Template:Mean cumulative function for recurrence data - Revision history

Nicolette Young at 23:02, 25 April 2012

2012-04-25T23:02:39Z

← Older revision		Revision as of 23:02, 25 April 2012
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	In a non-parametric analysis of recurrent event data, each population unit can be described by a cumulative history function for the cumulative number of recurrences. It is a staircase function that depicts the cumulative number of recurrences of a particular event, such as repairs over time. The figure below depicts a unit's cumulative history function.		In a non-parametric analysis of recurrent event data, each population unit can be described by a cumulative history function for the cumulative number of recurrences. It is a staircase function that depicts the cumulative number of recurrences of a particular event, such as repairs over time. The figure below depicts a unit's cumulative history function.

	[[Image:Lda11.1.png\|center\|~~400px~~]]		[[Image:Lda11.1.png\|center\|250px]]

	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 <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).

	[[Image:Lda11.2.png\|center\|~~400px~~]]		[[Image:Lda11.2.png\|center\|250px]]

	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.

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

2012-03-16T18:38:46Z

The Mean Cumulative Function (MCF)

← Older revision		Revision as of 18:38, 16 March 2012
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	In a non-parametric analysis of recurrent event data, each population unit can be described by a cumulative history function for the cumulative number of recurrences. It is a staircase function that depicts the cumulative number of recurrences of a particular event, such as repairs over time. The figure below depicts a unit's cumulative history function.		In a non-parametric analysis of recurrent event data, each population unit can be described by a cumulative history function for the cumulative number of recurrences. It is a staircase function that depicts the cumulative number of recurrences of a particular event, such as repairs over time. The figure below depicts a unit's cumulative history function.

	[[Image:Lda11.1.~~gif\|thumb~~\|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 <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).

	[[Image:Lda11.2.~~gif\|thumb~~\|center\|~~392px~~]]		[[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.

Kate Racaza at 17:14, 8 March 2012

2012-03-08T17:14:40Z

← Older revision		Revision as of 17:14, 8 March 2012
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	<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.

	'''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~~:'''~~		'''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

	::<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>

Kate Racaza at 17:10, 8 March 2012

2012-03-08T17:10:39Z

← Older revision		Revision as of 17:10, 8 March 2012
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	=== The Mean Cumulative Function (MCF) ~~for Recurrent Event Data Analysis~~ ===		=== The Mean Cumulative Function (MCF) ===

	In a non-parametric analysis of recurrent event data, each population unit can be described by a cumulative history function for the cumulative number of recurrences. It is a staircase function that depicts the cumulative number of recurrences of a particular event, such as repairs over time. The figure below depicts a unit's cumulative history function.		In a non-parametric analysis of recurrent event data, each population unit can be described by a cumulative history function for the cumulative number of recurrences. It is a staircase function that depicts the cumulative number of recurrences of a particular event, such as repairs over time. The figure below depicts a unit's cumulative history function.
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	<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.

	'''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:		'''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:'''

	::<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>

Kate Racaza at 16:49, 8 March 2012

2012-03-08T16:49:12Z

← Older revision		Revision as of 16:49, 8 March 2012
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	<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.

	'''3rd Step - Calculate the ~~<span class="texhtml">''M''</span><span class="texhtml" />C''''F''''''~~estimate, ~~<span class="texhtml" />~~:~~'''~~For each sample recurrence age ~~<~~span class="texhtml" ~~/>calculate the mean cumulative function estimate as follows:~~''~~'''M~~''<~~sup>* </sup~~>~~(''t'')'''~~''t''<sub></~~sub~~>,''		'''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:

	::<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>

Kate Racaza at 16:33, 8 March 2012

2012-03-08T16:33:05Z

← Older revision		Revision as of 16:33, 8 March 2012
Line 24:		Line 24:
	<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.

	'''3rd Step - Calculate the <span class="texhtml">''M''</span><span class="texhtml"></~~span>~~C''''F''''''estimate, <span class="texhtml"></~~span>~~:'''For each sample recurrence age <span class="texhtml"></~~span>~~ calculate the mean cumulative function estimate as follows:'''''M''<sup>* </sup>(''t'')'''''t''<sub></sub>,~~'''~~''		'''3rd Step - Calculate the <span class="texhtml">''M''</span><span class="texhtml" />C''''F''''''estimate, <span class="texhtml" />:'''For each sample recurrence age <span class="texhtml" />calculate the mean cumulative function estimate as follows:'''''M''<sup>* </sup>(''t'')'''''t''<sub></sub>,''

	::<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>
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	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> .

	<br>'''Example 1:''' {{Example: Recurrent Events Data Non-~~Parameteric~~ MCF Example}}		<br>'''Example 1:''' {{Example: Recurrent Events Data Non-parameteric MCF Example}}

Kate Racaza at 16:27, 8 March 2012

2012-03-08T16:27:49Z

← Older revision		Revision as of 16:27, 8 March 2012
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	=== The Mean Cumulative Function (MCF) for ~~Recurrence~~ Data ===		=== The Mean Cumulative Function (MCF) for Recurrent Event Data Analysis ===

	In non-parametric analysis of recurrent ~~events~~ data, each population unit can be described by a cumulative history function for the cumulative number of recurrences. It is a staircase function that depicts the cumulative number of recurrences of a particular event, such as repairs over time. The figure below depicts a unit's cumulative history function.		In a non-parametric analysis of recurrent event data, each population unit can be described by a cumulative history function for the cumulative number of recurrences. It is a staircase function that depicts the cumulative number of recurrences of a particular event, such as repairs over time. The figure below depicts a unit's cumulative history function.

