ANOVA for Designed Experiments - Revision history

Sharon Honecker: /* Box-Cox Method */

2017-08-26T03:29:28Z

Box-Cox Method

← Older revision		Revision as of 03:29, 26 August 2017
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	</table>		</table>


	~~[[Image:doet6.2.png\|thumb\|center\|400px\|Recommended Box-Cox power transformations.]]~~

Sharon Honecker: /* Box-Cox Method */

2017-08-26T03:28:53Z

Box-Cox Method

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 where <math>\dot{y}=\exp \left[ \frac{1}{n}\sum\limits_{i=1}^{n}y_{i}\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. 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 shown in the table below.
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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| y_{\min }\right|\times 1.1 & \text{Negative Response} \\
- & y(i)= & y(i)+1\text{                          Zero Response}
+y\left( i\right) =& y\left( i\right) +1 & \text{Zero Response}
-\end{align}\,\!</math>
+\end{array}\,\!</math>

Sharon Honecker: /* Box-Cox Method */

2017-08-26T02:56:40Z

Box-Cox Method

← Older revision		Revision as of 02:56, 26 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>


	where <math>\dot{y}=\exp \left[ \frac{1}{n}\sum\limits_{i=1}^{n}y_{i}\right]\,\!</math>.		where <math>\dot{y}=\exp \left[ \frac{1}{n}\sum\limits_{i=1}^{n}y_{i}\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. 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 table below.		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 shown in the table below.

Sharon Honecker: /* Box-Cox Method */

2017-08-26T02:52:10Z

Box-Cox Method

← Older revision		Revision as of 02:52, 26 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}y_{i}\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. 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 table below.		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 table below.

Sharon Honecker: /* Box-Cox Method */

2017-08-26T01:14:51Z

Box-Cox Method

← Older revision		Revision as of 01:14, 26 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 following 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> :


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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. 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 table below.


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	<center>[[Image:doe6.7.png\|thumb\|center\|400px\|Box-Cox power transformation plot for the data in the first table.]]</center>		<center>[[Image:doe6.7.png\|thumb\|center\|400px\|Box-Cox power transformation plot for the data in the first table.]]</center>


	==Experiments with Several Factors - Factorial Experiments==		==Experiments with Several Factors - Factorial Experiments==

Harry Guo at 17:17, 10 September 2013

2013-09-10T17:17:18Z

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	which is obviously incorrect since the sum of squares cannot be negative. For a detailed discussion on this refer to [[DOE References\|[Searle (1997)]]].		which is obviously incorrect since the sum of squares cannot be negative. For a detailed discussion on this refer to [[DOE References\|Searle(1997, 1971)]].

Harry Guo at 17:11, 10 September 2013

2013-09-10T17:11:04Z

← Older revision		Revision as of 17:11, 10 September 2013
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	which is obviously incorrect since the sum of squares cannot be negative. For a detailed discussion on this refer to [[~~EDAR Appendix F~~\|[23]]].		which is obviously incorrect since the sum of squares cannot be negative. For a detailed discussion on this refer to [[DOE References\|[Searle (1997)]]].

Harry Guo: /* Fitting ANOVA Models */

2013-01-30T17:17:04Z

Fitting ANOVA Models

← Older revision		Revision as of 17:17, 30 January 2013
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	The equation given above represents the "regression version" of the ANOVA model.		The equation given above represents the "regression version" of the ANOVA model.


			===Treat Numerical Factors as Qualitative or Quantitative ?===

	It can be seen from the equation given above that in an ANOVA model each factor is treated as a qualitative factor. In the present example the factor, lathe speed, is a quantitative factor with three levels. But the ANOVA model treats this factor as a qualitative factor with three levels. Therefore, two indicator variables, <math>{{x}_{1}}\,\!</math> and <math>{{x}_{2}}\,\!</math>, are required to represent this factor.		It can be seen from the equation given above that in an ANOVA model each factor is treated as a qualitative factor. In the present example the factor, lathe speed, is a quantitative factor with three levels. But the ANOVA model treats this factor as a qualitative factor with three levels. Therefore, two indicator variables, <math>{{x}_{1}}\,\!</math> and <math>{{x}_{2}}\,\!</math>, are required to represent this factor.

Harry Guo: /* Comparison of ANOVA and Regression Models */

2013-01-30T17:15:14Z

Comparison of ANOVA and Regression Models

← Older revision		Revision as of 17:15, 30 January 2013
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	The equation given above represents the "regression version" of the ANOVA model.		The equation given above represents the "regression version" of the ANOVA model.

	~~====Comparison of~~ ANOVA and ~~Regression Models====~~		It can be seen from the equation given above that in an ANOVA model each factor is treated as a qualitative factor. In the present example the factor, lathe speed, is a quantitative factor with three levels. But the ANOVA model treats this factor as a qualitative factor with three levels. Therefore, two indicator variables, <math>{{x}_{1}}\,\!</math> and <math>{{x}_{2}}\,\!</math>, are required to represent this factor.

	~~It can be seen from the equation given above~~ that in ~~an ANOVA~~ model ~~each factor is~~ treated as ~~a qualitative factor. In the present example the factor, lathe speed, is~~ a quantitative ~~factor with three levels. But the ANOVA model treats this factor as~~ a qualitative ~~factor with three levels~~. Therefore, two indicator variables, <math>{{x}_{1}}\,\!</math> and <math>{{x}_{2}}\,\!</math>, are required to represent this factor. Note that if a regression model were to be fitted to the data in the first table then a distinction is made between quantitative and qualitative factors. For regression models, the factor, lathe speed, would be used as a quantitative factor and represented with a single predictor variable. For example, if a first order ~~regression~~ model were to be fitted to the data in the first table, then the regression model would take the form <math>{{Y}_{ij}}={{\beta }_{0}}+{{\beta }_{1}}{{x}_{i1}}+{{\epsilon }_{ij}}\,\!</math>. If a second order regression model were to be fitted, the regression model would be <math>{{Y}_{ij}}={{\beta }_{0}}+{{\beta }_{1}}{{x}_{i1}}+{{\beta }_{2}}x_{i1}^{2}+{{\epsilon }_{ij}}\,\!</math>. Notice that unlike these regression models, the regression version of the ANOVA model does not make any assumption about the nature of relationship between the response and the factor being investigated.		Note that in a regression model a variable can either be treated as a quantitative or a qualitative variable. The factor, lathe speed, would be used as a quantitative factor and represented with a single predictor variable in a regression model. For example, if a first order model were to be fitted to the data in the first table, then the regression model would take the form <math>{{Y}_{ij}}={{\beta }_{0}}+{{\beta }_{1}}{{x}_{i1}}+{{\epsilon }_{ij}}\,\!</math>. If a second order regression model were to be fitted, the regression model would be <math>{{Y}_{ij}}={{\beta }_{0}}+{{\beta }_{1}}{{x}_{i1}}+{{\beta }_{2}}x_{i1}^{2}+{{\epsilon }_{ij}}\,\!</math>. Notice that unlike these regression models, the regression version of the ANOVA model does not make any assumption about the nature of relationship between the response and the factor being investigated.

	The choice of the two models for a particular data set depends on the objective of the experimenter. In the case of the data of the first table, the objective of the experimenter is to compare the levels of the factor to see if change in the levels leads to a significant change in the response. The objective is not to make predictions on the response for a given level of the factor. Therefore, the ANOVA model is used in this case. If the objective of the experimenter were prediction or optimization, the experimenter would use a regression model and focus on aspects such as the nature of relationship between the factor, lathe speed, and the response, surface finish, so that the regression model obtained is able to make accurate predictions.

			The choice of treating a particular factor as a quantitative or qualitative variable depends on the objective of the experimenter. In the case of the data of the first table, the objective of the experimenter is to compare the levels of the factor to see if change in the levels leads to a significant change in the response. The objective is not to make predictions on the response for a given level of the factor. Therefore, the factor is treated as a qualitative factor in this case. If the objective of the experimenter were prediction or optimization, the experimenter would focus on aspects such as the nature of relationship between the factor, lathe speed, and the response, surface finish, so that the factor should be modeled as a quantitative factor to make accurate predictions.

