Grouped Data Parameter Estimation - Revision history

Lisa Hacker at 21:45, 18 September 2023

2023-09-18T21:45:56Z

← Older revision		Revision as of 21:45, 18 September 2023
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	<noinclude>{{Banner Weibull Articles}}		<noinclude>{{Banner Weibull Articles}}
	''This article appears in the [~~[Appendix~~:~~_Special_Analysis_Methods#Grouped_Data_Analysis\|~~Life ~~Data Analysis Reference book]~~].''		''This article appears in the [https://help.reliasoft.com/reference/life_data_analysis Life data analysis reference].''

	</noinclude>		</noinclude>

Harry Guo: /* When Using Maximum Likelihood */

2014-01-10T21:19:58Z

When Using Maximum Likelihood

← Older revision		Revision as of 21:19, 10 January 2014
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	=== When Using Maximum Likelihood===		=== When Using Maximum Likelihood===
	When using maximum likelihood methods, each individual time is explicitly used in the calculation of the parameters~~; thus~~, there is no difference in the entry of a group of 10 units failing at 100 hours and 10 individual entries of 100 hours. This is inherent in the standard MLE method. In other words, no matter how the data were entered (i.e., as grouped or non-grouped) the results will be identical. When using maximum likelihood, we highly recommend entering redundant data in groups, as this significantly speeds up the calculations.		When using maximum likelihood methods, each individual time is explicitly used in the calculation of the parameters. Theoretically, there is no difference in the entry of a group of 10 units failing at 100 hours and 10 individual entries of 100 hours. This is inherent in the standard MLE method. In other words, no matter how the data were entered (i.e., as grouped or non-grouped) the results will be identical. However, due to the precision issues during the computation, the grouped and ungrouped data may give slightly different results. When using maximum likelihood, we highly recommend entering redundant data in groups, as this significantly speeds up the calculations.

Richard House at 18:21, 26 September 2012

2012-09-26T18:21:42Z

← Older revision		Revision as of 18:21, 26 September 2012
Line 9:		Line 9:

	=== The Mathematics===		=== The Mathematics===
	Median ranks are used to obtain an estimate of the unreliability, <math>Q({{T}_{j}}),\,\!</math> for each failure at a <math>50%</math> confidence level. In the case of grouped data, the ranks are estimated for each group of failures, instead of each failure.		Median ranks are used to obtain an estimate of the unreliability, <math>Q({{T}_{j}}),\,\!</math> for each failure at a <math>50%\,\!</math> confidence level. In the case of grouped data, the ranks are estimated for each group of failures, instead of each failure.
	For example, consider a group of 10 failures at 100 hours, 10 at 200 hours and 10 at 300 hours. Weibull++ estimates the median ranks (<math>Z\,\!</math> values) by solving the cumulative binomial equation with the appropriate values for order number and total number of test units.		For example, consider a group of 10 failures at 100 hours, 10 at 200 hours and 10 at 300 hours. Weibull++ estimates the median ranks (<math>Z\,\!</math> values) by solving the cumulative binomial equation with the appropriate values for order number and total number of test units.
	For 10 failures at 100 hours, the median rank, <math>Z,\,\!</math> is estimated by using:		For 10 failures at 100 hours, the median rank, <math>Z,\,\!</math> is estimated by using:
Line 16:		Line 16:
	N \\		N \\
	k \\		k \\
	\end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}}</math>		\end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}}\,\!</math>

	with:		with:
Line 22:		Line 22:
	::<math>\begin{align}		::<math>\begin{align}
	N=30,\text{ }J=10		N=30,\text{ }J=10
	\end{align}</math>		\end{align}\,\!</math>

	One <math>Z\,\!</math> is obtained for the group, to represent the probability of 10 failures occurring out of 30.		One <math>Z\,\!</math> is obtained for the group, to represent the probability of 10 failures occurring out of 30.
Line 31:		Line 31:
	N \\		N \\
	k \\		k \\
	\end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}}</math>		\end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}}\,\!</math>

	where:		where:
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	::<math>\begin{align}		::<math>\begin{align}
	N=30,\text{ }J=20		N=30,\text{ }J=20
	\end{align}</math>		\end{align}\,\!</math>

	This represents the probability of 20 failures out of 30.		This represents the probability of 20 failures out of 30.
Line 46:		Line 46:
	N \\		N \\
	k \\		k \\
	\end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}}</math>		\end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}}\,\!</math>

	where:		where:
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	::<math>\begin{align}		::<math>\begin{align}
	N=30,\text{ }J=30		N=30,\text{ }J=30
	\end{align}</math>		\end{align}\,\!</math>

	This represents the probability of 30 failures out of 30.		This represents the probability of 30 failures out of 30.

