General Full Factorial Designs - Revision history

Richard House: /* Residual Analysis */

2017-08-10T16:23:31Z

Residual Analysis

← Older revision		Revision as of 16:23, 10 August 2017
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	===Residual Analysis===		===Residual Analysis===
	As in the case of single factor experiments, plots of residuals can also be used to check for model adequacy in factorial experiments. [[ANOVA_for_Designed_Experiments#Box-Cox_Method\|Box-Cox transformations]] are also available in ~~DOE~~++ for factorial experiments.		As in the case of single factor experiments, plots of residuals can also be used to check for model adequacy in factorial experiments. [[ANOVA_for_Designed_Experiments#Box-Cox_Method\|Box-Cox transformations]] are also available in Weibull++ DOE folios for factorial experiments.

Richard House: /* Calculation of the Test Statistics */

2017-08-10T16:23:07Z

Calculation of the Test Statistics

← Older revision		Revision as of 16:23, 10 August 2017
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	Assuming that the desired significance level is 0.1, since <math>p\,\!</math> value > 0.1, we fail to reject <math>{{H}_{0}}:{{(\tau \delta )}_{ij}}=0\,\!</math> and conclude that the interaction between speed and fuel additive does not significantly affect the mileage of the sport utility vehicle. DOE++ displays this result in the ANOVA table, as shown in the following figure. In the absence of the interaction, the analysis of main effects becomes important.		Assuming that the desired significance level is 0.1, since <math>p\,\!</math> value > 0.1, we fail to reject <math>{{H}_{0}}:{{(\tau \delta )}_{ij}}=0\,\!</math> and conclude that the interaction between speed and fuel additive does not significantly affect the mileage of the sport utility vehicle. The DOE folio displays this result in the ANOVA table, as shown in the following figure. In the absence of the interaction, the analysis of main effects becomes important.

Richard House: /* Expression of the ANOVA Model as y = ΧΒ + ε */

2017-08-10T16:22:49Z

Expression of the ANOVA Model as y = ΧΒ + ε

← Older revision		Revision as of 16:22, 10 August 2017
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	Therefore, only two of the <math>{{\tau }_{i}}\,\!</math> effects are independent. Assuming that <math>{{\tau }_{1}}\,\!</math> and <math>{{\tau }_{2}}\,\!</math> are independent, <math>{{\tau }_{3}}=-({{\tau }_{1}}+{{\tau }_{2}})\,\!</math>. (The null hypothesis to test the significance of factor <math>A\,\!</math> can be rewritten using only the independent effects as <math>{{H}_{0}}:{{\tau }_{1}}={{\tau }_{2}}=0\,\!</math>.) DOE++ displays only the independent effects because only these effects are important to the analysis. The independent effects, <math>{{\tau }_{1}}\,\!</math> and <math>{{\tau }_{2}}\,\!</math>, are displayed as A[1] and A[2] respectively because these are the effects associated with factor <math>A\,\!</math> (speed).		Therefore, only two of the <math>{{\tau }_{i}}\,\!</math> effects are independent. Assuming that <math>{{\tau }_{1}}\,\!</math> and <math>{{\tau }_{2}}\,\!</math> are independent, <math>{{\tau }_{3}}=-({{\tau }_{1}}+{{\tau }_{2}})\,\!</math>. (The null hypothesis to test the significance of factor <math>A\,\!</math> can be rewritten using only the independent effects as <math>{{H}_{0}}:{{\tau }_{1}}={{\tau }_{2}}=0\,\!</math>.) The DOE folio displays only the independent effects because only these effects are important to the analysis. The independent effects, <math>{{\tau }_{1}}\,\!</math> and <math>{{\tau }_{2}}\,\!</math>, are displayed as A[1] and A[2] respectively because these are the effects associated with factor <math>A\,\!</math> (speed).
	Constraints on <math>{{\delta }_{j}}\,\!</math> are:		Constraints on <math>{{\delta }_{j}}\,\!</math> are:

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	Therefore, only one of the <math>{{\delta }_{j}}\,\!</math> effects are independent. Assuming that <math>{{\delta }_{1}}\,\!</math> is independent, <math>{{\delta }_{2}}=-{{\delta }_{1}}\,\!</math>. (The null hypothesis to test the significance of factor <math>B\,\!</math> can be rewritten using only the independent effect as <math>{{H}_{0}}:{{\delta }_{1}}=0\,\!</math>.) The independent effect <math>{{\delta }_{1}}\,\!</math> is displayed as B:B in DOE++.		Therefore, only one of the <math>{{\delta }_{j}}\,\!</math> effects are independent. Assuming that <math>{{\delta }_{1}}\,\!</math> is independent, <math>{{\delta }_{2}}=-{{\delta }_{1}}\,\!</math>. (The null hypothesis to test the significance of factor <math>B\,\!</math> can be rewritten using only the independent effect as <math>{{H}_{0}}:{{\delta }_{1}}=0\,\!</math>.) The independent effect <math>{{\delta }_{1}}\,\!</math> is displayed as B:B in the DOE folio.
	Constraints on <math>{{(\tau \delta )}_{ij}}\,\!</math> are:		Constraints on <math>{{(\tau \delta )}_{ij}}\,\!</math> are:

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	The last five equations given above represent four constraints, as only four of these five equations are independent. Therefore, only two out of the six <math>{{(\tau \delta )}_{ij}}\,\!</math> effects are independent. Assuming that <math>{{(\tau \delta )}_{11}}\,\!</math> and <math>{{(\tau \delta )}_{21}}\,\!</math> are independent, the other four effects can be expressed in terms of these effects. (The null hypothesis to test the significance of interaction <math>AB\,\!</math> can be rewritten using only the independent effects as <math>{{H}_{0}}:{{(\tau \delta )}_{11}}={{(\tau \delta )}_{21}}=0\,\!</math>.) The effects <math>{{(\tau \delta )}_{11}}\,\!</math> and <math>{{(\tau \delta )}_{21}}\,\!</math> are displayed as A[1]B and A[2]B respectively in DOE++.		The last five equations given above represent four constraints, as only four of these five equations are independent. Therefore, only two out of the six <math>{{(\tau \delta )}_{ij}}\,\!</math> effects are independent. Assuming that <math>{{(\tau \delta )}_{11}}\,\!</math> and <math>{{(\tau \delta )}_{21}}\,\!</math> are independent, the other four effects can be expressed in terms of these effects. (The null hypothesis to test the significance of interaction <math>AB\,\!</math> can be rewritten using only the independent effects as <math>{{H}_{0}}:{{(\tau \delta )}_{11}}={{(\tau \delta )}_{21}}=0\,\!</math>.) The effects <math>{{(\tau \delta )}_{11}}\,\!</math> and <math>{{(\tau \delta )}_{21}}\,\!</math> are displayed as A[1]B and A[2]B respectively in the DOE folio.

	The regression version of the ANOVA model can be obtained using indicator variables, similar to the case of the single factor experiment in [[One_Factor_Designs#Fitting_ANOVA_Models\|Fitting ANOVA Models]]. Since factor <math>A\,\!</math> has three levels, two indicator variables, <math>{{x}_{1}}\,\!</math> and <math>{{x}_{2}}\,\!</math>, are required which need to be coded as shown next:		The regression version of the ANOVA model can be obtained using indicator variables, similar to the case of the single factor experiment in [[One_Factor_Designs#Fitting_ANOVA_Models\|Fitting ANOVA Models]]. Since factor <math>A\,\!</math> has three levels, two indicator variables, <math>{{x}_{1}}\,\!</math> and <math>{{x}_{2}}\,\!</math>, are required which need to be coded as shown next:

Richard House: /* Analysis of General Factorial Experiments */

2017-08-10T16:20:48Z

Analysis of General Factorial Experiments

← Older revision		Revision as of 16:20, 10 August 2017
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	==Analysis of General Factorial Experiments==		==Analysis of General Factorial Experiments==

	In ~~DOE~~++, factorial experiments are referred to as ''factorial designs''. The experiments explained in this section are referred to as g''~~eneral~~ factorial designs''. This is done to distinguish these experiments from the other factorial designs supported by ~~DOE~~++ (see the figure below).		In Weibull++ DOE folios, factorial experiments are referred to as ''factorial designs''. The experiments explained in this section are referred to as ''general factorial designs''. This is done to distinguish these experiments from the other factorial designs supported by Weibull++ DOE folios (see the figure below).


