Design Evaluation and Power Study - Revision history

Richard House: /* Discussion on Power Calculation */

2017-08-10T17:13:40Z

Discussion on Power Calculation

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	All the above examples show how to calculate the power for a given amount of effect.When a power value is given, using the above method we also can calculate the corresponding effect. If the power is too low for an effect of interest, the sample size of the experiment must be increased in order to get a higher power value.		All the above examples show how to calculate the power for a given amount of effect.When a power value is given, using the above method we also can calculate the corresponding effect. If the power is too low for an effect of interest, the sample size of the experiment must be increased in order to get a higher power value.

	We discussed in detail how to define an “effect” for quantitative and qualitative factors, and how to use model coefficients to represent a given effect. The power in DOE++ is calculated based on this definition. Readers may find that power is calculated directly based on model coefficients (instead of the contrasts) in other software packages or books. However, for some cases, such as for the main and interaction effects of qualitative factors with multiple levels, the meaning of model coefficients is not very straightforward. Therefore, it is better to use the defined effect (or contrast) shown here to calculate the power, even though this calculation is much more complicated.		We discussed in detail how to define an “effect” for quantitative and qualitative factors, and how to use model coefficients to represent a given effect. The power in a DOE folio is calculated based on this definition. Readers may find that power is calculated directly based on model coefficients (instead of the contrasts) in other software packages or books. However, for some cases, such as for the main and interaction effects of qualitative factors with multiple levels, the meaning of model coefficients is not very straightforward. Therefore, it is better to use the defined effect (or contrast) shown here to calculate the power, even though this calculation is much more complicated.

	=Conclusion=		=Conclusion=

	In this chapter, we discussed how to evaluate an experiment design. Although the evaluation can be conducted either before or after conducting the experiment, it is always recommended to evaluate an experiment before performing it. A bad design will waste time and money. Readers should check the alias structure, the orthogonality and the power for important effects for an experiment before the tests.		In this chapter, we discussed how to evaluate an experiment design. Although the evaluation can be conducted either before or after conducting the experiment, it is always recommended to evaluate an experiment before performing it. A bad design will waste time and money. Readers should check the alias structure, the orthogonality and the power for important effects for an experiment before the tests.

Richard House: /* Power Study for Response Surface Method Designs */

2017-08-10T17:13:11Z

Power Study for Response Surface Method Designs

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	\|}		\|}

	The above design can be created in DOE++ using the following settings:		The above design can be created in a DOE folio using the following settings:

	[[Image: DesignEvluation_9.png\|center\|600px\|Settings for creating the RSM design]]		[[Image: DesignEvluation_9.png\|center\|600px\|Settings for creating the RSM design]]
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	\|}		\|}

	The results in DOE++ can be obtained from the design evaluation.		The results in a DOE folio can be obtained from the design evaluation.

	[[Image: DesignEvluation_10.png\|center\|300px\|Settings for creating the RSM design]]		[[Image: DesignEvluation_10.png\|center\|300px\|Settings for creating the RSM design]]

Richard House: /* Power Study for General Level Factorial Designs */

2017-08-10T17:12:08Z

Power Study for General Level Factorial Designs

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	There are 2 regression terms for each main effect, and 4 regression terms for the interaction effect. We will use the above equation to explain how the power for the main effects and interaction effects is calculated in DOE++. The following balanced design is used for the calculation:		There are 2 regression terms for each main effect, and 4 regression terms for the interaction effect. We will use the above equation to explain how the power for the main effects and interaction effects is calculated in the DOE folio. The following balanced design is used for the calculation:

Richard House: /* Power Study for 2 Level Factorial Designs */

2017-08-10T17:11:44Z

Power Study for 2 Level Factorial Designs

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	The settings and results in DOE++ are given below.		The settings and results in the DOE folio are given below.

Richard House: /* Power Calculation for Comparing Multiple Means: Unbalanced Designs */

2017-08-10T17:11:14Z

Power Calculation for Comparing Multiple Means: Unbalanced Designs

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	\end{align} \right)\,\!</math> optimization problems and use the smallest <math>\min ({{\phi }_{i}})\,\!</math> among all the solutions to calculate the power of the experiment. Clearly, the calculation will be very expensive.		\end{align} \right)\,\!</math> optimization problems and use the smallest <math>\min ({{\phi }_{i}})\,\!</math> among all the solutions to calculate the power of the experiment. Clearly, the calculation will be very expensive.