Richard House: /* When Using Rank Regression (Least Squares) */

2012-09-05T16:24:29Z

When Using Rank Regression (Least Squares)

← Older revision		Revision as of 16:24, 5 September 2012
Line 6:		Line 6:

	=== When Using Rank Regression (Least Squares)===		=== When Using Rank Regression (Least Squares)===
	When using grouped data, Weibull++ plots the data point corresponding to the highest rank position in each group. For example, given 3 groups of 10 units, each failing at 100, 200 and 300 hours respectively, the three plotted points will be the end point of each group, or the 10th rank position out of 30, the 20th rank position out of 30 and the 30th rank position out of 30. This procedure is identical to standard procedures for using grouped data [[Appendix: ~~Weibull References~~\|[19]]]. In cases where the grouped data are interval censored, it is assumed that the failures occurred at some time in the interval between the previous and current time to failure. In our example, this would be the same as saying that 10 units have failed in the interval between zero and 100 hours, another 10 units failed in the interval between 100 and 200 hours, and 10 more units failed in the interval from 200 to 300 hours. The rank regression analysis automatically takes this into account. If this assumption of interval failure is incorrect (i.e., 10 units failed at exactly 100 hours, 10 failed at exactly 200 hours and 10 failed at exactly 300 hours), then it is recommended that you enter the data as non-grouped when using rank regression, or select the '''Ungroup on Regression''' check box on the Analysis page of the folio's control panel.		When using grouped data, Weibull++ plots the data point corresponding to the highest rank position in each group. For example, given 3 groups of 10 units, each failing at 100, 200 and 300 hours respectively, the three plotted points will be the end point of each group, or the 10th rank position out of 30, the 20th rank position out of 30 and the 30th rank position out of 30. This procedure is identical to standard procedures for using grouped data, as discussed in Kececioglu [[Appendix:_Life_Data_Analysis_References\|[19]]]. In cases where the grouped data are interval censored, it is assumed that the failures occurred at some time in the interval between the previous and current time to failure. In our example, this would be the same as saying that 10 units have failed in the interval between zero and 100 hours, another 10 units failed in the interval between 100 and 200 hours, and 10 more units failed in the interval from 200 to 300 hours. The rank regression analysis automatically takes this into account. If this assumption of interval failure is incorrect (i.e., 10 units failed at exactly 100 hours, 10 failed at exactly 200 hours and 10 failed at exactly 300 hours), then it is recommended that you enter the data as non-grouped when using rank regression, or select the '''Ungroup on Regression''' check box on the Analysis page of the folio's control panel.

	=== The Mathematics===		=== The Mathematics===

Richard House: /* The Mathematics */

2012-08-20T05:08:15Z

The Mathematics

← Older revision		Revision as of 05:08, 20 August 2012
Line 9:		Line 9:

	=== The Mathematics===		=== The Mathematics===
	Median ranks are used to obtain an estimate of the unreliability, <math>Q({{T}_{j}}),</math> for each failure at a <math>50%</math> confidence level. In the case of grouped data, the ranks are estimated for each group of failures, instead of each failure.		Median ranks are used to obtain an estimate of the unreliability, <math>Q({{T}_{j}}),\,\!</math> for each failure at a <math>50%</math> confidence level. In the case of grouped data, the ranks are estimated for each group of failures, instead of each failure.
	For example, consider a group of 10 failures at 100 hours, 10 at 200 hours and 10 at 300 hours. Weibull++ estimates the median ranks (<math>Z</math> values) by solving the cumulative binomial equation with the appropriate values for order number and total number of test units.		For example, consider a group of 10 failures at 100 hours, 10 at 200 hours and 10 at 300 hours. Weibull++ estimates the median ranks (<math>Z\,\!</math> values) by solving the cumulative binomial equation with the appropriate values for order number and total number of test units.
	For 10 failures at 100 hours, the median rank, <math>Z,</math> is estimated by using:		For 10 failures at 100 hours, the median rank, <math>Z,\,\!</math> is estimated by using:

	::<math>0.50=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix}		::<math>0.50=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix}
Line 24:		Line 24:
	\end{align}</math>		\end{align}</math>

	One <math>Z</math> is obtained for the group, to represent the probability of 10 failures occurring out of 30.		One <math>Z\,\!</math> is obtained for the group, to represent the probability of 10 failures occurring out of 30.