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

Chris Kahn at 16:15, 28 January 2013

2013-01-28T16:15:16Z

← Older revision		Revision as of 16:15, 28 January 2013
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	The studies that enable the establishment of a cause-and-effect relationship are called ''experiments''. In experiments the response is investigated by studying only the effect of the factor(s) of interest and excluding all other effects that may provide alternative justifications to the observed change in response. This is done in two ways. First, the levels of the factors to be investigated are carefully selected and then strictly controlled during the execution of the experiment. The aspect of selecting what factor levels should be investigated in the experiment is called the ''design'' of the experiment. The second distinguishing feature of experiments is that observations in an experiment are recorded in a random order. By doing this, it is hoped that the effect of all other factors not being investigated in the experiment will get cancelled out so that the change in the response is the result of only the investigated factors. Using these two techniques, experiments tend to ensure that alternative justifications to observed changes in the response are voided, thereby enabling the establishment of a cause-and-effect relationship between the response and the investigated factors.		The studies that enable the establishment of a cause-and-effect relationship are called ''experiments''. In experiments the response is investigated by studying only the effect of the factor(s) of interest and excluding all other effects that may provide alternative justifications to the observed change in response. This is done in two ways. First, the levels of the factors to be investigated are carefully selected and then strictly controlled during the execution of the experiment. The aspect of selecting what factor levels should be investigated in the experiment is called the ''design'' of the experiment. The second distinguishing feature of experiments is that observations in an experiment are recorded in a random order. By doing this, it is hoped that the effect of all other factors not being investigated in the experiment will get cancelled out so that the change in the response is the result of only the investigated factors. Using these two techniques, experiments tend to ensure that alternative justifications to observed changes in the response are voided, thereby enabling the establishment of a cause-and-effect relationship between the response and the investigated factors.

	~~===~~Randomization~~===~~		'''Randomization'''

	The aspect of recording observations in an experiment in a random order is referred to as ''randomization''. Specifically, randomization is the process of assigning the various levels of the investigated factors to the experimental units in a random fashion. An experiment is said to be ''completely randomized'' if the probability of an experimental unit to be subjected to any level of a factor is equal for all the experimental units. The importance of randomization can be illustrated using an example. Consider an experiment where the effect of the speed of a lathe machine on the surface finish of a product is being investigated. In order to save time, the experimenter records surface finish values by running the lathe machine continuously and recording observations in the order of increasing speeds. The analysis of the experiment data shows that an increase in lathe speeds causes a decrease in the quality of surface finish. However the results of the experiment are disputed by the lathe operator who claims that he has been able to obtain better surface finish quality in the products by operating the lathe machine at higher speeds. It is later found that the faulty results were caused because of overheating of the tool used in the machine. Since the lathe was run continuously in the order of increased speeds the observations were recorded in the order of increased tool temperatures. This problem could have been avoided if the experimenter had randomized the experiment and taken reading at the various lathe speeds in a random fashion. This would require the experimenter to stop and restart the machine at every observation, thereby keeping the temperature of the tool within a reasonable range. Randomization would have ensured that the effect of heating of the machine tool is not included in the experiment.		The aspect of recording observations in an experiment in a random order is referred to as ''randomization''. Specifically, randomization is the process of assigning the various levels of the investigated factors to the experimental units in a random fashion. An experiment is said to be ''completely randomized'' if the probability of an experimental unit to be subjected to any level of a factor is equal for all the experimental units. The importance of randomization can be illustrated using an example. Consider an experiment where the effect of the speed of a lathe machine on the surface finish of a product is being investigated. In order to save time, the experimenter records surface finish values by running the lathe machine continuously and recording observations in the order of increasing speeds. The analysis of the experiment data shows that an increase in lathe speeds causes a decrease in the quality of surface finish. However the results of the experiment are disputed by the lathe operator who claims that he has been able to obtain better surface finish quality in the products by operating the lathe machine at higher speeds. It is later found that the faulty results were caused because of overheating of the tool used in the machine. Since the lathe was run continuously in the order of increased speeds the observations were recorded in the order of increased tool temperatures. This problem could have been avoided if the experimenter had randomized the experiment and taken reading at the various lathe speeds in a random fashion. This would require the experimenter to stop and restart the machine at every observation, thereby keeping the temperature of the tool within a reasonable range. Randomization would have ensured that the effect of heating of the machine tool is not included in the experiment.