Non-Parametric Life Data Analysis - Revision history

Nicolette Young: /* Confidence Bounds Example */

2015-12-23T17:15:47Z

Confidence Bounds Example

← Older revision		Revision as of 17:15, 23 December 2015
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	Once again, this type of problem is most readily solved by constructing a table similar to the following:		Once again, this type of problem is most readily solved by constructing a table similar to the following:

	[[Image:Ldat22.1.png\|center\|~~750px~~]]		[[Image:Ldat22.1.png\|center\|800px]]

	The following plot illustrates these results graphically:		The following plot illustrates these results graphically:

	[[Image:WB.17 nonparametric reliability plot.png\|center\|550px]] <br>		[[Image:WB.17 nonparametric reliability plot.png\|center\|550px]] <br>

Nicolette Young: /* Confidence Bounds Example */

2015-12-23T17:15:29Z

Confidence Bounds Example

← Older revision		Revision as of 17:15, 23 December 2015
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	Once again, this type of problem is most readily solved by constructing a table similar to the following:		Once again, this type of problem is most readily solved by constructing a table similar to the following:

	[[Image:Ldat22.1.png\|center\|~~700px~~]]		[[Image:Ldat22.1.png\|center\|750px]]

	The following plot illustrates these results graphically:		The following plot illustrates these results graphically:

	[[Image:WB.17 nonparametric reliability plot.png\|center\|~~500px~~]] <br>		[[Image:WB.17 nonparametric reliability plot.png\|center\|550px]] <br>

Harry Guo at 21:15, 12 December 2013

2013-12-12T21:15:16Z

← Older revision		Revision as of 21:15, 12 December 2013
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	Note that this is similar to the ~~Bernard~~'s approximation of the median ranks, as discussed in the [[Parameter Estimation]] chapter. The following non-parametric analysis methods are essentially variations of this concept.		Note that this is similar to the Benard's approximation of the median ranks, as discussed in the [[Parameter Estimation]] chapter. The following non-parametric analysis methods are essentially variations of this concept.

	== Kaplan-Meier Estimator ==		== Kaplan-Meier Estimator ==

Richard House: /* Non-Parametric Confidence Bounds */

2012-09-26T18:09:23Z

Non-Parametric Confidence Bounds

← Older revision		Revision as of 18:09, 26 September 2012
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	The variable <~~span class="texhtml"~~>''n~~''<sub>''~~i~~''</sub>~~</~~span~~> is defined by:		The variable <math>{{n}_{i}}\,\!</math> is defined by:

	::<math>{{n}_{i}}=n-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{s}_{j}}-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{r}_{j,}}\text{ }i=1,...,m\,\!</math>		::<math>{{n}_{i}}=n-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{s}_{j}}-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{r}_{j,}}\text{ }i=1,...,m\,\!</math>
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	::<math>w={{e}^{{{z}_{\alpha }}\cdot \tfrac{{{\widehat{se}}_{\widehat{R}}}}{\left[ \widehat{R}\cdot (1-\widehat{R}) \right]}}}\,\!</math>		::<math>w={{e}^{{{z}_{\alpha }}\cdot \tfrac{{{\widehat{se}}_{\widehat{R}}}}{\left[ \widehat{R}\cdot (1-\widehat{R}) \right]}}}\,\!</math>

	and <~~span class="texhtml"~~>α</~~span~~> is the desired confidence level for the 1-sided confidence bounds.		and <math>\alpha\,\!</math> is the desired confidence level for the 1-sided confidence bounds.

	===Confidence Bounds Example===		===Confidence Bounds Example===

Richard House: /* Actuarial-Simple Method */

2012-09-26T18:07:47Z

Actuarial-Simple Method

← Older revision		Revision as of 18:07, 26 September 2012
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	== Actuarial-Simple Method ==		== Actuarial-Simple Method ==
	The actuarial-simple method is an easy-to-use form of non-parametric data analysis that can be used for multiple censored data that are arranged in intervals. This method is based on calculating the number of failures in a time interval, <~~span class="texhtml"~~>''r~~''<sub>''~~j~~''</sub>~~,</~~span~~> versus the number of operating units in that time period, <~~span class="texhtml"~~>''n~~''<sub>''~~j~~''</sub>~~</~~span~~> . The equation for the reliability estimator for the standard actuarial method is given by:		The actuarial-simple method is an easy-to-use form of non-parametric data analysis that can be used for multiple censored data that are arranged in intervals. This method is based on calculating the number of failures in a time interval, <math>{{r}_{j}}\,\!</math> versus the number of operating units in that time period, <math>{{n}_{j}}\,\!</math>. The equation for the reliability estimator for the standard actuarial method is given by:

	::<math>\widehat{R}({{t}_{i}})=\underset{j=1}{\overset{i}{\mathop \prod }}\,\left( 1-\frac{{{r}_{j}}}{{{n}_{j}}} \right),\text{ }i=1,...,m\,\!</math>		::<math>\widehat{R}({{t}_{i}})=\underset{j=1}{\overset{i}{\mathop \prod }}\,\left( 1-\frac{{{r}_{j}}}{{{n}_{j}}} \right),\text{ }i=1,...,m\,\!</math>
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	The variable <~~span class="texhtml"~~>''n~~''<sub>''~~i~~''</sub>~~</~~span~~> is defined by:		The variable <math>{{n}_{i}}\,\!</math> is defined by:

	::<math>{{n}_{i}}=n-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{s}_{j}}-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{r}_{j,}}\text{ }i=1,...,m\,\!</math>		::<math>{{n}_{i}}=n-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{s}_{j}}-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{r}_{j,}}\text{ }i=1,...,m\,\!</math>

