Reliability DOE for Life Tests - Revision history

Richard House: /* Likelihood Ratio Test */

2017-08-10T16:11:28Z

Likelihood Ratio Test

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	Assuming that the desired significance is 0.1, since <math>p\ value<0.1\,\!</math>, <math>{{H}_{0}}:{{\theta }_{i}}=0\,\!</math> is rejected, it is concluded that, at a significance of 0.1, at least one of the parameters, <math>{{\beta }_{1}}\,\!</math> or <math>{{\beta }_{2}}\,\!</math>, is non-zero. Therefore, factor <math>A\,\!</math> affects the life of the product. This result is shown in the Likelihood Ratio Test table in the analysis results.		Assuming that the desired significance is 0.1, since <math>p\ value<0.1\,\!</math>, <math>{{H}_{0}}:{{\theta }_{i}}=0\,\!</math> is rejected, it is concluded that, at a significance of 0.1, at least one of the parameters, <math>{{\beta }_{1}}\,\!</math> or <math>{{\beta }_{2}}\,\!</math>, is non-zero. Therefore, factor <math>A\,\!</math> affects the life of the product. This result is shown in the Likelihood Ratio Test table in the analysis results.

	Additional results for single factor R-DOE analysis obtained from DOE ++ include information on the life characteristic and comparison of life characteristics at different levels of the factor.		Additional results for single factor R-DOE analysis obtained from the DOE folio include information on the life characteristic and comparison of life characteristics at different levels of the factor.

	===Life Characteristic Summary Results===		===Life Characteristic Summary Results===

Richard House: /* Life Comparisons Results */

2017-08-10T15:51:09Z

Life Comparisons Results

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	This is the value displayed by ~~DOE~~++. Knowing the pooled standard error the confidence interval on the difference can be calculated. The 90% confidence interval on the difference in (logarithmic) life between levels 1 and 2 of factor <math>A\,\!</math> is:		This is the value displayed by the Weibull++ DOE folio. Knowing the pooled standard error the confidence interval on the difference can be calculated. The 90% confidence interval on the difference in (logarithmic) life between levels 1 and 2 of factor <math>A\,\!</math> is:

Richard House: /* Example */

2017-08-10T15:50:48Z

Example

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	where <math>{{\beta }_{0}}\,\!</math> is the intercept, <math>{{\beta }_{1}}\,\!</math> is the effect coefficient for the first level of the factor ( <math>{{\beta }_{1}}\,\!</math> is represented as "A[1]" in ~~DOE~~++) and <math>{{\beta }_{2}}\,\!</math> is the effect coefficient for the second level of the factor ( <math>{{\beta }_{2}}\,\!</math> is represented as "A[2]" in ~~DOE~~++). Two indicator variables, <math>{{x}_{1}}\,\!</math> and <math>{{x}_{2}},\,\!</math> are the used to represent the three levels of factor <math>A\,\!</math> such that:		where <math>{{\beta }_{0}}\,\!</math> is the intercept, <math>{{\beta }_{1}}\,\!</math> is the effect coefficient for the first level of the factor ( <math>{{\beta }_{1}}\,\!</math> is represented as "A[1]" in Weibull++ DOE folios) and <math>{{\beta }_{2}}\,\!</math> is the effect coefficient for the second level of the factor ( <math>{{\beta }_{2}}\,\!</math> is represented as "A[2]" in Weibull++ DOE folios). Two indicator variables, <math>{{x}_{1}}\,\!</math> and <math>{{x}_{2}},\,\!</math> are the used to represent the three levels of factor <math>A\,\!</math> such that:

Richard House: /* Single Factor R-DOE Analyses */

2017-08-10T15:49:02Z

Single Factor R-DOE Analyses

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	=Single Factor R-DOE Analyses=		=Single Factor R-DOE Analyses=
	~~DOE~~++ also ~~allows~~ for the analysis of single factor R-DOE experiments. This analysis is similar to the analysis of single factor designed experiments mentioned in [[One Factor Designs]]. In single factor R-DOE analysis, the focus is on discovering whether change in the level of a factor affects reliability and how each of the factor levels are different from the other levels. The analysis models and calculations are similar to multi-factor R-DOE analysis.		Webibull++ DOE folios also allow for the analysis of single factor R-DOE experiments. This analysis is similar to the analysis of single factor designed experiments mentioned in [[One Factor Designs]]. In single factor R-DOE analysis, the focus is on discovering whether change in the level of a factor affects reliability and how each of the factor levels are different from the other levels. The analysis models and calculations are similar to multi-factor R-DOE analysis.

