Template:Kaplan-meier estimator - Revision history

Kate Racaza: Redirected page to Non-Parametric Life Data Analysis

2012-06-29T07:35:00Z

Redirected page to Non-Parametric Life Data Analysis

← Older revision		Revision as of 07:35, 29 June 2012
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	~~== Kaplan~~-~~Meier Estimator ==~~		#REDIRECT [[Non-Parametric Life Data Analysis]]

	The Kaplan-Meier estimator, also known as the product limit estimator, can be used to calculate values for non-parametric reliability for data sets with multiple failures and suspensions. The equation of the estimator is given by:

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

	~~where:~~

	~~::<math>\begin{align}~~
	~~& m= \text{the total number of data points} \\~~
	~~& n= \text{the total number of units}~~
	~~\end{align}</math>~~

	~~The variable <span class="texhtml">''n''<sub>''i''</sub></span> 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>~~

	~~where:~~

	~~::<math>\begin{align}~~
	~~& {{r}_{j}}= \text{the number of failures in the }{{j}^{th}}\text{ data group} \\~~
	~~& {{s}_{j}}= \text{the number of suspensions in the }{{j}^{th}}\text{ data group}~~
	~~\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> .

	~~<br>'''Example 1:''' {{Example: Kaplan-Meier Example}}~~

Kate Racaza at 22:08, 9 March 2012

2012-03-09T22:08:34Z

← Older revision		Revision as of 22:08, 9 March 2012
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	==Kaplan-Meier Estimator==		== Kaplan-Meier Estimator ==

	The Kaplan-Meier estimator, also known as the product limit estimator, can be used to calculate values for ~~nonparametric~~ reliability for data sets with multiple failures and suspensions. The equation of the estimator is given by:		The Kaplan-Meier estimator, also known as the product limit estimator, can be used to calculate values for non-parametric reliability for data sets with multiple failures and suspensions. The equation of the estimator is given by:

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

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

	where:		where:

	::<math>\begin{align}		::<math>\begin{align}
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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 <~~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~~> .		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> .


	'''Example 1:'''		<br>'''Example 1:''' {{Example: Kaplan-Meier Example}}
	{{Example: Kaplan-Meier Example}}

Harry Guo at 20:44, 20 February 2012

2012-02-20T20:44:02Z

← Older revision		Revision as of 20:44, 20 February 2012
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	::<math>\begin{align}		::<math>\begin{align}
	& m= & \text{the total number of data points} \\		& m= \text{the total number of data points} \\
	& n= & \text{the total number of units}		& n= \text{the total number of units}
	\end{align}</math>		\end{align}</math>

Harry Guo: /* Kaplan-Meier Estimator */

2012-02-20T20:43:50Z

Kaplan-Meier Estimator

← Older revision		Revision as of 20:43, 20 February 2012
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	::<math>\begin{align}		::<math>\begin{align}
	& {{r}_{j}}= & \text{the number of failures in the }{{j}^{th}}\text{ data group} \\		& {{r}_{j}}= \text{the number of failures in the }{{j}^{th}}\text{ data group} \\
	& {{s}_{j}}= & \text{the number of suspensions in the }{{j}^{th}}\text{ data group}		& {{s}_{j}}= \text{the number of suspensions in the }{{j}^{th}}\text{ data group}
	\end{align}</math>		\end{align}</math>

Harry Guo at 19:10, 20 February 2012

2012-02-20T19:10:13Z

← Older revision		Revision as of 19:10, 20 February 2012
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	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> .		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> .


	'''Example 1:'''		'''Example 1:'''
	{{Example: Kaplan-Meier Example}}		{{Example: Kaplan-Meier Example}}

Harry Guo: /* Kaplan-Meier Estimator */

2012-02-20T19:10:03Z

Kaplan-Meier Estimator

← Older revision		Revision as of 19:10, 20 February 2012
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	::<math>\widehat{R}({{t}_{i}})=\underset{j=1}{\overset{i}{\mathop \prod }}\,\frac{{{n}_{j}}-{{r}_{j}}}{{{n}_{j}}},\text{ }i=1,...,m</math>		::<math>\widehat{R}({{t}_{i}})=\underset{j=1}{\overset{i}{\mathop \prod }}\,\frac{{{n}_{j}}-{{r}_{j}}}{{{n}_{j}}},\text{ }i=1,...,m</math>

	:where:		where:

	::<math>\begin{align}		::<math>\begin{align}
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	::<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>

	:where:		where:

	::<math>\begin{align}		::<math>\begin{align}

Harry Guo at 18:56, 20 February 2012

2012-02-20T18:56:47Z

@@ Line 25: / Line 25: @@
 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> .
-====Example 9====
+'''Example 1:'''
-A group of 20 units are put on a life test with the following results.
+{{Example: Kaplan-Meier Example}}

Harry Guo at 18:48, 20 February 2012

2012-02-20T18:48:13Z

← Older revision		Revision as of 18:48, 20 February 2012
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	===Kaplan-Meier Estimator===		==Kaplan-Meier Estimator==

	The Kaplan-Meier estimator, also known as the product limit estimator, can be used to calculate values for nonparametric reliability for data sets with multiple failures and suspensions. The equation of the estimator is given by:		The Kaplan-Meier estimator, also known as the product limit estimator, can be used to calculate values for nonparametric reliability for data sets with multiple failures and suspensions. The equation of the estimator is given by:

Nicolette Young: Created page with '===Kaplan-Meier Estimator=== The Kaplan-Meier estimator, also known as the product limit estimator, can be used to calculate values for nonparametric reliability for data sets w…'

2012-01-05T00:01:26Z

Created page with '===Kaplan-Meier Estimator=== The Kaplan-Meier estimator, also known as the product limit estimator, can be used to calculate values for nonparametric reliability for data sets w…'

New page

===Kaplan-Meier Estimator===

The Kaplan-Meier estimator, also known as the product limit estimator, can be used to calculate values for nonparametric reliability for data sets with multiple failures and suspensions. The equation of the estimator is given by:

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

:where:

::<math>\begin{align}
& m= & \text{the total number of data points} \\
& n= & \text{the total number of units}
\end{align}</math>

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>

:where:

::<math>\begin{align}
& {{r}_{j}}= & \text{the number of failures in the }{{j}^{th}}\text{ data group} \\
& {{s}_{j}}= & \text{the number of suspensions in the }{{j}^{th}}\text{ data group}
\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 <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> .

====Example 9====
A group of 20 units are put on a life test with the following results.

<center><math>\begin{matrix}
Number & State & State \\
in State & (F or S) & End Time \\
3 & F & 9 \\
1 & S & 9 \\
1 & F & 11 \\
1 & S & 12 \\
1 & F & 13 \\
1 & S & 13 \\
1 & S & 15 \\
1 & F & 17 \\
1 & F & 21 \\
1 & S & 22 \\
1 & S & 24 \\
1 & S & 26 \\
1 & F & 28 \\
1 & F & 30 \\
1 & S & 32 \\
2 & S & 35 \\
1 & S & 39 \\
1 & S & 41 \\
\end{matrix}</math></center>

Use the Kaplan-Meier estimator to determine the reliability estimates for each failure time.

=====Solution to Example 9=====
Using the data and Eqn. (kapmeier), the following table can be constructed:

<center><math>\begin{matrix}
State & Number of & Number of & Available & {} & {} \\
End Time & Failures, {{r}_{i}} & Suspensions, {{s}_{i}} & Units, {{n}_{i}} & \tfrac{{{n}_{i}}-{{r}_{i}}}{{{n}_{i}}} & \mathop{}_{}^{}\tfrac{{{n}_{i}}-{{r}_{i}}}{{{n}_{i}}} \\
9 & 3 & 1 & 20 & 0.850 & 0.850 \\
11 & 1 & 0 & 16 & 0.938 & 0.797 \\
12 & 0 & 1 & 15 & 1.000 & 0.797 \\
13 & 1 & 1 & 14 & 0.929 & 0.740 \\
15 & 0 & 1 & 12 & 1.000 & 0.740 \\
17 & 1 & 0 & 11 & 0.909 & 0.673 \\
21 & 1 & 0 & 10 & 0.900 & 0.605 \\
22 & 0 & 1 & 9 & 1.000 & 0.605 \\
24 & 0 & 1 & 8 & 1.000 & 0.605 \\
26 & 0 & 1 & 7 & 1.000 & 0.605 \\
28 & 1 & 0 & 6 & 0.833 & 0.505 \\
30 & 1 & 0 & 5 & 0.800 & 0.404 \\
32 & 0 & 1 & 4 & 1.000 & 0.404 \\
35 & 0 & 1 & 3 & 1.000 & 0.404 \\
39 & 0 & 1 & 2 & 1.000 & 0.404 \\
41 & 0 & 1 & 1 & 1.000 & 0.404 \\
\end{matrix}</math></center>

As can be determined from the preceding table, the reliability estimates for the failure times are:

<center><math>\begin{matrix}
Failure Time & Reliability Est. \\
9 & 85.0% \\
11 & 79.7% \\
13 & 74.0% \\
17 & 67.3% \\
21 & 60.5% \\
28 & 50.5% \\
30 & 40.4% \\
\end{matrix}</math></center>