</math>, with <math>i=1\,\!</math> representing factor <math>A\,\!</math> and <math>i=2\,\!</math> representing factor <math>B\,\!</math>. Therefore:

::<math>L={{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}}\,\!</math>

The value of <math>{{\alpha }_{1}}\,\!</math> is one because the exponent of factor <math>A\,\!</math> in the confounded interaction <math>AB\,\!</math> is one. Similarly, the value of <math>{{\alpha }_{2}}\,\!</math> is one because the exponent of factor <math>B\,\!</math> in the confounded interaction <math>AB\,\!</math> is also one. Therefore, the defining contrast for this design can be written as:

::<math>\begin{align}
& L= & {{\alpha }_{1}}{{q}_{1}}+{{\alpha }_{2}}{{q}_{2}} \\
& = & 1\cdot {{q}_{1}}+1\cdot {{q}_{2}} \\
& = & {{q}_{1}}+{{q}_{2}}
\end{align}\,\!</math>

Once the defining contrast is known, it can be used to allocate treatments to the blocks. For the <math>{2}^{2}\,\!</math> design, there are four treatments <math>(1)\,\!</math>, <math>a\,\!</math>, <math>b\,\!</math> and <math>ab\,\!</math>. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. In order to decide which block the treatment <math>(1)\,\!</math> belongs to, the levels of factors <math>A\,\!</math> and <math>B\,\!</math> for this run are used. Since factor <math>A\,\!</math> is at the low level in this treatment, <math>{{q}_{1}}=0\,\!</math>. Similarly, since factor <math>B\,\!</math> is also at the low level in this treatment, <math>{{q}_{2}}=0\,\!</math>. Therefore:

::<math>\begin{align}
& L= & {{q}_{1}}+{{q}_{2}} \\
& = & 0+0=0\text{ (mod 2)}
\end{align}\,\!</math>

Note that the value of <math>L\,\!</math> used to decide the block allocation is "mod 2" of the original value. This value is obtained by taking the value of 1 for odd numbers and 0 otherwise. Based on the value of <math>L\,\!</math>, treatment <math>(1)\,\!</math> is assigned to block 1. Other treatments can be assigned using the following calculations:

::<math>\begin{align}
& (1): & \text{ }L=0+0=0=0\text{ (mod 2)} \\
& a: & \text{ }L=1+0=1=1\text{ (mod 2)} \\
& b: & \text{ }L=0+1=1=1\text{ (mod 2)} \\
& ab: & \text{ }L=1+1=2=0\text{ (mod 2)}
\end{align}\,\!</math>

Therefore, to confound the interaction <math>AB\,\!</math> with the block effect in the <math>{2}^{2}\,\!</math> incomplete block design, treatments <math>(1)\,\!</math> and <math>ab\,\!</math> (with <math>L=0\,\!</math>) should be assigned to block 2 and treatment combinations <math>a\,\!</math> and <math>b\,\!</math> (with <math>L=1\,\!</math>) should be assigned to block 1.

====Example: Two Level Factorial Design with Two Blocks====

This example illustrates how treatments can be allocated to two blocks for an unreplicated <math>{2}^{k}\,\!</math> design. Consider the unreplicated <math>{2}^{4}\,\!</math> design to investigate the four factors affecting the defects in automobile vinyl panels discussed in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments required for this experiment were run by two different operators with each operator conducting 8 runs. This experiment is an example of an incomplete block design. The analyst in charge of this experiment assumed that the interaction <math>ABCD\,\!</math> was not significant and decided to allocate treatments to the two operators so that the <math>ABCD\,\!</math> interaction was confounded with the block effect (the two operators are the blocks). The allocation scheme to assign treatments to the two operators can be obtained as follows.
 
The defining contrast for the <math>{2}^{4}\,\!</math> design where the <math>ABCD\,\!</math> interaction is confounded with the blocks is:

::<math>L={{q}_{1}}+{{q}_{2}}+{{q}_{3}}+{{q}_{4}}\,\!</math>

The treatments can be allocated to the two operators using the values of the defining contrast. Assume that <math>L=0\,\!</math> represents block 2 and <math>L=1\,\!</math> represents block 1. Then the value of the defining contrast for treatment <math>a\,\!</math> is:

::<math>a\ \ :\ \ \text{ }L=1+0+0+0=1=1\text{ (mod 2)}\,\!</math>

Therefore, treatment <math>a\,\!</math> should be assigned to Block 1 or the first operator. Similarly, for treatment <math>ab\,\!</math> we have:

::<math>ab\ \ :\ \ \text{ }L=1+1+0+0=2=0\text{ (mod 2)}\,\!</math>

[[Image:doe7.22.png|center|200px| Allocation of treatments to two blocks for the <math>2^4</math> design in the example by confounding interaction of <math>ABCD</math> with the blocks.]]

Therefore, <math>ab\,\!</math> should be assigned to Block 2 or the second operator. Other treatments can be allocated to the two operators in a similar manner to arrive at the allocation scheme shown in the figure below.
In DOE++, to confound the <math>ABCD\,\!</math> interaction for the <math>{2}^{4}\,\!</math> design into two blocks, the number of blocks are specified as shown in the figure below. Then the interaction <math>ABCD\,\!</math> is entered in the Block Generator window (second following figure) which is available using the Block Generator button in the following figure. The design generated by DOE++ is shown in the third of the following figures. This design matches the allocation scheme of the preceding figure.

[[Image:doe7_23.png|center|700px| Adding block properties for the experiment in the example.|link=]]

[[Image:doe7_24.png|center|700px|Specifying the interaction ABCD as the interaction to be confounded with the blocks for the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

[[Image:doe7_25.png|center|800px|Two block design for the experiment in the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

For the analysis of this design, the sum of squares for all effects are calculated assuming no blocking. Then, to account for blocking, the sum of squares corresponding to the <math>ABCD\,\!</math> interaction is considered as the sum of squares due to blocks and <math>ABCD\,\!</math>. In DOE++ this is done by displaying this sum of squares as the sum of squares due to the blocks. This is shown in the following figure where the sum of squares in question is obtained as 72.25 and is displayed against Block. The interaction ABCD, which is confounded with the blocks, is not displayed. Since the design is unreplicated, any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_26.png|center|793px|ANOVA table for the experiment of the [[Two_Level_Factorial_Experiments#Example_5| example]].|link=]]

===Unreplicated 2''k'' Designs in 2''p'' Blocks===

A single replicate of the <math>{2}^{k}\,\!</math> design can be run in up to <math>{2}^{p}\,\!</math> blocks where <math>p<k\,\!</math>. The number of effects confounded with the blocks equals the degrees of freedom associated with the block effect.

If two blocks are used (the block effect has two levels), then one (<math>2-1=1)\,\!</math> effect is confounded with the blocks. If four blocks are used, then three (<math>4-1=3\,\!</math>) effects are confounded with the blocks and so on. For example an unreplicated <math>{2}^{4}\,\!</math> design may be confounded in <math>{2}^{2}\,\!</math> (four) blocks using two contrasts, <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}}\,\!</math>. Let <math>AC\,\!</math> and <math>BD\,\!</math> be the effects to be confounded with the blocks. Corresponding to these two effects, the contrasts are respectively:

::<math>\begin{align}
& {{L}_{1}}= & {{q}_{1}}+{{q}_{3}} \\
& {{L}_{2}}= & {{q}_{2}}+{{q}_{4}}
\end{align}\,\!</math>

Based on the values of <math>{{L}_{1}}\,\!</math> and <math>{{L}_{2}},\,\!</math> the treatments can be assigned to the four blocks as follows:

<center><math>\begin{matrix}
\text{Block 4} & {} & \text{Block 3} & {} & \text{Block 2} & {} & \text{Block 1} \\
{{L}_{1}}=0,{{L}_{2}}=0 & {} & {{L}_{1}}=1,{{L}_{2}}=0 & {} & {{L}_{1}}=0,{{L}_{2}}=1 & {} & {{L}_{1}}=1,{{L}_{2}}=1 \\
{} & {} & {} & {} & {} & {} & {} \\
\left[ \begin{matrix}
(1) \\
ac \\
bd \\
abcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
a \\
c \\
abd \\
bcd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
b \\
abc \\
d \\
acd \\
\end{matrix} \right] & {} & \left[ \begin{matrix}
ab \\
bc \\
ad \\
cd \\
\end{matrix} \right] \\
\end{matrix}\,\!</math></center>

Since the block effect has three degrees of freedom, three effects are confounded with the block effect. In addition to <math>AC\,\!</math> and <math>BD\,\!</math>, the third effect confounded with the block effect is their generalized interaction, <math>(AC)(BD)=ABCD\,\!</math>.
In general, when an unreplicated <math>{2}^{k}\,\!</math> design is confounded in <math>{2}^{p}\,\!</math> blocks, <math>p\,\!</math> contrasts are needed (<math>{{L}_{1}},{{L}_{2}}...{{L}_{p}}\,\!</math>). <math>p\,\!</math> effects are selected to define these contrasts such that none of these effects are the generalized interaction of the others. The <math>{2}^{p}\,\!</math> blocks can then be assigned the treatments using the <math>p\,\!</math> contrasts. <math>{{2}^{p}}-(p+1)\,\!</math> effects, that are also confounded with the blocks, are then obtained as the generalized interaction of the <math>p\,\!</math> effects. In the statistical analysis of these designs, the sum of squares are computed as if no blocking were used. Then the block sum of squares is obtained by adding the sum of squares for all the effects confounded with the blocks.

====Example: 2 Level Factorial Design with Four Blocks====

This example illustrates how DOE++ obtains the sum of squares when treatments for an unreplicated <math>{2}^{k}\,\!</math> design are allocated among four blocks. Consider again the unreplicated <math>{2}^{4}\,\!</math> design used to investigate the defects in automobile vinyl panels presented in [[Two_Level_Factorial_Experiments#Normal_Probability_Plot_of_Effects| Normal Probability Plot of Effects]]. Assume that the 16 treatments needed to complete the experiment were run by four operators. Therefore, there are four blocks. Assume that the treatments were allocated to the blocks using the generators mentioned in the previous section, i.e., treatments were allocated among the four operators by confounding the effects, <math>AC\,\!</math> and <math>BD,\,\!</math> with the blocks. These effects can be specified as Block Generators as shown in the following figure. (The generalized interaction of these two effects, interaction <math>ABCD\,\!</math>, will also get confounded with the blocks.) The resulting design is shown in the second following figure and matches the allocation scheme obtained in the previous section.

[[Image:doe7_27.png|center|700px|Specifying the interactions AC and BD as block generators for the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

The sum of squares in this case can be obtained by calculating the sum of squares for each of the effects assuming there is no blocking. Once the individual sum of squares have been obtained, the block sum of squares can be calculated. The block sum of squares is the sum of the sum of squares of effects, <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, since these effects are confounded with the block effect. As shown in the second following figure, this sum of squares is 92.25 and is displayed against Block. The interactions <math>AC\,\!</math>, <math>BD\,\!</math> and <math>ABCD\,\!</math>, which are confounded with the blocks, are not displayed. Since the present design is unreplicated any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

[[Image:doe7_28.png|center|800px|Design for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

[[Image:doe7_29.png|center|793px|ANOVA table for the experiment in the [[Two_Level_Factorial_Experiments#Example_6| example]].|link=]]

==Variability Analysis==

For replicated two level factorial experiments, DOE++ provides the option of conducting variability analysis (using the Variability Analysis icon under the Data menu). The analysis is used to identify the treatment that results in the least amount of variation in the product or process being investigated. Variability analysis is conducted by treating the standard deviation of the response for each treatment of the experiment as an additional response. The standard deviation for a treatment is obtained by using the replicated response values at that treatment run. As an example, consider the <math>{2}^{3}\,\!</math> design shown in the following figure where each run is replicated four times. A variability analysis can be conducted for this design. DOE++ calculates eight standard deviation values corresponding to each treatment of the design (see second following figure). Then, the design is analyzed as an unreplicated <math>{2}^{3}\,\!</math> design with the standard deviations (displayed as Y Standard Deviation. in second following figure) as the response. The normal probability plot of effects identifies <math>AC\,\!</math> as the effect that influences variability (see third figure following). Based on the effect coefficients obtained in the fourth figure following, the model for Y Std. is:

::<math>\begin{align}
& \text{Y Std}\text{.}= & 0.6779+0.2491\cdot AC \\
& = & 0.6779+0.2491{{x}_{1}}{{x}_{3}}
\end{align}\,\!</math>

Based on the model, the experimenter has two choices to minimize variability (by minimizing Y Std.). The first choice is that <math>{{x}_{1}}\,\!</math> should be <math>1\,\!</math> (i.e., <math>A\,\!</math> should be set at the high level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the low level). The second choice is that <math>{{x}_{1}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>A\,\!</math> should be set at the low level) and <math>{{x}_{3}}\,\!</math> should be <math>-1\,\!</math> (i.e., <math>C\,\!</math> should be set at the high level). The experimenter can select the most feasible choice.

[[Image:doe7.30.png|center|391px|A <math>2^3\,\!</math> design with four replicated response values that can be used to conduct a variability analysis.]]
 

[[Image:doe7_31.png|center|727px|Variability analysis in DOE++.|link=]]
 

[[Image:doe7_32.png|center|650px|Normal probability plot of effects for the variability analysis example.|link=]]
 

[[Image:doe7_33.png|center|727px|Effect coefficients for the variability analysis example.|link=]]

 

==Two Level Fractional Factorial Designs==

As the number of factors in a two level factorial design increases, the number of runs for even a single replicate of the <math>{2}^{k}\,\!</math> design becomes very large. For example, a single replicate of an eight factor two level experiment would require 256 runs. Fractional factorial designs can be used in these cases to draw out valuable conclusions from fewer runs. The basis of fractional factorial designs is the ''sparsity of effects'' principle.[[DOE_References|[Wu, 2000]]] The principle states that, most of the time, responses are affected by a small number of main effects and lower order interactions, while higher order interactions are relatively unimportant. Fractional factorial designs are used as screening experiments during the initial stages of experimentation. At these stages, a large number of factors have to be investigated and the focus is on the main effects and two factor interactions. These designs obtain information about main effects and lower order interactions with fewer experiment runs by confounding these effects with unimportant higher order interactions. As an example, consider a <math>{2}^{8}\,\!</math> design that requires 256 runs. This design allows for the investigation of 8 main effects and 28 two factor interactions. However, 219 degrees of freedom are devoted to three factor or higher order interactions. This full factorial design can prove to be very inefficient when these higher order interactions can be assumed to be unimportant. Instead, a fractional design can be used here to identify the important factors that can then be investigated more thoroughly in subsequent experiments. In unreplicated fractional factorial designs, no degrees of freedom are available to calculate the error sum of squares and the techniques mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks| Unreplicated <math>2^k</math> designs]] should be employed for the analysis of these designs.

==Half-fraction Designs==

A half-fraction of the <math>{2}^{k}\,\!</math> design involves running only half of the treatments of the full factorial design. For example, consider a <math>{2}^{3}\,\!</math> design that requires eight runs in all. The design matrix for this design is shown in the figure (a) below. A half-fraction of this design is the design in which only four of the eight treatments are run. The fraction is denoted as <math>{2}^{3-1}\,\!</math> with the "<math>-1\,\!</math>" in the index denoting a half-fraction. Assume that the treatments chosen for the half-fraction design are the ones where the interaction <math>ABC\,\!</math> is at the high level (i.e., only those rows are chosen from the following figure (a) where the column for <math>ABC\,\!</math> has entries of 1). The resulting <math>{2}^{3-1}\,\!</math> design has a design matrix as shown in figure (b) below.

[[Image:doe7.34.png|center|330px|Half-fractions of the <math>2^3\,\!</math> design. (a) shows the full factorial <math>2^3\,\!</math> design, (b) shows the <math>2^{3-1}\,\!</math> design with the defining relation <math>I=ABC\,\!</math> and (c) shows the <math>{2}^{3-1}\,\!</math> design with the defining relation <math>I=-ABC\,\!</math>.]]

In the <math>{2}^{3-1}\,\!</math> design of figure (b), since the interaction <math>ABC\,\!</math> is always included at the same level (the high level represented by 1), it is not possible to measure this interaction effect. The effect, <math>ABC\,\!</math>, is called the ''generator'' or ''word'' for this design. It can be noted that, in the design matrix of the following figure (b), the column corresponding to the intercept, <math>I\,\!</math>, and column corresponding to the interaction <math>ABC\,\!</math>, are identical. The identical columns are written as <math>I=ABC\,\!</math> and this equation is called the ''defining relation'' for the design. In DOE++, the present <math>{2}^{3-1}\,\!</math> design can be obtained by specifying the design properties as shown in the following figure.

[[Image:doe7_35.png|center|700px|Design properties for the <math>2^{3-1}\,\!</math> design.|link=]]

The defining relation, <math>I=ABC\,\!</math>, is entered in the Fraction Generator window as shown next.

[[Image:doe7_36.png|center|700px|Specifying the defining relation for the <math>2^{3-1}\,\!</math> design.|link=]]

Note that in the figure following that, the defining relation is specified as <math>C=AB\,\!</math>. This relation is obtained by multiplying the defining relation, <math>I=ABC\,\!</math>, by the last factor, <math>C\,\!</math>, of the design.

===Calculation of Effects===

Using the four runs of the <math>{2}^{3-1}\,\!</math> design in figure (b) discussed above, the main effects can be calculated as follows:

::<math>\begin{align}
& A= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& B= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& C= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

where <math>a\,\!</math>, <math>b\,\!</math>, <math>c\,\!</math> and <math>abc\,\!</math> are the treatments included in the <math>{2}^{3-1}\,\!</math> design.

Similarly, the two factor interactions can also be obtained as:

::<math>\begin{align}
& BC= & \frac{(a+abc)}{2}-\frac{(b+c)}{2}=\frac{1}{2}(a-b-c+abc) \\
& AC= & \frac{(b+abc)}{2}-\frac{(a+c)}{2}=\frac{1}{2}(-a+b-c+abc) \\
& AB= & \frac{(c+abc)}{2}-\frac{(a+b)}{2}=\frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

The equations for <math>A\,\!</math> and <math>BC\,\!</math> above result in the same effect values showing that effects <math>A\,\!</math> and <math>BC\,\!</math> are confounded in the present <math>{2}^{3-1}\,\!</math> design. Thus, the quantity, <math>\tfrac{1}{2}(a-b-c+abc),\,\!</math> estimates <math>A+BC\,\!</math> (i.e., both the main effect <math>A\,\!</math> and the two-factor interaction <math>BC\,\!</math>). The effects, <math>A\,\!</math> and <math>BC,\,\!</math> are called ''aliases''. From the remaining equations given above, it can be seen that the other aliases for this design are <math>B\,\!</math> and <math>AC\,\!</math>, and <math>C\,\!</math> and <math>AB\,\!</math>. Therefore, the equations to calculate the effects in the present <math>{2}^{3-1}\,\!</math> design can be written as follows:

::<math>\begin{align}
& A+BC= & \frac{1}{2}(a-b-c+abc) \\
& B+AC= & \frac{1}{2}(-a+b-c+abc) \\
& C+AB= & \frac{1}{2}(-a-b+c+abc)
\end{align}\,\!</math>

===Calculation of Aliases===

Aliases for a fractional factorial design can be obtained using the defining relation for the design. The defining relation for the present <math>{2}^{3-1}\,\!</math> design is:

::<math>I=ABC\,\!</math>

Multiplying both sides of the previous equation by the main effect, <math>A,\,\!</math> gives the alias effect of <math>A\,\!</math> :

::<math>\begin{align}
& A\cdot I= & A\cdot ABC \\
& A= & {{A}^{2}}BC \\
& A= & BC
\end{align}\,\!</math>

Note that in calculating the alias effects, any effect multiplied by <math>I\,\!</math> remains the same (<math>A\cdot I=A\,\!</math>), while an effect multiplied by itself results in <math>I\,\!</math> (<math>{{A}^{2}}=I\,\!</math>). Other aliases can also be obtained:

::<math>\begin{align}
& B\cdot I= & B\cdot ABC \\
& B= & A{{B}^{2}}C \\
& B= & AC
\end{align}\,\!</math>

:and:

::<math>\begin{align}
& C\cdot I= & C\cdot ABC \\
& C= & AB{{C}^{2}} \\
& C= & AB
\end{align}\,\!</math>

===Fold-over Design===

If it can be assumed for this design that the two-factor interactions are unimportant, then in the absence of <math>BC\,\!</math>, <math>AC\,\!</math> and <math>AB\,\!</math>, the equations for (A+BC), (B+AC) and (C+AB) can be used to estimate the main effects, <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math>, respectively. However, if such an assumption is not applicable, then to uncouple the main effects from their two factor aliases, the alternate fraction that contains runs having <math>ABC\,\!</math> at the lower level should be run. The design matrix for this design is shown in the preceding figure (c). The defining relation for this design is <math>I=-ABC\,\!</math> because the four runs for this design are obtained by selecting the rows of the preceding figure (a) for which the value of the <math>ABC\,\!</math> column is <math>-1\,\!</math>. The aliases for this fraction can be obtained as explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]] as <math>A=-BC\,\!</math>, <math>B=-AC\,\!</math> and <math>C=-AB\,\!</math>. The effects for this design can be calculated as:

::<math>\begin{align}
& A-BC= & \frac{1}{2}(ab+ac-(1)-bc) \\
& B-AC= & \frac{1}{2}(ab-ac+(1)-bc) \\
& C-AB= & \frac{1}{2}(-ab+ac-(1)+bc)
\end{align}\,\!</math>

These equations can be combined with the equations for (A+BC), (B+AC) and (C+AB) to obtain the de-aliased main effects and two factor interactions. For example, adding equations (A+BC) and (A-BC) returns the main effect <math>A\,\!</math>.

::<math>\begin{align}
& 2A= & \frac{1}{2}(a-b-c+abc)+ \\
& & \frac{1}{2}(ab+ac-(1)-bc)
\end{align}\,\!</math>

The process of augmenting a fractional factorial design by a second fraction of the same size by simply reversing the signs (of all effect columns except <math>I\,\!</math>) is called ''folding over''. The combined design is referred to as a ''fold-over design''.

==Quarter and Smaller Fraction Designs==

At times, the number of runs even for a half-fraction design are very large. In these cases, smaller fractions are used. A quarter-fraction design, denoted as <math>{2}^{k-2}\,\!</math>, consists of a fourth of the runs of the full factorial design. Quarter-fraction designs require two defining relations. The first defining relation returns the half-fraction or the <math>{2}^{k-1}\,\!</math> design. The second defining relation selects half of the runs of the <math>{2}^{k-1}\,\!</math> design to give the quarter-fraction. For example, consider the <math>{2}^{4}\,\!</math> design. To obtain a <math>{2}^{4-2}\,\!</math> design from this design, first a half-fraction of this design is obtained by using a defining relation. Assume that the defining relation used is <math>I=ABCD\,\!</math>. The design matrix for the resulting <math>{2}^{4-1}\,\!</math> design is shown in figure (a) below. Now, a quarter-fraction can be obtained from the <math>{2}^{4-1}\,\!</math> design shown in figure (a) below using a second defining relation <math>I=AD\,\!</math>. The resulting <math>{2}^{4-2}\,\!</math> design obtained is shown in figure (b) below.

[[Image:doe7.37.png|center|465px|Fractions of the <math>2^4\,\!</math> design - Figure (a) shows the <math>2^{4-1}</math> design with the defining relation <math>I=ABCD\,\!</math> and (b) shows the <math>2^{4-2}\,\!</math> design with the defining relation <math>I=ABCD=AD=BC\,\!</math>.]]

The complete defining relation for this <math>{2}^{4-2}\,\!</math> design is:

::<math>I=ABCD=AD=BC\,\!</math>

Note that the effect, <math>BC,\,\!</math> in the defining relation is the generalized interaction of <math>ABCD\,\!</math> and <math>AD\,\!</math> and is obtained using <math>(ABCD)(AD)={{A}^{2}}BC{{D}^{2}}=BC\,\!</math>. In general, a <math>{2}^{k-p}\,\!</math> fractional factorial design requires <math>p\,\!</math> independent generators. The defining relation for the design consists of the <math>p\,\!</math> independent generators and their <math>{2}^{p}\,\!</math> - (<math>p\,\!</math> +1) generalized interactions.

===Calculation of Aliases===

The alias structure for the present <math>{2}^{4-2}\,\!</math> design can be obtained using the defining relation of equation (I=ABCD=AD=BC) following the procedure explained in [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction Designs]]. For example, multiplying the defining relation by <math>A\,\!</math> returns the effects aliased with the main effect, <math>A\,\!</math>, as follows:

::<math>\begin{align}
& A\cdot I= & A\cdot ABCD=A\cdot AD=A\cdot BC \\
& A= & {{A}^{2}}BCD={{A}^{2}}D=ABC \\
& A= & BCD=D=ABC
\end{align}\,\!</math>

Therefore, in the present <math>{2}^{4-2}\,\!</math> design, it is not possible to distinguish between effects <math>A\,\!</math>, <math>D\,\!</math>, <math>BCD\,\!</math> and <math>ABC\,\!</math>. Similarly, multiplying the defining relation by <math>B\,\!</math> and <math>AB\,\!</math> returns the effects that are aliased with these effects:

::<math>\begin{align}
& B= & ACD=ABD=C \\
& AB= & CD=AD=AC
\end{align}\,\!</math>

Other aliases can be obtained in a similar way. It can be seen that each effect in this design has three aliases. In general, each effect in a <math>{2}^{k-p}\,\!</math> design has <math>{2}^{p-1}\,\!</math> aliases.
The aliases for the <math>{2}^{4-2}\,\!</math> design show that in this design the main effects are aliased with each other (<math>A\,\!</math> is aliased with <math>D\,\!</math> and <math>B\,\!</math> is aliased with <math>C\,\!</math>). Therefore, this design is not a useful design and is not available in DOE++. It is important to ensure that main effects and lower order interactions of interest are not aliased in a fractional factorial design. This is known by looking at the resolution of the fractional factorial design.

==Design Resolution==

The resolution of a fractional factorial design is defined as the number of factors in the lowest order effect in the defining relation. For example, in the defining relation <math>I=ABCD=AD=BC\,\!</math> of the previous <math>{2}^{4-2}\,\!</math> design, the lowest-order effect is either <math>AD\,\!</math> or <math>BC,\,\!</math> containing two factors. Therefore, the resolution of this design is equal to two. The resolution of a fractional factorial design is represented using Roman numerals. For example, the previously mentioned <math>{2}^{4-2}\,\!</math> design with a resolution of two can be represented as 2 <math>_{\text{II}}^{4-2}\,\!</math>. The resolution provides information about the confounding in the design as explained next:
 
 
#'''Resolution III Designs''' In these designs, the lowest order effect in the defining relation has three factors (e.g., a <math>{2}^{5-2}\,\!</math> design with the defining relation <math>I=ABDE=ABC=CDE\,\!</math>). In resolution III designs, no main effects are aliased with any other main effects, but main effects are aliased with two factor interactions. In addition, some two factor interactions are aliased with each other.
#'''Resolution IV Designs''' In these designs, the lowest order effect in the defining relation has four factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABDE\,\!</math>). In resolution IV designs, no main effects are aliased with any other main effects or two factor interactions. However, some main effects are aliased with three factor interactions and the two factor interactions are aliased with each other.
#'''Resolution V Designs''' In these designs the lowest order effect in the defining relation has five factors (e.g., a <math>{2}^{5-1}\,\!</math> design with the defining relation <math>I=ABCDE\,\!</math>). In resolution V designs, no main effects or two factor interactions are aliased with any other main effects or two factor interactions. However, some main effects are aliased with four factor interactions and the two factor interactions are aliased with three factor interactions.