	[[Image:Lda11.1.gif\|thumb\|center\|400px]]		[[Image:Lda11.1.gif\|thumb\|center\|400px]]
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	<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.

	'''3rd Step - Calculate the <span class="texhtml">''M~~''''C''''F~~''</span> ~~estimate,~~ <span class="texhtml">''M''~~<sup> * </sup>(~~''t'')</span>:''' For each sample recurrence age <span class="texhtml">''t''<~~sub~~>''i''</sub>,</~~span~~> ~~calculate the mean cumulative function estimate as follows:~~		'''3rd Step - Calculate the <span class="texhtml">''M''</span><span class="texhtml"></span>C''''F''''''estimate, <span class="texhtml"></span>:'''For each sample recurrence age <span class="texhtml"></span> calculate the mean cumulative function estimate as follows:'''''M''<sup>* </sup>(''t'')'''''t''<sub></sub>,'''''

	::<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>

Kate Racaza at 15:45, 8 March 2012

2012-03-08T15:45:59Z

← Older revision		Revision as of 15:45, 8 March 2012
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	===The Mean Cumulative Function (MCF) for Recurrence Data===		=== The Mean Cumulative Function (MCF) for Recurrence Data ===
	In non-parametric analysis of recurrent events data, each population unit can be described by a cumulative history function for the cumulative number of recurrences. It is a staircase function that depicts the cumulative number of recurrences of a particular event, such as repairs over time. The figure below depicts a unit's cumulative history function.

	~~[[Image:lda11.1~~.~~gif\|thumb\|center\|400px\|Cumulative~~ number of ~~failures~~. ]]		In non-parametric analysis of recurrent events data, each population unit can be described by a cumulative history function for the cumulative number of recurrences. It is a staircase function that depicts the cumulative number of recurrences of a particular event, such as repairs over time. The figure below depicts a unit's cumulative history function.

	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:Lda11.1.gif\|thumb\|center\|400px]]

	[[Image:~~lda11~~.2.gif\|thumb\|center\|400px~~\|Illustration of MCF and population distribution at age ''t''.~~ ]]

	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~~> .		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).
	When the histories are censored, the following steps are applied.
			[[Image:Lda11.2.gif\|thumb\|center\|392px]]

			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:'''

	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, <~~math~~>{{r~~}_{~~i}}</~~math~~> , of units that passed through age <~~math~~>{{t~~}_{~~i}}</~~math~~> :'''		'''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> :'''

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

	<~~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.		<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.

	'''3rd Step - Calculate the <~~math~~>~~MCF~~</~~math~~> estimate, <~~math~~>{{M~~}^{~~*}}(t)</~~math~~>:'''		'''3rd Step - Calculate the <span class="texhtml">''M''''C''''F''</span> estimate, <span class="texhtml">''M''<sup> * </sup>(''t'')</span>:''' 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, <~~math~~>{{t~~}_{~~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> .


	'''Example 1:'''		<br>'''Example 1:''' {{Example: Recurrent Events Data Non-Parameteric MCF Example}}
	{{Example: Recurrent Events Data Non-~~Parametric~~ MCF Example}}

Kate Racaza: /* The Mean Cumulative Function (MCF) for Recurrence Data */

2012-03-08T00:36:00Z

The Mean Cumulative Function (MCF) for Recurrence Data

← Older revision		Revision as of 00:36, 8 March 2012
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	'''Example 1:'''		'''Example 1:'''
	{{Example: Recurrent Events Data Non-~~Parameteric~~ MCF Example}}		{{Example: Recurrent Events Data Non-Parametric MCF Example}}

Kate Racaza: /* The Mean Cumulative Function (MCF) for Recurrence Data */

2012-03-08T00:25:31Z

The Mean Cumulative Function (MCF) for Recurrence Data

← Older revision		Revision as of 00:25, 8 March 2012
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	When the histories are censored, the following steps are applied.		When the histories are censored, the following steps are applied.

	'''1st Step - Order ~~All Ages~~:'''		'''1st Step - Order all ages:'''

	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, 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~~, <math>{{r}_{i}}</math> , of ~~Units~~ that ~~Passed Through Age~~ <math>{{t}_{i}}</math> :'''		'''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}
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	<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.		<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 <math>MCF</math> ~~Estimate~~, <math>{{M}^{*}}(t)</math>:'''		'''3rd Step - Calculate the <math>MCF</math> estimate, <math>{{M}^{*}}(t)</math>:'''
	For each sample recurrence age <math>{{t}_{i}},</math> 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:

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	'''Example 1:'''		'''Example 1:'''
	{{Example: Recurrent Events Data Non-~~parameteric~~ MCF Example}}		{{Example: Recurrent Events Data Non-Parameteric MCF Example}}