	For 10 failures at 200 hours, <math>Z</math> is estimated by using:		For 10 failures at 200 hours, <math>Z\,\!</math> is estimated by using:

	::<math>0.50=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix}		::<math>0.50=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix}
Line 41:		Line 41:
	This represents the probability of 20 failures out of 30.		This represents the probability of 20 failures out of 30.

	For 10 failures at 300 hours, <math>Z</math> is estimated by using:		For 10 failures at 300 hours, <math>Z\,\!</math> is estimated by using:

	::<math>0.50=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix}		::<math>0.50=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix}

Melinda Caroline: /* When Using Rank Regression (Least Squares) */

2012-08-16T06:28:03Z

When Using Rank Regression (Least Squares)

← Older revision		Revision as of 06:28, 16 August 2012
Line 6:		Line 6:

	=== When Using Rank Regression (Least Squares)===		=== When Using Rank Regression (Least Squares)===
	When using grouped data, Weibull++ plots the data point corresponding to the highest rank position in each group. For example, given 3 groups of 10 units, each failing at 100, 200, and 300 hours respectively, the three plotted points will be the end point of each group, or the 10th rank position out of 30, the 20th rank position out of 30 and the 30th rank position out of 30. This procedure is identical to standard procedures for using grouped data [[Appendix: Weibull References\|[19]]]. In cases where the grouped data are interval censored, it is assumed that the failures occurred at some time in the interval between the previous and current time to failure. In our example, this would be the same as saying that 10 units have failed in the interval between zero and 100 hours, another 10 units failed in the interval between 100 and 200 hours, and 10 more units failed in the interval from 200 to 300 hours. The rank regression analysis automatically takes this into account. If this assumption of interval failure is incorrect (i.e., 10 units failed at exactly 100 hours, 10 failed at exactly 200 hours and 10 failed at exactly 300 hours), then it is recommended that you enter the data as non-grouped when using rank regression, or select the '''Ungroup on Regression''' check box on the Analysis page of the folio's control panel.		When using grouped data, Weibull++ plots the data point corresponding to the highest rank position in each group. For example, given 3 groups of 10 units, each failing at 100, 200 and 300 hours respectively, the three plotted points will be the end point of each group, or the 10th rank position out of 30, the 20th rank position out of 30 and the 30th rank position out of 30. This procedure is identical to standard procedures for using grouped data [[Appendix: Weibull References\|[19]]]. In cases where the grouped data are interval censored, it is assumed that the failures occurred at some time in the interval between the previous and current time to failure. In our example, this would be the same as saying that 10 units have failed in the interval between zero and 100 hours, another 10 units failed in the interval between 100 and 200 hours, and 10 more units failed in the interval from 200 to 300 hours. The rank regression analysis automatically takes this into account. If this assumption of interval failure is incorrect (i.e., 10 units failed at exactly 100 hours, 10 failed at exactly 200 hours and 10 failed at exactly 300 hours), then it is recommended that you enter the data as non-grouped when using rank regression, or select the '''Ungroup on Regression''' check box on the Analysis page of the folio's control panel.

	=== The Mathematics===		=== The Mathematics===

Kate Racaza: moved updated content in Template back to this page, added W++ banner and transcluded content to Life Data Analysis Reference book

2012-08-10T03:55:41Z

moved updated content in Template back to this page, added W++ banner and transcluded content to Life Data Analysis Reference book

← Older revision		Revision as of 03:55, 10 August 2012
Line 1:		Line 1:
	#~~REDIRECT~~ [[Appendix:~~_Special_Analysis_Methods~~]]		<noinclude>{{Banner Weibull Articles}}
			''This article appears in the [[Appendix:_Special_Analysis_Methods#Grouped_Data_Analysis\|Life Data Analysis Reference book]].''