Fractional factorial designs with the highest resolution possible should be selected because the higher the resolution of the design, the less severe the degree of confounding. In general, designs with a resolution less than III are never used because in these designs some of the main effects are aliased with each other. The table below shows fractional factorial designs with the highest available resolution for three to ten factor designs along with their defining relations.

[[Image:doet7.3.png|center|474px|Highest resolution designs available for fractional factorial designs with 3 to 10 factors.]]

All of the two level fractional factorial designs available in DOE++ are shown next.

[[Image:doe7.38.png|center|471px|Two level fractional factorial designs available in DOE++ and their resolutions.]]

===Minimum Aberration Designs===
At times, different designs with the same resolution but different aliasing may be available. The best design to select in such a case is the minimum aberration design. For example, all <math>{2}^{7-2}\,\!</math> designs in the fourth table have a resolution of four (since the generator with the minimum number of factors in each design has four factors). Design <math>1\,\!</math> has three generators of length four (<math>ABCF,\,\!</math> <math>BCDG,\,\!</math> <math>ADFG\,\!</math>). Design <math>2\,\!</math> has two generators of length four (<math>ABCF,\,\!</math> <math>ADEG\,\!</math>). Design <math>3\,\!</math> has one generator of length four (<math>CEFG\,\!</math>). Therefore, design <math>3\,\!</math> has the least number of generators with the minimum length of four. Design <math>3\,\!</math> is called the minimum aberration design. It can be seen that the alias structure for design <math>3\,\!</math> is less involved compared to the other designs. For details refer to [[DOE_References|[Wu, 2000]]].

[[Image:doet7.4.png|center|432px|Three <math>2_{IV}^{7-2}\,\!</math> designs with different defining relations.]]

====Example====

The design of an automobile fuel cone is thought to be affected by six factors in the manufacturing process: cavity temperature (factor <math>A\,\!</math>), core temperature (factor <math>B\,\!</math>), melt temperature (factor <math>C\,\!</math>), hold pressure (factor <math>D\,\!</math>), injection speed (factor <math>E\,\!</math>) and cool time (factor <math>F\,\!</math>). The manufacturer of the fuel cone is unable to run the <math>{2}^{6}=64\,\!</math> runs required to complete one replicate for a two level full factorial experiment with six factors. Instead, they decide to run a fractional factorial design. Considering that three factor and higher order interactions are likely to be inactive, the manufacturer selects a <math>{2}^{6-2}\,\!</math> design that will require only 16 runs. The manufacturer chooses the resolution IV design which will ensure that all main effects are free from aliasing (assuming three factor and higher order interactions are absent). However, in this design the two factor interactions may be aliased with each other. It is decided that, if important two factor interactions are found to be present, additional experiment trials may be conducted to separate the aliased effects. The performance of the fuel cone is measured on a scale of 1 to 15. In DOE++, the design for this experiment is set up using the properties shown in the following figure. The Fraction Generators for the design, <math>E=ABC\,\!</math> and <math>F=BCD\,\!</math>, are the same as the defaults used in DOE++. The resulting <math>{2}^{6-2}\,\!</math> design and the corresponding response values are shown in the following two figures.

[[Image:doe7_39.png|center|644px|Design properties for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

[[Image:doe7_40.png|center|534px|Experiment design for the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The complete alias structure for the 2 <math>_{\text{IV}}^{6-2}\,\!</math> design is shown next.

<center><math>I=ABCE=ADEF=BCDF\,\!</math></center>

 
<center><math>\begin{align}
& A= & BCE=DEF=ABCDF \\
& B= & ACE=CDF=ABDEF \\
& C= & ABE=BDF=ACDEF \\
& D= & AEF=BCF=ABCDE \\
& E= & ABC=ADF=BCDEF \\
& F= & ADE=BCD=ABCEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& AB= & CE=ACDF=BDEF \\
& AC= & BE=ABDF=CDEF \\
& AD= & EF=ABCF=BCDE \\
& AE= & BC=DF=ABCDEF \\
& AF= & DE=ABCD=BCEF \\
& BD= & CF=ABEF=ACDE \\
& BF= & CD=ABDE=ACEF
\end{align}\,\!</math></center>

<center><math>\begin{align}
& ABD= & ACF=BEF=CDE \\
& ABF= & ACD=BDE=CEF
\end{align}\,\!</math></center>

In DOE++, the alias structure is displayed in the Design Summary and as part of the Design Evaluation result, as shown next:

[[Image:doe7_41.png|center|700px|Alias structure for the experiment design in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

The normal probability plot of effects for this unreplicated design shows the main effects of factors <math>C\,\!</math> and <math>D\,\!</math> and the interaction effect, <math>BF\,\!</math>, to be significant (see the following figure).

[[Image:doe7_42.png|center|650px|Normal probability plot of effects for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

From the alias structure, it can be seen that for the present design interaction effect, <math>BF,\,\!</math> is confounded with <math>CD\,\!</math>. Therefore, the actual source of this effect cannot be known on the basis of the present experiment. However because neither factor <math>B\,\!</math> nor <math>F\,\!</math> is found to be significant there is an indication the observed effect is likely due to interaction, <math>CD\,\!</math>. To confirm this, a follow-up <math>{2}^{2}\,\!</math> experiment is run involving only factors <math>B\,\!</math> and <math>F\,\!</math>. The interaction, <math>BF\,\!</math>, is found to be inactive, leading to the conclusion that the interaction effect in the original experiment is effect, <math>CD\,\!</math>. Given these results, the fitted regression model for the fuel cone design as per the coefficients obtained from DOE++ is shown next.

::<math>\hat{y}=7.6875+C+2\cdot D+2.1875\cdot CD\,\!</math>

[[Image:doe7_43.png|center|700px|Effect coefficients for the experiment in the [[Two_Level_Factorial_Experiments#Example_7| example]].|link=]]

==Projection==

Projection refers to the reduction of a fractional factorial design to a full factorial design by dropping out some of the factors of the design. Any fractional factorial design of resolution, <math>R,\,\!</math> can be reduced to complete factorial designs in any subset of <math>R-1\,\!</math> factors. For example, consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. The resolution of this design is four. Therefore, this design can be reduced to full factorial designs in any three (<math>4-1=3\,\!</math>) of the original seven factors (by dropping the remaining four of factors). Further, a fractional factorial design can also be reduced to a full factorial design in any <math>R\,\!</math> of the original factors, as long as these <math>R\,\!</math> factors are not part of the generator in the defining relation. Again consider the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. This design can be reduced to a full factorial design in four factors provided these four factors do not appear together as a generator in the defining relation. The complete defining relation for this design is:

::<math>\begin{align}
& I= & ABCE=BCDF=ACDG=ADEF=ABFG=BDEG=CEFG
\end{align}\,\!</math>

Therefore, there are seven four factor combinations out of the 35 (<math>(_{7}^{4})=35\,\!</math>) possible four-factor combinations that are used as generators in the defining relation. The designs with the remaining 28 four factor combinations would be full factorial 16-run designs. For example, factors <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math> do not occur as a generator in the defining relation of the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design. If the remaining factors, <math>E\,\!</math>, <math>F\,\!</math> and <math>G\,\!</math>, are dropped, the 2 <math>_{\text{IV}}^{7-3}\,\!</math> design will reduce to a full factorial design in <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math> and <math>D\,\!</math>.

==Resolution III Designs==

At times, the factors to be investigated in screening experiments are so large that even running a fractional factorial design is impractical. This can be partially solved by using resolution III fractional factorial designs in the cases where three factor and higher order interactions can be assumed to be unimportant. Resolution III designs, such as the 2 <math>_{\text{III}}^{3-1}\,\!</math> design, can be used to estimate <math>k\,\!</math> main effects using just <math>k+1\,\!</math> runs. In these designs, the main effects are aliased with two factor interactions. Once the results from these designs are obtained, and knowing that three factor and higher order interactions are unimportant, the experimenter can decide if there is a need to run a fold-over design to de-alias the main effects from the two factor interactions. Thus, the 2 <math>_{\text{III}}^{3-1}\,\!</math> design can be used to investigate three factors in four runs, the 2 <math>_{\text{III}}^{7-4}\,\!</math> design can be used to investigate seven factors in eight runs, the 2 <math>_{\text{III}}^{15-11}\,\!</math> design can be used to investigate fifteen factors in sixteen runs and so on.

====Example====

{{:Resolution_III_Design_Example}}

==Alias Matrix==
In [[Two_Level_Factorial_Experiments#Half-fraction_Designs| Half-fraction designs]] and [[Two_Level_Factorial_Experiments#Quarter_and_Smaller_Fraction_Designs| Quarter and Smaller Fraction Designs]], the alias structure for fractional factorial designs was obtained using the defining relation. However, this method of obtaining the alias structure is not very efficient when the alias structure is very complex or when partial aliasing is involved. One of the ways to obtain the alias structure for any design, regardless of its complexity, is to use the alias matrix. The alias matrix for a design is calculated using <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> where <math>{{X}_{1}}\,\!</math> is the portion of the design matrix, <math>X,\,\!</math> that contains the effects for which the aliases need to be calculated, and <math>{{X}_{2}}\,\!</math> contains the remaining columns of the design matrix, other than those included in <math>{{X}_{1}}\,\!</math>.

To illustrate the use of the alias matrix, consider the design matrix for the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design (using the defining relation <math>I=ABCD\,\!</math>) shown next:

[[Image:Chapter7__879.png|center|link=]]

The alias structure for this design can be obtained by defining <math>{{X}_{1}}\,\!</math> using eight columns since the 2 <math>_{\text{IV}}^{4-1}\,\!</math> design estimates eight effects. If the first eight columns of <math>X\,\!</math> are used then <math>{{X}_{1}}\,\!</math> is:

[[Image:Chapter7__884.png|center|link=]]

<math>{{X}_{2}}\,\!</math> is obtained using the remaining columns as:

[[Image:Chapter7__886.png|center|link=]]

Then the alias matrix <math>{{(X_{1}^{\prime }{{X}_{1}})}^{-1}}X_{1}^{\prime }{{X}_{2}}\,\!</math> is:

[[Image:Chapter7__888.png|center|link=]]

The alias relations can be easily obtained by observing the alias matrix as:

<center><math>\begin{align}
& I= & ABCD \\
& A= & BCD \\
& B= & ACD \\
& AB= & CD \\
& C= & ABD \\
& AC= & BD \\
& BC= & AD \\
& D= & ABC
\end{align}\,\!</math></center>

Highly Fractional Factorial Designs

2017-08-10T17:23:16Z

Richard House: /* Example */

{{Template:Doebook|9}}
This chapter discusses factorial designs that are commonly used in designed experiments, but are not necessarily limited to two level factors. These designs are the [[Highly_Fractional_Factorial_Designs#Plackett-Burman_Designs|Plackett-Burman designs]] and [[Highly_Fractional_Factorial_Designs#Taguchi.27s_Orthogonal_Arrays|Taguchi's orthogonal arrays]].

==Plackett-Burman Designs==

It was mentioned in [[Two_Level_Factorial_Experiments| Two Level Factorial Experiments]] that resolution III designs can be used as highly fractional designs to investigate <math>k\,\!</math> main effects using <math>k+1\,\!</math> runs (provided that three factor and higher order interaction effects are not important to the experimenter). A limitation with these designs is that all runs in these designs have to be a power of 2. The valid runs for these designs are 4, 8, 16, 32, etc. Therefore, the next design after the 2 <math>_{\text{III}}^{3-1}\,\!</math> design with 4 runs is the 2 <math>_{\text{III}}^{7-4}\,\!</math> design with 8 runs, and the design after this is the 2 <math>_{\text{III}}^{15-11}\,\!</math> design with 32 runs and so on, as shown in the next table.

[[Image:doet8.1.png|center|487px|Highly fractional designs to investigate main effects.]]

Plackett-Burman designs solve this problem. These designs were proposed by R. L. Plackett and J.P. Burman (1946). They allow the estimation of <math>k\,\!</math> main effects using <math>k+1\,\!</math> runs. In these designs, runs are a multiple of 4 (i.e., 4, 8, 12, 16, 20 and so on). When the runs are a power of 2, the designs correspond to the resolution III two factor fractional factorial designs. Although Plackett-Burman designs are all two level orthogonal designs, the alias structure for these designs is complicated when runs are not a power of 2.

As an example, consider the 12-run Plackett-Burman design shown in the figure below.

[[Image:doe8.1.png|center|400px|12-run Plackett-Burman design.]]

If 11 main effects are to be estimated using this design, then each of these main effects is partially aliased with all other two factor interactions not containing that main effect. For example, the main effect <math>A\,\!</math> is partially aliased with all two factor interactions except <math>AB\,\!</math>, <math>AC\,\!</math>, <math>AD\,\!</math>, <math>AE\,\!</math>, <math>AF\,\!</math>, <math>AG\,\!</math>, <math>AH\,\!</math>, <math>AJ\,\!</math>, <math>AK\,\!</math> and <math>AL\,\!</math>. There are 45 such two factor interactions that are aliased with <math>A\,\!</math>.

::<math>\begin{align}
& A= & A-\frac{1}{3}BC-\frac{1}{3}BD+\frac{1}{3}CD-\frac{1}{3}BE-\frac{1}{3}CE+\frac{1}{3}DE+... \\
& & ...+\frac{1}{3}EL-\frac{1}{3}FL-\frac{1}{3}GL+\frac{1}{3}HL+\frac{1}{3}JL-\frac{1}{3}KL
\end{align}\,\!</math>

Due to the complex aliasing, Plackett-Burman designs involving a large number of factors should be used with care. Some of the Plackett-Burman designs available in DOE++ are included in [[Plackett-Burman_Designs|Appendix C]].

==Taguchi's Orthogonal Arrays==

Taguchi's orthogonal arrays are highly fractional orthogonal designs proposed by Dr. Genichi Taguchi, a Japanese industrialist. These designs can be used to estimate main effects using only a few experimental runs. These designs are not only applicable to two level factorial experiments; they can also investigate main effects when factors have more than two levels. Designs are also available to investigate main effects for certain mixed level experiments where the factors included do not have the same number of levels. As in the case of Placket-Burman designs, these designs require the experimenter to assume that interaction effects are unimportant and can be ignored. A few of Taguchi's orthogonal arrays available in DOE++ are included in [[Taguchi_Orthogonal_Arrays|Appendix D]].

Some of Taguchi's arrays, with runs that are a power of 2, are similar to the corresponding 2 <math>_{\text{III}}^{k-f}\,\!</math> designs. For example, consider the L4 array shown in figure (a) below. The L4 array is denoted as L4(2^3) in a DOE folio. L4 means the array requires 4 runs. 2^3 indicates that the design estimates up to three main effects at 2 levels each. The L4 array can be used to estimate three main effects using four runs provided that the two factor and three factor interactions can be ignored. Figure (b) below shows the 2 <math>_{\text{III}}^{3-1}\,\!</math> design (defining relation <math>I=-ABC\,\!</math> ) which also requires four runs and can be used to estimate up to three main effects, assuming that all two factor and three factor interactions are unimportant. A comparison between the two designs shows that the columns in the two designs are the same except for the arrangement of the columns. In figure (c) below, columns of the L4 array are marked with the name of the effect from the corresponding column of the 2 <math>_{\text{III}}^{3-1}\,\!</math> design.

[[Image:doe8.3.png|center|400px|Taguchi's L4 orthogonal array - Figure (a) shows the design, (b) shows the <math>2_{III}^{3-1} \,\!</math> design with the defining relation <math>I=-ABC \,\!</math> and (c) marks the columns of the L4 array with the corresponding columns of the design in (b).]]

Similarly, consider the L8(2^7) array shown in figure (a) below. This design can be used to estimate up to seven main effects using eight runs. This array is again similar to the 2 <math>_{\text{III}}^{7-4}\,\!</math> design shown in figure (b) below, except that the aliasing between the columns of the two designs differs in sign for some of the columns (see figure (c)).

[[Image:doe8.4.png|center|400px|Taguchi's L8 orthogonal array - Figure (a) shows the design, (b) shows the <math>2_{III}^{7-4} \,\!</math> design with the defining relation <math>I=ABD=ACE=BCF=ABCG \,\!</math> and (c) marks the columns of the L8 array with the corresponding columns of the design in (b).]]

The L8 array can also be used as a full factorial three factor experiment design in the same way as a <math>2^{3}\,\!</math> design. However, the orthogonal arrays should be used carefully in such cases, taking into consideration the alias relationships between the columns of the array. For the L8 array, figure (c) above shows that the third column of the array is the product of the first two columns. If the L8 array is used as a two level full factorial design in the place of a 2 <math>^{3}\,\!</math> design, and if the main effects are assigned to the first three columns, the main effect assigned to the third column will be aliased with the two factor interaction of the first two main effects. The proper assignment of the main effects to the columns of the L8 array requires the experimenter to assign the three main effects to the first, second and fourth columns. These columns are sometimes referred to as the ''preferred columns'' for the L8 array. To know the preferred columns for any of the orthogonal arrays, the alias relationships between the array columns must be known. The alias relations between the main effects and two factor interactions of the columns for the L8 array are shown in the next table.

[[Image:doet8.2.png|center|400px|Alias relations for the L8 array.]]

The cell value in any (<math>i,j\,\!</math>) cell of the table gives the column number of the two factor interaction for the <math>i\,\!</math>th row and <math>j\,\!</math>th column. For example, to know which column is confounded with the interaction of the first and second columns, look at the value in the ( <math>1,2\,\!</math> ) cell. The value of 3 indicates that the third column is the same as the product of the first and second columns. The alias relations for some of Taguchi's orthogonal arrays are available in [[Alias_Relations_for_Taguchi_Orthogonal_Arrays|Appendix E]].

===Example===

Recall the experiment to investigate factors affecting the surface finish of automobile brake drums discussed in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]]. The three factors investigated in the experiment were honing pressure (factor A), number of strokes (factor B) and cycle time (factor C). Assume that you used Taguchi's L8 orthogonal array to investigate the three factors instead of the <math>{2}^{3}\,\!</math> design that was used in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]]. Based on the discussion in the previous section, the preferred columns for the L8 array are the first, second and fourth columns. Therefore, the three factors should be assigned to these columns. The three factors are assigned to these columns based on the figure (c) above, so that you can easily compare results obtained from the L8 array to the ones included in [[Two_Level_Factorial_Experiments| Two Level Factorial Experiments]]. Based on this assignment, the L8 array for the two replicates, along with the respective response values, should be as shown in the third table. Note that to run the experiment using the L8 array, you would use only the first, the second and the fourth column to set the three factors.

[[Image:doet8.3.png|center|639px|Using Taguchi's L8 array to investigate factors affecting the surface finish of automobile brake drums.]]

The experiment design for this example can be set using the properties shown in the figure below.

[[Image:doe8_5.png|center|700px|Design properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example| example]].|link=]]

Note that for this design, the factor properties are set up as shown in the design summary.

[[Image:doe8_6.png|center|700px|Factor properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example| example]].|link=]]

The resulting design along with the response values is shown in the figure below.

[[Image:doe8_7.png|center|650px|Experiment design for the [[Highly_Fractional_Factorial_Designs#Example| example]].|link=]]

And the results from the DOE folio for the design are shown in the next figure.

[[Image:doe8_8_1.png|center|700px|Results for the experiment in the example.|link=]]

The results identify honing pressure, number of strokes, and the interaction between honing pressure and cycle time to be significant effects. This is identical to the conclusion obtained from the <math>{2}^{3}\,\!</math> design used in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]].

=== Preferred Columns in Taguchi OA===
One of the difficulties of using Taguchi OA is to assign factors to the appropriate columns of the array. For example, take a simple Taguchi OA L8(2^7), which can be used for experiments with up to 7 factors. If you have only 3 factors, which 3 columns in this array should be used? DOE++ provides a simple utility to help users utilize Taguchi OA more effectively by assigning factors to the appropriate columns.
Let’s use Taguchi OA L8(2^7) as an example. The design table for this array is:

[[Image:DOEtableChapter9.png|center]]

This is a fractional factorial design for 7 factors. For any fractional factorial design, the first thing we need to do is check its alias structure. In general, the alias structures for Taguchi OAs are very complicated. People usually use the following table to represent the alias relations between each factor. For the above orthogonal array, the alias table is:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|1|| 2|| 3|| 4|| 5|| 6|| 7
|-
|2x3|| 1x3|| 1x2|| 1x5|| 1x4|| 1x7|| 1x6
|-
|4x5|| 4x6|| 4x7|| 2x6|| 2x7|| 2x4|| 2x5
|-
|6x7|| 5x7|| 5x6|| 3x7|| 3x6|| 3x5|| 3x4
|}

In the above table, an Arabic number is used to represent a factor. For instance, “1” represents the factor assigned to the 1st column in the array. “2x3” represents the interaction effect of the two factors assigned to column 2 and 3. Each column in the above alias table lists all the 2-way interaction effects that are aliased with the main effect of the factor assigned to this column. For example, for the 1st column, the main effect of the factor assigned to it is aliased with interaction effects of 2x3, 4x5 and 6x7.
If an experiment has only 3 factors and these 3 factors A, B and C are assigned to the first 3 columns of the above L8(2^7) array, then the design table will be:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|Run|| A (Column 1)|| B (Column 2)|| C (Column 3)
|-
|1|| 1|| 1|| 1
|-
|2|| 1|| 1|| 1
|-
|3|| 1|| 2|| 2
|-
|4|| 1|| 2|| 2
|-
|5|| 2|| 1|| 2
|-
|6|| 2|| 1|| 2
|-
|7|| 2|| 2|| 1
|-
|8|| 2|| 2|| 1
|}

The alias structure for the above table is:

{| style="text-align:center;" cellpadding="2" align="center"
|-
|[I] = I – ABC
|-
|[A] = A – BC
|-
|[B] = B – AC
|-
|[C] = C – AB
|}

This is a resolution 3 design. All the main effects are aliased with 2-way interactions. There are many ways to choose 3 columns from the 7 columns of L8(2^7). If the 3 factors are assigned to column 1, 2, and 4, then the design table is:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|Run|| A (Column 1)|| B (Column 2)|| C (Column 4)
|-
|1|| 1|| 1|| 1
|-
|2|| 1|| 1|| 1
|-
|3|| 1|| 2|| 2
|-
|4|| 1|| 2|| 2
|-
|5|| 2|| 1|| 2
|-
|6|| 2|| 1|| 2
|-
|7|| 2|| 2|| 1
|-
|8|| 2|| 2|| 1
|}

For experiments using the above design table, all the effects will be alias free. Therefore, this design is much better than the previous one which used column 1, 2, and 3 of L8(2^7). Although both designs have the same number of runs, more information can be obtained from this design since it is alias free.

Clearly, it is very important to assign factors to the right columns when applying Taguchi OA. DOE++ can help users automatically choose the right columns when the number of factors is less than the number of columns in a Taguchi OA. The selection is based on the specified model terms by users. Let’s use an example to explain this.

====Example====
Design an experiment with 3 qualitative factors. Factors A and B have 2 levels; factor C has 4 levels. The experimenters are interested in all the main effects and the interaction effect AC.

Based on this requirement, Taguchi OA L16(2^6*4^3) can be used since it can handle both 2 level and 4 level factors. It has 9 columns. The first 6 columns are used for 2 level factors and the last 3 columns are used for 4 level factors. We need to assign factor A and B to two of the first 6 columns, and assign factor C to one of the last 3 columns.

In a Weibull++ DOE folio, we can choose L16(2^6*4^3) in the Additional Settings.

[[Image:doe8_9.png|center|842px|Selecting the Taguchi OA design type.|link=]]

Click '''Specify Interaction Terms''' to specify the interaction terms that are of interest to the experimenters.

[[Image:doe8_10.png|center|420px|Specifying the interaction terms of interest.|link=]]

Based on the specified interaction effects, DOE++ will assign each factor to the appropriate column. In this case, they are column 1, 3, and 7 as shown below.

[[Image:doe8_11.png|center|842px|The interaction terms of interest have been specified.|link=]]

However, for a given Taguchi OA, it may not be possible to estimate all the specified interaction terms. If not all the requirements can be satisfied, the DOE folio will assign factors to columns that result in the least number of aliased effects. In this case, users should either use another Taguchi OA or other design types. The following example is one of these cases.

====Example====
Design an experiment for a test with 4 qualitative factors. Factors A and B have 2 levels; C and D have 4 levels. We are interested in all the main effects and the interaction effects AC and BD.

Assume again we want to use Taguchi OA L16(2^6*4^3). Click '''Specify Interaction Terms''' as shown above to specify the interaction effects that you want to estimate in the experiment.

[[Image:doe8_12.png|center|420px|Specifying the interaction terms of interest.|link=]]

When you click '''OK''', you will see a message warning that some of the specified interaction effects are aliased with main effects. These means that it is not possible to clearly estimate all the main effects and the specified interaction effects AC and BD.

This can be explained by checking the alias table of an L16(2^6*4^3) design as given below.

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|1|| 2|| 3|| 4|| 5|| 6|| 7|| 8|| 9
|-
|2x7|| 1x7|| 1x9|| 1x5|| 1x4|| 1x8|| 1x2|| 1x6|| 1x3
|-
|3x9|| 3x6|| 2x6|| 2x8|| 2x9|| 2x3|| 1x8|| 1x7|| 1x7
|-
|4x5|| 4x8|| 4x7|| 3x7|| 3x8|| 4x9|| 1x9|| 1x9|| 1x8
|-
|6x8|| 5x9|| 5x8|| 6x9|| 6x7|| 5x7|| 2x8|| 2x4|| 2x5
|-
|7x8|| 7x8|| 7x8|| 7x8|| 7x8|| 7x8|| 2x9|| 2x7|| 2x7
|-
|7x9|| 7x9|| 7x9|| 7x9|| 7x9|| 7x9|| 3x8|| 2x9|| 2x8
|-
|8x9|| 8x9|| 8x9|| 8x9|| 8x9|| 8x9|| 3x9|| 3x5|| 3x7
|-
| || || || || || ||4x8|| 3x7|| 3x8
|-
| || || || || || ||4x9|| 3x9|| 4x6
|-
| || || || || || ||5x6|| 4x7|| 4x7
|-
| || || || || || ||5x8|| 4x9|| 4x8
|-
| || || || || || ||5x9|| 5x7|| 5x7
|-
| || || || || || ||6x8|| 5x9|| 5x8
|-
| || || || || || ||6x9|| 6x7|| 6x7
|-
| || || || || || ||8x9|| 6x9|| 6x8
|-
| || || || || || || || 7x9|| 7x8
|}

From this table, we can see that it is impossible to clearly estimate both AC and BD. Factors C and D can only be assigned to the last three columns since they are 4 level factors. Assume we assign factor C to column 7 and factor D to column 8. Factor B (2 level) will be in one of the columns from 1 to 6. Therefore, effect BD will be one of the effects highlighted in the table, where the first term of the interaction is between 1 and 6 and the last term is 8 (i.e., factor D).

The above alias table shows that factor C is aliased with one of those highlighted effects. Thus, no matter which of the first six columns is assigned to factor B, the main effect C will be aliased with interaction effect BD. This is also true if C is assigned to column 8 or 9. Therefore, if L16(2^6*4^3) is used, there is no way to clearly estimate all the main effects and the interaction effects AC and BD. Another Taguchi OA or other design types such as a general full factorial design should be used. A more efficient way might be to create an optimal custom design that can clearly estimate all the specified terms. For more detail, please refer to the chapter on [[Optimal Custom Designs]].

Highly Fractional Factorial Designs

2017-08-10T17:22:18Z

Richard House: /* Example */

{{Template:Doebook|9}}
This chapter discusses factorial designs that are commonly used in designed experiments, but are not necessarily limited to two level factors. These designs are the [[Highly_Fractional_Factorial_Designs#Plackett-Burman_Designs|Plackett-Burman designs]] and [[Highly_Fractional_Factorial_Designs#Taguchi.27s_Orthogonal_Arrays|Taguchi's orthogonal arrays]].

==Plackett-Burman Designs==

It was mentioned in [[Two_Level_Factorial_Experiments| Two Level Factorial Experiments]] that resolution III designs can be used as highly fractional designs to investigate <math>k\,\!</math> main effects using <math>k+1\,\!</math> runs (provided that three factor and higher order interaction effects are not important to the experimenter). A limitation with these designs is that all runs in these designs have to be a power of 2. The valid runs for these designs are 4, 8, 16, 32, etc. Therefore, the next design after the 2 <math>_{\text{III}}^{3-1}\,\!</math> design with 4 runs is the 2 <math>_{\text{III}}^{7-4}\,\!</math> design with 8 runs, and the design after this is the 2 <math>_{\text{III}}^{15-11}\,\!</math> design with 32 runs and so on, as shown in the next table.

[[Image:doet8.1.png|center|487px|Highly fractional designs to investigate main effects.]]

Plackett-Burman designs solve this problem. These designs were proposed by R. L. Plackett and J.P. Burman (1946). They allow the estimation of <math>k\,\!</math> main effects using <math>k+1\,\!</math> runs. In these designs, runs are a multiple of 4 (i.e., 4, 8, 12, 16, 20 and so on). When the runs are a power of 2, the designs correspond to the resolution III two factor fractional factorial designs. Although Plackett-Burman designs are all two level orthogonal designs, the alias structure for these designs is complicated when runs are not a power of 2.

As an example, consider the 12-run Plackett-Burman design shown in the figure below.

[[Image:doe8.1.png|center|400px|12-run Plackett-Burman design.]]

If 11 main effects are to be estimated using this design, then each of these main effects is partially aliased with all other two factor interactions not containing that main effect. For example, the main effect <math>A\,\!</math> is partially aliased with all two factor interactions except <math>AB\,\!</math>, <math>AC\,\!</math>, <math>AD\,\!</math>, <math>AE\,\!</math>, <math>AF\,\!</math>, <math>AG\,\!</math>, <math>AH\,\!</math>, <math>AJ\,\!</math>, <math>AK\,\!</math> and <math>AL\,\!</math>. There are 45 such two factor interactions that are aliased with <math>A\,\!</math>.

::<math>\begin{align}
& A= & A-\frac{1}{3}BC-\frac{1}{3}BD+\frac{1}{3}CD-\frac{1}{3}BE-\frac{1}{3}CE+\frac{1}{3}DE+... \\
& & ...+\frac{1}{3}EL-\frac{1}{3}FL-\frac{1}{3}GL+\frac{1}{3}HL+\frac{1}{3}JL-\frac{1}{3}KL
\end{align}\,\!</math>

Due to the complex aliasing, Plackett-Burman designs involving a large number of factors should be used with care. Some of the Plackett-Burman designs available in DOE++ are included in [[Plackett-Burman_Designs|Appendix C]].

==Taguchi's Orthogonal Arrays==

Taguchi's orthogonal arrays are highly fractional orthogonal designs proposed by Dr. Genichi Taguchi, a Japanese industrialist. These designs can be used to estimate main effects using only a few experimental runs. These designs are not only applicable to two level factorial experiments; they can also investigate main effects when factors have more than two levels. Designs are also available to investigate main effects for certain mixed level experiments where the factors included do not have the same number of levels. As in the case of Placket-Burman designs, these designs require the experimenter to assume that interaction effects are unimportant and can be ignored. A few of Taguchi's orthogonal arrays available in DOE++ are included in [[Taguchi_Orthogonal_Arrays|Appendix D]].

Some of Taguchi's arrays, with runs that are a power of 2, are similar to the corresponding 2 <math>_{\text{III}}^{k-f}\,\!</math> designs. For example, consider the L4 array shown in figure (a) below. The L4 array is denoted as L4(2^3) in a DOE folio. L4 means the array requires 4 runs. 2^3 indicates that the design estimates up to three main effects at 2 levels each. The L4 array can be used to estimate three main effects using four runs provided that the two factor and three factor interactions can be ignored. Figure (b) below shows the 2 <math>_{\text{III}}^{3-1}\,\!</math> design (defining relation <math>I=-ABC\,\!</math> ) which also requires four runs and can be used to estimate up to three main effects, assuming that all two factor and three factor interactions are unimportant. A comparison between the two designs shows that the columns in the two designs are the same except for the arrangement of the columns. In figure (c) below, columns of the L4 array are marked with the name of the effect from the corresponding column of the 2 <math>_{\text{III}}^{3-1}\,\!</math> design.

[[Image:doe8.3.png|center|400px|Taguchi's L4 orthogonal array - Figure (a) shows the design, (b) shows the <math>2_{III}^{3-1} \,\!</math> design with the defining relation <math>I=-ABC \,\!</math> and (c) marks the columns of the L4 array with the corresponding columns of the design in (b).]]

Similarly, consider the L8(2^7) array shown in figure (a) below. This design can be used to estimate up to seven main effects using eight runs. This array is again similar to the 2 <math>_{\text{III}}^{7-4}\,\!</math> design shown in figure (b) below, except that the aliasing between the columns of the two designs differs in sign for some of the columns (see figure (c)).

[[Image:doe8.4.png|center|400px|Taguchi's L8 orthogonal array - Figure (a) shows the design, (b) shows the <math>2_{III}^{7-4} \,\!</math> design with the defining relation <math>I=ABD=ACE=BCF=ABCG \,\!</math> and (c) marks the columns of the L8 array with the corresponding columns of the design in (b).]]

The L8 array can also be used as a full factorial three factor experiment design in the same way as a <math>2^{3}\,\!</math> design. However, the orthogonal arrays should be used carefully in such cases, taking into consideration the alias relationships between the columns of the array. For the L8 array, figure (c) above shows that the third column of the array is the product of the first two columns. If the L8 array is used as a two level full factorial design in the place of a 2 <math>^{3}\,\!</math> design, and if the main effects are assigned to the first three columns, the main effect assigned to the third column will be aliased with the two factor interaction of the first two main effects. The proper assignment of the main effects to the columns of the L8 array requires the experimenter to assign the three main effects to the first, second and fourth columns. These columns are sometimes referred to as the ''preferred columns'' for the L8 array. To know the preferred columns for any of the orthogonal arrays, the alias relationships between the array columns must be known. The alias relations between the main effects and two factor interactions of the columns for the L8 array are shown in the next table.

[[Image:doet8.2.png|center|400px|Alias relations for the L8 array.]]

The cell value in any (<math>i,j\,\!</math>) cell of the table gives the column number of the two factor interaction for the <math>i\,\!</math>th row and <math>j\,\!</math>th column. For example, to know which column is confounded with the interaction of the first and second columns, look at the value in the ( <math>1,2\,\!</math> ) cell. The value of 3 indicates that the third column is the same as the product of the first and second columns. The alias relations for some of Taguchi's orthogonal arrays are available in [[Alias_Relations_for_Taguchi_Orthogonal_Arrays|Appendix E]].

===Example===

Recall the experiment to investigate factors affecting the surface finish of automobile brake drums discussed in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]]. The three factors investigated in the experiment were honing pressure (factor A), number of strokes (factor B) and cycle time (factor C). Assume that you used Taguchi's L8 orthogonal array to investigate the three factors instead of the <math>{2}^{3}\,\!</math> design that was used in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]]. Based on the discussion in the previous section, the preferred columns for the L8 array are the first, second and fourth columns. Therefore, the three factors should be assigned to these columns. The three factors are assigned to these columns based on the figure (c) above, so that you can easily compare results obtained from the L8 array to the ones included in [[Two_Level_Factorial_Experiments| Two Level Factorial Experiments]]. Based on this assignment, the L8 array for the two replicates, along with the respective response values, should be as shown in the third table. Note that to run the experiment using the L8 array, you would use only the first, the second and the fourth column to set the three factors.

[[Image:doet8.3.png|center|639px|Using Taguchi's L8 array to investigate factors affecting the surface finish of automobile brake drums.]]

The experiment design for this example can be set using the properties shown in the figure below.

[[Image:doe8_5.png|center|700px|Design properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example| example]].|link=]]

Note that for this design, the factor properties are set up as shown in the design summary.

[[Image:doe8_6.png|center|700px|Factor properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example| example]].|link=]]

The resulting design along with the response values is shown in the figure below.

[[Image:doe8_7.png|center|650px|Experiment design for the [[Highly_Fractional_Factorial_Designs#Example| example]].|link=]]

And the results from the DOE folio for the design are shown in the next figure.

[[Image:doe8_8_1.png|center|700px|Results for the experiment in the example.|link=]]

The results identify honing pressure, number of strokes, and the interaction between honing pressure and cycle time to be significant effects. This is identical to the conclusion obtained from the <math>{2}^{3}\,\!</math> design used in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]].

=== Preferred Columns in Taguchi OA===
One of the difficulties of using Taguchi OA is to assign factors to the appropriate columns of the array. For example, take a simple Taguchi OA L8(2^7), which can be used for experiments with up to 7 factors. If you have only 3 factors, which 3 columns in this array should be used? DOE++ provides a simple utility to help users utilize Taguchi OA more effectively by assigning factors to the appropriate columns.
Let’s use Taguchi OA L8(2^7) as an example. The design table for this array is:

[[Image:DOEtableChapter9.png|center]]

This is a fractional factorial design for 7 factors. For any fractional factorial design, the first thing we need to do is check its alias structure. In general, the alias structures for Taguchi OAs are very complicated. People usually use the following table to represent the alias relations between each factor. For the above orthogonal array, the alias table is:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|1|| 2|| 3|| 4|| 5|| 6|| 7
|-
|2x3|| 1x3|| 1x2|| 1x5|| 1x4|| 1x7|| 1x6
|-
|4x5|| 4x6|| 4x7|| 2x6|| 2x7|| 2x4|| 2x5
|-
|6x7|| 5x7|| 5x6|| 3x7|| 3x6|| 3x5|| 3x4
|}

In the above table, an Arabic number is used to represent a factor. For instance, “1” represents the factor assigned to the 1st column in the array. “2x3” represents the interaction effect of the two factors assigned to column 2 and 3. Each column in the above alias table lists all the 2-way interaction effects that are aliased with the main effect of the factor assigned to this column. For example, for the 1st column, the main effect of the factor assigned to it is aliased with interaction effects of 2x3, 4x5 and 6x7.
If an experiment has only 3 factors and these 3 factors A, B and C are assigned to the first 3 columns of the above L8(2^7) array, then the design table will be:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|Run|| A (Column 1)|| B (Column 2)|| C (Column 3)
|-
|1|| 1|| 1|| 1
|-
|2|| 1|| 1|| 1
|-
|3|| 1|| 2|| 2
|-
|4|| 1|| 2|| 2
|-
|5|| 2|| 1|| 2
|-
|6|| 2|| 1|| 2
|-
|7|| 2|| 2|| 1
|-
|8|| 2|| 2|| 1
|}

The alias structure for the above table is:

{| style="text-align:center;" cellpadding="2" align="center"
|-
|[I] = I – ABC
|-
|[A] = A – BC
|-
|[B] = B – AC
|-
|[C] = C – AB
|}

This is a resolution 3 design. All the main effects are aliased with 2-way interactions. There are many ways to choose 3 columns from the 7 columns of L8(2^7). If the 3 factors are assigned to column 1, 2, and 4, then the design table is:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|Run|| A (Column 1)|| B (Column 2)|| C (Column 4)
|-
|1|| 1|| 1|| 1
|-
|2|| 1|| 1|| 1
|-
|3|| 1|| 2|| 2
|-
|4|| 1|| 2|| 2
|-
|5|| 2|| 1|| 2
|-
|6|| 2|| 1|| 2
|-
|7|| 2|| 2|| 1
|-
|8|| 2|| 2|| 1
|}

For experiments using the above design table, all the effects will be alias free. Therefore, this design is much better than the previous one which used column 1, 2, and 3 of L8(2^7). Although both designs have the same number of runs, more information can be obtained from this design since it is alias free.

Clearly, it is very important to assign factors to the right columns when applying Taguchi OA. DOE++ can help users automatically choose the right columns when the number of factors is less than the number of columns in a Taguchi OA. The selection is based on the specified model terms by users. Let’s use an example to explain this.

====Example====
Design an experiment with 3 qualitative factors. Factors A and B have 2 levels; factor C has 4 levels. The experimenters are interested in all the main effects and the interaction effect AC.

Based on this requirement, Taguchi OA L16(2^6*4^3) can be used since it can handle both 2 level and 4 level factors. It has 9 columns. The first 6 columns are used for 2 level factors and the last 3 columns are used for 4 level factors. We need to assign factor A and B to two of the first 6 columns, and assign factor C to one of the last 3 columns.

In DOE++, we can choose L16(2^6*4^3) in the Additional Settings.

[[Image:doe8_9.png|center|842px|Selecting the Taguchi OA design type.|link=]]

Click '''Specify Interaction Terms''' to specify the interaction terms that are of interest to the experimenters.

[[Image:doe8_10.png|center|420px|Specifying the interaction terms of interest.|link=]]

Based on the specified interaction effects, DOE++ will assign each factor to the appropriate column. In this case, they are column 1, 3, and 7 as shown below.

[[Image:doe8_11.png|center|842px|The interaction terms of interest have been specified.|link=]]

However, for a given Taguchi OA, it may not be possible to estimate all the specified interaction terms. If not all the requirements can be satisfied, DOE++ will assign factors to columns that result in the least number of aliased effects. In this case, users should either use another Taguchi OA or other design types. The following example is one of these cases.

====Example====
Design an experiment for a test with 4 qualitative factors. Factors A and B have 2 levels; C and D have 4 levels. We are interested in all the main effects and the interaction effects AC and BD.

Assume again we want to use Taguchi OA L16(2^6*4^3). Click '''Specify Interaction Terms''' as shown above to specify the interaction effects that you want to estimate in the experiment.

[[Image:doe8_12.png|center|420px|Specifying the interaction terms of interest.|link=]]

When you click '''OK''', you will see a message warning that some of the specified interaction effects are aliased with main effects. These means that it is not possible to clearly estimate all the main effects and the specified interaction effects AC and BD.

This can be explained by checking the alias table of an L16(2^6*4^3) design as given below.

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|1|| 2|| 3|| 4|| 5|| 6|| 7|| 8|| 9
|-
|2x7|| 1x7|| 1x9|| 1x5|| 1x4|| 1x8|| 1x2|| 1x6|| 1x3
|-
|3x9|| 3x6|| 2x6|| 2x8|| 2x9|| 2x3|| 1x8|| 1x7|| 1x7
|-
|4x5|| 4x8|| 4x7|| 3x7|| 3x8|| 4x9|| 1x9|| 1x9|| 1x8
|-
|6x8|| 5x9|| 5x8|| 6x9|| 6x7|| 5x7|| 2x8|| 2x4|| 2x5
|-
|7x8|| 7x8|| 7x8|| 7x8|| 7x8|| 7x8|| 2x9|| 2x7|| 2x7
|-
|7x9|| 7x9|| 7x9|| 7x9|| 7x9|| 7x9|| 3x8|| 2x9|| 2x8
|-
|8x9|| 8x9|| 8x9|| 8x9|| 8x9|| 8x9|| 3x9|| 3x5|| 3x7
|-
| || || || || || ||4x8|| 3x7|| 3x8
|-
| || || || || || ||4x9|| 3x9|| 4x6
|-
| || || || || || ||5x6|| 4x7|| 4x7
|-
| || || || || || ||5x8|| 4x9|| 4x8
|-
| || || || || || ||5x9|| 5x7|| 5x7
|-
| || || || || || ||6x8|| 5x9|| 5x8
|-
| || || || || || ||6x9|| 6x7|| 6x7
|-
| || || || || || ||8x9|| 6x9|| 6x8
|-
| || || || || || || || 7x9|| 7x8
|}

From this table, we can see that it is impossible to clearly estimate both AC and BD. Factors C and D can only be assigned to the last three columns since they are 4 level factors. Assume we assign factor C to column 7 and factor D to column 8. Factor B (2 level) will be in one of the columns from 1 to 6. Therefore, effect BD will be one of the effects highlighted in the table, where the first term of the interaction is between 1 and 6 and the last term is 8 (i.e., factor D).

The above alias table shows that factor C is aliased with one of those highlighted effects. Thus, no matter which of the first six columns is assigned to factor B, the main effect C will be aliased with interaction effect BD. This is also true if C is assigned to column 8 or 9. Therefore, if L16(2^6*4^3) is used, there is no way to clearly estimate all the main effects and the interaction effects AC and BD. Another Taguchi OA or other design types such as a general full factorial design should be used. A more efficient way might be to create an optimal custom design that can clearly estimate all the specified terms. For more detail, please refer to the chapter on [[Optimal Custom Designs]].

Highly Fractional Factorial Designs

2017-08-10T17:21:51Z

Richard House: /* Taguchi's Orthogonal Arrays */

{{Template:Doebook|9}}
This chapter discusses factorial designs that are commonly used in designed experiments, but are not necessarily limited to two level factors. These designs are the [[Highly_Fractional_Factorial_Designs#Plackett-Burman_Designs|Plackett-Burman designs]] and [[Highly_Fractional_Factorial_Designs#Taguchi.27s_Orthogonal_Arrays|Taguchi's orthogonal arrays]].

==Plackett-Burman Designs==

It was mentioned in [[Two_Level_Factorial_Experiments| Two Level Factorial Experiments]] that resolution III designs can be used as highly fractional designs to investigate <math>k\,\!</math> main effects using <math>k+1\,\!</math> runs (provided that three factor and higher order interaction effects are not important to the experimenter). A limitation with these designs is that all runs in these designs have to be a power of 2. The valid runs for these designs are 4, 8, 16, 32, etc. Therefore, the next design after the 2 <math>_{\text{III}}^{3-1}\,\!</math> design with 4 runs is the 2 <math>_{\text{III}}^{7-4}\,\!</math> design with 8 runs, and the design after this is the 2 <math>_{\text{III}}^{15-11}\,\!</math> design with 32 runs and so on, as shown in the next table.

[[Image:doet8.1.png|center|487px|Highly fractional designs to investigate main effects.]]

Plackett-Burman designs solve this problem. These designs were proposed by R. L. Plackett and J.P. Burman (1946). They allow the estimation of <math>k\,\!</math> main effects using <math>k+1\,\!</math> runs. In these designs, runs are a multiple of 4 (i.e., 4, 8, 12, 16, 20 and so on). When the runs are a power of 2, the designs correspond to the resolution III two factor fractional factorial designs. Although Plackett-Burman designs are all two level orthogonal designs, the alias structure for these designs is complicated when runs are not a power of 2.

As an example, consider the 12-run Plackett-Burman design shown in the figure below.

[[Image:doe8.1.png|center|400px|12-run Plackett-Burman design.]]

If 11 main effects are to be estimated using this design, then each of these main effects is partially aliased with all other two factor interactions not containing that main effect. For example, the main effect <math>A\,\!</math> is partially aliased with all two factor interactions except <math>AB\,\!</math>, <math>AC\,\!</math>, <math>AD\,\!</math>, <math>AE\,\!</math>, <math>AF\,\!</math>, <math>AG\,\!</math>, <math>AH\,\!</math>, <math>AJ\,\!</math>, <math>AK\,\!</math> and <math>AL\,\!</math>. There are 45 such two factor interactions that are aliased with <math>A\,\!</math>.

::<math>\begin{align}
& A= & A-\frac{1}{3}BC-\frac{1}{3}BD+\frac{1}{3}CD-\frac{1}{3}BE-\frac{1}{3}CE+\frac{1}{3}DE+... \\
& & ...+\frac{1}{3}EL-\frac{1}{3}FL-\frac{1}{3}GL+\frac{1}{3}HL+\frac{1}{3}JL-\frac{1}{3}KL
\end{align}\,\!</math>

Due to the complex aliasing, Plackett-Burman designs involving a large number of factors should be used with care. Some of the Plackett-Burman designs available in DOE++ are included in [[Plackett-Burman_Designs|Appendix C]].

==Taguchi's Orthogonal Arrays==

Taguchi's orthogonal arrays are highly fractional orthogonal designs proposed by Dr. Genichi Taguchi, a Japanese industrialist. These designs can be used to estimate main effects using only a few experimental runs. These designs are not only applicable to two level factorial experiments; they can also investigate main effects when factors have more than two levels. Designs are also available to investigate main effects for certain mixed level experiments where the factors included do not have the same number of levels. As in the case of Placket-Burman designs, these designs require the experimenter to assume that interaction effects are unimportant and can be ignored. A few of Taguchi's orthogonal arrays available in DOE++ are included in [[Taguchi_Orthogonal_Arrays|Appendix D]].

Some of Taguchi's arrays, with runs that are a power of 2, are similar to the corresponding 2 <math>_{\text{III}}^{k-f}\,\!</math> designs. For example, consider the L4 array shown in figure (a) below. The L4 array is denoted as L4(2^3) in a DOE folio. L4 means the array requires 4 runs. 2^3 indicates that the design estimates up to three main effects at 2 levels each. The L4 array can be used to estimate three main effects using four runs provided that the two factor and three factor interactions can be ignored. Figure (b) below shows the 2 <math>_{\text{III}}^{3-1}\,\!</math> design (defining relation <math>I=-ABC\,\!</math> ) which also requires four runs and can be used to estimate up to three main effects, assuming that all two factor and three factor interactions are unimportant. A comparison between the two designs shows that the columns in the two designs are the same except for the arrangement of the columns. In figure (c) below, columns of the L4 array are marked with the name of the effect from the corresponding column of the 2 <math>_{\text{III}}^{3-1}\,\!</math> design.

[[Image:doe8.3.png|center|400px|Taguchi's L4 orthogonal array - Figure (a) shows the design, (b) shows the <math>2_{III}^{3-1} \,\!</math> design with the defining relation <math>I=-ABC \,\!</math> and (c) marks the columns of the L4 array with the corresponding columns of the design in (b).]]

Similarly, consider the L8(2^7) array shown in figure (a) below. This design can be used to estimate up to seven main effects using eight runs. This array is again similar to the 2 <math>_{\text{III}}^{7-4}\,\!</math> design shown in figure (b) below, except that the aliasing between the columns of the two designs differs in sign for some of the columns (see figure (c)).

[[Image:doe8.4.png|center|400px|Taguchi's L8 orthogonal array - Figure (a) shows the design, (b) shows the <math>2_{III}^{7-4} \,\!</math> design with the defining relation <math>I=ABD=ACE=BCF=ABCG \,\!</math> and (c) marks the columns of the L8 array with the corresponding columns of the design in (b).]]

The L8 array can also be used as a full factorial three factor experiment design in the same way as a <math>2^{3}\,\!</math> design. However, the orthogonal arrays should be used carefully in such cases, taking into consideration the alias relationships between the columns of the array. For the L8 array, figure (c) above shows that the third column of the array is the product of the first two columns. If the L8 array is used as a two level full factorial design in the place of a 2 <math>^{3}\,\!</math> design, and if the main effects are assigned to the first three columns, the main effect assigned to the third column will be aliased with the two factor interaction of the first two main effects. The proper assignment of the main effects to the columns of the L8 array requires the experimenter to assign the three main effects to the first, second and fourth columns. These columns are sometimes referred to as the ''preferred columns'' for the L8 array. To know the preferred columns for any of the orthogonal arrays, the alias relationships between the array columns must be known. The alias relations between the main effects and two factor interactions of the columns for the L8 array are shown in the next table.

[[Image:doet8.2.png|center|400px|Alias relations for the L8 array.]]

The cell value in any (<math>i,j\,\!</math>) cell of the table gives the column number of the two factor interaction for the <math>i\,\!</math>th row and <math>j\,\!</math>th column. For example, to know which column is confounded with the interaction of the first and second columns, look at the value in the ( <math>1,2\,\!</math> ) cell. The value of 3 indicates that the third column is the same as the product of the first and second columns. The alias relations for some of Taguchi's orthogonal arrays are available in [[Alias_Relations_for_Taguchi_Orthogonal_Arrays|Appendix E]].

===Example===

Recall the experiment to investigate factors affecting the surface finish of automobile brake drums discussed in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]]. The three factors investigated in the experiment were honing pressure (factor A), number of strokes (factor B) and cycle time (factor C). Assume that you used Taguchi's L8 orthogonal array to investigate the three factors instead of the <math>{2}^{3}\,\!</math> design that was used in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]]. Based on the discussion in the previous section, the preferred columns for the L8 array are the first, second and fourth columns. Therefore, the three factors should be assigned to these columns. The three factors are assigned to these columns based on the figure (c) above, so that you can easily compare results obtained from the L8 array to the ones included in [[Two_Level_Factorial_Experiments| Two Level Factorial Experiments]]. Based on this assignment, the L8 array for the two replicates, along with the respective response values, should be as shown in the third table. Note that to run the experiment using the L8 array, you would use only the first, the second and the fourth column to set the three factors.

[[Image:doet8.3.png|center|639px|Using Taguchi's L8 array to investigate factors affecting the surface finish of automobile brake drums.]]

The experiment design for this example can be set using the properties shown in the figure below.

[[Image:doe8_5.png|center|700px|Design properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example| example]].|link=]]

Note that for this design, the factor properties are set up as shown in the design summary.

[[Image:doe8_6.png|center|700px|Factor properties for the experiment in the [[Highly_Fractional_Factorial_Designs#Example| example]].|link=]]

The resulting design along with the response values is shown in the figure below.