	In DOE++, instead of calculating the exact solution, we use the optimal <math>\beta \,\!</math> for a balanced design to calculate the approximated power for an unbalanced design. It can be seen that the optimal <math>\beta \,\!</math> for a balanced design also can satisfy all the constraints for an unbalanced design. Therefore, the approximated power is always higher than the unknown true power when the design is unbalanced.		In a DOE folio, instead of calculating the exact solution, we use the optimal <math>\beta \,\!</math> for a balanced design to calculate the approximated power for an unbalanced design. It can be seen that the optimal <math>\beta \,\!</math> for a balanced design also can satisfy all the constraints for an unbalanced design. Therefore, the approximated power is always higher than the unknown true power when the design is unbalanced.


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	In practical cases, the above method can be applied to quickly check the power of a design. If the calculated power cannot meet the required value, the true power definitely will not meet the requirement, since the calculated power using this procedure is always equal to (for balanced designs) or larger than (for unbalanced designs) the true value.		In practical cases, the above method can be applied to quickly check the power of a design. If the calculated power cannot meet the required value, the true power definitely will not meet the requirement, since the calculated power using this procedure is always equal to (for balanced designs) or larger than (for unbalanced designs) the true value.

	The result in DOE++ for this example is given as:		The result in the DOE folio for this example is given as:

Richard House: /* Power Calculation for Comparing Multiple Means: Balanced Designs */

2017-08-10T17:10:21Z

Power Calculation for Comparing Multiple Means: Balanced Designs

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	Since the power is greater than 80%, the sample size of 15 is sufficient. Otherwise, the sample size should be increased in order to achieve the desired power requirement. The settings and results in DOE++ are given below.		Since the power is greater than 80%, the sample size of 15 is sufficient. Otherwise, the sample size should be increased in order to achieve the desired power requirement. The settings and results in the DOE folio are given below.

Richard House: /* Power Calculation for Comparing Multiple Means: Balanced Designs */

2017-08-10T17:09:16Z

Power Calculation for Comparing Multiple Means: Balanced Designs

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	paired comparisons. In this case, what is the power of detecting a given difference among these comparisons?		paired comparisons. In this case, what is the power of detecting a given difference among these comparisons?

	''In DOE++, power for a multiple level factor is defined as follows: given the largest difference among all the level means is'' <math>\Delta \,\!</math>'', power is the smallest probability of detecting this difference at a given significance level.''		''In a DOE folio, the power for a multiple level factor is defined as follows: given the largest difference among all the level means is'' <math>\Delta \,\!</math>'', power is the smallest probability of detecting this difference at a given significance level.''

	For example, if a factor has 4 levels and		For example, if a factor has 4 levels and

Richard House: /* Power Calculation for Comparing Two Means */

2017-08-10T17:07:55Z

Power Calculation for Comparing Two Means

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	. So the power is 1-0.08609 =0.91391.		. So the power is 1-0.08609 =0.91391.

	In DOE++, the '''''Effect''''' for the power calculation is entered as the multiple of the standard deviation of error. So effect of 30 is <math>30/S=30/12.4499=\text{2}.\text{4}0\text{9658}\,\!</math> standard deviation. This information is illustrated below.		In a DOE folio, the '''''Effect''''' for the power calculation is entered as the multiple of the standard deviation of error. So effect of 30 is <math>30/S=30/12.4499=\text{2}.\text{4}0\text{9658}\,\!</math> standard deviation. This information is illustrated below.

	[[Image: DesignEvluation_2.png\|center\|317px]]		[[Image: DesignEvluation_2.png\|center\|317px]]

Richard House: /* Optimality */

2017-08-10T17:06:58Z

Optimality

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	The determinant of <math>X'X\,\!</math> and the trace of <math>{{\left( X'X \right)}^{-1}}\,\!</math>		The determinant of <math>X'X\,\!</math> and the trace of <math>{{\left( X'X \right)}^{-1}}\,\!</math>
	are given in the design evaluation in ~~Version 9 of~~ DOE++. ''V''-optimality is not yet included.		are given in the design evaluation in the DOE folio. ''V''-optimality is not yet included.

	=Power Study =		=Power Study =

Richard House: /* Power Calculation for Comparing Two Means */

2016-03-11T21:00:52Z

Power Calculation for Comparing Two Means

← Older revision		Revision as of 21:00, 11 March 2016
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	. So the power is 1-0.08609 =0.91391.		. So the power is 1-0.08609 =0.91391.

	In DOE++, the '''''Effect''''' for the power calculation is entered as the multiple of the standard deviation of error. So effect of 30 is <math>30/S=30/12.4499=\text{2}.\text{4}0\text{9658}</math> standard deviation. This information is illustrated below.		In DOE++, the '''''Effect''''' for the power calculation is entered as the multiple of the standard deviation of error. So effect of 30 is <math>30/S=30/12.4499=\text{2}.\text{4}0\text{9658}\,\!</math> standard deviation. This information is illustrated below.

	[[Image: DesignEvluation_2.png\|center\|317px]]		[[Image: DesignEvluation_2.png\|center\|317px]]