			</noinclude>
			The grouped data type in Weibull++ is used for tests where there are groups of units having the same time-to-failure, or units are grouped together in intervals, or there are groups of units suspended at the same time. However, you must be cautious in using the different parameter estimation methods because different methods treat grouped data in different ways. ReliaSoft designed Weibull++ to treat grouped data in different ways to maximize the options available to you.

			=== When Using Rank Regression (Least Squares)===
			When using grouped data, Weibull++ plots the data point corresponding to the highest rank position in each group. For example, given 3 groups of 10 units, each failing at 100, 200, and 300 hours respectively, the three plotted points will be the end point of each group, or the 10th rank position out of 30, the 20th rank position out of 30 and the 30th rank position out of 30. This procedure is identical to standard procedures for using grouped data [[Appendix: Weibull References\|[19]]]. In cases where the grouped data are interval censored, it is assumed that the failures occurred at some time in the interval between the previous and current time to failure. In our example, this would be the same as saying that 10 units have failed in the interval between zero and 100 hours, another 10 units failed in the interval between 100 and 200 hours, and 10 more units failed in the interval from 200 to 300 hours. The rank regression analysis automatically takes this into account. If this assumption of interval failure is incorrect (i.e., 10 units failed at exactly 100 hours, 10 failed at exactly 200 hours and 10 failed at exactly 300 hours), then it is recommended that you enter the data as non-grouped when using rank regression, or select the '''Ungroup on Regression''' check box on the Analysis page of the folio's control panel.

			=== The Mathematics===
			Median ranks are used to obtain an estimate of the unreliability, <math>Q({{T}_{j}}),</math> for each failure at a <math>50%</math> confidence level. In the case of grouped data, the ranks are estimated for each group of failures, instead of each failure.
			For example, consider a group of 10 failures at 100 hours, 10 at 200 hours and 10 at 300 hours. Weibull++ estimates the median ranks (<math>Z</math> values) by solving the cumulative binomial equation with the appropriate values for order number and total number of test units.
			For 10 failures at 100 hours, the median rank, <math>Z,</math> is estimated by using:

			::<math>0.50=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix}
			N \\
			k \\
			\end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}}</math>

			with:

			::<math>\begin{align}
			N=30,\text{ }J=10
			\end{align}</math>

			One <math>Z</math> is obtained for the group, to represent the probability of 10 failures occurring out of 30.

			For 10 failures at 200 hours, <math>Z</math> is estimated by using:

			::<math>0.50=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix}
			N \\
			k \\
			\end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}}</math>

			where:

			::<math>\begin{align}
			N=30,\text{ }J=20
			\end{align}</math>

			This represents the probability of 20 failures out of 30.

			For 10 failures at 300 hours, <math>Z</math> is estimated by using:

			::<math>0.50=\underset{k=j}{\overset{N}{\mathop \sum }}\,\left( \begin{matrix}
			N \\
			k \\
			\end{matrix} \right){{Z}^{k}}{{\left( 1-Z \right)}^{N-k}}</math>

			where:

			::<math>\begin{align}
			N=30,\text{ }J=30
			\end{align}</math>

			This represents the probability of 30 failures out of 30.

			=== When Using Maximum Likelihood===
			When using maximum likelihood methods, each individual time is explicitly used in the calculation of the parameters; thus, there is no difference in the entry of a group of 10 units failing at 100 hours and 10 individual entries of 100 hours. This is inherent in the standard MLE method. In other words, no matter how the data were entered (i.e., as grouped or non-grouped) the results will be identical. When using maximum likelihood, we highly recommend entering redundant data in groups, as this significantly speeds up the calculations.

Kate Racaza: Redirected page to Appendix: Special Analysis Methods

2012-07-02T09:11:43Z

Redirected page to Appendix: Special Analysis Methods

← Older revision		Revision as of 09:11, 2 July 2012
Line 1:		Line 1:
	#REDIRECT [[~~Template~~:~~Grouped Data Parameter Estimation~~]]		#REDIRECT [[Appendix:_Special_Analysis_Methods]]

Harry Guo: moved Grouped Data Parameter Estimation to Template:Grouped Data Parameter Estimation

2012-02-07T00:23:21Z

moved Grouped Data Parameter Estimation to Template:Grouped Data Parameter Estimation

New page

#REDIRECT [[Template:Grouped Data Parameter Estimation]]