Richard House: /* Kaplan-Meier Estimator */

2012-09-26T18:06:37Z

Kaplan-Meier Estimator

← Older revision		Revision as of 18:06, 26 September 2012
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	The variable <~~span class="texhtml"~~>''n~~''<sub>''~~i~~''</sub>~~</~~span~~> is defined by:		The variable <math>{{n}_{i}}\,\!</math> is defined by:

	::<math>{{n}_{i}}=n-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{s}_{j}}-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{r}_{j,}}\text{ }i=1,...,m\,\!</math>		::<math>{{n}_{i}}=n-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{s}_{j}}-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{r}_{j,}}\text{ }i=1,...,m\,\!</math>
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	Note that the reliability estimate is only calculated for times at which one or more failures occurred. For the sake of calculating the value of <~~span class="texhtml"~~>''n~~''<sub>''~~j~~''</sub>~~</~~span~~> at time values that have failures and suspensions, it is assumed that the suspensions occur slightly after the failures, so that the suspended units are considered to be operating and included in the count of <~~span class="texhtml"~~>''n~~''<sub>''~~j~~''</sub>~~</~~span~~> .		Note that the reliability estimate is only calculated for times at which one or more failures occurred. For the sake of calculating the value of <math>{{n}_{j}}\,\!</math> at time values that have failures and suspensions, it is assumed that the suspensions occur slightly after the failures, so that the suspended units are considered to be operating and included in the count of <math>{{n}_{j}}\,\!</math>.

	===Kaplan-Meier Example===		===Kaplan-Meier Example===

Richard House at 18:04, 26 September 2012

2012-09-26T18:04:25Z

Show changes

Lisa Hacker: /* Non-Parametric Confidence Bounds */

2012-09-07T22:51:51Z

Non-Parametric Confidence Bounds

← Older revision		Revision as of 22:51, 7 September 2012
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	Confidence bounds for non-parametric reliability estimates can be calculated using a method similar to that of parametric confidence bounds. The difficulty in dealing with nonparametric data lies in the estimation of the variance. To estimate the variance for non-parametric data, Weibull++ uses Greenwood's formula [[Appendix:_Life_Data_Analysis_References\|[27]]]:		Confidence bounds for non-parametric reliability estimates can be calculated using a method similar to that of parametric confidence bounds. The difficulty in dealing with nonparametric data lies in the estimation of the variance. To estimate the variance for non-parametric data, Weibull++ uses Greenwood's formula [[Appendix:_Life_Data_Analysis_References\|[27]]]:

	::<math>\widehat{Var}(\~~widehat~~{R}({{t}_{i}}))={{\left[ \~~widehat~~{R}({{t}_{i}}) \right]}^{2}}\cdot \underset{j=1}{\overset{i}{\mathop \sum }}\,\frac{\tfrac{{{r}_{j}}}{{{n}_{j}}}}{{{n}_{j}}\cdot \left( 1-\tfrac{{{r}_{j}}}{{{n}_{j}}} \right)}</math>		::<math>\widehat{Var}(\hat{R}({{t}_{i}}))={{\left[ \hat{R}({{t}_{i}}) \right]}^{2}}\cdot \underset{j=1}{\overset{i}{\mathop \sum }}\,\frac{\tfrac{{{r}_{j}}}{{{n}_{j}}}}{{{n}_{j}}\cdot \left( 1-\tfrac{{{r}_{j}}}{{{n}_{j}}} \right)}</math>

	where:		where:

Richard House: /* Non-Parametric Confidence Bounds */

2012-09-05T17:24:42Z

Non-Parametric Confidence Bounds

← Older revision		Revision as of 17:24, 5 September 2012
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	== Non-Parametric Confidence Bounds ==		== Non-Parametric Confidence Bounds ==
	Confidence bounds for non-parametric reliability estimates can be calculated using a method similar to that of parametric confidence bounds. The difficulty in dealing with nonparametric data lies in the estimation of the variance. To estimate the variance for non-parametric data, Weibull++ uses Greenwood's formula [[Appendix: ~~Weibull References~~\|[27]]]:		Confidence bounds for non-parametric reliability estimates can be calculated using a method similar to that of parametric confidence bounds. The difficulty in dealing with nonparametric data lies in the estimation of the variance. To estimate the variance for non-parametric data, Weibull++ uses Greenwood's formula [[Appendix:_Life_Data_Analysis_References\|[27]]]:

	::<math>\widehat{Var}(\widehat{R}({{t}_{i}}))={{\left[ \widehat{R}({{t}_{i}}) \right]}^{2}}\cdot \underset{j=1}{\overset{i}{\mathop \sum }}\,\frac{\tfrac{{{r}_{j}}}{{{n}_{j}}}}{{{n}_{j}}\cdot \left( 1-\tfrac{{{r}_{j}}}{{{n}_{j}}} \right)}</math>		::<math>\widehat{Var}(\widehat{R}({{t}_{i}}))={{\left[ \widehat{R}({{t}_{i}}) \right]}^{2}}\cdot \underset{j=1}{\overset{i}{\mathop \sum }}\,\frac{\tfrac{{{r}_{j}}}{{{n}_{j}}}}{{{n}_{j}}\cdot \left( 1-\tfrac{{{r}_{j}}}{{{n}_{j}}} \right)}</math>

Melinda Caroline: /* Confidence Bounds Example */

2012-08-16T02:10:23Z

Confidence Bounds Example

← Older revision		Revision as of 02:10, 16 August 2012
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	===Confidence Bounds Example===		===Confidence Bounds Example===

	Determine the 1-sided confidence bounds for the reliability estimates in Simple~~-Actuarial~~ example, with a 95% confidence level.		Determine the 1-sided confidence bounds for the reliability estimates in the Actuarial-Simple example, with a 95% confidence level.

	'''Solution'''		'''Solution'''