One Factor Designs - Revision history

Sharon Honecker: /* Box-Cox Method */ replaced recommended transformation picture with html table

2017-08-29T00:07:39Z

Box-Cox Method: replaced recommended transformation picture with html table

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 where <math>\dot{y}=\exp \left[ \frac{1}{n}\sum\limits_{i=1}^{n}\ln \left( y_{i}\right) \right]\,\!</math>.
-Once all <math>{{Y}^{\lambda }}\,\!</math> values are obtained for a value of <math>\lambda \,\!</math>, the corresponding <math>S{{S}_{E}}\,\!</math> for these values is obtained using <math>{{y}^{\lambda \prime }}[I-H]{{y}^{\lambda }}\,\!</math>. The process is repeated for a number of <math>\lambda \,\!</math> values to obtain a plot of <math>S{{S}_{E}}\,\!</math> against <math>\lambda \,\!</math>. Then the value of <math>\lambda \,\!</math> corresponding to the minimum <math>S{{S}_{E}}\,\!</math> is selected as the required transformation for the given data. The DOE folio plots <math>\ln S{{S}_{E}}\,\!</math> values against <math>\lambda \,\!</math> values because the range of <math>S{{S}_{E}}\,\!</math> values is large and if this is not done, all values cannot be displayed on the same plot. The range of search for the best <math>\lambda \,\!</math> value in the software is from <math>-5\,\!</math> to <math>5\,\!</math>, because larger values of of <math>\lambda \,\!</math> are usually not meaningful. The DOE folio also displays a recommended transformation based on the best <math>\lambda \,\!</math> value obtained as per the second table.
+Once all <math>{{Y}^{\lambda }}\,\!</math> values are obtained for a value of <math>\lambda \,\!</math>, the corresponding <math>S{{S}_{E}}\,\!</math> for these values is obtained using <math>{{y}^{\lambda \prime }}[I-H]{{y}^{\lambda }}\,\!</math>. The process is repeated for a number of <math>\lambda \,\!</math> values to obtain a plot of <math>S{{S}_{E}}\,\!</math> against <math>\lambda \,\!</math>. Then the value of <math>\lambda \,\!</math> corresponding to the minimum <math>S{{S}_{E}}\,\!</math> is selected as the required transformation for the given data. The DOE folio plots <math>\ln S{{S}_{E}}\,\!</math> values against <math>\lambda \,\!</math> values because the range of <math>S{{S}_{E}}\,\!</math> values is large and if this is not done, all values cannot be displayed on the same plot. The range of search for the best <math>\lambda \,\!</math> value in the software is from <math>-5\,\!</math> to <math>5\,\!</math>, because larger values of of <math>\lambda \,\!</math> are usually not meaningful. The DOE folio also displays a recommended transformation based on the best <math>\lambda \,\!</math> value obtained as shown in the following table.
-[[Image:doet6.2.png|center|433px|Recommended Box-Cox power transformations.|link=]]
+<center><table border = "1" cellpadding = "5" cellspacing = "1">

Sharon Honecker: /* Box-Cox Method */

2017-08-28T19:12:43Z

Box-Cox Method

← Older revision		Revision as of 19:12, 28 August 2017
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	===Box-Cox Method===		===Box-Cox Method===

	Transformations on the response may be used when residual plots for an experiment show a pattern. This indicates that the equality of variance does not hold for the residuals of the given model. The Box-Cox method can be used to automatically identify a suitable power transformation for the data based on the relationship:		Transformations on the response may be used when residual plots for an experiment show a pattern. This indicates that the equality of variance does not hold for the residuals of the given model. The Box-Cox method can be used to automatically identify a suitable power transformation for the data based on the following relationship:

Sharon Honecker: /* Box-Cox Method */ fixed alignment in equation for zero and negative responses

2017-08-28T19:12:23Z

Box-Cox Method: fixed alignment in equation for zero and negative responses

← Older revision		Revision as of 19:12, 28 August 2017
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	::<math>\begin{~~align~~}		::<math>\begin{array}{rll}
	y(i)& = y(i)+\left~~\| {{y}_~~{\min }} \right\|\times 1.1 & \text{Negative Response} \\		y\left( i\right) = & y\left( i\right) +\left\vert y_{\min }\right\vert
	y(i)& = y(i)+1 & \text{Zero Response}		\times 1.1 & \text{Negative Response} \\
	\end{~~align~~}\,\!</math>		y\left( i\right) = & y\left( i\right) +1 & \text{Zero Response}
			\end{array}\,\!</math>

Sharon Honecker: /* Box-Cox Method */ fixed equation for geometric mean

2017-08-28T19:08:56Z

Box-Cox Method: fixed equation for geometric mean

← Older revision		Revision as of 19:08, 28 August 2017
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	where <math>\dot{y}={{\~~ln }^~~{-1}}~~[(1/n)~~\~~mathop~~{}_{}^{}\~~ln y~~]\,\!</math>.		where <math>\dot{y}=\exp \left[ \frac{1}{n}\sum\limits_{i=1}^{n}\ln \left( y_{i}\right) \right]\,\!</math>.
	Once all <math>{{Y}^{\lambda }}\,\!</math> values are obtained for a value of <math>\lambda \,\!</math>, the corresponding <math>S{{S}_{E}}\,\!</math> for these values is obtained using <math>{{y}^{\lambda \prime }}[I-H]{{y}^{\lambda }}\,\!</math>. The process is repeated for a number of <math>\lambda \,\!</math> values to obtain a plot of <math>S{{S}_{E}}\,\!</math> against <math>\lambda \,\!</math>. Then the value of <math>\lambda \,\!</math> corresponding to the minimum <math>S{{S}_{E}}\,\!</math> is selected as the required transformation for the given data. The DOE folio plots <math>\ln S{{S}_{E}}\,\!</math> values against <math>\lambda \,\!</math> values because the range of <math>S{{S}_{E}}\,\!</math> values is large and if this is not done, all values cannot be displayed on the same plot. The range of search for the best <math>\lambda \,\!</math> value in the software is from <math>-5\,\!</math> to <math>5\,\!</math>, because larger values of of <math>\lambda \,\!</math> are usually not meaningful. The DOE folio also displays a recommended transformation based on the best <math>\lambda \,\!</math> value obtained as per the second table.		Once all <math>{{Y}^{\lambda }}\,\!</math> values are obtained for a value of <math>\lambda \,\!</math>, the corresponding <math>S{{S}_{E}}\,\!</math> for these values is obtained using <math>{{y}^{\lambda \prime }}[I-H]{{y}^{\lambda }}\,\!</math>. The process is repeated for a number of <math>\lambda \,\!</math> values to obtain a plot of <math>S{{S}_{E}}\,\!</math> against <math>\lambda \,\!</math>. Then the value of <math>\lambda \,\!</math> corresponding to the minimum <math>S{{S}_{E}}\,\!</math> is selected as the required transformation for the given data. The DOE folio plots <math>\ln S{{S}_{E}}\,\!</math> values against <math>\lambda \,\!</math> values because the range of <math>S{{S}_{E}}\,\!</math> values is large and if this is not done, all values cannot be displayed on the same plot. The range of search for the best <math>\lambda \,\!</math> value in the software is from <math>-5\,\!</math> to <math>5\,\!</math>, because larger values of of <math>\lambda \,\!</math> are usually not meaningful. The DOE folio also displays a recommended transformation based on the best <math>\lambda \,\!</math> value obtained as per the second table.

Sharon Honecker: /* Box-Cox Method */

2017-08-28T19:03:53Z

Box-Cox Method

← Older revision		Revision as of 19:03, 28 August 2017
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	===Box-Cox Method===		===Box-Cox Method===

	Transformations on the response may be used when residual plots for an experiment show a pattern. This indicates that the equality of variance does not hold for the residuals of the given model. The Box-Cox method can be used to automatically identify a suitable power transformation for the data based on the ~~relation~~:		Transformations on the response may be used when residual plots for an experiment show a pattern. This indicates that the equality of variance does not hold for the residuals of the given model. The Box-Cox method can be used to automatically identify a suitable power transformation for the data based on the relationship:


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	<math>\lambda \,\!</math> is determined using the given data such that <math>S{{S}_{E}}\,\!</math> is minimized. The values of <math>{{Y}^{\lambda }}\,\!</math> are not used as is because of issues related to calculation or comparison of <math>S{{S}_{E}}\,\!</math> values for different values of <math>\lambda \,\!</math>. For example, for <math>\lambda =0\,\!</math> all response values will become 1. Therefore, the following ~~relation~~ is used to obtain <math>{{Y}^{\lambda }}\,\!</math> :		<math>\lambda \,\!</math> is determined using the given data such that <math>S{{S}_{E}}\,\!</math> is minimized. The values of <math>{{Y}^{\lambda }}\,\!</math> are not used as is because of issues related to calculation or comparison of <math>S{{S}_{E}}\,\!</math> values for different values of <math>\lambda \,\!</math>. For example, for <math>\lambda =0\,\!</math> all response values will become 1. Therefore, the following relationship is used to obtain <math>{{Y}^{\lambda }}\,\!</math> :