[[Image:doe8_7.png|center|650px|Experiment design for the [[Highly_Fractional_Factorial_Designs#Example| example]].|link=]]

And the results from DOE++ for the design are shown in the next figure.

[[Image:doe8_8_1.png|center|700px|Results for the experiment in the example.|link=]]

The results identify honing pressure, number of strokes, and the interaction between honing pressure and cycle time to be significant effects. This is identical to the conclusion obtained from the <math>{2}^{3}\,\!</math> design used in [[Two_Level_Factorial_Experiments|Two Level Factorial Experiments]].

=== Preferred Columns in Taguchi OA===
One of the difficulties of using Taguchi OA is to assign factors to the appropriate columns of the array. For example, take a simple Taguchi OA L8(2^7), which can be used for experiments with up to 7 factors. If you have only 3 factors, which 3 columns in this array should be used? DOE++ provides a simple utility to help users utilize Taguchi OA more effectively by assigning factors to the appropriate columns.
Let’s use Taguchi OA L8(2^7) as an example. The design table for this array is:

[[Image:DOEtableChapter9.png|center]]

This is a fractional factorial design for 7 factors. For any fractional factorial design, the first thing we need to do is check its alias structure. In general, the alias structures for Taguchi OAs are very complicated. People usually use the following table to represent the alias relations between each factor. For the above orthogonal array, the alias table is:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|1|| 2|| 3|| 4|| 5|| 6|| 7
|-
|2x3|| 1x3|| 1x2|| 1x5|| 1x4|| 1x7|| 1x6
|-
|4x5|| 4x6|| 4x7|| 2x6|| 2x7|| 2x4|| 2x5
|-
|6x7|| 5x7|| 5x6|| 3x7|| 3x6|| 3x5|| 3x4
|}

In the above table, an Arabic number is used to represent a factor. For instance, “1” represents the factor assigned to the 1st column in the array. “2x3” represents the interaction effect of the two factors assigned to column 2 and 3. Each column in the above alias table lists all the 2-way interaction effects that are aliased with the main effect of the factor assigned to this column. For example, for the 1st column, the main effect of the factor assigned to it is aliased with interaction effects of 2x3, 4x5 and 6x7.
If an experiment has only 3 factors and these 3 factors A, B and C are assigned to the first 3 columns of the above L8(2^7) array, then the design table will be:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|Run|| A (Column 1)|| B (Column 2)|| C (Column 3)
|-
|1|| 1|| 1|| 1
|-
|2|| 1|| 1|| 1
|-
|3|| 1|| 2|| 2
|-
|4|| 1|| 2|| 2
|-
|5|| 2|| 1|| 2
|-
|6|| 2|| 1|| 2
|-
|7|| 2|| 2|| 1
|-
|8|| 2|| 2|| 1
|}

The alias structure for the above table is:

{| style="text-align:center;" cellpadding="2" align="center"
|-
|[I] = I – ABC
|-
|[A] = A – BC
|-
|[B] = B – AC
|-
|[C] = C – AB
|}

This is a resolution 3 design. All the main effects are aliased with 2-way interactions. There are many ways to choose 3 columns from the 7 columns of L8(2^7). If the 3 factors are assigned to column 1, 2, and 4, then the design table is:

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|Run|| A (Column 1)|| B (Column 2)|| C (Column 4)
|-
|1|| 1|| 1|| 1
|-
|2|| 1|| 1|| 1
|-
|3|| 1|| 2|| 2
|-
|4|| 1|| 2|| 2
|-
|5|| 2|| 1|| 2
|-
|6|| 2|| 1|| 2
|-
|7|| 2|| 2|| 1
|-
|8|| 2|| 2|| 1
|}

For experiments using the above design table, all the effects will be alias free. Therefore, this design is much better than the previous one which used column 1, 2, and 3 of L8(2^7). Although both designs have the same number of runs, more information can be obtained from this design since it is alias free.

Clearly, it is very important to assign factors to the right columns when applying Taguchi OA. DOE++ can help users automatically choose the right columns when the number of factors is less than the number of columns in a Taguchi OA. The selection is based on the specified model terms by users. Let’s use an example to explain this.

====Example====
Design an experiment with 3 qualitative factors. Factors A and B have 2 levels; factor C has 4 levels. The experimenters are interested in all the main effects and the interaction effect AC.

Based on this requirement, Taguchi OA L16(2^6*4^3) can be used since it can handle both 2 level and 4 level factors. It has 9 columns. The first 6 columns are used for 2 level factors and the last 3 columns are used for 4 level factors. We need to assign factor A and B to two of the first 6 columns, and assign factor C to one of the last 3 columns.

In DOE++, we can choose L16(2^6*4^3) in the Additional Settings.

[[Image:doe8_9.png|center|842px|Selecting the Taguchi OA design type.|link=]]

Click '''Specify Interaction Terms''' to specify the interaction terms that are of interest to the experimenters.

[[Image:doe8_10.png|center|420px|Specifying the interaction terms of interest.|link=]]

Based on the specified interaction effects, DOE++ will assign each factor to the appropriate column. In this case, they are column 1, 3, and 7 as shown below.

[[Image:doe8_11.png|center|842px|The interaction terms of interest have been specified.|link=]]

However, for a given Taguchi OA, it may not be possible to estimate all the specified interaction terms. If not all the requirements can be satisfied, DOE++ will assign factors to columns that result in the least number of aliased effects. In this case, users should either use another Taguchi OA or other design types. The following example is one of these cases.

====Example====
Design an experiment for a test with 4 qualitative factors. Factors A and B have 2 levels; C and D have 4 levels. We are interested in all the main effects and the interaction effects AC and BD.

Assume again we want to use Taguchi OA L16(2^6*4^3). Click '''Specify Interaction Terms''' as shown above to specify the interaction effects that you want to estimate in the experiment.

[[Image:doe8_12.png|center|420px|Specifying the interaction terms of interest.|link=]]

When you click '''OK''', you will see a message warning that some of the specified interaction effects are aliased with main effects. These means that it is not possible to clearly estimate all the main effects and the specified interaction effects AC and BD.

This can be explained by checking the alias table of an L16(2^6*4^3) design as given below.

{| style="text-align:center;" cellpadding="2" border="1" align="center"
|-
|1|| 2|| 3|| 4|| 5|| 6|| 7|| 8|| 9
|-
|2x7|| 1x7|| 1x9|| 1x5|| 1x4|| 1x8|| 1x2|| 1x6|| 1x3
|-
|3x9|| 3x6|| 2x6|| 2x8|| 2x9|| 2x3|| 1x8|| 1x7|| 1x7
|-
|4x5|| 4x8|| 4x7|| 3x7|| 3x8|| 4x9|| 1x9|| 1x9|| 1x8
|-
|6x8|| 5x9|| 5x8|| 6x9|| 6x7|| 5x7|| 2x8|| 2x4|| 2x5
|-
|7x8|| 7x8|| 7x8|| 7x8|| 7x8|| 7x8|| 2x9|| 2x7|| 2x7
|-
|7x9|| 7x9|| 7x9|| 7x9|| 7x9|| 7x9|| 3x8|| 2x9|| 2x8
|-
|8x9|| 8x9|| 8x9|| 8x9|| 8x9|| 8x9|| 3x9|| 3x5|| 3x7
|-
| || || || || || ||4x8|| 3x7|| 3x8
|-
| || || || || || ||4x9|| 3x9|| 4x6
|-
| || || || || || ||5x6|| 4x7|| 4x7
|-
| || || || || || ||5x8|| 4x9|| 4x8
|-
| || || || || || ||5x9|| 5x7|| 5x7
|-
| || || || || || ||6x8|| 5x9|| 5x8
|-
| || || || || || ||6x9|| 6x7|| 6x7
|-
| || || || || || ||8x9|| 6x9|| 6x8
|-
| || || || || || || || 7x9|| 7x8
|}

From this table, we can see that it is impossible to clearly estimate both AC and BD. Factors C and D can only be assigned to the last three columns since they are 4 level factors. Assume we assign factor C to column 7 and factor D to column 8. Factor B (2 level) will be in one of the columns from 1 to 6. Therefore, effect BD will be one of the effects highlighted in the table, where the first term of the interaction is between 1 and 6 and the last term is 8 (i.e., factor D).

The above alias table shows that factor C is aliased with one of those highlighted effects. Thus, no matter which of the first six columns is assigned to factor B, the main effect C will be aliased with interaction effect BD. This is also true if C is assigned to column 8 or 9. Therefore, if L16(2^6*4^3) is used, there is no way to clearly estimate all the main effects and the interaction effects AC and BD. Another Taguchi OA or other design types such as a general full factorial design should be used. A more efficient way might be to create an optimal custom design that can clearly estimate all the specified terms. For more detail, please refer to the chapter on [[Optimal Custom Designs]].

Response Surface Methods for Optimization

2017-08-10T17:19:17Z

Richard House: /* Desirability Functions */

{{Template:Doebook|10}}
The experiment designs mentioned in [[Two_Level_Factorial_Experiments| Two Level Factorial Experiments]] and [[Highly_Fractional_Factorial_Designs| Highly Fractional Factorial Designs]] help the experimenter identify factors that affect the response. Once the important factors have been identified, the next step is to determine the settings for these factors that result in the optimum value of the response. The optimum value of the response may either be a maximum value or a minimum value, depending upon the product or process in question. For example, if the response in an experiment is the yield from a chemical process, then the objective might be to find the settings of the factors affecting the yield so that the yield is maximized. On the other hand, if the response in an experiment is the number of defects, then the goal would be to find the factor settings that minimize the number of defects. Methodologies that help the experimenter reach the goal of optimum response are referred to as ''response surface methods''. These methods are exclusively used to examine the "surface," or the relationship between the response and the factors affecting the response. Regression models are used for the analysis of the response, as the focus now is on the nature of the relationship between the response and the factors, rather than identification of the important factors.

Response surface methods usually involve the following steps:

#The experimenter needs to move from the present operating conditions to the vicinity of the operating conditions where the response is optimum. This is done using the ''method of steepest ascent'' in the case of maximizing the response. The same method can be used to minimize the response and is then referred to as the ''method of steepest descent''.
#Once in the vicinity of the optimum response the experimenter needs to fit a more elaborate model between the response and the factors. Special experiment designs, referred to as [[Response_Surface_Methods_for_Optimization#RSM_Designs|''RSM designs'']], are used to accomplish this. The fitted model is used to arrive at the best operating conditions that result in either a maximum or minimum response.
#It is possible that a number of responses may have to be optimized at the same time. For example, an experimenter may want to maximize strength, while keeping the number of defects to a minimum. The optimum settings for each of the responses in such cases may lead to conflicting settings for the factors. A balanced setting has to be found that gives the most appropriate values for all the responses. Desirability functions are useful in these cases.

==Method of Steepest Ascent==
The first step in obtaining the optimum response settings, after the important factors have been identified, is to explore the region around the current operating conditions to decide what direction needs to be taken to move towards the optimum region. Usually, a first order regression model (containing just the main effects and no interaction terms) is sufficient at the current operating conditions because the operating conditions are normally far from the optimum response settings. The experimenter needs to move from the current operating conditions to the optimum region in the most efficient way by using the minimum number of experiments. This is done using the method of steepest ascent. In this method, the contour plot of the first order model is used to decide the settings for the next experiment, in order to move towards the optimum conditions. Consider a process where the response has been found to be a function of two factors. To explore the region around the current operating conditions, the experimenter fits the following first order model between the response and the two factors:

::<math>y={{\beta }_{0}}+{{\beta }_{1}}{{x}_{1}}+{{\beta }_{2}}{{x}_{2}}+\epsilon \,\!</math>

The response surface plot for the model, along with the contours, is shown in the figure below. It can be seen in the figure that in order to maximize the response, the most efficient direction in which to move the experiment is along the line perpendicular to the contours. This line, also referred to as the path of steepest ascent, is the line along which the rate of increase of the response is maximum. The steps along this line to move towards the optimum region are proportional to the regression coefficients, <math>{{\beta }_{j}},\,\!</math> of the fitted first order model.

[[Image:doe9_1.png|center|504px|Path of steepest ascent for the model <math>y=\beta_0+\beta_1 x_1+\beta_2 x_2+\epsilon \,\!</math>.]]

Experiments are conducted along each step of the path of steepest ascent until an increase in the response is not seen. Then, a new first order model is fit at the region of the maximum response. If the first order model shows a lack of fit, then this indicates that the experimenter has reached the vicinity of the optimum. RSM designs are then used explore the region thoroughly and obtain the point of the maximum response. If the first order model does not show a lack of fit, then a new path of steepest ascent is determined and the process is repeated.

===Example===
The yield from a chemical process is found to be affected by two factors: reaction temperature and reaction time. The current reaction temperature is 230 <math>F\,\!</math> and the reaction time is 65 minutes. The experimenter wants to determine the settings of the two factors such that maximum yield can be obtained from the process. To explore the region around the current operating conditions, the experimenter decides to use a single replicate of the <math>2^{2}\,\!</math> design. The range of the factors for this design are chosen to be (225, 235) <math>F\,\!</math> for the reaction temperature and (55, 75) minutes for the reaction time. The unreplicated <math>2^{2}\,\!</math> design is also augmented with five runs at the center point to estimate the error sum of squares, <math>S{{S}_{E}}\,\!</math>, and check for model adequacy. The response values obtained for this design are shown next.

[[Image:doe9.2.png|center|256px|The <math>2^{2}\,\!</math> design augmented with five center points to explore the region around current operating conditions for a chemical process.|link=]]

In a DOE folio, this design can be set up using the properties shown next.

[[Image:doe9_3.png|center|809px|Design properties for the <math>2^{2}\,\!</math> design to explore the current operating conditions.|link=]]

The resulting design and the analysis results are shown next.

[[Image:doe9_4.png|center|778px|The <math>2^{2}\,\!</math> experiment design in to explore the current operating conditions.|link=]]

[[Image:doe9_5.png|center|774px|Results for the <math>2^{2}\,\!</math> experiment to explore the current operating conditions.|link=]]

Note that the results shown are in terms of the coded values of the factors (taking -1 as the value of the lower settings for reaction temperature and reaction time and +1 as the value for the higher settings for these two factors). The results show that the factors, <math>A\,\!</math> (temperature) and <math>B\,\!</math> (time), affect the response significantly but their interaction does not affect the response. Therefore the interaction term can be dropped from the model for this experiment. The results also show that Curvature is not a significant factor. This indicates that the first order model is adequate for the experiment at the current operating conditions. Using these two conclusions, the model for the current operating conditions, in terms the coded variables is:

:<math>\hat{y}=36.6375+1.1635x_{1}+0.4875x_{2}\,\!</math>

where <math>\hat{y}\,\!</math> represents the yield and <math>x_{1}\,\!</math> and <math>x_{2}\,\!</math> are the predictor variables for the two factors, <math>A\,\!</math> and <math>B\,\!</math>, respectively. To further confirm the adequacy of the model of the equation given above, the experiment can be analyzed again after dropping the interaction term, <math>AB\,\!</math>. The results are shown next.

[[Image:doe9_6.png|center|774px|Results for the <math>2^{2}\,\!</math> experiment after the interaction term is dropped from the model.|link=]]

The results show that the lack-of-fit for this model (because of the deficiency created in the model by the absence of the interaction term) is not significant, confirming that the model is adequate.

==Path of Steepest Ascent==
The contour plot for the model used in the above example is shown next.

[[Image:doe9_7.png|center|650px|Results for the <math>2^{2}\,\!</math> experiment after the interaction term is dropped from the model.|link=]]

The regression coefficients for the model are <math>\hat{\beta}_{1}= 1.1625\,\!</math> and <math>\hat{\beta}_{2}= 0.4875\,\!</math>. To move towards the optimum, the experimenter needs to move along the path of steepest ascent, which lies perpendicular to the contours. This path is the line through the center point of the current operating conditions (<math>x_{1}=0\,\!</math>, <math>x_{2}=0\,\!</math>) with a slope of <math>\hat{\beta}_{2}/\hat{\beta}_{1}\,\!</math>. Therefore, in terms of the coded variables, the experiment should be moved 1.1625 units in the <math>x_{1}\,\!</math> direction for every 0.4875 units in the <math>x_{2}\,\!</math> direction. To move along this path, the experimenter decides to use a step-size of 10 minutes for the reaction time, <math>x_{2}\,\!</math>. The coded value for this step size can be obtained as follows. Recall from [[Multiple_Linear_Regression_Analysis|Multiple Linear Regression Analysis]] that the relationship between coded and actual values is:

::<math>coded\ value=\frac{actual\ value - mean}{half\ of\ range}</math>

or

::<math>coded\ value=\frac{step\ size\ in\ actual\ value}{half\ of\ range}</math>

Thus, for a step-size of 10 minutes, the equivalent step size in coded value for <math>x_{2}\,\!</math> is:

::<math>=\frac{10}{half\ of\ range\ of\ x_{2}}\,\!</math>
::<math>=\frac{10}{(75-55)/2}\,\!</math>
::<math>=1\,\!</math>

In terms of the coded variables, the path of steepest ascent requires a move of 1.1625 units in the <math>x_{1}\,\!</math> direction for every 0.4875 units in the <math>x_{2}\,\!</math> direction. The step-size for <math>x_{1}\,\!</math>, in terms of the coded value corresponding to any step-size in <math>x_{2}\,\!</math>, is:

::<math>=\frac{\hat{\beta_{1}}}{\hat{\beta_{2}}}\cdot(step\ size\ in\ x_{2})\,\!</math>

Therefore, the step-size for the reaction temperature, <math>x_{1}\,\!</math>, in terms of the coded variables is:

::<math>=\frac{\hat{\beta_{1}}}{\hat{\beta_{2}}}\cdot\Delta x_{2}\,\!</math>
::<math>=\frac{1.1625}{0.4875}\cdot1\,\!</math>
::<math>=\frac{\hat{\beta_{1}}}{\hat{\beta_{2}}}\cdot\Delta x_{2}\,\!</math>

This corresponds to a step of approximately 12 <math>F\,\!</math> for temperature in terms of the actual value as shown next:

::<math>\begin{align}
step\ size\ in\ actual\ value = & (coded\ value)\cdot(half\ of\ the\ range\ of\ x_{1})\\
= & 2.39\cdot10/2\\
= & 11.95
\end{align}\,\!</math>

Using a step of 12 <math>F\,\!</math> and 10 minutes, the experimenter conducts experiments until no further increase is observed in the yield. The yield values at each step are shown in the table given next.

[[Image:doet9.1.png|center|341px|Response values at each step of the path of steepest ascent for the experiment to investigate the yield of a chemical process. Units for factor levels and the response have been omitted.]]

The yield starts decreasing after the reaction temperature of 350 <math>F\,\!</math> and the reaction time of 165 minutes, indicating that this point may lie close to the optimum region. To analyze the vicinity of this point, a <math>2^{2}\,\!</math> design augmented by five center points is selected. The range of exploration is chosen to be 345 to 355 <math>F\,\!</math> for reaction temperature and 155 to 175 minutes for reaction time. The response values recorded are shown next.

[[Image:doe9.8.png|center|348px|The <math>2^{2}\,\!</math> design augmented with five center points to explore the region of maximum response obtained from the path of steepest ascent. Note that the center point of this design is the new origin.]]

The results for this design are shown next.

[[Image:doe9_9.png|center|760px|Results for the <math>2^{2}\,\!</math> experiment to explore the region of maximum response.|link=]]

In the results, Curvature is displayed as a significant factor. This indicates that the first order model is not adequate for this region of the experiment and a higher order model is required. As a result, the methodology of steepest ascent can no longer be used. The presence of curvature indicates that the experiment region may be close to the optimum. Special designs that allow the use of second order models are needed at this point.

==RSM Designs==
A second order model is generally used to approximate the response once it is realized that the experiment is close to the optimum response region where a first order model is no longer adequate. The second order model is usually sufficient for the optimum region, as third order and higher effects are seldom important. The second order regression model takes the following form for <math>k\,\!</math> factors:

::<math>\begin{align}
y= & {{\beta }_{0}}+{{\beta }_{1}}{{x}_{1}}+{{\beta }_{2}}{{x}_{2}}+...+{{\beta }_{k}}{{x}_{k}}+ \\
& {{\beta }_{11}}x_{1}^{2}+{{\beta }_{22}}x_{2}^{2}+...+{{\beta }_{kk}}x_{k}^{2}+{{\beta }_{12}}{{x}_{1}}{{x}_{2}}+...+{{\beta }_{k-1,k}}{{x}_{k-1}}{{x}_{k}}+\epsilon
\end{align}\,\!</math>

The model contains <math>p=(k+1)(k+2)/2\,\!</math> regression parameters that include coefficients for main effects (<math>{{\beta }_{1}},{{\beta }_{2}}...{{\beta }_{k}}\,\!</math>), coefficients for quadratic main effects (<math>{{\beta }_{11}},{{\beta }_{22}}...{{\beta }_{kk}}\,\!</math>) and coefficients for two factor interaction effects (<math>{{\beta }_{12}},{{\beta }_{13}}\,\!</math>... <math>{{\beta }_{k-1,k}}\,\!</math>). A full factorial design with all factors at three levels would provide estimation of all the required regression parameters. However, full factorial three level designs are expensive to use as the number of runs increases rapidly with the number of factors. For example, a three factor full factorial design with each factor at three levels would require <math>{{3}^{3}}=27\,\!</math> runs while a design with four factors would require <math>{{3}^{4}}=81\,\!</math> runs. Additionally, these designs will estimate a number of higher order effects which are usually not of much importance to the experimenter. Therefore, for the purpose of analysis of response surfaces, special designs are used that help the experimenter fit the second order model to the response with the use of a minimum number of runs. Examples of these designs are the central composite and Box-Behnken designs.

===Central Composite Designs===
Central composite designs are two level full factorial (<math>2^{k}\,\!</math>) or fractional factorial (<math>2^{k-f}\,\!</math>) designs augmented by a number of center points and other chosen runs. These designs are such that they allow the estimation of all the regression parameters required to fit a second order model to a given response.

The simplest of the central composite designs can be used to fit a second order model to a response with two factors. The design consists of a <math>2^{2}\,\!</math> full factorial design augmented by a few runs at the center point (such a design is shown in figure (a) given below). A central composite design is obtained when runs at points (<math>-1,0\,\!</math>), (<math>1,0\,\!</math>), (<math>0,-1\,\!</math>) and (<math>0,1\,\!</math>) are added to this design. These points are referred to as ''axial points'' or ''star points'' and represent runs where all but one of the factors are set at their mid-levels. The number of axial points in a central composite design having <math>k\,\!</math> factors is <math>2k\,\!</math>. The distance of the axial points from the center point is denoted by <math>\alpha \,\!</math> and is always specified in terms of coded values. For example, the central composite design in figure (b) given below has <math>\alpha =1\,\!</math>, while for the design of figure (c) <math>\alpha =1.414\,\!</math>.

[[Image:doe9.10.png|center|227px|Central composite designs: (a) shows the <math>2^2\,\!</math> design with center point runs, (b) shows the two factor central composite design with <math>\alpha=1\,\!</math> and (c) shows the two factor central composite design with <math>\alpha=\sqrt{2}\,\!</math>.]]

It can be noted that when <math>\alpha >1\,\!</math>, each factor is run at five levels (<math>-\alpha \,\!</math>, <math>-1\,\!</math>, <math>0\,\!</math>, <math>1\,\!</math> and <math>\alpha \,\!</math>) instead of the three levels of <math>-1\,\!</math>, <math>0\,\!</math> and <math>1\,\!</math>. The reason for running central composite designs with <math>\alpha >1\,\!</math> is to have a rotatable design, which is explained next.

====Rotatability====
A central composite design is said to be ''rotatable'' if the variance of any predicted value of the response, <math>{{\hat{y}}_{p}}\,\!</math>, for any level of the factors depends only on the distance of the point from the center of the design, regardless of the direction. In other words, a rotatable central composite design provides constant variance of the estimated response corresponding to all new observation points that are at the same distance from the center point of the design (in terms of the coded variables).

The variance of the predicted response at any point, <math>{{x}_{p}}\,\!</math>, is given as follows:

::<math>V[{{\hat{y}}_{p}}]={{\sigma }^{2}}x_{p}^{\prime }{{({{X}^{\prime }}X)}^{-1}}{{x}_{p}}\,\!</math>

The contours of <math>V[{{\hat{y}}_{p}}]\,\!</math> for the central composite design in figure (c) above are shown in the figure below. The contours are concentric circles indicating that the central composite design of figure (c) is rotatable. Rotatability is a desirable property because the experimenter does not have any prior information about the location of the optimum. Therefore, a design that provides equal precision of estimation in all directions would be preferred. Such a design will assure the experimenter that no matter what direction is taken to search for the optimum, he/she will be able to estimate the response value with equal precision. A central composite design is rotatable if the value of <math>\alpha \,\!</math> for the design satisfies the following equation:

::<math>\alpha ={{\left[ \frac{{{2}^{k-f}}({{n}_{f}})}{{{n}_{s}}} \right]}^{1/4}}\,\!</math>

where <math>{{n}_{f}}\,\!</math> is the number of replicates of the runs in the original factorial design and <math>{{n}_{s}}\,\!</math> is the number of replicates of the runs at the axial points. For example, a central composite design with two factors, having a single replicate of the original factorial design, and a single replicate of all the axial points, would be rotatable for the following <math>\alpha \,\!</math> value:

::<math>\begin{align}
\alpha = & {{\left[ \frac{{{2}^{k-f}}({{n}_{f}})}{{{n}_{s}}} \right]}^{1/4}} \\
= & {{\left[ \frac{{{2}^{2}}(1)}{1} \right]}^{1/4}} \\
= & \sqrt{2} \\
= & 1.414
\end{align}\,\!</math>

Thus, a central composite design in two factors, having a single replicate of the original <math>2^{2}\,\!</math> design and axial points, and with <math>\alpha =1.414\,\!</math>, is a rotatable design. This design is shown in figure (c) above.

[[Image:doe9.11.png|center|266px|The countours of <math>V[\hat{y}_p] \,\!</math> for the rotatable two factor central composite design.]]

====Spherical Design====
A central composite design is said to be ''spherical'' if all factorial and axial points are at same distance from the center of the design. Spherical central composite designs are obtained by setting <math>\alpha =\sqrt{k}\,\!</math>. For example, the rotatable design in the figure above (c) is also a spherical design because for this design <math>\alpha =\sqrt{k}=\sqrt{2}=1.414\,\!</math>.

====Face-centered Design====
Central composite designs in which the axial points represent the mid levels for all but one of the factors are also referred to as ''face-centered'' central composite designs. For these designs, <math>\alpha =1\,\!</math> and all factors are run at three levels, which are <math>-1\,\!</math>, <math>0\,\!</math> and <math>1,\,\!</math> in terms of the coded values (see the figure below).

[[Image:doe9.12.png|center|300px|Face-centered central composite design for three factors.]]

===Box-Behnken Designs===
In [[Highly_Fractional_Factorial_Designs| Highly Fractional Factorial Designs]], highly fractional designs introduced by Plackett and Burman were discussed. Plackett-Burman designs are used to estimate main effects in the case of two level fractional factorial experiments using very few runs. [[DOE_References|[G. E. P. Box and D. W. Behnken (1960)]]] introduced similar designs for three level factors that are widely used in response surface methods to fit second-order models to the response. The designs are referred to as Box-Behnken designs. The designs were developed by the combination of two level factorial designs with incomplete block designs. For example, the figure below shows the Box-Behnken design for three factors. The design is obtained by the combination of <math>2^{2}\,\!</math> design with a balanced incomplete block design having three treatments and three blocks (for details see [[DOE_References|[Box 1960, Montgomery 2001]]]).

[[Image:doe9.13.png|center|400px|Box-Behnken design for three factors: (a) shows the geometric representation and (b) shows the design.]]

The advantages of Box-Behnken designs include the fact that they are all spherical designs and require factors to be run at only three levels. The designs are also rotatable or nearly rotatable. Some of these designs also provide orthogonal blocking. Thus, if there is a need to separate runs into blocks for the Box-Behnken design, then designs are available that allow blocks to be used in such a way that the estimation of the regression parameters for the factor effects are not affected by the blocks. In other words, in these designs the block effects are orthogonal to the other factor effects. Yet another advantage of these designs is that there are no runs where all factors are at either the <math>+1\,\!</math> or <math>-1\,\!</math> levels. For example, in the figure below the representation of the Box-Behnken design for three factors clearly shows that there are no runs at the corner points. This could be advantageous when the corner points represent runs that are expensive or inconvenient because they lie at the end of the range of the factor levels. A few of the Box-Behnken designs available in DOE++ are presented in [[Box-Behnken_Designs|Appendix F]].

====Example====
Continuing with the example in [[Response_Surface_Methods_for_Optimization#Method_of_Steepest_Ascent| Method of Steepest Ascent]], the first order model was found to be inadequate for the region near the optimum. Once the experimenter realized that the first order model was not adequate (for the region with a reaction temperature of 350 <math>F\,\!</math> and reaction time of 165 minutes), it was decided to augment the experiment with axial runs to be able to complete a central composite design and fit a second order model to the response. Notice the advantage of using a central composite design, as the experimenter only had to add the axial runs to the <math>2^{2}\,\!</math> design with center point runs, and did not have to begin a new experiment. The experimenter decided to use <math>\alpha =1.4142\,\!</math> to get a rotatable design. The obtained response values are shown in the figure below.

[[Image:doe9.14.png|center|280px|Response values for the two factor central composite design in the [[Response_Surface_Methods_for_Optimization#Example_2| example]].]]

Such a design can be set up in a Weibull++ DOE folio using the properties shown in the figure below.

[[Image:doe9_15.png|center|744px|Properties for the central composite design in the [[Response_Surface_Methods_for_Optimization#Example_2| example]].|link=]]

The resulting design is shown in the figure shown next.

[[Image:doe9_16.png|center|800px|Central composite design for the experiment in the [[Response_Surface_Methods_for_Optimization#Example_2| example]].|link=]]

Results from the analysis of the design are shown in the next figure.

[[Image:doe9_17.png|center|744px|Results for the central composite design in the [[Response_Surface_Methods_for_Optimization#Example_2| example]].|link=]]

The results in the figure above show that the main effects, <math>A\,\!</math> and <math>B\,\!</math>, the interaction, <math>AB\,\!</math>, and the quadratic main effects, <math>{{A}^{2}}\,\!</math> and <math>{{B}^{2}}\,\!</math>, (represented as AA and BB in the figure) are significant. The lack-of-fit test also shows that the second order model with these terms is adequate and a higher order model is not needed. Using these results, the model for the experiment in terms of the coded values is:

::<math>\hat{y}=94.91+0.74{{x}_{1}}+1.53{{x}_{2}}+0.45{{x}_{1}}{{x}_{2}}-1.52x_{1}^{2}-2.08x_{2}^{2}\,\!</math>

The response surface and the contour plot for this model, in terms of the actual variables, are shown in the below figures (a) and (b), respectively.

[[Image:doe9_18_a.png|center|650px|link=]]

[[Image:doe9_18_b.png|center|650px|Response surface and countour plot for the experiment in the [[Response_Surface_Methods_for_Optimization#Example_2| example]].|link=]]

==Analysis of the Second Order Model==
Once a second order model is fit to the response, the next step is to locate the point of maximum or minimum response. The second order model for <math>k\,\!</math> factors can be written as:

::<math>\hat{y}={{\hat{\beta }}_{0}}+\underset{i=1}{\overset{k}{\mathop{\sum }}}\,{{\hat{\beta }}_{i}}{{x}_{i}}+\underset{i=1}{\overset{k}{\mathop{\sum }}}\,{{\hat{\beta }}_{ii}}x_{i}^{2}+\underset{i=1}{\overset{k}{\mathop{\sum }}}\,\underset{j=1}{\overset{i-1}{\mathop{\sum }}}\,{{\hat{\beta }}_{ij}}{{x}_{i}}{{x}_{j}}\,\!</math>

The point for which the response, <math>\hat{y}\,\!</math>, is optimized is the point at which the partial derivatives, <math>\partial \hat{y}/\partial {{x}_{1}}\,\!</math>, <math>\partial \hat{y}/\partial {{x}_{2}}\,\!</math>, <math>...\partial \hat{y}/\partial {{x}_{k}},\,\!</math> are all equal to zero. This point is called the ''stationary point''. The stationary point may be a point of maximum response, minimum response or a saddle point. These three conditions are shown in the following figures (a), (b) and (c) respectively.

[[Image:doe9.19.png|center|300px|Types of second order response surfaces and their contour plots. (a) shows the surface with a maximum point, (b) shows the surface with a minimum point and (c) shows the surface with a saddle point.]]

Notice that these conditions are easy to identify, in the case of two factor experiments, by the inspection of the contour plots. However, when more than two factors exist in an experiment, then the general mathematical solution for the location of the stationary point has to be used. The equation given above can be written in matrix notation as:

::<math>\hat{y}={{\hat{\beta }}_{0}}+{{x}^{\prime }}b+{{x}^{\prime }}Bx\,\!</math>

where:

::<math>x=\left[ \begin{matrix}
{{x}_{1}} \\
{{x}_{2}} \\
. \\
. \\
. \\
{{x}_{k}} \\
\end{matrix} \right]\text{ }b=\left[ \begin{matrix}
{{{\hat{\beta }}}_{1}} \\
{{{\hat{\beta }}}_{2}} \\
. \\
. \\
. \\
{{{\hat{\beta }}}_{k}} \\
\end{matrix} \right]\text{ and }B=\left[ \begin{matrix}
{{{\hat{\beta }}}_{11}} & {{{\hat{\beta }}}_{12}}/2 & . & . & . & {{{\hat{\beta }}}_{1k}}/2 \\
{} & {{{\hat{\beta }}}_{22}} & . & . & . & {{{\hat{\beta }}}_{2k}}/2 \\
{} & {} & . & . & . & . \\
{} & {} & {} & . & . & . \\
{} & {} & {} & {} & . & . \\
\text{sym}\text{.} & {} & {} & {} & {} & {{{\hat{\beta }}}_{kk}} \\
\end{matrix} \right]\,\!</math>

Then the stationary point can be determined as follows:

::<math>\frac{\partial \hat{y}}{\partial x}=b+2Bx=0\,\!</math>

Thus, the stationary point is:

::<math>{{x}_{s}}=-\frac{1}{2}{{B}^{-1}}b\,\!</math>

The optimum response is the response corresponding to <math>{{x}_{s}}\,\!</math>. The optimum response can be obtained to get:

::<math>{{\hat{y}}_{s}}={{\hat{\beta }}_{0}}+\frac{1}{2}x_{s}^{\prime }b\,\!</math>

Once the stationary point is known, it is necessary to determine if it is a maximum or minimum or saddle point. To do this, the second order model has to be transformed to the canonical form. This is done by transforming the model to a new coordinate system such that the origin lies at the stationary point and the axes are parallel to the principal axes of the fitted response surface, shown next.

[[Image:doe9.20.png|center|247px|The second order model in canonical form.]]

The resulting model equation then takes the following form:

::<math>\hat{y}={{\hat{y}}_{s}}+{{\lambda }_{1}}w_{1}^{2}+{{\lambda }_{2}}w_{2}^{2}+...+{{\lambda }_{k}}w_{k}^{2}\,\!</math>

where the <math>{{w}_{i}}\,\!</math> s are the transformed independent variables, and <math>{{\lambda }_{i}}\,\!</math>s are constants that are also the eigenvalues of the matrix <math>B\,\!</math>. The nature of the stationary point is known by looking at the signs of the <math>{{\lambda }_{i}}\,\!</math>s. If the <math>{{\lambda }_{i}}\,\!</math> s are all negative, then <math>{{x}_{s}}\,\!</math> is a point of maximum response. If the <math>{{\lambda }_{i}}\,\!</math>s are all positive then <math>{{x}_{s}}\,\!</math> is a point of minimum response. If the <math>{{\lambda }_{i}}\,\!</math>s have different signs, then <math>{{x}_{s}}\,\!</math> is a saddle point.

====Example====
Continuing with the [[Response_Surface_Methods_for_Optimization#Example_2| example]] in [[Response_Surface_Methods_for_Optimization#Method_of_Steepest_Ascent| Method of Steepest Ascent]], the second order model fitted to the response, in terms of the coded variables, was obtained as:

::<math>\hat{y}=94.91+0.74{{x}_{1}}+1.53{{x}_{2}}+0.45{{x}_{1}}{{x}_{2}}-1.52x_{1}^{2}-2.08x_{2}^{2}\,\!</math>

Then the <math>b\,\!</math> and <math>B\,\!</math> matrices for this model are:

::<math>b=\left[ \begin{matrix}
0.74 \\
1.53 \\
\end{matrix} \right]\text{ }B=\left[ \begin{matrix}
-1.52 & 0.45 \\
0.45 & -2.08 \\
\end{matrix} \right]\,\!</math>
 

The stationary point is:

<math>\begin{align}
{{x}_{s}}= & -\frac{1}{2}{{B}^{-1}}b \\
= & -\frac{1}{2}{{\left[ \begin{matrix}
-1.52 & 0.45 \\
0.45 & -2.08 \\
\end{matrix} \right]}^{-1}}\left[ \begin{matrix}
0.74 \\
1.53 \\
\end{matrix} \right] \\
= & \left[ \begin{matrix}
0.3 \\
0.4 \\
\end{matrix} \right]
\end{align}\,\!</math>

Then, in terms of the actual values, the stationary point can be found as:

::<math>\begin{align}
\text{Reaction Temperature:} & 0.3=\frac{T-350}{(355-345)/2} \\
T= & 351.5\text{ }F
\end{align}\,\!</math>

::<math>\begin{align}
\text{Reaction Time:} & 0.4=\frac{t-165}{(175-155)/2} \\
t= & 169.0\text{ minutes}
\end{align}\,\!</math>

To find the nature of the stationary point the eigenvalues of the <math>B\,\!</math> matrix can be obtained as follows using the determinant of the matrix <math>B-\lambda I\,\!</math>:

::<math>\begin{align}
\left| B-\lambda I \right|= & 0 \\
\left| \begin{matrix}
-1.52-\lambda & 0.45 \\
0.45 & -2.08-\lambda \\
\end{matrix} \right|= & 0
\end{align}\,\!</math>

This gives us:

::<math>{{\lambda }^{2}}+3.61\lambda +2.9743=0\,\!</math>

Solving the quadratic equation in <math>\lambda \,\!</math> returns the eigenvalues <math>{{\lambda }_{1}}=-1.2723\,\!</math> and <math>{{\lambda }_{2}}=-2.3377\,\!</math>. Since both the eigenvalues are negative, it can be concluded that the stationary point is a point of maximum response. The predicted value of the maximum response can be obtained as:

::<math>\begin{align}
{{{\hat{y}}}_{s}}= & {{{\hat{\beta }}}_{0}}+\frac{1}{2}x_{s}^{\prime }b \\
= & 94.91+\frac{1}{2}{{\left[ \begin{matrix}
0.3 \\
0.4 \\
\end{matrix} \right]}^{\prime }}\left[ \begin{matrix}
0.74 \\
1.53 \\
\end{matrix} \right] \\
= & 95.3
\end{align}\,\!</math>

In a DOE folio, the maximum response can be obtained by entering the required values as shown in the figure below. In the figure, the goal is to maximize the response and the limits of the search range for maximizing the response are entered as 90 and 100. The value of the maximum response and the corresponding values of the factors obtained are shown in the second figure following. These values match the values calculated in this example.

[[Image:doe9_21.png|center|543px|Settings to obtain the maximum value of the response in the [[Response_Surface_Methods_for_Optimization#Example_3| example]].|link=]]

[[Image:doe9_22.png|center|650px|Plot of the maximum response in the [[Response_Surface_Methods_for_Optimization#Example_3| example]] against the factors, temperature and time.|link=]]

==Multiple Responses==
In many cases, the experimenter has to optimize a number of responses at the same time. For the example in [[Response_Surface_Methods_for_Optimization#Method_of_Steepest_Ascent| Method of Steepest Ascent]], assume that the experimenter has to also consider two other responses: cost of the product (which should be minimized) and the pH of the product (which should be close to 7 so that the product is neither acidic nor basic). The data is presented in the figure below.

[[Image:doe9.23.png|center|400px|Data for the additional responses of cost and pH for the example to investigate the yield of a chemical process.]]

The problem in dealing with multiple responses is that now there might be conflicting objectives because of the different requirements of each of the responses. The experimenter needs to come up with a solution that satisfies each of the requirements as much as possible without compromising too much on any of the requirements. The approach used in a Weibull++ DOE folio to deal with optimization of multiple responses involves the use of desirability functions that are discussed next (for details see [[DOE_References|[Derringer and Suich, 1980]]]).

===Desirability Functions===
Under this approach, each <math>i\,\!</math>th response is assigned a desirability function, <math>{{d}_{i}}\,\!</math>, where the value of <math>{{d}_{i}}\,\!</math> varies between 0 and 1. The function, <math>{{d}_{i}},\,\!</math> is defined differently based on the objective of the response. If the response is to be maximized, as in the case of the previous example where the yield had to be maximized, then <math>{{d}_{i}}\,\!</math> is defined as follows:

::<math>{{d}_{i}}=\left\{ \begin{matrix}
0 & {{y}_{i}}<L \\
{{\left( \frac{{{y}_{i}}-L}{T-L} \right)}^{\omega }} & L\le {{y}_{i}}\le T \\
1 & {{y}_{i}}>T \\
\end{matrix} \right.</math>

where <math>T\,\!</math> represents the target value of the <math>i\,\!</math>th response, <math>{{y}_{i}}\,\!</math>, <math>L\,\!</math> represents the acceptable lower limit value for this response and <math>\omega \,\!</math> represents the weight. When <math>\omega =1\,\!</math> the function <math>{{d}_{i}}\,\!</math> is linear. If <math>\omega >1\,\!</math> then more importance is placed on achieving the target for the response, <math>{{y}_{i}}\,\!</math>. When <math>\omega <1\,\!</math>, less weight is assigned to achieving the target for the response, <math>{{y}_{i}}\,\!</math>. A graphical representation is shown in figure (a) below.

[[Image:doe9.24.png|center|400px|Desirability function plots for different response optimizations: (a) the goal is to maximize the response, (b) the goal is to minimize the response and (c) the goal is to get the response to a target value.]]

If the response is to be minimized, as in the case when the response is cost, <math>{{d}_{i}}\,\!</math> is defined as follows:

::<math>{{d}_{i}}=\left\{ \begin{matrix}
1 & {{y}_{i}}<T \\
{{\left( \frac{U-{{y}_{i}}}{U-T} \right)}^{\omega }} & T\le {{y}_{i}}\le U \\
0 & {{y}_{i}}>U \\
\end{matrix} \right.</math>

Here <math>U\,\!</math> represents the acceptable upper limit for the response (see figure (b) above).

There may be times when the experimenter wants the response to be neither maximized nor minimized, but instead stay as close to a specified target as possible. For example, in the case where the experimenter wants the product to be neither acidic nor basic, there is a requirement to keep the pH close to the neutral value of 7. In such cases, the desirability function is defined as follows (see figure (c) above):

::<math>{{d}_{i}}=\left\{ \begin{matrix}
0 & {{y}_{i}}<L \\
{{\left( \frac{{{y}_{i}}-L}{T-L} \right)}^{{{\omega }_{1}}}} & L\le {{y}_{i}}\le T \\
{{\left( \frac{U-{{y}_{i}}}{U-T} \right)}^{{{\omega }_{2}}}} & T\le {{y}_{i}}\le U \\
0 & {{y}_{i}}>U \\
\end{matrix} \right.</math>

Once a desirability function is defined for each of the responses, assuming that there are <math>m\,\!</math> responses, an overall desirability function is obtained as follows:

::<math>D={{(d_{1}^{{{r}_{1}}}\cdot d_{2}^{{{r}_{2}}}\cdot ...\cdot d_{m}^{{{r}_{m}}})}^{1/({{r}_{1}}+{{r}_{2}}+...+{{r}_{m}})}}\,\!</math>

where the <math>{{r}_{i}}\,\!</math> s represent the importance of each response. The greater the value of <math>{{r}_{i}}\,\!</math>, the more important the response with respect to the other responses. The objective is to now find the settings that return the maximum value of <math>D\,\!</math>.

To illustrate the use of desirability functions, consider the previous example with the three responses of yield, cost and pH. The response surfaces for the two additional responses of cost and pH are shown next in the figures (a) and (b), respectively.

[[Image:doe9_25.png|center|500px|Response surfaces for (a) cost and (b) pH.]]

In terms of actual variables, the models obtained for all three responses are as shown next:

::<math>\begin{align}
\text{Yield: } & {{{\hat{y}}}_{1}}=-7480.24+41.24{{x}_{1}}+3.88{{x}_{2}}+0.009{{x}_{1}}{{x}_{2}} \\
& -0.060825x_{1}^{2}-0.020831x_{2}^{2} \\
\text{Cost: } & {{{\hat{y}}}_{2}}=15560.49-82.48{{x}_{1}}-7.76{{x}_{2}}-0.02{{x}_{1}}{{x}_{2}} \\
& +0.12165x_{1}^{2}+0.041663x_{2}^{2} \\
\text{pH: } & {{{\hat{y}}}_{3}}=38.87-0.0709{{x}_{1}}-0.041{{x}_{2}}
\end{align}\,\!</math>

Assume that the experimenter wants to have a target yield value of 95, although any value of yield greater than 94 is acceptable. Then the desirability function for yield is:

::<math>{{d}_{1}}=\left\{ \begin{matrix}
0 & {{y}_{1}}<94 \\
{{\left( \frac{{{y}_{1}}-94}{95-94} \right)}^{\omega }} & 94\le {{y}_{1}}\le 95 \\
1 & {{y}_{1}}>95 \\
\end{matrix} \right.</math>

For the cost, assume that the experimenter wants to lower the cost to 400, although any cost value below 415 is acceptable. Then the desirability function for cost is:

::<math>{{d}_{2}}=\left\{ \begin{matrix}
1 & {{y}_{2}}<400 \\
{{\left( \frac{415-{{y}_{2}}}{415-400} \right)}^{\omega }} & 400\le {{y}_{2}}\le 415 \\
0 & {{y}_{2}}>415 \\
\end{matrix} \right.</math>

For the pH, a target of 7 is desired but values between 6.9 and 7.1 are also acceptable. Thus, the desirability function here is:

::<math>{{d}_{3}}=\left\{ \begin{matrix}
0 & {{y}_{3}}<L \\
{{\left( \frac{{{y}_{3}}-6.9}{7-6.9} \right)}^{{{\omega }_{1}}}} & 6.9\le {{y}_{3}}\le 7 \\
{{\left( \frac{7.1-{{y}_{3}}}{7.1-7} \right)}^{{{\omega }_{2}}}} & 7\le {{y}_{3}}\le 7.1 \\
0 & {{y}_{3}}>7.1 \\
\end{matrix} \right.</math>

Notice that in the previous equations all weights used (<math>{{\omega }_{i}}\,\!</math> s) are 1. Thus, all three desirability functions are linear. The overall desirability function, assuming equal importance (<math>{{r}_{1}}={{r}_{2}}={{r}_{3}}=1\,\!</math>) for all the responses, is:

::<math>\begin{align}
D= & {{({{d}_{1}}\cdot {{d}_{2}}\cdot {{d}_{3}})}^{1/3}} \\
= & f({{{\hat{y}}}_{1}},{{{\hat{y}}}_{2}},{{{\hat{y}}}_{3}})
\end{align}\,\!</math>

The objective of the experimenter is to find the settings of <math>{{x}_{1}}\,\!</math> and <math>{{x}_{2}}\,\!</math> such that the overall desirability, <math>D\,\!</math>, is maximum. In a DOE folio, the settings for the desirability functions for each of the three responses can be entered as shown in the next figure.

[[Image:doe9_26.png|center|750px|Optimization settings for the three responses of yield, cost, and pH.]]

Based on these settings, the Weibull++ DOE folio solves this optimization problem to obtain the following solution:

[[Image:doe9_27.png|center|650px|Optimum solution from DOE++ for the three responses of yield, cost, and pH.]]

::<math>\begin{align}
Temperature= & 351.5\text{ }F\text{ } \\
Time= & 169\text{ }minutes
\end{align}\,\!</math>

The overall desirability achieved with this solution can be calculated easily. The values of each of the response for these settings are:

::<math>\begin{align}
\text{ }{{{\hat{y}}}_{1}}= & -7480.24+41.24{{x}_{1}}+3.88{{x}_{2}}+0.009{{x}_{1}}{{x}_{2}} \\
& -0.060825x_{1}^{2}-0.020831x_{2}^{2} \\
= & -7480.24+41.24(351.5)+3.88(169)+0.009(351.3)(169) \\
& -0.060825{{(351.5)}^{2}}-0.020831{{(169)}^{2}} \\
= & 95.3 \\
\text{ }{{{\hat{y}}}_{2}}= & 15560.49-82.48{{x}_{1}}-7.76{{x}_{2}}-0.02{{x}_{1}}{{x}_{2}} \\
& +0.12165x_{1}^{2}+0.041663x_{2}^{2} \\
= & 15560.49-82.48(351.5)-7.76(169)-0.02(351.5)(169) \\
& +0.12165{{(351.5)}^{2}}+0.041663{{(169)}^{2}} \\
= & 409.35 \\
\text{ }{{{\hat{y}}}_{3}}= & 38.87-0.0709{{x}_{1}}-0.041{{x}_{2}} \\
= & 38.87-0.0709(351.5)-0.041(169) \\
= & 7.00
\end{align}\,\!</math>

Based on the response values, the individual desirability functions are:

::<math>\begin{align}
{{d}_{1}}= & 1\text{ (since }{{{\hat{y}}}_{1}}>95\text{)} \\
{{d}_{2}}= & \left( \frac{415-{{{\hat{y}}}_{2}}}{415-400} \right)\text{ (since }400\le {{y}_{2}}\le 415\text{)} \\
= & \left( \frac{415-409.35}{415-400} \right) \\
= & 0.3767 \\
{{d}_{3}}= & 1\text{ (since }{{{\hat{y}}}_{3}}\approx 7\text{)}
\end{align}\,\!</math>

Then the overall desirability is:

::<math>\begin{align}
D= & {{({{d}_{1}}\cdot {{d}_{2}}\cdot {{d}_{3}})}^{1/3}} \\
= & {{(1\cdot 0.3767\cdot 1)}^{1/3}} \\
= & 0.72
\end{align}\,\!</math>

This is the same as the Global Desirability displayed by the Weibull++ DOE folio in the figure above. At times, a number of solutions may be obtained from the DOE folio, and it is up to the experimenter to choose the most feasible one.

Response Surface Methods for Optimization

2017-08-10T17:18:03Z

Richard House: /* Multiple Responses */

{{Template:Doebook|10}}
The experiment designs mentioned in [[Two_Level_Factorial_Experiments| Two Level Factorial Experiments]] and [[Highly_Fractional_Factorial_Designs| Highly Fractional Factorial Designs]] help the experimenter identify factors that affect the response. Once the important factors have been identified, the next step is to determine the settings for these factors that result in the optimum value of the response. The optimum value of the response may either be a maximum value or a minimum value, depending upon the product or process in question. For example, if the response in an experiment is the yield from a chemical process, then the objective might be to find the settings of the factors affecting the yield so that the yield is maximized. On the other hand, if the response in an experiment is the number of defects, then the goal would be to find the factor settings that minimize the number of defects. Methodologies that help the experimenter reach the goal of optimum response are referred to as ''response surface methods''. These methods are exclusively used to examine the "surface," or the relationship between the response and the factors affecting the response. Regression models are used for the analysis of the response, as the focus now is on the nature of the relationship between the response and the factors, rather than identification of the important factors.

Response surface methods usually involve the following steps:

#The experimenter needs to move from the present operating conditions to the vicinity of the operating conditions where the response is optimum. This is done using the ''method of steepest ascent'' in the case of maximizing the response. The same method can be used to minimize the response and is then referred to as the ''method of steepest descent''.
#Once in the vicinity of the optimum response the experimenter needs to fit a more elaborate model between the response and the factors. Special experiment designs, referred to as [[Response_Surface_Methods_for_Optimization#RSM_Designs|''RSM designs'']], are used to accomplish this. The fitted model is used to arrive at the best operating conditions that result in either a maximum or minimum response.
#It is possible that a number of responses may have to be optimized at the same time. For example, an experimenter may want to maximize strength, while keeping the number of defects to a minimum. The optimum settings for each of the responses in such cases may lead to conflicting settings for the factors. A balanced setting has to be found that gives the most appropriate values for all the responses. Desirability functions are useful in these cases.

==Method of Steepest Ascent==
The first step in obtaining the optimum response settings, after the important factors have been identified, is to explore the region around the current operating conditions to decide what direction needs to be taken to move towards the optimum region. Usually, a first order regression model (containing just the main effects and no interaction terms) is sufficient at the current operating conditions because the operating conditions are normally far from the optimum response settings. The experimenter needs to move from the current operating conditions to the optimum region in the most efficient way by using the minimum number of experiments. This is done using the method of steepest ascent. In this method, the contour plot of the first order model is used to decide the settings for the next experiment, in order to move towards the optimum conditions. Consider a process where the response has been found to be a function of two factors. To explore the region around the current operating conditions, the experimenter fits the following first order model between the response and the two factors:

::<math>y={{\beta }_{0}}+{{\beta }_{1}}{{x}_{1}}+{{\beta }_{2}}{{x}_{2}}+\epsilon \,\!</math>

The response surface plot for the model, along with the contours, is shown in the figure below. It can be seen in the figure that in order to maximize the response, the most efficient direction in which to move the experiment is along the line perpendicular to the contours. This line, also referred to as the path of steepest ascent, is the line along which the rate of increase of the response is maximum. The steps along this line to move towards the optimum region are proportional to the regression coefficients, <math>{{\beta }_{j}},\,\!</math> of the fitted first order model.

[[Image:doe9_1.png|center|504px|Path of steepest ascent for the model <math>y=\beta_0+\beta_1 x_1+\beta_2 x_2+\epsilon \,\!</math>.]]

Experiments are conducted along each step of the path of steepest ascent until an increase in the response is not seen. Then, a new first order model is fit at the region of the maximum response. If the first order model shows a lack of fit, then this indicates that the experimenter has reached the vicinity of the optimum. RSM designs are then used explore the region thoroughly and obtain the point of the maximum response. If the first order model does not show a lack of fit, then a new path of steepest ascent is determined and the process is repeated.

===Example===
The yield from a chemical process is found to be affected by two factors: reaction temperature and reaction time. The current reaction temperature is 230 <math>F\,\!</math> and the reaction time is 65 minutes. The experimenter wants to determine the settings of the two factors such that maximum yield can be obtained from the process. To explore the region around the current operating conditions, the experimenter decides to use a single replicate of the <math>2^{2}\,\!</math> design. The range of the factors for this design are chosen to be (225, 235) <math>F\,\!</math> for the reaction temperature and (55, 75) minutes for the reaction time. The unreplicated <math>2^{2}\,\!</math> design is also augmented with five runs at the center point to estimate the error sum of squares, <math>S{{S}_{E}}\,\!</math>, and check for model adequacy. The response values obtained for this design are shown next.

[[Image:doe9.2.png|center|256px|The <math>2^{2}\,\!</math> design augmented with five center points to explore the region around current operating conditions for a chemical process.|link=]]

In a DOE folio, this design can be set up using the properties shown next.

[[Image:doe9_3.png|center|809px|Design properties for the <math>2^{2}\,\!</math> design to explore the current operating conditions.|link=]]

The resulting design and the analysis results are shown next.

[[Image:doe9_4.png|center|778px|The <math>2^{2}\,\!</math> experiment design in to explore the current operating conditions.|link=]]

[[Image:doe9_5.png|center|774px|Results for the <math>2^{2}\,\!</math> experiment to explore the current operating conditions.|link=]]

Note that the results shown are in terms of the coded values of the factors (taking -1 as the value of the lower settings for reaction temperature and reaction time and +1 as the value for the higher settings for these two factors). The results show that the factors, <math>A\,\!</math> (temperature) and <math>B\,\!</math> (time), affect the response significantly but their interaction does not affect the response. Therefore the interaction term can be dropped from the model for this experiment. The results also show that Curvature is not a significant factor. This indicates that the first order model is adequate for the experiment at the current operating conditions. Using these two conclusions, the model for the current operating conditions, in terms the coded variables is:

:<math>\hat{y}=36.6375+1.1635x_{1}+0.4875x_{2}\,\!</math>

where <math>\hat{y}\,\!</math> represents the yield and <math>x_{1}\,\!</math> and <math>x_{2}\,\!</math> are the predictor variables for the two factors, <math>A\,\!</math> and <math>B\,\!</math>, respectively. To further confirm the adequacy of the model of the equation given above, the experiment can be analyzed again after dropping the interaction term, <math>AB\,\!</math>. The results are shown next.

[[Image:doe9_6.png|center|774px|Results for the <math>2^{2}\,\!</math> experiment after the interaction term is dropped from the model.|link=]]

The results show that the lack-of-fit for this model (because of the deficiency created in the model by the absence of the interaction term) is not significant, confirming that the model is adequate.

==Path of Steepest Ascent==
The contour plot for the model used in the above example is shown next.

[[Image:doe9_7.png|center|650px|Results for the <math>2^{2}\,\!</math> experiment after the interaction term is dropped from the model.|link=]]

The regression coefficients for the model are <math>\hat{\beta}_{1}= 1.1625\,\!</math> and <math>\hat{\beta}_{2}= 0.4875\,\!</math>. To move towards the optimum, the experimenter needs to move along the path of steepest ascent, which lies perpendicular to the contours. This path is the line through the center point of the current operating conditions (<math>x_{1}=0\,\!</math>, <math>x_{2}=0\,\!</math>) with a slope of <math>\hat{\beta}_{2}/\hat{\beta}_{1}\,\!</math>. Therefore, in terms of the coded variables, the experiment should be moved 1.1625 units in the <math>x_{1}\,\!</math> direction for every 0.4875 units in the <math>x_{2}\,\!</math> direction. To move along this path, the experimenter decides to use a step-size of 10 minutes for the reaction time, <math>x_{2}\,\!</math>. The coded value for this step size can be obtained as follows. Recall from [[Multiple_Linear_Regression_Analysis|Multiple Linear Regression Analysis]] that the relationship between coded and actual values is:

::<math>coded\ value=\frac{actual\ value - mean}{half\ of\ range}</math>

or

::<math>coded\ value=\frac{step\ size\ in\ actual\ value}{half\ of\ range}</math>

Thus, for a step-size of 10 minutes, the equivalent step size in coded value for <math>x_{2}\,\!</math> is:

::<math>=\frac{10}{half\ of\ range\ of\ x_{2}}\,\!</math>
::<math>=\frac{10}{(75-55)/2}\,\!</math>
::<math>=1\,\!</math>

In terms of the coded variables, the path of steepest ascent requires a move of 1.1625 units in the <math>x_{1}\,\!</math> direction for every 0.4875 units in the <math>x_{2}\,\!</math> direction. The step-size for <math>x_{1}\,\!</math>, in terms of the coded value corresponding to any step-size in <math>x_{2}\,\!</math>, is:

::<math>=\frac{\hat{\beta_{1}}}{\hat{\beta_{2}}}\cdot(step\ size\ in\ x_{2})\,\!</math>

Therefore, the step-size for the reaction temperature, <math>x_{1}\,\!</math>, in terms of the coded variables is:

::<math>=\frac{\hat{\beta_{1}}}{\hat{\beta_{2}}}\cdot\Delta x_{2}\,\!</math>
::<math>=\frac{1.1625}{0.4875}\cdot1\,\!</math>
::<math>=\frac{\hat{\beta_{1}}}{\hat{\beta_{2}}}\cdot\Delta x_{2}\,\!</math>

This corresponds to a step of approximately 12 <math>F\,\!</math> for temperature in terms of the actual value as shown next:

::<math>\begin{align}
step\ size\ in\ actual\ value = & (coded\ value)\cdot(half\ of\ the\ range\ of\ x_{1})\\
= & 2.39\cdot10/2\\
= & 11.95
\end{align}\,\!</math>

Using a step of 12 <math>F\,\!</math> and 10 minutes, the experimenter conducts experiments until no further increase is observed in the yield. The yield values at each step are shown in the table given next.

[[Image:doet9.1.png|center|341px|Response values at each step of the path of steepest ascent for the experiment to investigate the yield of a chemical process. Units for factor levels and the response have been omitted.]]

The yield starts decreasing after the reaction temperature of 350 <math>F\,\!</math> and the reaction time of 165 minutes, indicating that this point may lie close to the optimum region. To analyze the vicinity of this point, a <math>2^{2}\,\!</math> design augmented by five center points is selected. The range of exploration is chosen to be 345 to 355 <math>F\,\!</math> for reaction temperature and 155 to 175 minutes for reaction time. The response values recorded are shown next.

[[Image:doe9.8.png|center|348px|The <math>2^{2}\,\!</math> design augmented with five center points to explore the region of maximum response obtained from the path of steepest ascent. Note that the center point of this design is the new origin.]]

The results for this design are shown next.

[[Image:doe9_9.png|center|760px|Results for the <math>2^{2}\,\!</math> experiment to explore the region of maximum response.|link=]]

In the results, Curvature is displayed as a significant factor. This indicates that the first order model is not adequate for this region of the experiment and a higher order model is required. As a result, the methodology of steepest ascent can no longer be used. The presence of curvature indicates that the experiment region may be close to the optimum. Special designs that allow the use of second order models are needed at this point.

==RSM Designs==
A second order model is generally used to approximate the response once it is realized that the experiment is close to the optimum response region where a first order model is no longer adequate. The second order model is usually sufficient for the optimum region, as third order and higher effects are seldom important. The second order regression model takes the following form for <math>k\,\!</math> factors:

::<math>\begin{align}
y= & {{\beta }_{0}}+{{\beta }_{1}}{{x}_{1}}+{{\beta }_{2}}{{x}_{2}}+...+{{\beta }_{k}}{{x}_{k}}+ \\
& {{\beta }_{11}}x_{1}^{2}+{{\beta }_{22}}x_{2}^{2}+...+{{\beta }_{kk}}x_{k}^{2}+{{\beta }_{12}}{{x}_{1}}{{x}_{2}}+...+{{\beta }_{k-1,k}}{{x}_{k-1}}{{x}_{k}}+\epsilon
\end{align}\,\!</math>

The model contains <math>p=(k+1)(k+2)/2\,\!</math> regression parameters that include coefficients for main effects (<math>{{\beta }_{1}},{{\beta }_{2}}...{{\beta }_{k}}\,\!</math>), coefficients for quadratic main effects (<math>{{\beta }_{11}},{{\beta }_{22}}...{{\beta }_{kk}}\,\!</math>) and coefficients for two factor interaction effects (<math>{{\beta }_{12}},{{\beta }_{13}}\,\!</math>... <math>{{\beta }_{k-1,k}}\,\!</math>). A full factorial design with all factors at three levels would provide estimation of all the required regression parameters. However, full factorial three level designs are expensive to use as the number of runs increases rapidly with the number of factors. For example, a three factor full factorial design with each factor at three levels would require <math>{{3}^{3}}=27\,\!</math> runs while a design with four factors would require <math>{{3}^{4}}=81\,\!</math> runs. Additionally, these designs will estimate a number of higher order effects which are usually not of much importance to the experimenter. Therefore, for the purpose of analysis of response surfaces, special designs are used that help the experimenter fit the second order model to the response with the use of a minimum number of runs. Examples of these designs are the central composite and Box-Behnken designs.

===Central Composite Designs===
Central composite designs are two level full factorial (<math>2^{k}\,\!</math>) or fractional factorial (<math>2^{k-f}\,\!</math>) designs augmented by a number of center points and other chosen runs. These designs are such that they allow the estimation of all the regression parameters required to fit a second order model to a given response.

The simplest of the central composite designs can be used to fit a second order model to a response with two factors. The design consists of a <math>2^{2}\,\!</math> full factorial design augmented by a few runs at the center point (such a design is shown in figure (a) given below). A central composite design is obtained when runs at points (<math>-1,0\,\!</math>), (<math>1,0\,\!</math>), (<math>0,-1\,\!</math>) and (<math>0,1\,\!</math>) are added to this design. These points are referred to as ''axial points'' or ''star points'' and represent runs where all but one of the factors are set at their mid-levels. The number of axial points in a central composite design having <math>k\,\!</math> factors is <math>2k\,\!</math>. The distance of the axial points from the center point is denoted by <math>\alpha \,\!</math> and is always specified in terms of coded values. For example, the central composite design in figure (b) given below has <math>\alpha =1\,\!</math>, while for the design of figure (c) <math>\alpha =1.414\,\!</math>.

[[Image:doe9.10.png|center|227px|Central composite designs: (a) shows the <math>2^2\,\!</math> design with center point runs, (b) shows the two factor central composite design with <math>\alpha=1\,\!</math> and (c) shows the two factor central composite design with <math>\alpha=\sqrt{2}\,\!</math>.]]

It can be noted that when <math>\alpha >1\,\!</math>, each factor is run at five levels (<math>-\alpha \,\!</math>, <math>-1\,\!</math>, <math>0\,\!</math>, <math>1\,\!</math> and <math>\alpha \,\!</math>) instead of the three levels of <math>-1\,\!</math>, <math>0\,\!</math> and <math>1\,\!</math>. The reason for running central composite designs with <math>\alpha >1\,\!</math> is to have a rotatable design, which is explained next.

====Rotatability====
A central composite design is said to be ''rotatable'' if the variance of any predicted value of the response, <math>{{\hat{y}}_{p}}\,\!</math>, for any level of the factors depends only on the distance of the point from the center of the design, regardless of the direction. In other words, a rotatable central composite design provides constant variance of the estimated response corresponding to all new observation points that are at the same distance from the center point of the design (in terms of the coded variables).

The variance of the predicted response at any point, <math>{{x}_{p}}\,\!</math>, is given as follows:

::<math>V[{{\hat{y}}_{p}}]={{\sigma }^{2}}x_{p}^{\prime }{{({{X}^{\prime }}X)}^{-1}}{{x}_{p}}\,\!</math>

The contours of <math>V[{{\hat{y}}_{p}}]\,\!</math> for the central composite design in figure (c) above are shown in the figure below. The contours are concentric circles indicating that the central composite design of figure (c) is rotatable. Rotatability is a desirable property because the experimenter does not have any prior information about the location of the optimum. Therefore, a design that provides equal precision of estimation in all directions would be preferred. Such a design will assure the experimenter that no matter what direction is taken to search for the optimum, he/she will be able to estimate the response value with equal precision. A central composite design is rotatable if the value of <math>\alpha \,\!</math> for the design satisfies the following equation:

::<math>\alpha ={{\left[ \frac{{{2}^{k-f}}({{n}_{f}})}{{{n}_{s}}} \right]}^{1/4}}\,\!</math>

where <math>{{n}_{f}}\,\!</math> is the number of replicates of the runs in the original factorial design and <math>{{n}_{s}}\,\!</math> is the number of replicates of the runs at the axial points. For example, a central composite design with two factors, having a single replicate of the original factorial design, and a single replicate of all the axial points, would be rotatable for the following <math>\alpha \,\!</math> value:

::<math>\begin{align}
\alpha = & {{\left[ \frac{{{2}^{k-f}}({{n}_{f}})}{{{n}_{s}}} \right]}^{1/4}} \\
= & {{\left[ \frac{{{2}^{2}}(1)}{1} \right]}^{1/4}} \\
= & \sqrt{2} \\
= & 1.414
\end{align}\,\!</math>

Thus, a central composite design in two factors, having a single replicate of the original <math>2^{2}\,\!</math> design and axial points, and with <math>\alpha =1.414\,\!</math>, is a rotatable design. This design is shown in figure (c) above.

[[Image:doe9.11.png|center|266px|The countours of <math>V[\hat{y}_p] \,\!</math> for the rotatable two factor central composite design.]]

====Spherical Design====
A central composite design is said to be ''spherical'' if all factorial and axial points are at same distance from the center of the design. Spherical central composite designs are obtained by setting <math>\alpha =\sqrt{k}\,\!</math>. For example, the rotatable design in the figure above (c) is also a spherical design because for this design <math>\alpha =\sqrt{k}=\sqrt{2}=1.414\,\!</math>.

====Face-centered Design====
Central composite designs in which the axial points represent the mid levels for all but one of the factors are also referred to as ''face-centered'' central composite designs. For these designs, <math>\alpha =1\,\!</math> and all factors are run at three levels, which are <math>-1\,\!</math>, <math>0\,\!</math> and <math>1,\,\!</math> in terms of the coded values (see the figure below).

[[Image:doe9.12.png|center|300px|Face-centered central composite design for three factors.]]

===Box-Behnken Designs===
In [[Highly_Fractional_Factorial_Designs| Highly Fractional Factorial Designs]], highly fractional designs introduced by Plackett and Burman were discussed. Plackett-Burman designs are used to estimate main effects in the case of two level fractional factorial experiments using very few runs. [[DOE_References|[G. E. P. Box and D. W. Behnken (1960)]]] introduced similar designs for three level factors that are widely used in response surface methods to fit second-order models to the response. The designs are referred to as Box-Behnken designs. The designs were developed by the combination of two level factorial designs with incomplete block designs. For example, the figure below shows the Box-Behnken design for three factors. The design is obtained by the combination of <math>2^{2}\,\!</math> design with a balanced incomplete block design having three treatments and three blocks (for details see [[DOE_References|[Box 1960, Montgomery 2001]]]).

[[Image:doe9.13.png|center|400px|Box-Behnken design for three factors: (a) shows the geometric representation and (b) shows the design.]]

The advantages of Box-Behnken designs include the fact that they are all spherical designs and require factors to be run at only three levels. The designs are also rotatable or nearly rotatable. Some of these designs also provide orthogonal blocking. Thus, if there is a need to separate runs into blocks for the Box-Behnken design, then designs are available that allow blocks to be used in such a way that the estimation of the regression parameters for the factor effects are not affected by the blocks. In other words, in these designs the block effects are orthogonal to the other factor effects. Yet another advantage of these designs is that there are no runs where all factors are at either the <math>+1\,\!</math> or <math>-1\,\!</math> levels. For example, in the figure below the representation of the Box-Behnken design for three factors clearly shows that there are no runs at the corner points. This could be advantageous when the corner points represent runs that are expensive or inconvenient because they lie at the end of the range of the factor levels. A few of the Box-Behnken designs available in DOE++ are presented in [[Box-Behnken_Designs|Appendix F]].

====Example====
Continuing with the example in [[Response_Surface_Methods_for_Optimization#Method_of_Steepest_Ascent| Method of Steepest Ascent]], the first order model was found to be inadequate for the region near the optimum. Once the experimenter realized that the first order model was not adequate (for the region with a reaction temperature of 350 <math>F\,\!</math> and reaction time of 165 minutes), it was decided to augment the experiment with axial runs to be able to complete a central composite design and fit a second order model to the response. Notice the advantage of using a central composite design, as the experimenter only had to add the axial runs to the <math>2^{2}\,\!</math> design with center point runs, and did not have to begin a new experiment. The experimenter decided to use <math>\alpha =1.4142\,\!</math> to get a rotatable design. The obtained response values are shown in the figure below.

[[Image:doe9.14.png|center|280px|Response values for the two factor central composite design in the [[Response_Surface_Methods_for_Optimization#Example_2| example]].]]

Such a design can be set up in a Weibull++ DOE folio using the properties shown in the figure below.

[[Image:doe9_15.png|center|744px|Properties for the central composite design in the [[Response_Surface_Methods_for_Optimization#Example_2| example]].|link=]]

The resulting design is shown in the figure shown next.

[[Image:doe9_16.png|center|800px|Central composite design for the experiment in the [[Response_Surface_Methods_for_Optimization#Example_2| example]].|link=]]

Results from the analysis of the design are shown in the next figure.

[[Image:doe9_17.png|center|744px|Results for the central composite design in the [[Response_Surface_Methods_for_Optimization#Example_2| example]].|link=]]

The results in the figure above show that the main effects, <math>A\,\!</math> and <math>B\,\!</math>, the interaction, <math>AB\,\!</math>, and the quadratic main effects, <math>{{A}^{2}}\,\!</math> and <math>{{B}^{2}}\,\!</math>, (represented as AA and BB in the figure) are significant. The lack-of-fit test also shows that the second order model with these terms is adequate and a higher order model is not needed. Using these results, the model for the experiment in terms of the coded values is:

::<math>\hat{y}=94.91+0.74{{x}_{1}}+1.53{{x}_{2}}+0.45{{x}_{1}}{{x}_{2}}-1.52x_{1}^{2}-2.08x_{2}^{2}\,\!</math>

The response surface and the contour plot for this model, in terms of the actual variables, are shown in the below figures (a) and (b), respectively.

[[Image:doe9_18_a.png|center|650px|link=]]

[[Image:doe9_18_b.png|center|650px|Response surface and countour plot for the experiment in the [[Response_Surface_Methods_for_Optimization#Example_2| example]].|link=]]

==Analysis of the Second Order Model==
Once a second order model is fit to the response, the next step is to locate the point of maximum or minimum response. The second order model for <math>k\,\!</math> factors can be written as:

::<math>\hat{y}={{\hat{\beta }}_{0}}+\underset{i=1}{\overset{k}{\mathop{\sum }}}\,{{\hat{\beta }}_{i}}{{x}_{i}}+\underset{i=1}{\overset{k}{\mathop{\sum }}}\,{{\hat{\beta }}_{ii}}x_{i}^{2}+\underset{i=1}{\overset{k}{\mathop{\sum }}}\,\underset{j=1}{\overset{i-1}{\mathop{\sum }}}\,{{\hat{\beta }}_{ij}}{{x}_{i}}{{x}_{j}}\,\!</math>

The point for which the response, <math>\hat{y}\,\!</math>, is optimized is the point at which the partial derivatives, <math>\partial \hat{y}/\partial {{x}_{1}}\,\!</math>, <math>\partial \hat{y}/\partial {{x}_{2}}\,\!</math>, <math>...\partial \hat{y}/\partial {{x}_{k}},\,\!</math> are all equal to zero. This point is called the ''stationary point''. The stationary point may be a point of maximum response, minimum response or a saddle point. These three conditions are shown in the following figures (a), (b) and (c) respectively.

[[Image:doe9.19.png|center|300px|Types of second order response surfaces and their contour plots. (a) shows the surface with a maximum point, (b) shows the surface with a minimum point and (c) shows the surface with a saddle point.]]

Notice that these conditions are easy to identify, in the case of two factor experiments, by the inspection of the contour plots. However, when more than two factors exist in an experiment, then the general mathematical solution for the location of the stationary point has to be used. The equation given above can be written in matrix notation as:

::<math>\hat{y}={{\hat{\beta }}_{0}}+{{x}^{\prime }}b+{{x}^{\prime }}Bx\,\!</math>

where:

::<math>x=\left[ \begin{matrix}
{{x}_{1}} \\
{{x}_{2}} \\
. \\
. \\
. \\
{{x}_{k}} \\
\end{matrix} \right]\text{ }b=\left[ \begin{matrix}
{{{\hat{\beta }}}_{1}} \\
{{{\hat{\beta }}}_{2}} \\
. \\
. \\
. \\
{{{\hat{\beta }}}_{k}} \\
\end{matrix} \right]\text{ and }B=\left[ \begin{matrix}
{{{\hat{\beta }}}_{11}} & {{{\hat{\beta }}}_{12}}/2 & . & . & . & {{{\hat{\beta }}}_{1k}}/2 \\
{} & {{{\hat{\beta }}}_{22}} & . & . & . & {{{\hat{\beta }}}_{2k}}/2 \\
{} & {} & . & . & . & . \\
{} & {} & {} & . & . & . \\
{} & {} & {} & {} & . & . \\
\text{sym}\text{.} & {} & {} & {} & {} & {{{\hat{\beta }}}_{kk}} \\
\end{matrix} \right]\,\!</math>

Then the stationary point can be determined as follows:

::<math>\frac{\partial \hat{y}}{\partial x}=b+2Bx=0\,\!</math>

Thus, the stationary point is:

::<math>{{x}_{s}}=-\frac{1}{2}{{B}^{-1}}b\,\!</math>

The optimum response is the response corresponding to <math>{{x}_{s}}\,\!</math>. The optimum response can be obtained to get:

::<math>{{\hat{y}}_{s}}={{\hat{\beta }}_{0}}+\frac{1}{2}x_{s}^{\prime }b\,\!</math>

Once the stationary point is known, it is necessary to determine if it is a maximum or minimum or saddle point. To do this, the second order model has to be transformed to the canonical form. This is done by transforming the model to a new coordinate system such that the origin lies at the stationary point and the axes are parallel to the principal axes of the fitted response surface, shown next.

[[Image:doe9.20.png|center|247px|The second order model in canonical form.]]

The resulting model equation then takes the following form:

::<math>\hat{y}={{\hat{y}}_{s}}+{{\lambda }_{1}}w_{1}^{2}+{{\lambda }_{2}}w_{2}^{2}+...+{{\lambda }_{k}}w_{k}^{2}\,\!</math>

where the <math>{{w}_{i}}\,\!</math> s are the transformed independent variables, and <math>{{\lambda }_{i}}\,\!</math>s are constants that are also the eigenvalues of the matrix <math>B\,\!</math>. The nature of the stationary point is known by looking at the signs of the <math>{{\lambda }_{i}}\,\!</math>s. If the <math>{{\lambda }_{i}}\,\!</math> s are all negative, then <math>{{x}_{s}}\,\!</math> is a point of maximum response. If the <math>{{\lambda }_{i}}\,\!</math>s are all positive then <math>{{x}_{s}}\,\!</math> is a point of minimum response. If the <math>{{\lambda }_{i}}\,\!</math>s have different signs, then <math>{{x}_{s}}\,\!</math> is a saddle point.

====Example====
Continuing with the [[Response_Surface_Methods_for_Optimization#Example_2| example]] in [[Response_Surface_Methods_for_Optimization#Method_of_Steepest_Ascent| Method of Steepest Ascent]], the second order model fitted to the response, in terms of the coded variables, was obtained as:

::<math>\hat{y}=94.91+0.74{{x}_{1}}+1.53{{x}_{2}}+0.45{{x}_{1}}{{x}_{2}}-1.52x_{1}^{2}-2.08x_{2}^{2}\,\!</math>

Then the <math>b\,\!</math> and <math>B\,\!</math> matrices for this model are:

::<math>b=\left[ \begin{matrix}
0.74 \\
1.53 \\
\end{matrix} \right]\text{ }B=\left[ \begin{matrix}
-1.52 & 0.45 \\
0.45 & -2.08 \\
\end{matrix} \right]\,\!</math>
 

The stationary point is:

<math>\begin{align}
{{x}_{s}}= & -\frac{1}{2}{{B}^{-1}}b \\
= & -\frac{1}{2}{{\left[ \begin{matrix}
-1.52 & 0.45 \\
0.45 & -2.08 \\
\end{matrix} \right]}^{-1}}\left[ \begin{matrix}
0.74 \\
1.53 \\
\end{matrix} \right] \\
= & \left[ \begin{matrix}
0.3 \\
0.4 \\
\end{matrix} \right]
\end{align}\,\!</math>

Then, in terms of the actual values, the stationary point can be found as:

::<math>\begin{align}
\text{Reaction Temperature:} & 0.3=\frac{T-350}{(355-345)/2} \\
T= & 351.5\text{ }F
\end{align}\,\!</math>

::<math>\begin{align}
\text{Reaction Time:} & 0.4=\frac{t-165}{(175-155)/2} \\
t= & 169.0\text{ minutes}
\end{align}\,\!</math>

To find the nature of the stationary point the eigenvalues of the <math>B\,\!</math> matrix can be obtained as follows using the determinant of the matrix <math>B-\lambda I\,\!</math>:

::<math>\begin{align}
\left| B-\lambda I \right|= & 0 \\
\left| \begin{matrix}
-1.52-\lambda & 0.45 \\
0.45 & -2.08-\lambda \\
\end{matrix} \right|= & 0
\end{align}\,\!</math>

This gives us:

::<math>{{\lambda }^{2}}+3.61\lambda +2.9743=0\,\!</math>

Solving the quadratic equation in <math>\lambda \,\!</math> returns the eigenvalues <math>{{\lambda }_{1}}=-1.2723\,\!</math> and <math>{{\lambda }_{2}}=-2.3377\,\!</math>. Since both the eigenvalues are negative, it can be concluded that the stationary point is a point of maximum response. The predicted value of the maximum response can be obtained as:

::<math>\begin{align}
{{{\hat{y}}}_{s}}= & {{{\hat{\beta }}}_{0}}+\frac{1}{2}x_{s}^{\prime }b \\
= & 94.91+\frac{1}{2}{{\left[ \begin{matrix}
0.3 \\
0.4 \\
\end{matrix} \right]}^{\prime }}\left[ \begin{matrix}
0.74 \\
1.53 \\
\end{matrix} \right] \\
= & 95.3
\end{align}\,\!</math>

In a DOE folio, the maximum response can be obtained by entering the required values as shown in the figure below. In the figure, the goal is to maximize the response and the limits of the search range for maximizing the response are entered as 90 and 100. The value of the maximum response and the corresponding values of the factors obtained are shown in the second figure following. These values match the values calculated in this example.

[[Image:doe9_21.png|center|543px|Settings to obtain the maximum value of the response in the [[Response_Surface_Methods_for_Optimization#Example_3| example]].|link=]]

[[Image:doe9_22.png|center|650px|Plot of the maximum response in the [[Response_Surface_Methods_for_Optimization#Example_3| example]] against the factors, temperature and time.|link=]]

==Multiple Responses==
In many cases, the experimenter has to optimize a number of responses at the same time. For the example in [[Response_Surface_Methods_for_Optimization#Method_of_Steepest_Ascent| Method of Steepest Ascent]], assume that the experimenter has to also consider two other responses: cost of the product (which should be minimized) and the pH of the product (which should be close to 7 so that the product is neither acidic nor basic). The data is presented in the figure below.

[[Image:doe9.23.png|center|400px|Data for the additional responses of cost and pH for the example to investigate the yield of a chemical process.]]

The problem in dealing with multiple responses is that now there might be conflicting objectives because of the different requirements of each of the responses. The experimenter needs to come up with a solution that satisfies each of the requirements as much as possible without compromising too much on any of the requirements. The approach used in a Weibull++ DOE folio to deal with optimization of multiple responses involves the use of desirability functions that are discussed next (for details see [[DOE_References|[Derringer and Suich, 1980]]]).

===Desirability Functions===
Under this approach, each <math>i\,\!</math>th response is assigned a desirability function, <math>{{d}_{i}}\,\!</math>, where the value of <math>{{d}_{i}}\,\!</math> varies between 0 and 1. The function, <math>{{d}_{i}},\,\!</math> is defined differently based on the objective of the response. If the response is to be maximized, as in the case of the previous example where the yield had to be maximized, then <math>{{d}_{i}}\,\!</math> is defined as follows:

::<math>{{d}_{i}}=\left\{ \begin{matrix}
0 & {{y}_{i}}<L \\
{{\left( \frac{{{y}_{i}}-L}{T-L} \right)}^{\omega }} & L\le {{y}_{i}}\le T \\
1 & {{y}_{i}}>T \\
\end{matrix} \right.</math>

where <math>T\,\!</math> represents the target value of the <math>i\,\!</math>th response, <math>{{y}_{i}}\,\!</math>, <math>L\,\!</math> represents the acceptable lower limit value for this response and <math>\omega \,\!</math> represents the weight. When <math>\omega =1\,\!</math> the function <math>{{d}_{i}}\,\!</math> is linear. If <math>\omega >1\,\!</math> then more importance is placed on achieving the target for the response, <math>{{y}_{i}}\,\!</math>. When <math>\omega <1\,\!</math>, less weight is assigned to achieving the target for the response, <math>{{y}_{i}}\,\!</math>. A graphical representation is shown in figure (a) below.

[[Image:doe9.24.png|center|400px|Desirability function plots for different response optimizations: (a) the goal is to maximize the response, (b) the goal is to minimize the response and (c) the goal is to get the response to a target value.]]

If the response is to be minimized, as in the case when the response is cost, <math>{{d}_{i}}\,\!</math> is defined as follows:

::<math>{{d}_{i}}=\left\{ \begin{matrix}
1 & {{y}_{i}}<T \\
{{\left( \frac{U-{{y}_{i}}}{U-T} \right)}^{\omega }} & T\le {{y}_{i}}\le U \\
0 & {{y}_{i}}>U \\
\end{matrix} \right.</math>

Here <math>U\,\!</math> represents the acceptable upper limit for the response (see figure (b) above).

There may be times when the experimenter wants the response to be neither maximized nor minimized, but instead stay as close to a specified target as possible. For example, in the case where the experimenter wants the product to be neither acidic nor basic, there is a requirement to keep the pH close to the neutral value of 7. In such cases, the desirability function is defined as follows (see figure (c) above):

::<math>{{d}_{i}}=\left\{ \begin{matrix}
0 & {{y}_{i}}<L \\
{{\left( \frac{{{y}_{i}}-L}{T-L} \right)}^{{{\omega }_{1}}}} & L\le {{y}_{i}}\le T \\
{{\left( \frac{U-{{y}_{i}}}{U-T} \right)}^{{{\omega }_{2}}}} & T\le {{y}_{i}}\le U \\
0 & {{y}_{i}}>U \\
\end{matrix} \right.</math>

Once a desirability function is defined for each of the responses, assuming that there are <math>m\,\!</math> responses, an overall desirability function is obtained as follows:

::<math>D={{(d_{1}^{{{r}_{1}}}\cdot d_{2}^{{{r}_{2}}}\cdot ...\cdot d_{m}^{{{r}_{m}}})}^{1/({{r}_{1}}+{{r}_{2}}+...+{{r}_{m}})}}\,\!</math>

where the <math>{{r}_{i}}\,\!</math> s represent the importance of each response. The greater the value of <math>{{r}_{i}}\,\!</math>, the more important the response with respect to the other responses. The objective is to now find the settings that return the maximum value of <math>D\,\!</math>.

To illustrate the use of desirability functions, consider the previous example with the three responses of yield, cost and pH. The response surfaces for the two additional responses of cost and pH are shown next in the figures (a) and (b), respectively.

[[Image:doe9_25.png|center|500px|Response surfaces for (a) cost and (b) pH.]]

In terms of actual variables, the models obtained for all three responses are as shown next:

::<math>\begin{align}
\text{Yield: } & {{{\hat{y}}}_{1}}=-7480.24+41.24{{x}_{1}}+3.88{{x}_{2}}+0.009{{x}_{1}}{{x}_{2}} \\
& -0.060825x_{1}^{2}-0.020831x_{2}^{2} \\
\text{Cost: } & {{{\hat{y}}}_{2}}=15560.49-82.48{{x}_{1}}-7.76{{x}_{2}}-0.02{{x}_{1}}{{x}_{2}} \\
& +0.12165x_{1}^{2}+0.041663x_{2}^{2} \\
\text{pH: } & {{{\hat{y}}}_{3}}=38.87-0.0709{{x}_{1}}-0.041{{x}_{2}}
\end{align}\,\!</math>

Assume that the experimenter wants to have a target yield value of 95, although any value of yield greater than 94 is acceptable. Then the desirability function for yield is:

::<math>{{d}_{1}}=\left\{ \begin{matrix}
0 & {{y}_{1}}<94 \\
{{\left( \frac{{{y}_{1}}-94}{95-94} \right)}^{\omega }} & 94\le {{y}_{1}}\le 95 \\
1 & {{y}_{1}}>95 \\
\end{matrix} \right.</math>

For the cost, assume that the experimenter wants to lower the cost to 400, although any cost value below 415 is acceptable. Then the desirability function for cost is:

::<math>{{d}_{2}}=\left\{ \begin{matrix}
1 & {{y}_{2}}<400 \\
{{\left( \frac{415-{{y}_{2}}}{415-400} \right)}^{\omega }} & 400\le {{y}_{2}}\le 415 \\
0 & {{y}_{2}}>415 \\
\end{matrix} \right.</math>

For the pH, a target of 7 is desired but values between 6.9 and 7.1 are also acceptable. Thus, the desirability function here is:

::<math>{{d}_{3}}=\left\{ \begin{matrix}
0 & {{y}_{3}}<L \\
{{\left( \frac{{{y}_{3}}-6.9}{7-6.9} \right)}^{{{\omega }_{1}}}} & 6.9\le {{y}_{3}}\le 7 \\
{{\left( \frac{7.1-{{y}_{3}}}{7.1-7} \right)}^{{{\omega }_{2}}}} & 7\le {{y}_{3}}\le 7.1 \\
0 & {{y}_{3}}>7.1 \\
\end{matrix} \right.</math>

Notice that in the previous equations all weights used (<math>{{\omega }_{i}}\,\!</math> s) are 1. Thus, all three desirability functions are linear. The overall desirability function, assuming equal importance (<math>{{r}_{1}}={{r}_{2}}={{r}_{3}}=1\,\!</math>) for all the responses, is:

::<math>\begin{align}
D= & {{({{d}_{1}}\cdot {{d}_{2}}\cdot {{d}_{3}})}^{1/3}} \\
= & f({{{\hat{y}}}_{1}},{{{\hat{y}}}_{2}},{{{\hat{y}}}_{3}})
\end{align}\,\!</math>

The objective of the experimenter is to find the settings of <math>{{x}_{1}}\,\!</math> and <math>{{x}_{2}}\,\!</math> such that the overall desirability, <math>D\,\!</math>, is maximum. In DOE++, the settings for the desirability functions for each of the three responses can be entered as shown in the next figure.

[[Image:doe9_26.png|center|750px|Optimization settings for the three responses of yield, cost, and pH.]]

Based on these settings, DOE++ solves this optimization problem to obtain the following solution:

[[Image:doe9_27.png|center|650px|Optimum solution from DOE++ for the three responses of yield, cost, and pH.]]

::<math>\begin{align}
Temperature= & 351.5\text{ }F\text{ } \\
Time= & 169\text{ }minutes
\end{align}\,\!</math>

The overall desirability achieved with this solution can be calculated easily. The values of each of the response for these settings are:

::<math>\begin{align}
\text{ }{{{\hat{y}}}_{1}}= & -7480.24+41.24{{x}_{1}}+3.88{{x}_{2}}+0.009{{x}_{1}}{{x}_{2}} \\
& -0.060825x_{1}^{2}-0.020831x_{2}^{2} \\
= & -7480.24+41.24(351.5)+3.88(169)+0.009(351.3)(169) \\
& -0.060825{{(351.5)}^{2}}-0.020831{{(169)}^{2}} \\
= & 95.3 \\
\text{ }{{{\hat{y}}}_{2}}= & 15560.49-82.48{{x}_{1}}-7.76{{x}_{2}}-0.02{{x}_{1}}{{x}_{2}} \\
& +0.12165x_{1}^{2}+0.041663x_{2}^{2} \\
= & 15560.49-82.48(351.5)-7.76(169)-0.02(351.5)(169) \\
& +0.12165{{(351.5)}^{2}}+0.041663{{(169)}^{2}} \\
= & 409.35 \\
\text{ }{{{\hat{y}}}_{3}}= & 38.87-0.0709{{x}_{1}}-0.041{{x}_{2}} \\
= & 38.87-0.0709(351.5)-0.041(169) \\
= & 7.00
\end{align}\,\!</math>

Based on the response values, the individual desirability functions are:

::<math>\begin{align}
{{d}_{1}}= & 1\text{ (since }{{{\hat{y}}}_{1}}>95\text{)} \\
{{d}_{2}}= & \left( \frac{415-{{{\hat{y}}}_{2}}}{415-400} \right)\text{ (since }400\le {{y}_{2}}\le 415\text{)} \\
= & \left( \frac{415-409.35}{415-400} \right) \\
= & 0.3767 \\
{{d}_{3}}= & 1\text{ (since }{{{\hat{y}}}_{3}}\approx 7\text{)}
\end{align}\,\!</math>

Then the overall desirability is:

::<math>\begin{align}
D= & {{({{d}_{1}}\cdot {{d}_{2}}\cdot {{d}_{3}})}^{1/3}} \\
= & {{(1\cdot 0.3767\cdot 1)}^{1/3}} \\
= & 0.72
\end{align}\,\!</math>

This is the same as the Global Desirability displayed by DOE++ in the figure above. At times, a number of solutions may be obtained from DOE++, and it is up to the experimenter to choose the most feasible one.

Response Surface Methods for Optimization

2017-08-10T17:17:16Z

Richard House: /* Example */

{{Template:Doebook|10}}
The experiment designs mentioned in [[Two_Level_Factorial_Experiments| Two Level Factorial Experiments]] and [[Highly_Fractional_Factorial_Designs| Highly Fractional Factorial Designs]] help the experimenter identify factors that affect the response. Once the important factors have been identified, the next step is to determine the settings for these factors that result in the optimum value of the response. The optimum value of the response may either be a maximum value or a minimum value, depending upon the product or process in question. For example, if the response in an experiment is the yield from a chemical process, then the objective might be to find the settings of the factors affecting the yield so that the yield is maximized. On the other hand, if the response in an experiment is the number of defects, then the goal would be to find the factor settings that minimize the number of defects. Methodologies that help the experimenter reach the goal of optimum response are referred to as ''response surface methods''. These methods are exclusively used to examine the "surface," or the relationship between the response and the factors affecting the response. Regression models are used for the analysis of the response, as the focus now is on the nature of the relationship between the response and the factors, rather than identification of the important factors.

Response surface methods usually involve the following steps:

#The experimenter needs to move from the present operating conditions to the vicinity of the operating conditions where the response is optimum. This is done using the ''method of steepest ascent'' in the case of maximizing the response. The same method can be used to minimize the response and is then referred to as the ''method of steepest descent''.
#Once in the vicinity of the optimum response the experimenter needs to fit a more elaborate model between the response and the factors. Special experiment designs, referred to as [[Response_Surface_Methods_for_Optimization#RSM_Designs|''RSM designs'']], are used to accomplish this. The fitted model is used to arrive at the best operating conditions that result in either a maximum or minimum response.
#It is possible that a number of responses may have to be optimized at the same time. For example, an experimenter may want to maximize strength, while keeping the number of defects to a minimum. The optimum settings for each of the responses in such cases may lead to conflicting settings for the factors. A balanced setting has to be found that gives the most appropriate values for all the responses. Desirability functions are useful in these cases.

==Method of Steepest Ascent==
The first step in obtaining the optimum response settings, after the important factors have been identified, is to explore the region around the current operating conditions to decide what direction needs to be taken to move towards the optimum region. Usually, a first order regression model (containing just the main effects and no interaction terms) is sufficient at the current operating conditions because the operating conditions are normally far from the optimum response settings. The experimenter needs to move from the current operating conditions to the optimum region in the most efficient way by using the minimum number of experiments. This is done using the method of steepest ascent. In this method, the contour plot of the first order model is used to decide the settings for the next experiment, in order to move towards the optimum conditions. Consider a process where the response has been found to be a function of two factors. To explore the region around the current operating conditions, the experimenter fits the following first order model between the response and the two factors:

::<math>y={{\beta }_{0}}+{{\beta }_{1}}{{x}_{1}}+{{\beta }_{2}}{{x}_{2}}+\epsilon \,\!</math>

The response surface plot for the model, along with the contours, is shown in the figure below. It can be seen in the figure that in order to maximize the response, the most efficient direction in which to move the experiment is along the line perpendicular to the contours. This line, also referred to as the path of steepest ascent, is the line along which the rate of increase of the response is maximum. The steps along this line to move towards the optimum region are proportional to the regression coefficients, <math>{{\beta }_{j}},\,\!</math> of the fitted first order model.

[[Image:doe9_1.png|center|504px|Path of steepest ascent for the model <math>y=\beta_0+\beta_1 x_1+\beta_2 x_2+\epsilon \,\!</math>.]]

Experiments are conducted along each step of the path of steepest ascent until an increase in the response is not seen. Then, a new first order model is fit at the region of the maximum response. If the first order model shows a lack of fit, then this indicates that the experimenter has reached the vicinity of the optimum. RSM designs are then used explore the region thoroughly and obtain the point of the maximum response. If the first order model does not show a lack of fit, then a new path of steepest ascent is determined and the process is repeated.

===Example===
The yield from a chemical process is found to be affected by two factors: reaction temperature and reaction time. The current reaction temperature is 230 <math>F\,\!</math> and the reaction time is 65 minutes. The experimenter wants to determine the settings of the two factors such that maximum yield can be obtained from the process. To explore the region around the current operating conditions, the experimenter decides to use a single replicate of the <math>2^{2}\,\!</math> design. The range of the factors for this design are chosen to be (225, 235) <math>F\,\!</math> for the reaction temperature and (55, 75) minutes for the reaction time. The unreplicated <math>2^{2}\,\!</math> design is also augmented with five runs at the center point to estimate the error sum of squares, <math>S{{S}_{E}}\,\!</math>, and check for model adequacy. The response values obtained for this design are shown next.

[[Image:doe9.2.png|center|256px|The <math>2^{2}\,\!</math> design augmented with five center points to explore the region around current operating conditions for a chemical process.|link=]]

In a DOE folio, this design can be set up using the properties shown next.

[[Image:doe9_3.png|center|809px|Design properties for the <math>2^{2}\,\!</math> design to explore the current operating conditions.|link=]]

The resulting design and the analysis results are shown next.

[[Image:doe9_4.png|center|778px|The <math>2^{2}\,\!</math> experiment design in to explore the current operating conditions.|link=]]

[[Image:doe9_5.png|center|774px|Results for the <math>2^{2}\,\!</math> experiment to explore the current operating conditions.|link=]]

Note that the results shown are in terms of the coded values of the factors (taking -1 as the value of the lower settings for reaction temperature and reaction time and +1 as the value for the higher settings for these two factors). The results show that the factors, <math>A\,\!</math> (temperature) and <math>B\,\!</math> (time), affect the response significantly but their interaction does not affect the response. Therefore the interaction term can be dropped from the model for this experiment. The results also show that Curvature is not a significant factor. This indicates that the first order model is adequate for the experiment at the current operating conditions. Using these two conclusions, the model for the current operating conditions, in terms the coded variables is:

:<math>\hat{y}=36.6375+1.1635x_{1}+0.4875x_{2}\,\!</math>

where <math>\hat{y}\,\!</math> represents the yield and <math>x_{1}\,\!</math> and <math>x_{2}\,\!</math> are the predictor variables for the two factors, <math>A\,\!</math> and <math>B\,\!</math>, respectively. To further confirm the adequacy of the model of the equation given above, the experiment can be analyzed again after dropping the interaction term, <math>AB\,\!</math>. The results are shown next.

[[Image:doe9_6.png|center|774px|Results for the <math>2^{2}\,\!</math> experiment after the interaction term is dropped from the model.|link=]]

The results show that the lack-of-fit for this model (because of the deficiency created in the model by the absence of the interaction term) is not significant, confirming that the model is adequate.

==Path of Steepest Ascent==
The contour plot for the model used in the above example is shown next.

[[Image:doe9_7.png|center|650px|Results for the <math>2^{2}\,\!</math> experiment after the interaction term is dropped from the model.|link=]]

The regression coefficients for the model are <math>\hat{\beta}_{1}= 1.1625\,\!</math> and <math>\hat{\beta}_{2}= 0.4875\,\!</math>. To move towards the optimum, the experimenter needs to move along the path of steepest ascent, which lies perpendicular to the contours. This path is the line through the center point of the current operating conditions (<math>x_{1}=0\,\!</math>, <math>x_{2}=0\,\!</math>) with a slope of <math>\hat{\beta}_{2}/\hat{\beta}_{1}\,\!</math>. Therefore, in terms of the coded variables, the experiment should be moved 1.1625 units in the <math>x_{1}\,\!</math> direction for every 0.4875 units in the <math>x_{2}\,\!</math> direction. To move along this path, the experimenter decides to use a step-size of 10 minutes for the reaction time, <math>x_{2}\,\!</math>. The coded value for this step size can be obtained as follows. Recall from [[Multiple_Linear_Regression_Analysis|Multiple Linear Regression Analysis]] that the relationship between coded and actual values is:

::<math>coded\ value=\frac{actual\ value - mean}{half\ of\ range}</math>

or

::<math>coded\ value=\frac{step\ size\ in\ actual\ value}{half\ of\ range}</math>

Thus, for a step-size of 10 minutes, the equivalent step size in coded value for <math>x_{2}\,\!</math> is:

::<math>=\frac{10}{half\ of\ range\ of\ x_{2}}\,\!</math>
::<math>=\frac{10}{(75-55)/2}\,\!</math>
::<math>=1\,\!</math>

In terms of the coded variables, the path of steepest ascent requires a move of 1.1625 units in the <math>x_{1}\,\!</math> direction for every 0.4875 units in the <math>x_{2}\,\!</math> direction. The step-size for <math>x_{1}\,\!</math>, in terms of the coded value corresponding to any step-size in <math>x_{2}\,\!</math>, is:

::<math>=\frac{\hat{\beta_{1}}}{\hat{\beta_{2}}}\cdot(step\ size\ in\ x_{2})\,\!</math>

Therefore, the step-size for the reaction temperature, <math>x_{1}\,\!</math>, in terms of the coded variables is:

::<math>=\frac{\hat{\beta_{1}}}{\hat{\beta_{2}}}\cdot\Delta x_{2}\,\!</math>
::<math>=\frac{1.1625}{0.4875}\cdot1\,\!</math>
::<math>=\frac{\hat{\beta_{1}}}{\hat{\beta_{2}}}\cdot\Delta x_{2}\,\!</math>

This corresponds to a step of approximately 12 <math>F\,\!</math> for temperature in terms of the actual value as shown next:

::<math>\begin{align}
step\ size\ in\ actual\ value = & (coded\ value)\cdot(half\ of\ the\ range\ of\ x_{1})\\
= & 2.39\cdot10/2\\
= & 11.95
\end{align}\,\!</math>

Using a step of 12 <math>F\,\!</math> and 10 minutes, the experimenter conducts experiments until no further increase is observed in the yield. The yield values at each step are shown in the table given next.

[[Image:doet9.1.png|center|341px|Response values at each step of the path of steepest ascent for the experiment to investigate the yield of a chemical process. Units for factor levels and the response have been omitted.]]

The yield starts decreasing after the reaction temperature of 350 <math>F\,\!</math> and the reaction time of 165 minutes, indicating that this point may lie close to the optimum region. To analyze the vicinity of this point, a <math>2^{2}\,\!</math> design augmented by five center points is selected. The range of exploration is chosen to be 345 to 355 <math>F\,\!</math> for reaction temperature and 155 to 175 minutes for reaction time. The response values recorded are shown next.

[[Image:doe9.8.png|center|348px|The <math>2^{2}\,\!</math> design augmented with five center points to explore the region of maximum response obtained from the path of steepest ascent. Note that the center point of this design is the new origin.]]

The results for this design are shown next.

[[Image:doe9_9.png|center|760px|Results for the <math>2^{2}\,\!</math> experiment to explore the region of maximum response.|link=]]

In the results, Curvature is displayed as a significant factor. This indicates that the first order model is not adequate for this region of the experiment and a higher order model is required. As a result, the methodology of steepest ascent can no longer be used. The presence of curvature indicates that the experiment region may be close to the optimum. Special designs that allow the use of second order models are needed at this point.

==RSM Designs==
A second order model is generally used to approximate the response once it is realized that the experiment is close to the optimum response region where a first order model is no longer adequate. The second order model is usually sufficient for the optimum region, as third order and higher effects are seldom important. The second order regression model takes the following form for <math>k\,\!</math> factors:

::<math>\begin{align}
y= & {{\beta }_{0}}+{{\beta }_{1}}{{x}_{1}}+{{\beta }_{2}}{{x}_{2}}+...+{{\beta }_{k}}{{x}_{k}}+ \\
& {{\beta }_{11}}x_{1}^{2}+{{\beta }_{22}}x_{2}^{2}+...+{{\beta }_{kk}}x_{k}^{2}+{{\beta }_{12}}{{x}_{1}}{{x}_{2}}+...+{{\beta }_{k-1,k}}{{x}_{k-1}}{{x}_{k}}+\epsilon
\end{align}\,\!</math>

The model contains <math>p=(k+1)(k+2)/2\,\!</math> regression parameters that include coefficients for main effects (<math>{{\beta }_{1}},{{\beta }_{2}}...{{\beta }_{k}}\,\!</math>), coefficients for quadratic main effects (<math>{{\beta }_{11}},{{\beta }_{22}}...{{\beta }_{kk}}\,\!</math>) and coefficients for two factor interaction effects (<math>{{\beta }_{12}},{{\beta }_{13}}\,\!</math>... <math>{{\beta }_{k-1,k}}\,\!</math>). A full factorial design with all factors at three levels would provide estimation of all the required regression parameters. However, full factorial three level designs are expensive to use as the number of runs increases rapidly with the number of factors. For example, a three factor full factorial design with each factor at three levels would require <math>{{3}^{3}}=27\,\!</math> runs while a design with four factors would require <math>{{3}^{4}}=81\,\!</math> runs. Additionally, these designs will estimate a number of higher order effects which are usually not of much importance to the experimenter. Therefore, for the purpose of analysis of response surfaces, special designs are used that help the experimenter fit the second order model to the response with the use of a minimum number of runs. Examples of these designs are the central composite and Box-Behnken designs.

===Central Composite Designs===
Central composite designs are two level full factorial (<math>2^{k}\,\!</math>) or fractional factorial (<math>2^{k-f}\,\!</math>) designs augmented by a number of center points and other chosen runs. These designs are such that they allow the estimation of all the regression parameters required to fit a second order model to a given response.

The simplest of the central composite designs can be used to fit a second order model to a response with two factors. The design consists of a <math>2^{2}\,\!</math> full factorial design augmented by a few runs at the center point (such a design is shown in figure (a) given below). A central composite design is obtained when runs at points (<math>-1,0\,\!</math>), (<math>1,0\,\!</math>), (<math>0,-1\,\!</math>) and (<math>0,1\,\!</math>) are added to this design. These points are referred to as ''axial points'' or ''star points'' and represent runs where all but one of the factors are set at their mid-levels. The number of axial points in a central composite design having <math>k\,\!</math> factors is <math>2k\,\!</math>. The distance of the axial points from the center point is denoted by <math>\alpha \,\!</math> and is always specified in terms of coded values. For example, the central composite design in figure (b) given below has <math>\alpha =1\,\!</math>, while for the design of figure (c) <math>\alpha =1.414\,\!</math>.

[[Image:doe9.10.png|center|227px|Central composite designs: (a) shows the <math>2^2\,\!</math> design with center point runs, (b) shows the two factor central composite design with <math>\alpha=1\,\!</math> and (c) shows the two factor central composite design with <math>\alpha=\sqrt{2}\,\!</math>.]]

It can be noted that when <math>\alpha >1\,\!</math>, each factor is run at five levels (<math>-\alpha \,\!</math>, <math>-1\,\!</math>, <math>0\,\!</math>, <math>1\,\!</math> and <math>\alpha \,\!</math>) instead of the three levels of <math>-1\,\!</math>, <math>0\,\!</math> and <math>1\,\!</math>. The reason for running central composite designs with <math>\alpha >1\,\!</math> is to have a rotatable design, which is explained next.

====Rotatability====
A central composite design is said to be ''rotatable'' if the variance of any predicted value of the response, <math>{{\hat{y}}_{p}}\,\!</math>, for any level of the factors depends only on the distance of the point from the center of the design, regardless of the direction. In other words, a rotatable central composite design provides constant variance of the estimated response corresponding to all new observation points that are at the same distance from the center point of the design (in terms of the coded variables).

The variance of the predicted response at any point, <math>{{x}_{p}}\,\!</math>, is given as follows:

::<math>V[{{\hat{y}}_{p}}]={{\sigma }^{2}}x_{p}^{\prime }{{({{X}^{\prime }}X)}^{-1}}{{x}_{p}}\,\!</math>

The contours of <math>V[{{\hat{y}}_{p}}]\,\!</math> for the central composite design in figure (c) above are shown in the figure below. The contours are concentric circles indicating that the central composite design of figure (c) is rotatable. Rotatability is a desirable property because the experimenter does not have any prior information about the location of the optimum. Therefore, a design that provides equal precision of estimation in all directions would be preferred. Such a design will assure the experimenter that no matter what direction is taken to search for the optimum, he/she will be able to estimate the response value with equal precision. A central composite design is rotatable if the value of <math>\alpha \,\!</math> for the design satisfies the following equation:

::<math>\alpha ={{\left[ \frac{{{2}^{k-f}}({{n}_{f}})}{{{n}_{s}}} \right]}^{1/4}}\,\!</math>

where <math>{{n}_{f}}\,\!</math> is the number of replicates of the runs in the original factorial design and <math>{{n}_{s}}\,\!</math> is the number of replicates of the runs at the axial points. For example, a central composite design with two factors, having a single replicate of the original factorial design, and a single replicate of all the axial points, would be rotatable for the following <math>\alpha \,\!</math> value:

::<math>\begin{align}
\alpha = & {{\left[ \frac{{{2}^{k-f}}({{n}_{f}})}{{{n}_{s}}} \right]}^{1/4}} \\
= & {{\left[ \frac{{{2}^{2}}(1)}{1} \right]}^{1/4}} \\
= & \sqrt{2} \\
= & 1.414
\end{align}\,\!</math>

Thus, a central composite design in two factors, having a single replicate of the original <math>2^{2}\,\!</math> design and axial points, and with <math>\alpha =1.414\,\!</math>, is a rotatable design. This design is shown in figure (c) above.

[[Image:doe9.11.png|center|266px|The countours of <math>V[\hat{y}_p] \,\!</math> for the rotatable two factor central composite design.]]

====Spherical Design====
A central composite design is said to be ''spherical'' if all factorial and axial points are at same distance from the center of the design. Spherical central composite designs are obtained by setting <math>\alpha =\sqrt{k}\,\!</math>. For example, the rotatable design in the figure above (c) is also a spherical design because for this design <math>\alpha =\sqrt{k}=\sqrt{2}=1.414\,\!</math>.

====Face-centered Design====
Central composite designs in which the axial points represent the mid levels for all but one of the factors are also referred to as ''face-centered'' central composite designs. For these designs, <math>\alpha =1\,\!</math> and all factors are run at three levels, which are <math>-1\,\!</math>, <math>0\,\!</math> and <math>1,\,\!</math> in terms of the coded values (see the figure below).

[[Image:doe9.12.png|center|300px|Face-centered central composite design for three factors.]]

===Box-Behnken Designs===
In [[Highly_Fractional_Factorial_Designs| Highly Fractional Factorial Designs]], highly fractional designs introduced by Plackett and Burman were discussed. Plackett-Burman designs are used to estimate main effects in the case of two level fractional factorial experiments using very few runs. [[DOE_References|[G. E. P. Box and D. W. Behnken (1960)]]] introduced similar designs for three level factors that are widely used in response surface methods to fit second-order models to the response. The designs are referred to as Box-Behnken designs. The designs were developed by the combination of two level factorial designs with incomplete block designs. For example, the figure below shows the Box-Behnken design for three factors. The design is obtained by the combination of <math>2^{2}\,\!</math> design with a balanced incomplete block design having three treatments and three blocks (for details see [[DOE_References|[Box 1960, Montgomery 2001]]]).

[[Image:doe9.13.png|center|400px|Box-Behnken design for three factors: (a) shows the geometric representation and (b) shows the design.]]

The advantages of Box-Behnken designs include the fact that they are all spherical designs and require factors to be run at only three levels. The designs are also rotatable or nearly rotatable. Some of these designs also provide orthogonal blocking. Thus, if there is a need to separate runs into blocks for the Box-Behnken design, then designs are available that allow blocks to be used in such a way that the estimation of the regression parameters for the factor effects are not affected by the blocks. In other words, in these designs the block effects are orthogonal to the other factor effects. Yet another advantage of these designs is that there are no runs where all factors are at either the <math>+1\,\!</math> or <math>-1\,\!</math> levels. For example, in the figure below the representation of the Box-Behnken design for three factors clearly shows that there are no runs at the corner points. This could be advantageous when the corner points represent runs that are expensive or inconvenient because they lie at the end of the range of the factor levels. A few of the Box-Behnken designs available in DOE++ are presented in [[Box-Behnken_Designs|Appendix F]].

====Example====
Continuing with the example in [[Response_Surface_Methods_for_Optimization#Method_of_Steepest_Ascent| Method of Steepest Ascent]], the first order model was found to be inadequate for the region near the optimum. Once the experimenter realized that the first order model was not adequate (for the region with a reaction temperature of 350 <math>F\,\!</math> and reaction time of 165 minutes), it was decided to augment the experiment with axial runs to be able to complete a central composite design and fit a second order model to the response. Notice the advantage of using a central composite design, as the experimenter only had to add the axial runs to the <math>2^{2}\,\!</math> design with center point runs, and did not have to begin a new experiment. The experimenter decided to use <math>\alpha =1.4142\,\!</math> to get a rotatable design. The obtained response values are shown in the figure below.

[[Image:doe9.14.png|center|280px|Response values for the two factor central composite design in the [[Response_Surface_Methods_for_Optimization#Example_2| example]].]]

Such a design can be set up in a Weibull++ DOE folio using the properties shown in the figure below.

[[Image:doe9_15.png|center|744px|Properties for the central composite design in the [[Response_Surface_Methods_for_Optimization#Example_2| example]].|link=]]

The resulting design is shown in the figure shown next.

[[Image:doe9_16.png|center|800px|Central composite design for the experiment in the [[Response_Surface_Methods_for_Optimization#Example_2| example]].|link=]]

Results from the analysis of the design are shown in the next figure.

[[Image:doe9_17.png|center|744px|Results for the central composite design in the [[Response_Surface_Methods_for_Optimization#Example_2| example]].|link=]]

The results in the figure above show that the main effects, <math>A\,\!</math> and <math>B\,\!</math>, the interaction, <math>AB\,\!</math>, and the quadratic main effects, <math>{{A}^{2}}\,\!</math> and <math>{{B}^{2}}\,\!</math>, (represented as AA and BB in the figure) are significant. The lack-of-fit test also shows that the second order model with these terms is adequate and a higher order model is not needed. Using these results, the model for the experiment in terms of the coded values is:

::<math>\hat{y}=94.91+0.74{{x}_{1}}+1.53{{x}_{2}}+0.45{{x}_{1}}{{x}_{2}}-1.52x_{1}^{2}-2.08x_{2}^{2}\,\!</math>

The response surface and the contour plot for this model, in terms of the actual variables, are shown in the below figures (a) and (b), respectively.

[[Image:doe9_18_a.png|center|650px|link=]]

[[Image:doe9_18_b.png|center|650px|Response surface and countour plot for the experiment in the [[Response_Surface_Methods_for_Optimization#Example_2| example]].|link=]]

==Analysis of the Second Order Model==
Once a second order model is fit to the response, the next step is to locate the point of maximum or minimum response. The second order model for <math>k\,\!</math> factors can be written as:

::<math>\hat{y}={{\hat{\beta }}_{0}}+\underset{i=1}{\overset{k}{\mathop{\sum }}}\,{{\hat{\beta }}_{i}}{{x}_{i}}+\underset{i=1}{\overset{k}{\mathop{\sum }}}\,{{\hat{\beta }}_{ii}}x_{i}^{2}+\underset{i=1}{\overset{k}{\mathop{\sum }}}\,\underset{j=1}{\overset{i-1}{\mathop{\sum }}}\,{{\hat{\beta }}_{ij}}{{x}_{i}}{{x}_{j}}\,\!</math>

The point for which the response, <math>\hat{y}\,\!</math>, is optimized is the point at which the partial derivatives, <math>\partial \hat{y}/\partial {{x}_{1}}\,\!</math>, <math>\partial \hat{y}/\partial {{x}_{2}}\,\!</math>, <math>...\partial \hat{y}/\partial {{x}_{k}},\,\!</math> are all equal to zero. This point is called the ''stationary point''. The stationary point may be a point of maximum response, minimum response or a saddle point. These three conditions are shown in the following figures (a), (b) and (c) respectively.

[[Image:doe9.19.png|center|300px|Types of second order response surfaces and their contour plots. (a) shows the surface with a maximum point, (b) shows the surface with a minimum point and (c) shows the surface with a saddle point.]]

Notice that these conditions are easy to identify, in the case of two factor experiments, by the inspection of the contour plots. However, when more than two factors exist in an experiment, then the general mathematical solution for the location of the stationary point has to be used. The equation given above can be written in matrix notation as:

::<math>\hat{y}={{\hat{\beta }}_{0}}+{{x}^{\prime }}b+{{x}^{\prime }}Bx\,\!</math>

where:

::<math>x=\left[ \begin{matrix}
{{x}_{1}} \\
{{x}_{2}} \\
. \\
. \\
. \\
{{x}_{k}} \\
\end{matrix} \right]\text{ }b=\left[ \begin{matrix}
{{{\hat{\beta }}}_{1}} \\
{{{\hat{\beta }}}_{2}} \\
. \\
. \\
. \\
{{{\hat{\beta }}}_{k}} \\
\end{matrix} \right]\text{ and }B=\left[ \begin{matrix}
{{{\hat{\beta }}}_{11}} & {{{\hat{\beta }}}_{12}}/2 & . & . & . & {{{\hat{\beta }}}_{1k}}/2 \\
{} & {{{\hat{\beta }}}_{22}} & . & . & . & {{{\hat{\beta }}}_{2k}}/2 \\
{} & {} & . & . & . & . \\
{} & {} & {} & . & . & . \\
{} & {} & {} & {} & . & . \\
\text{sym}\text{.} & {} & {} & {} & {} & {{{\hat{\beta }}}_{kk}} \\
\end{matrix} \right]\,\!</math>

Then the stationary point can be determined as follows:

::<math>\frac{\partial \hat{y}}{\partial x}=b+2Bx=0\,\!</math>

Thus, the stationary point is:

::<math>{{x}_{s}}=-\frac{1}{2}{{B}^{-1}}b\,\!</math>

The optimum response is the response corresponding to <math>{{x}_{s}}\,\!</math>. The optimum response can be obtained to get:

::<math>{{\hat{y}}_{s}}={{\hat{\beta }}_{0}}+\frac{1}{2}x_{s}^{\prime }b\,\!</math>

Once the stationary point is known, it is necessary to determine if it is a maximum or minimum or saddle point. To do this, the second order model has to be transformed to the canonical form. This is done by transforming the model to a new coordinate system such that the origin lies at the stationary point and the axes are parallel to the principal axes of the fitted response surface, shown next.

[[Image:doe9.20.png|center|247px|The second order model in canonical form.]]

The resulting model equation then takes the following form:

::<math>\hat{y}={{\hat{y}}_{s}}+{{\lambda }_{1}}w_{1}^{2}+{{\lambda }_{2}}w_{2}^{2}+...+{{\lambda }_{k}}w_{k}^{2}\,\!</math>

where the <math>{{w}_{i}}\,\!</math> s are the transformed independent variables, and <math>{{\lambda }_{i}}\,\!</math>s are constants that are also the eigenvalues of the matrix <math>B\,\!</math>. The nature of the stationary point is known by looking at the signs of the <math>{{\lambda }_{i}}\,\!</math>s. If the <math>{{\lambda }_{i}}\,\!</math> s are all negative, then <math>{{x}_{s}}\,\!</math> is a point of maximum response. If the <math>{{\lambda }_{i}}\,\!</math>s are all positive then <math>{{x}_{s}}\,\!</math> is a point of minimum response. If the <math>{{\lambda }_{i}}\,\!</math>s have different signs, then <math>{{x}_{s}}\,\!</math> is a saddle point.

====Example====
Continuing with the [[Response_Surface_Methods_for_Optimization#Example_2| example]] in [[Response_Surface_Methods_for_Optimization#Method_of_Steepest_Ascent| Method of Steepest Ascent]], the second order model fitted to the response, in terms of the coded variables, was obtained as:

::<math>\hat{y}=94.91+0.74{{x}_{1}}+1.53{{x}_{2}}+0.45{{x}_{1}}{{x}_{2}}-1.52x_{1}^{2}-2.08x_{2}^{2}\,\!</math>

Then the <math>b\,\!</math> and <math>B\,\!</math> matrices for this model are:

::<math>b=\left[ \begin{matrix}
0.74 \\
1.53 \\
\end{matrix} \right]\text{ }B=\left[ \begin{matrix}
-1.52 & 0.45 \\
0.45 & -2.08 \\
\end{matrix} \right]\,\!</math>
 

The stationary point is:

<math>\begin{align}
{{x}_{s}}= & -\frac{1}{2}{{B}^{-1}}b \\
= & -\frac{1}{2}{{\left[ \begin{matrix}
-1.52 & 0.45 \\
0.45 & -2.08 \\
\end{matrix} \right]}^{-1}}\left[ \begin{matrix}
0.74 \\
1.53 \\
\end{matrix} \right] \\
= & \left[ \begin{matrix}
0.3 \\
0.4 \\
\end{matrix} \right]
\end{align}\,\!</math>

Then, in terms of the actual values, the stationary point can be found as:

::<math>\begin{align}
\text{Reaction Temperature:} & 0.3=\frac{T-350}{(355-345)/2} \\
T= & 351.5\text{ }F
\end{align}\,\!</math>

::<math>\begin{align}
\text{Reaction Time:} & 0.4=\frac{t-165}{(175-155)/2} \\
t= & 169.0\text{ minutes}
\end{align}\,\!</math>

To find the nature of the stationary point the eigenvalues of the <math>B\,\!</math> matrix can be obtained as follows using the determinant of the matrix <math>B-\lambda I\,\!</math>:

::<math>\begin{align}
\left| B-\lambda I \right|= & 0 \\
\left| \begin{matrix}
-1.52-\lambda & 0.45 \\
0.45 & -2.08-\lambda \\
\end{matrix} \right|= & 0
\end{align}\,\!</math>

This gives us:

::<math>{{\lambda }^{2}}+3.61\lambda +2.9743=0\,\!</math>

Solving the quadratic equation in <math>\lambda \,\!</math> returns the eigenvalues <math>{{\lambda }_{1}}=-1.2723\,\!</math> and <math>{{\lambda }_{2}}=-2.3377\,\!</math>. Since both the eigenvalues are negative, it can be concluded that the stationary point is a point of maximum response. The predicted value of the maximum response can be obtained as:

::<math>\begin{align}
{{{\hat{y}}}_{s}}= & {{{\hat{\beta }}}_{0}}+\frac{1}{2}x_{s}^{\prime }b \\
= & 94.91+\frac{1}{2}{{\left[ \begin{matrix}
0.3 \\
0.4 \\
\end{matrix} \right]}^{\prime }}\left[ \begin{matrix}
0.74 \\
1.53 \\
\end{matrix} \right] \\
= & 95.3
\end{align}\,\!</math>

In a DOE folio, the maximum response can be obtained by entering the required values as shown in the figure below. In the figure, the goal is to maximize the response and the limits of the search range for maximizing the response are entered as 90 and 100. The value of the maximum response and the corresponding values of the factors obtained are shown in the second figure following. These values match the values calculated in this example.

[[Image:doe9_21.png|center|543px|Settings to obtain the maximum value of the response in the [[Response_Surface_Methods_for_Optimization#Example_3| example]].|link=]]

[[Image:doe9_22.png|center|650px|Plot of the maximum response in the [[Response_Surface_Methods_for_Optimization#Example_3| example]] against the factors, temperature and time.|link=]]

==Multiple Responses==
In many cases, the experimenter has to optimize a number of responses at the same time. For the example in [[Response_Surface_Methods_for_Optimization#Method_of_Steepest_Ascent| Method of Steepest Ascent]], assume that the experimenter has to also consider two other responses: cost of the product (which should be minimized) and the pH of the product (which should be close to 7 so that the product is neither acidic nor basic). The data is presented in the figure below.

[[Image:doe9.23.png|center|400px|Data for the additional responses of cost and pH for the example to investigate the yield of a chemical process.]]

The problem in dealing with multiple responses is that now there might be conflicting objectives because of the different requirements of each of the responses. The experimenter needs to come up with a solution that satisfies each of the requirements as much as possible without compromising too much on any of the requirements. The approach used in DOE++ to deal with optimization of multiple responses involves the use of desirability functions that are discussed next (for details see [[DOE_References|[Derringer and Suich, 1980]]]).

===Desirability Functions===
Under this approach, each <math>i\,\!</math>th response is assigned a desirability function, <math>{{d}_{i}}\,\!</math>, where the value of <math>{{d}_{i}}\,\!</math> varies between 0 and 1. The function, <math>{{d}_{i}},\,\!</math> is defined differently based on the objective of the response. If the response is to be maximized, as in the case of the previous example where the yield had to be maximized, then <math>{{d}_{i}}\,\!</math> is defined as follows:

::<math>{{d}_{i}}=\left\{ \begin{matrix}
0 & {{y}_{i}}<L \\
{{\left( \frac{{{y}_{i}}-L}{T-L} \right)}^{\omega }} & L\le {{y}_{i}}\le T \\
1 & {{y}_{i}}>T \\
\end{matrix} \right.</math>

where <math>T\,\!</math> represents the target value of the <math>i\,\!</math>th response, <math>{{y}_{i}}\,\!</math>, <math>L\,\!</math> represents the acceptable lower limit value for this response and <math>\omega \,\!</math> represents the weight. When <math>\omega =1\,\!</math> the function <math>{{d}_{i}}\,\!</math> is linear. If <math>\omega >1\,\!</math> then more importance is placed on achieving the target for the response, <math>{{y}_{i}}\,\!</math>. When <math>\omega <1\,\!</math>, less weight is assigned to achieving the target for the response, <math>{{y}_{i}}\,\!</math>. A graphical representation is shown in figure (a) below.

[[Image:doe9.24.png|center|400px|Desirability function plots for different response optimizations: (a) the goal is to maximize the response, (b) the goal is to minimize the response and (c) the goal is to get the response to a target value.]]

If the response is to be minimized, as in the case when the response is cost, <math>{{d}_{i}}\,\!</math> is defined as follows:

::<math>{{d}_{i}}=\left\{ \begin{matrix}
1 & {{y}_{i}}<T \\
{{\left( \frac{U-{{y}_{i}}}{U-T} \right)}^{\omega }} & T\le {{y}_{i}}\le U \\
0 & {{y}_{i}}>U \\
\end{matrix} \right.</math>

Here <math>U\,\!</math> represents the acceptable upper limit for the response (see figure (b) above).

There may be times when the experimenter wants the response to be neither maximized nor minimized, but instead stay as close to a specified target as possible. For example, in the case where the experimenter wants the product to be neither acidic nor basic, there is a requirement to keep the pH close to the neutral value of 7. In such cases, the desirability function is defined as follows (see figure (c) above):

::<math>{{d}_{i}}=\left\{ \begin{matrix}
0 & {{y}_{i}}<L \\
{{\left( \frac{{{y}_{i}}-L}{T-L} \right)}^{{{\omega }_{1}}}} & L\le {{y}_{i}}\le T \\
{{\left( \frac{U-{{y}_{i}}}{U-T} \right)}^{{{\omega }_{2}}}} & T\le {{y}_{i}}\le U \\
0 & {{y}_{i}}>U \\
\end{matrix} \right.</math>

Once a desirability function is defined for each of the responses, assuming that there are <math>m\,\!</math> responses, an overall desirability function is obtained as follows:

::<math>D={{(d_{1}^{{{r}_{1}}}\cdot d_{2}^{{{r}_{2}}}\cdot ...\cdot d_{m}^{{{r}_{m}}})}^{1/({{r}_{1}}+{{r}_{2}}+...+{{r}_{m}})}}\,\!</math>

where the <math>{{r}_{i}}\,\!</math> s represent the importance of each response. The greater the value of <math>{{r}_{i}}\,\!</math>, the more important the response with respect to the other responses. The objective is to now find the settings that return the maximum value of <math>D\,\!</math>.

To illustrate the use of desirability functions, consider the previous example with the three responses of yield, cost and pH. The response surfaces for the two additional responses of cost and pH are shown next in the figures (a) and (b), respectively.

[[Image:doe9_25.png|center|500px|Response surfaces for (a) cost and (b) pH.]]

In terms of actual variables, the models obtained for all three responses are as shown next:

::<math>\begin{align}
\text{Yield: } & {{{\hat{y}}}_{1}}=-7480.24+41.24{{x}_{1}}+3.88{{x}_{2}}+0.009{{x}_{1}}{{x}_{2}} \\
& -0.060825x_{1}^{2}-0.020831x_{2}^{2} \\
\text{Cost: } & {{{\hat{y}}}_{2}}=15560.49-82.48{{x}_{1}}-7.76{{x}_{2}}-0.02{{x}_{1}}{{x}_{2}} \\
& +0.12165x_{1}^{2}+0.041663x_{2}^{2} \\
\text{pH: } & {{{\hat{y}}}_{3}}=38.87-0.0709{{x}_{1}}-0.041{{x}_{2}}
\end{align}\,\!</math>

Assume that the experimenter wants to have a target yield value of 95, although any value of yield greater than 94 is acceptable. Then the desirability function for yield is:

::<math>{{d}_{1}}=\left\{ \begin{matrix}
0 & {{y}_{1}}<94 \\
{{\left( \frac{{{y}_{1}}-94}{95-94} \right)}^{\omega }} & 94\le {{y}_{1}}\le 95 \\
1 & {{y}_{1}}>95 \\
\end{matrix} \right.</math>

For the cost, assume that the experimenter wants to lower the cost to 400, although any cost value below 415 is acceptable. Then the desirability function for cost is:

::<math>{{d}_{2}}=\left\{ \begin{matrix}
1 & {{y}_{2}}<400 \\
{{\left( \frac{415-{{y}_{2}}}{415-400} \right)}^{\omega }} & 400\le {{y}_{2}}\le 415 \\
0 & {{y}_{2}}>415 \\
\end{matrix} \right.</math>

For the pH, a target of 7 is desired but values between 6.9 and 7.1 are also acceptable. Thus, the desirability function here is:

::<math>{{d}_{3}}=\left\{ \begin{matrix}
0 & {{y}_{3}}<L \\
{{\left( \frac{{{y}_{3}}-6.9}{7-6.9} \right)}^{{{\omega }_{1}}}} & 6.9\le {{y}_{3}}\le 7 \\
{{\left( \frac{7.1-{{y}_{3}}}{7.1-7} \right)}^{{{\omega }_{2}}}} & 7\le {{y}_{3}}\le 7.1 \\
0 & {{y}_{3}}>7.1 \\
\end{matrix} \right.</math>

Notice that in the previous equations all weights used (<math>{{\omega }_{i}}\,\!</math> s) are 1. Thus, all three desirability functions are linear. The overall desirability function, assuming equal importance (<math>{{r}_{1}}={{r}_{2}}={{r}_{3}}=1\,\!</math>) for all the responses, is:

::<math>\begin{align}
D= & {{({{d}_{1}}\cdot {{d}_{2}}\cdot {{d}_{3}})}^{1/3}} \\
= & f({{{\hat{y}}}_{1}},{{{\hat{y}}}_{2}},{{{\hat{y}}}_{3}})
\end{align}\,\!</math>

The objective of the experimenter is to find the settings of <math>{{x}_{1}}\,\!</math> and <math>{{x}_{2}}\,\!</math> such that the overall desirability, <math>D\,\!</math>, is maximum. In DOE++, the settings for the desirability functions for each of the three responses can be entered as shown in the next figure.

[[Image:doe9_26.png|center|750px|Optimization settings for the three responses of yield, cost, and pH.]]

Based on these settings, DOE++ solves this optimization problem to obtain the following solution:

[[Image:doe9_27.png|center|650px|Optimum solution from DOE++ for the three responses of yield, cost, and pH.]]

::<math>\begin{align}
Temperature= & 351.5\text{ }F\text{ } \\
Time= & 169\text{ }minutes
\end{align}\,\!</math>

The overall desirability achieved with this solution can be calculated easily. The values of each of the response for these settings are:

::<math>\begin{align}
\text{ }{{{\hat{y}}}_{1}}= & -7480.24+41.24{{x}_{1}}+3.88{{x}_{2}}+0.009{{x}_{1}}{{x}_{2}} \\
& -0.060825x_{1}^{2}-0.020831x_{2}^{2} \\
= & -7480.24+41.24(351.5)+3.88(169)+0.009(351.3)(169) \\
& -0.060825{{(351.5)}^{2}}-0.020831{{(169)}^{2}} \\
= & 95.3 \\
\text{ }{{{\hat{y}}}_{2}}= & 15560.49-82.48{{x}_{1}}-7.76{{x}_{2}}-0.02{{x}_{1}}{{x}_{2}} \\
& +0.12165x_{1}^{2}+0.041663x_{2}^{2} \\
= & 15560.49-82.48(351.5)-7.76(169)-0.02(351.5)(169) \\
& +0.12165{{(351.5)}^{2}}+0.041663{{(169)}^{2}} \\
= & 409.35 \\
\text{ }{{{\hat{y}}}_{3}}= & 38.87-0.0709{{x}_{1}}-0.041{{x}_{2}} \\
= & 38.87-0.0709(351.5)-0.041(169) \\
= & 7.00
\end{align}\,\!</math>

Based on the response values, the individual desirability functions are:

::<math>\begin{align}
{{d}_{1}}= & 1\text{ (since }{{{\hat{y}}}_{1}}>95\text{)} \\
{{d}_{2}}= & \left( \frac{415-{{{\hat{y}}}_{2}}}{415-400} \right)\text{ (since }400\le {{y}_{2}}\le 415\text{)} \\
= & \left( \frac{415-409.35}{415-400} \right) \\
= & 0.3767 \\
{{d}_{3}}= & 1\text{ (since }{{{\hat{y}}}_{3}}\approx 7\text{)}
\end{align}\,\!</math>

Then the overall desirability is:

::<math>\begin{align}
D= & {{({{d}_{1}}\cdot {{d}_{2}}\cdot {{d}_{3}})}^{1/3}} \\
= & {{(1\cdot 0.3767\cdot 1)}^{1/3}} \\
= & 0.72
\end{align}\,\!</math>

This is the same as the Global Desirability displayed by DOE++ in the figure above. At times, a number of solutions may be obtained from DOE++, and it is up to the experimenter to choose the most feasible one.