Non-Parametric Recurrent Event Data Analysis - Revision history

Lisa Hacker at 18:53, 18 September 2023

2023-09-18T18:53:40Z

← Older revision		Revision as of 18:53, 18 September 2023
Line 1:		Line 1:
	<noinclude>{{Banner Weibull Articles}}		<noinclude>{{Banner Weibull Articles}}
	''This article appears in the [~~[Recurrent_Event_Data_Analysis#Non-Parametric_Recurrent_Event_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>

Lisa Hacker: /* Example: Mean Cumulative Function */

2023-05-23T17:50:02Z

Example: Mean Cumulative Function

← Older revision		Revision as of 17:50, 23 May 2023
Line 67:		Line 67:

	{{:Non_Parametric_RDA_MCF_Example}}		{{:Non_Parametric_RDA_MCF_Example}}

	~~<div class="noprint">~~
	~~{{:Non-Parametric_RDA_Transmission_Example}}~~
	~~</div>~~

Nicolette Young: /* The Mean Cumulative Function (MCF) */

2015-12-23T17:25:57Z

The Mean Cumulative Function (MCF)

← Older revision		Revision as of 17:25, 23 December 2015
Line 18:		Line 18:
	The non-parametric model for a population of units is described as the population of cumulative history functions (curves). It is the population of all staircase functions of every unit in the population. At age t, the units have a distribution of their cumulative number of events. That is, a fraction of the population has accumulated 0 recurrences, another fraction has accumulated 1 recurrence, another fraction has accumulated 2 recurrences, etc. This distribution differs at different ages <math>t\,\!</math>, and has a mean <math>M(t)\,\!</math> called the mean cumulative function (MCF). The <math>M(t)\,\!</math> is the point-wise average of all population cumulative history functions (see figure below).		The non-parametric model for a population of units is described as the population of cumulative history functions (curves). It is the population of all staircase functions of every unit in the population. At age t, the units have a distribution of their cumulative number of events. That is, a fraction of the population has accumulated 0 recurrences, another fraction has accumulated 1 recurrence, another fraction has accumulated 2 recurrences, etc. This distribution differs at different ages <math>t\,\!</math>, and has a mean <math>M(t)\,\!</math> called the mean cumulative function (MCF). The <math>M(t)\,\!</math> is the point-wise average of all population cumulative history functions (see figure below).

	[[Image:Lda11.2.png\|center\|~~400px~~]]		[[Image:Lda11.2.png\|center\|450px]]

	For the case of uncensored data, the mean cumulative function <math>M{{(t)}_{i}}\ \,\!</math> values at different recurrence ages <math>{{t}_{i}}\,\!</math> are estimated by calculating the average of the cumulative number of recurrences of events for each unit in the population at <math>{{t}_{i}}\,\!</math>. When the histories are censored, the following steps are applied.		For the case of uncensored data, the mean cumulative function <math>M{{(t)}_{i}}\ \,\!</math> values at different recurrence ages <math>{{t}_{i}}\,\!</math> are estimated by calculating the average of the cumulative number of recurrences of events for each unit in the population at <math>{{t}_{i}}\,\!</math>. When the histories are censored, the following steps are applied.

Kate Racaza: /* Example: Mean Cumulative Function */

2015-12-01T18:29:40Z

Example: Mean Cumulative Function

← Older revision		Revision as of 18:29, 1 December 2015
Line 68:		Line 68:
	{{:Non_Parametric_RDA_MCF_Example}}		{{:Non_Parametric_RDA_MCF_Example}}

			<div class="noprint">
	{{:Non-Parametric_RDA_Transmission_Example}}		{{:Non-Parametric_RDA_Transmission_Example}}
			</div>

Harry Guo: /* Confidence Limits for the MCF */

2014-11-18T17:39:44Z

Confidence Limits for the MCF

← Older revision		Revision as of 17:39, 18 November 2014
Line 62:		Line 62:
	\end{align}\,\!</math>		\end{align}\,\!</math>

	In case there are multiple events at the same time <math>{{t}_{i}}\,\!</math>, <math>{{d}_{ji}}\,\!</math> is calculated sequentially for each event. For each event, only one <math>{{d}_{ji}}\,\!</math> can take value of 1. Once all the events are calculated, the final MCF and its variance are the values for time <math>{{t}_{i}}\,\!</math>. This is illustrated in the following example.		In case there are multiple events at the same time <math>{{t}_{i}}\,\!</math>, <math>{{d}_{ji}}\,\!</math> is calculated sequentially for each event. For each event, only one <math>{{d}_{ji}}\,\!</math> can take value of 1. Once all the events at <math>{{t}_{i}}\,\!</math> are calculated, the final calculated MCF and its variance are the values for time <math>{{t}_{i}}\,\!</math>. This is illustrated in the following example.

	==Example: Mean Cumulative Function==		==Example: Mean Cumulative Function==

Harry Guo: /* Confidence Limits for the MCF */

2014-11-18T17:28:52Z

Confidence Limits for the MCF

← Older revision		Revision as of 17:28, 18 November 2014
Line 61:		Line 61:
	& {{d}_{ji}}= 0\text{ if the }{{j}^{\text{th }}}\text{unit did not have an event reoccur at age }{{t}_{i}}		& {{d}_{ji}}= 0\text{ if the }{{j}^{\text{th }}}\text{unit did not have an event reoccur at age }{{t}_{i}}
	\end{align}\,\!</math>		\end{align}\,\!</math>

			In case there are multiple events at the same time <math>{{t}_{i}}\,\!</math>, <math>{{d}_{ji}}\,\!</math> is calculated sequentially for each event. For each event, only one <math>{{d}_{ji}}\,\!</math> can take value of 1. Once all the events are calculated, the final MCF and its variance are the values for time <math>{{t}_{i}}\,\!</math>. This is illustrated in the following example.

	==Example: Mean Cumulative Function==		==Example: Mean Cumulative Function==

Richard House: /* The Mean Cumulative Function (MCF) */

2012-09-26T21:43:49Z

The Mean Cumulative Function (MCF)

Richard House at 21:32, 26 September 2012

2012-09-26T21:32:42Z

Richard House at 17:13, 5 September 2012

2012-09-05T17:13:34Z

← Older revision		Revision as of 17:13, 5 September 2012
Line 3:		Line 3:

	</noinclude>		</noinclude>
	Non-parametric RDA provides a non-parametric graphical estimate of the mean cumulative number or cost of recurrence per unit versus age. In the reliability field, the Mean Cumulative Function (MCF) can be used to: ~~[[Appendix: Weibull References\|[31]]]~~		Non-parametric RDA provides a non-parametric graphical estimate of the mean cumulative number or cost of recurrence per unit versus age. As discussed in Nelson [[Appendix:_Life_Data_Analysis_References\|[31]]], in the reliability field, the Mean Cumulative Function (MCF) can be used to:

	:*Evaluate whether the population repair (or cost) rate increases or decreases with age (this is useful for product retirement and burn-in decisions).		:*Evaluate whether the population repair (or cost) rate increases or decreases with age (this is useful for product retirement and burn-in decisions).

Lisa Hacker at 02:10, 16 August 2012

2012-08-16T02:10:05Z

← Older revision		Revision as of 02:10, 16 August 2012
Line 65:		Line 65:

	{{:Non_Parametric_RDA_MCF_Example}}		{{:Non_Parametric_RDA_MCF_Example}}

	{{:Non-Parametric_RDA_Transmission_Example}}		{{:Non-Parametric_RDA_Transmission_Example}}

← Older revision		Revision as of 21:43, 26 September 2012
Line 16:		Line 16:
	[[Image:Lda11.1.png\|center\|400px]]		[[Image:Lda11.1.png\|center\|400px]]

	The non-parametric model for a population of units is described as the population of cumulative history functions (curves). It is the population of all staircase functions of every unit in the population. At age t, the units have a distribution of their cumulative number of events. That is, a fraction of the population has accumulated 0 recurrences, another fraction has accumulated 1 recurrence, another fraction has accumulated 2 recurrences, etc. This distribution differs at different ages <~~span class="texhtml"~~>''t''</~~span~~> , and has a mean <~~span class="texhtml"~~>''M''(''t'')</~~span~~> called the mean cumulative function (MCF). The <~~span class="texhtml"~~>''M''(''t'')</~~span~~> is the point-wise average of all population cumulative history functions (see figure below).		The non-parametric model for a population of units is described as the population of cumulative history functions (curves). It is the population of all staircase functions of every unit in the population. At age t, the units have a distribution of their cumulative number of events. That is, a fraction of the population has accumulated 0 recurrences, another fraction has accumulated 1 recurrence, another fraction has accumulated 2 recurrences, etc. This distribution differs at different ages <math>t\,\!</math>, and has a mean <math>M(t)\,\!</math> called the mean cumulative function (MCF). The <math>M(t)\,\!</math> is the point-wise average of all population cumulative history functions (see figure below).

	[[Image:Lda11.2.png\|center\|400px]]		[[Image:Lda11.2.png\|center\|400px]]

	For the case of uncensored data, the mean cumulative function <math>M{{(t)}_{i}}\ \,\!</math> values at different recurrence ages <~~span class="texhtml"~~>''t~~''<sub>''~~i~~''</sub>~~</~~span~~> are estimated by calculating the average of the cumulative number of recurrences of events for each unit in the population at <~~span class="texhtml"~~>''t~~''<sub>''~~i~~''</sub>~~</~~span~~> . When the histories are censored, the following steps are applied.		For the case of uncensored data, the mean cumulative function <math>M{{(t)}_{i}}\ \,\!</math> values at different recurrence ages <math>{{t}_{i}}\,\!</math> are estimated by calculating the average of the cumulative number of recurrences of events for each unit in the population at <math>{{t}_{i}}\,\!</math>. When the histories are censored, the following steps are applied.

	'''1st Step - Order all ages:'''		'''1st Step - Order all ages:'''
Line 26:		Line 26:
	Order all recurrence and censoring ages from smallest to largest. If a recurrence age for a unit is the same as its censoring (suspension) age, then the recurrence age goes first. If multiple units have a common recurrence or censoring age, then these units could be put in a certain order or be sorted randomly.		Order all recurrence and censoring ages from smallest to largest. If a recurrence age for a unit is the same as its censoring (suspension) age, then the recurrence age goes first. If multiple units have a common recurrence or censoring age, then these units could be put in a certain order or be sorted randomly.

	'''2nd Step - Calculate the number, <~~span class="texhtml"~~>''r~~''<sub>''~~i~~''</sub>~~</~~span~~> , of units that passed through age <~~span class="texhtml"~~>''t~~''<sub>''~~i~~''</sub>~~</~~span~~> :'''		'''2nd Step - Calculate the number, <math>{{r}_{i}}\,\!</math>, of units that passed through age <math>{{t}_{i}}\,\!</math> :'''

	::<math>\begin{align}		::<math>\begin{align}
Line 33:		Line 33:
	\end{align}\,\!</math>		\end{align}\,\!</math>

	<~~span class="texhtml"~~>''N''</~~span~~> is the total number of units and <~~span class="texhtml"~~>''r~~''<sub>~~1~~</sub>~~ = ''N''</~~span~~> at the first observed age which could be a recurrence or suspension.		<math>N\,\!</math> is the total number of units and <math>{{r}_{1}} = N\,\!</math> at the first observed age which could be a recurrence or suspension.

	'''3rd Step - Calculate the MCF estimate, M*(t):'''		'''3rd Step - Calculate the MCF estimate, M*(t):'''

	For each sample recurrence age <~~span class="texhtml"~~>''t~~''<sub>''~~i~~''</sub>~~</~~span~~>, calculate the mean cumulative function estimate as follows		For each sample recurrence age <math>{{t}_{i}}\,\!</math>, calculate the mean cumulative function estimate as follows

	::<math>{{M}^{}}({{t}_{i}})=\frac{1}{{{r}_{i}}}+{{M}^{}}({{t}_{i-1}})\,\!</math>		::<math>{{M}^{}}({{t}_{i}})=\frac{1}{{{r}_{i}}}+{{M}^{}}({{t}_{i-1}})\,\!</math>

	where <math>{{M}^{*}}(t)=\tfrac{1}{{{r}_{1}}}\,\!</math> at the earliest observed recurrence age, <~~span class="texhtml"~~>''t~~''<sub>~~1</~~sub></span~~> .		where <math>{{M}^{*}}(t)=\tfrac{1}{{{r}_{1}}}\,\!</math> at the earliest observed recurrence age, <math>{{t}_{1}}\,\!</math>.

	===Confidence Limits for the MCF===		===Confidence Limits for the MCF===
Line 55:		Line 55:
	::<math>Var[{{M}^{}}({{t}_{i}})]=Var[{{M}^{}}({{t}_{i-1}})]+\frac{1}{r_{i}^{2}}\left[ \underset{j\in {{R}_{i}}}{\overset{}{\mathop \sum }}\,{{\left( {{d}_{ji}}-\frac{1}{{{r}_{i}}} \right)}^{2}} \right]\,\!</math>		::<math>Var[{{M}^{}}({{t}_{i}})]=Var[{{M}^{}}({{t}_{i-1}})]+\frac{1}{r_{i}^{2}}\left[ \underset{j\in {{R}_{i}}}{\overset{}{\mathop \sum }}\,{{\left( {{d}_{ji}}-\frac{1}{{{r}_{i}}} \right)}^{2}} \right]\,\!</math>

	where <math>{r}_{i}\,\!</math> is defined in the equation of the survivals, <math>{{R}_{i}}\,\!</math> is the set of the units that have not been suspended by <math>i\,\!</math> and <math>{{d}_{ji}}\,\!</math> is defined as follows:		where <math>{r}_{i}\,\!</math> is defined in the equation of the survivals, <math>{{R}_{i}}\,\!</math> is the set of the units that have not been suspended by <math>i\,\!</math> and <math>{{d}_{ji}}\,\!</math> is defined as follows:

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

← Older revision		Revision as of 21:32, 26 September 2012
Line 20:		Line 20:
	[[Image:Lda11.2.png\|center\|400px]]		[[Image:Lda11.2.png\|center\|400px]]

	For the case of uncensored data, the mean cumulative function <math>M{{(t)}_{i}}\ </math> values at different recurrence ages <span class="texhtml">''t''<sub>''i''</sub></span> are estimated by calculating the average of the cumulative number of recurrences of events for each unit in the population at <span class="texhtml">''t''<sub>''i''</sub></span> . When the histories are censored, the following steps are applied.		For the case of uncensored data, the mean cumulative function <math>M{{(t)}_{i}}\ \,\!</math> values at different recurrence ages <span class="texhtml">''t''<sub>''i''</sub></span> are estimated by calculating the average of the cumulative number of recurrences of events for each unit in the population at <span class="texhtml">''t''<sub>''i''</sub></span> . When the histories are censored, the following steps are applied.

	'''1st Step - Order all ages:'''		'''1st Step - Order all ages:'''
Line 31:		Line 31:
	& {{r}_{i}}= & {{r}_{i-1}}\quad \quad \text{if }{{t}_{i}}\text{ is a recurrence age} \\		& {{r}_{i}}= & {{r}_{i-1}}\quad \quad \text{if }{{t}_{i}}\text{ is a recurrence age} \\
	& {{r}_{i}}= & {{r}_{i-1}}-1\text{ if }{{t}_{i}}\text{ is a censoring age}		& {{r}_{i}}= & {{r}_{i-1}}-1\text{ if }{{t}_{i}}\text{ is a censoring age}
	\end{align}</math>		\end{align}\,\!</math>

	<span class="texhtml">''N''</span> is the total number of units and <span class="texhtml">''r''<sub>1</sub> = ''N''</span> at the first observed age which could be a recurrence or suspension.		<span class="texhtml">''N''</span> is the total number of units and <span class="texhtml">''r''<sub>1</sub> = ''N''</span> at the first observed age which could be a recurrence or suspension.
Line 39:		Line 39:
	For each sample recurrence age <span class="texhtml">''t''<sub>''i''</sub></span>, calculate the mean cumulative function estimate as follows		For each sample recurrence age <span class="texhtml">''t''<sub>''i''</sub></span>, calculate the mean cumulative function estimate as follows

	::<math>{{M}^{}}({{t}_{i}})=\frac{1}{{{r}_{i}}}+{{M}^{}}({{t}_{i-1}})</math>		::<math>{{M}^{}}({{t}_{i}})=\frac{1}{{{r}_{i}}}+{{M}^{}}({{t}_{i-1}})\,\!</math>

	where <math>{{M}^{*}}(t)=\tfrac{1}{{{r}_{1}}}</math> at the earliest observed recurrence age, <span class="texhtml">''t''<sub>1</sub></span> .		where <math>{{M}^{*}}(t)=\tfrac{1}{{{r}_{1}}}\,\!</math> at the earliest observed recurrence age, <span class="texhtml">''t''<sub>1</sub></span> .