	==Example==		==Example==

Richard House: /* Using R-DOE to Determine the Best Factor Settings */

2017-08-10T15:44:47Z

Using R-DOE to Determine the Best Factor Settings

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	The short duration of the experiment and failure times were probably because the lights were tested under conditions which resulted in stress higher than normal conditions. The failure of the lights was assumed to follow the lognormal distribution.		The short duration of the experiment and failure times were probably because the lights were tested under conditions which resulted in stress higher than normal conditions. The failure of the lights was assumed to follow the lognormal distribution.

	The analysis results from ~~DOE~~++ for this experiment are shown next.		The analysis results from the Weibull++ DOE folio for this experiment are shown next.

Richard House: /* Model Diagnostics */

2017-08-10T15:43:38Z

Model Diagnostics

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	=Model Diagnostics=		=Model Diagnostics=
	Residual plots can be used to check if the model obtained, based on the MLE estimates, is a good fit to the data. ~~DOE~~++ ~~uses~~ standardized residuals for R-DOE analyses. If the data follows the lognormal distribution, then standardized residuals are calculated using the following equation:		Residual plots can be used to check if the model obtained, based on the MLE estimates, is a good fit to the data. Weibull++ DOE folios use standardized residuals for R-DOE analyses. If the data follows the lognormal distribution, then standardized residuals are calculated using the following equation:

Richard House: /* Likelihood Ratio Test */

2017-08-10T15:43:06Z

Likelihood Ratio Test

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	[[Image:doe11_3.png\|center\|650px\|Likelihood ratio test results from ~~DOE~~++ for the experiment in the [[Reliability_DOE_for_Life_Tests#Example\|example]].]]		[[Image:doe11_3.png\|center\|650px\|Likelihood ratio test results from Webibull++ for the experiment in the [[Reliability_DOE_for_Life_Tests#Example\|example]].]]

	==Fisher Matrix Bounds on Parameters==		==Fisher Matrix Bounds on Parameters==

Richard House: /* Fisher Matrix Bounds on Parameters */

2017-08-10T15:42:21Z

Fisher Matrix Bounds on Parameters

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	The previous calculation results are displayed as MLE Information in the results obtained from DOE++ as shown next.		The previous calculation results are displayed as MLE Information in the results obtained from the DOE folio as shown next.


	[[Image:doe11_4.png\|center\|650px\|MLE information from ~~DOE~~++.]]		[[Image:doe11_4.png\|center\|650px\|MLE information from Weibull++.]]

Richard House: /* Likelihood Ratio Test */

2017-08-10T15:40:30Z

Likelihood Ratio Test

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	Since <math>p\,\!</math> <math>value<0.1\,\!</math>, <math>{{H}_{0}}:{{\beta }_{2}}=0\,\!</math> is rejected and it is concluded that factor <math>B\,\!</math> affects the life of the product. The previous calculation results are displayed as the Likelihood Ratio Test Table in the results obtained from DOE++ as shown next.		Since <math>p\,\!</math> <math>value<0.1\,\!</math>, <math>{{H}_{0}}:{{\beta }_{2}}=0\,\!</math> is rejected and it is concluded that factor <math>B\,\!</math> affects the life of the product. The previous calculation results are displayed as the Likelihood Ratio Test Table in the results obtained from the DOE folio as shown next.

Richard House: /* Example */

2017-08-10T15:39:53Z

Example

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	where <math>\mu _{i}^{\prime }\,\!</math> is the mean of the natural logarithm of the times-to-failure at the <math>i\,\!</math> th treatment combination (<math>i=1,2,3,4\,\!</math>), <math>{{\beta }_{1}}\,\!</math> is the effect coefficient for factor <math>A\,\!</math> and <math>{{\beta }_{2}}\,\!</math> is the effect coefficient for factor <math>B\,\!</math>. The analysis for this case is carried out in DOE++ by excluding the interaction <math>AB\,\!</math> from the analysis.		where <math>\mu _{i}^{\prime }\,\!</math> is the mean of the natural logarithm of the times-to-failure at the <math>i\,\!</math> th treatment combination (<math>i=1,2,3,4\,\!</math>), <math>{{\beta }_{1}}\,\!</math> is the effect coefficient for factor <math>A\,\!</math> and <math>{{\beta }_{2}}\,\!</math> is the effect coefficient for factor <math>B\,\!</math>. The analysis for this case is carried out in a DOE folio by excluding the interaction <math>AB\,\!</math> from the analysis.

	The following hypotheses need to be tested in this example:		The following hypotheses need to be tested in this example: