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(Replaced content with 'Disclaimers <math> A &=&Average% \text{ }response\text{ }at\text{ }A_{\text{high}}-Average\text{ }response% \text{ }at\text{ }A_{\text{low}} \\ &=&\frac{45+55}{2}-\frac{25…')
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Disclaimers
Disclaimers
ctor on the response depends on the level of the other factor(s).}
<math>
One-factor-at-a-time experiments (where each factor is investigated
separately by keeping all the remaining factors constant) do not reveal the
interaction effects between the factors. Further, in one-factor-at-a-time
experiments full randomization is not possible.


To illustrate factorial experiments consider an experiment where the
response is investigated for two factors, $A$ and $B$. Assume that the
response is studied at two levels of factor $A$ with $A_{%
\text{low}}$ representing the lower level of $A$ and $A_{\text{high}}$
representing the higher level of $A$. Similarly, let $B_{\text{low}}$\ and $%
B_{\text{high}}$ represent the two levels of factor $B$ that are being
investigated in this experiment. Since there are two factors with two
levels, a total of $2\times 2=4$ combinations exist ($A_{\text{low}}$-$B_{%
\text{low}}$, $\ A_{\text{low}}$-$B_{\text{high}}$, $A_{\text{high}}$-$B_{%
\text{low}}$, $\ A_{\text{high}}$-$B_{\text{high}}$). Thus, four runs are
required for each replicate if a factorial experiment is to be carried out
in this case. Assume that the response values for each of these four
possible combinations are obtained as shown in Table 6.3.\FRAME{dtbpFU}{%
2.2762in}{1.2246in}{0pt}{\Qcb{Table 6.3: Two-factor factorial experiment.}}{%
}{Figure}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "T";width
2.2762in;height 1.2246in;depth 0pt;original-width 3.2119in;original-height
1.7097in;cropleft "0";croptop "1";cropright "1";cropbottom "0";tempfilename
'Ch6TwoFactExTable1.wmf';tempfile-properties "XNPR";}}


\subsection{Investigating Factor Effects}


The effect of factor $A$ on the response can be obtained by taking the
difference between the average response when $A$ is high and the average
response when $A$ is low. The change in the response due to a change in the
level of a factor is called the \emph{main effect} of the factor%
\index{Main effect}. The main effect of $A$ as per the response values in
Table 6.3 is:%
\begin{eqnarray*}
A &=&Average%
A &=&Average%
\text{ }response\text{ }at\text{ }A_{\text{high}}-Average\text{ }response%
\text{ }response\text{ }at\text{ }A_{\text{high}}-Average\text{ }response%
\text{ }at\text{ }A_{\text{low}} \\
\text{ }at\text{ }A_{\text{low}} \\
&=&\frac{45+55}{2}-\frac{25+35}{2} \\
&=&50-30 \\
&=&20
</math>

Revision as of 00:19, 11 December 2009

Disclaimers [math]\displaystyle{ A &=&Average% \text{ }response\text{ }at\text{ }A_{\text{high}}-Average\text{ }response% \text{ }at\text{ }A_{\text{low}} \\ &=&\frac{45+55}{2}-\frac{25+35}{2} \\ &=&50-30 \\ &=&20 }[/math]

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