	===Confidence Limits for the MCF===		===Confidence Limits for the MCF===
	Upper and lower confidence limits for <math>M({{t}_{i}})</math> are:		Upper and lower confidence limits for <math>M({{t}_{i}})\,\!</math> are:

	::<math>\begin{align}		::<math>\begin{align}
	& {{M}_{U}}({{t}_{i}})= {{M}^{}}({{t}_{i}}).{{e}^{\tfrac{{{K}_{\alpha }}.\sqrt{Var[{{M}^{}}({{t}_{i}})]}}{{{M}^{*}}({{t}_{i}})}}} \\		& {{M}_{U}}({{t}_{i}})= {{M}^{}}({{t}_{i}}).{{e}^{\tfrac{{{K}_{\alpha }}.\sqrt{Var[{{M}^{}}({{t}_{i}})]}}{{{M}^{*}}({{t}_{i}})}}} \\
	& {{M}_{L}}({{t}_{i}})= \frac{{{M}^{}}({{t}_{i}})}{{{e}^{\tfrac{{{K}_{\alpha }}.\sqrt{Var[{{M}^{}}({{t}_{i}})]}}{{{M}^{*}}({{t}_{i}})}}}}		& {{M}_{L}}({{t}_{i}})= \frac{{{M}^{}}({{t}_{i}})}{{{e}^{\tfrac{{{K}_{\alpha }}.\sqrt{Var[{{M}^{}}({{t}_{i}})]}}{{{M}^{*}}({{t}_{i}})}}}}
	\end{align}</math>		\end{align}\,\!</math>

	where <math>\alpha </math> ( <math>50%<\alpha <100%</math> ) is confidence level, <math>{{K}_{\alpha }}</math> is the <math>\alpha </math> standard normal percentile and <math>Var[{{M}^{*}}({{t}_{i}})]</math> is the variance of the MCF estimate at recurrence age <math>{{t}_{i}}</math> . The variance is calculated as follows:		where <math>\alpha \,\!</math> ( <math>50%<\alpha <100%\,\!</math> ) is confidence level, <math>{{K}_{\alpha }}\,\!</math> is the <math>\alpha \,\!</math> standard normal percentile and <math>Var[{{M}^{*}}({{t}_{i}})]\,\!</math> is the variance of the MCF estimate at recurrence age <math>{{t}_{i}}\,\!</math>. The variance is calculated as follows:

	::<math>Var[{{M}^{}}({{t}_{i}})]=Var[{{M}^{}}({{t}_{i-1}})]+\frac{1}{r_{i}^{2}}\left[ \underset{j\in {{R}_{i}}}{\overset{}{\mathop \sum }}\,{{\left( {{d}_{ji}}-\frac{1}{{{r}_{i}}} \right)}^{2}} \right]</math>		::<math>Var[{{M}^{}}({{t}_{i}})]=Var[{{M}^{}}({{t}_{i-1}})]+\frac{1}{r_{i}^{2}}\left[ \underset{j\in {{R}_{i}}}{\overset{}{\mathop \sum }}\,{{\left( {{d}_{ji}}-\frac{1}{{{r}_{i}}} \right)}^{2}} \right]\,\!</math>

	where <math>{r}_{i}</math> is defined in the equation of the survivals, <math>{{R}_{i}}</math> is the set of the units that have not been suspended by <math>i</math> and <math>{{d}_{ji}}</math> is defined as follows:		where <math>{r}_{i}\,\!</math> is defined in the equation of the survivals, <math>{{R}_{i}}\,\!</math> is the set of the units that have not been suspended by <math>i\,\!</math> and <math>{{d}_{ji}}\,\!</math> is defined as follows:

	::<math>\begin{align}		::<math>\begin{align}
	& {{d}_{ji}}= 1\text{ if the }{{j}^{\text{th }}}\text{unit had an event recurrence at age }{{t}_{i}} \\		& {{d}_{ji}}= 1\text{ if the }{{j}^{\text{th }}}\text{unit had an event recurrence at age }{{t}_{i}} \\
	& {{d}_{ji}}= 0\text{ if the }{{j}^{\text{th }}}\text{unit did not have an event reoccur at age }{{t}_{i}}		& {{d}_{ji}}= 0\text{ if the }{{j}^{\text{th }}}\text{unit did not have an event reoccur at age }{{t}_{i}}
	\end{align}</math>		\end{align}\,\!</math>

	==Example: Mean Cumulative Function==		==Example: Mean Cumulative Function==