	[[Image:doe6_10.png\|center\|800px\|Factorial experiments available in ~~DOE~~++.\|link=]]		[[Image:doe6_10.png\|center\|800px\|Factorial experiments available in Weibull++.\|link=]]

Sharon Honecker: /* Calculation of the Test Statistics */ changed "sports utility vehicle" to "sport utility vehicle"

2016-09-28T23:44:29Z

Calculation of the Test Statistics: changed "sports utility vehicle" to "sport utility vehicle"

← Older revision		Revision as of 23:44, 28 September 2016
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	Assuming that the desired significance level is 0.1, since <math>p\,\!</math> value > 0.1, we fail to reject <math>{{H}_{0}}:{{(\tau \delta )}_{ij}}=0\,\!</math> and conclude that the interaction between speed and fuel additive does not significantly affect the mileage of the ~~sports~~ utility vehicle. DOE++ displays this result in the ANOVA table, as shown in the following figure. In the absence of the interaction, the analysis of main effects becomes important.		Assuming that the desired significance level is 0.1, since <math>p\,\!</math> value > 0.1, we fail to reject <math>{{H}_{0}}:{{(\tau \delta )}_{ij}}=0\,\!</math> and conclude that the interaction between speed and fuel additive does not significantly affect the mileage of the sport utility vehicle. DOE++ displays this result in the ANOVA table, as shown in the following figure. In the absence of the interaction, the analysis of main effects becomes important.

Sharon Honecker: /* Example */ changed "sports utility vehicle" to "sport utility vehicle" in a second place

2016-09-28T23:43:59Z

Example: changed "sports utility vehicle" to "sport utility vehicle" in a second place

← Older revision		Revision as of 23:43, 28 September 2016
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	In the figure, the factor Speed is represented as factor <math>A\,\!</math> and the factor Fuel Additive is represented as factor <math>B\,\!</math>. The experimenter would like to investigate if speed, fuel additive or the interaction between speed and fuel additive affects the mileage of the ~~sports~~ utility vehicle. In other words, the following hypotheses need to be tested:		In the figure, the factor Speed is represented as factor <math>A\,\!</math> and the factor Fuel Additive is represented as factor <math>B\,\!</math>. The experimenter would like to investigate if speed, fuel additive or the interaction between speed and fuel additive affects the mileage of the sport utility vehicle. In other words, the following hypotheses need to be tested:

Sharon Honecker: /* Example */ changed "sports utility vehicle" to "sport utility vehicle"

2016-09-26T23:21:12Z

Example: changed "sports utility vehicle" to "sport utility vehicle"

← Older revision		Revision as of 23:21, 26 September 2016
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	====Example====		====Example====

	Consider an experiment to investigate the effect of speed and type of fuel additive used on the mileage of a ~~sports~~ utility vehicle. Three speeds and two types of fuel additives are investigated. Each of the treatment combinations are replicated three times. The mileage values observed are displayed in the table below.		Consider an experiment to investigate the effect of speed and type of fuel additive used on the mileage of a sport utility vehicle. Three speeds and two types of fuel additives are investigated. Each of the treatment combinations are replicated three times. The mileage values observed are displayed in the table below.

Richard House: /* Calculation of the Test Statistics */

2016-02-12T17:22:01Z

Calculation of the Test Statistics

← Older revision		Revision as of 17:22, 12 February 2016
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	[[Image:doe6_112.png\|center\|645px\|Analysis results for the experiment in the above table.]]		[[Image:doe6_112.png\|center\|645px\|Analysis results for the experiment in the above table.\|link=]]

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

Richard House: /* Example */

2016-02-12T16:58:50Z

Example

← Older revision		Revision as of 16:58, 12 February 2016
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	[[Image:doet6.5.png\|center\|400px\|Mileage data for different speeds and fuel additive types.]]		[[Image:doet6.5.png\|center\|400px\|Mileage data for different speeds and fuel additive types.\|link=]]


	The experimental design for the data is shown in the figure below.		The experimental design for the data is shown in the figure below.

	[[Image:doe6_11.png\|center\|639px\|Experimental design for the Mileage Test]]		[[Image:doe6_11.png\|center\|639px\|Experimental design for the Mileage Test\|link=]]

Richard House: /* Analysis of General Factorial Experiments */

2016-01-08T21:02:41Z

Analysis of General Factorial Experiments

← Older revision		Revision as of 21:02, 8 January 2016
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	[[Image:doe6_10.png\|center\|~~650px~~\|Factorial experiments available in DOE++.]]		[[Image:doe6_10.png\|center\|800px\|Factorial experiments available in DOE++.\|link=]]