Sharon Honecker: /* Box-Cox Method */ Fixed formatting on Y^lambda equation

2017-08-28T19:03:27Z

Box-Cox Method: Fixed formatting on Y^lambda equation

← Older revision		Revision as of 19:03, 28 August 2017
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	::<math>{{Y}^{\lambda }}=\{\begin{~~matrix~~}		::<math>Y^{\lambda }=\left\{
	\frac{~~{{Y}~~^{\lambda }}-1}{\lambda ~~{{{~~\dot{y}}}^{\lambda -1}}} \\		\begin{array}{cc}
	\dot{y}\ln y \\		\frac{y^{\lambda }-1}{\lambda \dot{y}^{\lambda -1}} & \lambda \neq 0 \\
	~~\end{matrix}\text{ }\begin{matrix}~~		\dot{y}\ln y & \lambda =0
	\lambda ~~\ne~~ 0~~\begin{matrix}~~		\end{array}
	\\		\right.\,\!</math>
	\\
	\end{~~matrix~~} \\
	\~~lambda =0 \\~~
	~~\end{matrix}~~\,\!</math>

Dlaird at 18:41, 28 August 2017

2017-08-28T18:41:46Z

← Older revision		Revision as of 18:41, 28 August 2017
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	{{Template:Doebook\|5}}		{{Template:Doebook\|5}}

	As explained in [[Simple_Linear_Regression_Analysis\| Simple Linear Regression Analysis]] and [[Multiple_Linear_Regression_Analysis\| Multiple Linear Regression Analysis]], the analysis of observational studies involves the use of regression models. The analysis of experimental studies involves the use of analysis of variance (ANOVA) models. For a comparison of the two models see [[One_Factor_Designs#Fitting_ANOVA_Models\|Fitting ANOVA Models]]. In single factor experiments, ANOVA models are used to compare the mean response values at different levels of the factor. Each level of the factor is investigated to see if the response is significantly different from the response at other levels of the factor. The analysis of single factor experiments is often referred to as ''one-way ANOVA''.		As explained in [[Simple_Linear_Regression_Analysis\| Simple Linear Regression Analysis]] and [[Multiple_Linear_Regression_Analysis\| Multiple Linear Regression Analysis]], the analysis of observational studies involves the use of regression models. The analysis of experimental studies involves the use of analysis of variance (ANOVA) models. For a comparison of the two models see [[One_Factor_Designs#Fitting_ANOVA_Models\|Fitting ANOVA Models]]. In single factor experiments, ANOVA models are used to compare the mean response values at different levels of the factor. Each level of the factor is investigated to see if the response is significantly different from the response at other levels of the factor. The analysis of single factor experiments is often referred to as ''one-way ANOVA''.

Dlaird at 18:40, 28 August 2017

2017-08-28T18:40:18Z

← Older revision		Revision as of 18:40, 28 August 2017
Line 1:		Line 1:
	{{Template:Doebook\|5}}		{{Template:Doebook\|5}}

	As explained in [[Simple_Linear_Regression_Analysis\| Simple Linear Regression Analysis]] and [[Multiple_Linear_Regression_Analysis\| Multiple Linear Regression Analysis]], the analysis of observational studies involves the use of regression models. The analysis of experimental studies involves the use of analysis of variance (ANOVA) models. For a comparison of the two models see [[One_Factor_Designs#Fitting_ANOVA_Models\|Fitting ANOVA Models]]. In single factor experiments, ANOVA models are used to compare the mean response values at different levels of the factor. Each level of the factor is investigated to see if the response is significantly different from the response at other levels of the factor. The analysis of single factor experiments is often referred to as ''one-way ANOVA''.		As explained in [[Simple_Linear_Regression_Analysis\| Simple Linear Regression Analysis]] and [[Multiple_Linear_Regression_Analysis\| Multiple Linear Regression Analysis]], the analysis of observational studies involves the use of regression models. The analysis of experimental studies involves the use of analysis of variance (ANOVA) models. For a comparison of the two models see [[One_Factor_Designs#Fitting_ANOVA_Models\|Fitting ANOVA Models]]. In single factor experiments, ANOVA models are used to compare the mean response values at different levels of the factor. Each level of the factor is investigated to see if the response is significantly different from the response at other levels of the factor. The analysis of single factor experiments is often referred to as ''one-way ANOVA''.

Richard House: /* Box-Cox Method */

2017-08-10T16:19:14Z

Box-Cox Method

← Older revision		Revision as of 16:19, 10 August 2017
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	The required limits for <math>\lambda \,\!</math> are the two values of <math>\lambda \,\!</math> corresponding to the value <math>S{{S}^{*}}\,\!</math> (on the plot of <math>S{{S}_{E}}\,\!</math> against <math>\lambda \,\!</math>). If the limits for <math>\lambda \,\!</math> do not include the value of one, then the transformation is applicable for the given data.		The required limits for <math>\lambda \,\!</math> are the two values of <math>\lambda \,\!</math> corresponding to the value <math>S{{S}^{*}}\,\!</math> (on the plot of <math>S{{S}_{E}}\,\!</math> against <math>\lambda \,\!</math>). If the limits for <math>\lambda \,\!</math> do not include the value of one, then the transformation is applicable for the given data.
	Note that the power transformations are not defined for response values that are negative or zero. DOE++ deals with negative and zero response values using the following equations (that involve addition of a suitable quantity to all of the response values if a zero or negative response value is encountered).		Note that the power transformations are not defined for response values that are negative or zero. The DOE folio deals with negative and zero response values using the following equations (that involve addition of a suitable quantity to all of the response values if a zero or negative response value is encountered).

Richard House: /* Box-Cox Method */

2017-08-10T16:18:24Z

Box-Cox Method

← Older revision		Revision as of 16:18, 10 August 2017
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	where <math>\dot{y}={{\ln }^{-1}}[(1/n)\mathop{}_{}^{}\ln y]\,\!</math>.		where <math>\dot{y}={{\ln }^{-1}}[(1/n)\mathop{}_{}^{}\ln y]\,\!</math>.
	Once all <math>{{Y}^{\lambda }}\,\!</math> values are obtained for a value of <math>\lambda \,\!</math>, the corresponding <math>S{{S}_{E}}\,\!</math> for these values is obtained using <math>{{y}^{\lambda \prime }}[I-H]{{y}^{\lambda }}\,\!</math>. The process is repeated for a number of <math>\lambda \,\!</math> values to obtain a plot of <math>S{{S}_{E}}\,\!</math> against <math>\lambda \,\!</math>. Then the value of <math>\lambda \,\!</math> corresponding to the minimum <math>S{{S}_{E}}\,\!</math> is selected as the required transformation for the given data. DOE++ plots <math>\ln S{{S}_{E}}\,\!</math> values against <math>\lambda \,\!</math> values because the range of <math>S{{S}_{E}}\,\!</math> values is large and if this is not done, all values cannot be displayed on the same plot. The range of search for the best <math>\lambda \,\!</math> value in the software is from <math>-5\,\!</math> to <math>5\,\!</math>, because larger values of of <math>\lambda \,\!</math> are usually not meaningful. DOE++ also displays a recommended transformation based on the best <math>\lambda \,\!</math> value obtained as per the second table.		Once all <math>{{Y}^{\lambda }}\,\!</math> values are obtained for a value of <math>\lambda \,\!</math>, the corresponding <math>S{{S}_{E}}\,\!</math> for these values is obtained using <math>{{y}^{\lambda \prime }}[I-H]{{y}^{\lambda }}\,\!</math>. The process is repeated for a number of <math>\lambda \,\!</math> values to obtain a plot of <math>S{{S}_{E}}\,\!</math> against <math>\lambda \,\!</math>. Then the value of <math>\lambda \,\!</math> corresponding to the minimum <math>S{{S}_{E}}\,\!</math> is selected as the required transformation for the given data. The DOE folio plots <math>\ln S{{S}_{E}}\,\!</math> values against <math>\lambda \,\!</math> values because the range of <math>S{{S}_{E}}\,\!</math> values is large and if this is not done, all values cannot be displayed on the same plot. The range of search for the best <math>\lambda \,\!</math> value in the software is from <math>-5\,\!</math> to <math>5\,\!</math>, because larger values of of <math>\lambda \,\!</math> are usually not meaningful. The DOE folio also displays a recommended transformation based on the best <math>\lambda \,\!</math> value obtained as per the second table.


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	A plot of <math>\ln S{{S}_{E}}\,\!</math> for various <math>\lambda \,\!</math> values, as obtained from DOE++, is shown in the following figure. The value of <math>\lambda \,\!</math> that gives the minimum <math>S{{S}_{E}}\,\!</math> is identified as 0.7841. The <math>S{{S}_{E}}\,\!</math> value corresponding to this value of <math>\lambda \,\!</math> is 73.74. A 90% confidence interval on this <math>\lambda \,\!</math> value is calculated as follows. <math>S{{S}^{*}}\,\!</math> can be obtained as shown next:		A plot of <math>\ln S{{S}_{E}}\,\!</math> for various <math>\lambda \,\!</math> values, as obtained from the DOE folio, is shown in the following figure. The value of <math>\lambda \,\!</math> that gives the minimum <math>S{{S}_{E}}\,\!</math> is identified as 0.7841. The <math>S{{S}_{E}}\,\!</math> value corresponding to this value of <math>\lambda \,\!</math> is 73.74. A 90% confidence interval on this <math>\lambda \,\!</math> value is calculated as follows. <math>S{{S}^{*}}\,\!</math> can be obtained as shown next: