Template:Confidence limits for the MCF - Revision history

Lisa Hacker: Redirected page to Non-Parametric Recurrent Event Data Analysis

2012-08-16T02:11:14Z

Redirected page to Non-Parametric Recurrent Event Data Analysis

← Older revision		Revision as of 02:11, 16 August 2012
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	~~===Confidence Limits for the MCF===~~		#REDIRECT [[Non-Parametric Recurrent Event Data Analysis]]
	~~Upper and lower confidence limits for <math>M({{t}_{i}})</math> are:~~

	~~::<math>\begin{align}~~
	~~& {{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}})}}}}~~
	~~\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:

	~~::<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:

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


	~~'''Example 2:'''~~
	~~{{Example: Recurrent Events Data~~ Non-~~parameteric MCF Bound Example}}~~


	~~'''Example 3:'''~~
	~~{{Example:~~ Recurrent ~~Events~~ Data ~~Non-parameteric Transmission Example}}~~

Kate Racaza: /* Confidence Limits for the MCF */

2012-03-16T23:43:05Z

Confidence Limits for the MCF

← Older revision		Revision as of 23:43, 16 March 2012
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	===Confidence Limits for the MCF===		===Confidence Limits for the MCF===
	Upper and lower ~~conifidence~~ 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}

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

2012-03-05T16:59:20Z

Confidence Limits for the MCF

← Older revision		Revision as of 16:59, 5 March 2012
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	::<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</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}

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

2012-03-05T16:57:55Z

Confidence Limits for the MCF

← Older revision		Revision as of 16:57, 5 March 2012
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	::<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</math> is defined in ~~Eqn.(R)~~, <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</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}

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

2012-02-22T18:26:55Z

Confidence Limits for the MCF

← Older revision		Revision as of 18:26, 22 February 2012
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	::<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>

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	::<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>

Harry Guo at 17:42, 22 February 2012

2012-02-22T17:42:52Z

← Older revision		Revision as of 17:42, 22 February 2012
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	& {{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 2:'''		'''Example 2:'''

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

2012-02-22T17:42:31Z

Confidence Limits for the MCF

← Older revision		Revision as of 17:42, 22 February 2012
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	\end{align}</math>		\end{align}</math>

	{{~~non~~-~~parametric rda examples~~}}		'''Example 2:'''
			{{Example: Recurrent Events Data Non-parameteric MCF Bound Example}}


			'''Example 3:'''
			{{Example: Recurrent Events Data Non-parameteric Transmission Example}}

Nicolette Young at 21:01, 17 January 2012

2012-01-17T21:01:16Z

Show changes

Nicolette Young: Created page with '===Confidence Limits for the MCF=== Upper and lower conifidence limits for $M({{t}_{i}})$ are: :: $\begin{align} & {{M}_{U}}({{t}_{i}})= & {{M}^{*}}({{t}_{i}}…'$

2012-01-04T23:27:02Z

Created page with '===Confidence Limits for the MCF=== Upper and lower conifidence limits for <math>M({{t}_{i}})</math> are: ::<math>\begin{align} & {{M}_{U}}({{t}_{i}})= & {{M}^{*}}({{t}_{i}}…'

New page

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

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

::<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</math> is defined in Eqn.(R), <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}
& {{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}}
\end{align}</math>

====Example 2====
Using the data in Example 1, estimate the 95% confidence bounds.
=====Solution to Example 2=====
Using Eq. MCF Var the following table of variance values can be obtained:

{|align="center" border="1"
|-
!ID
!Months
!State
!<math>r_i</math>
!<math>Var_i</math>
|-
|1 || 5||F || 5|| <math>(\tfrac{1}{5})^2[(1-\tfrac{1}{5})^2+4(0-\tfrac{1}{5})^2]=0.032</math>
|-
|2 || 6||F || 5|| <math>0.032+(\tfrac{1}{5})^2[(1-\tfrac{1}{5})^2+4(0-\tfrac{1}{5})^2]=0.064</math>
|-
| 1|| 10||F ||5 || <math>0.064+(\tfrac{1}{5})^2[(1-\tfrac{1}{5})^2+4(0-\tfrac{1}{5})^2]=0.096</math>
|-
|3 ||12 || F||5 || <math>0.096+(\tfrac{1}{5})^2[(1-\tfrac{1}{5})^2+4(0-\tfrac{1}{5})^2]=0.128</math>

|-
| 2|| 13||F ||5 || <math>0.128+(\tfrac{1}{5})^2[(1-\tfrac{1}{5})^2+4(0-\tfrac{1}{5})^2]=0.160</math>
|-
| 4||13 || F||5 || <math>0.160+(\tfrac{1}{5})^2[(1-\tfrac{1}{5})^2+4(0-\tfrac{1}{5})^2]=0.192</math>
|-
| 1||15 ||F ||5 ||<math>0.192+(\tfrac{1}{5})^2[(1-\tfrac{1}{5})^2+4(0-\tfrac{1}{5})^2]=0.224</math>
|-
| 4||15 || F||5 ||<math>0.224+(\tfrac{1}{5})^2[(1-\tfrac{1}{5})^2+4(0-\tfrac{1}{5})^2]=0.256</math>
|-
| 5|| 16||F ||5 || <math>0.256+(\tfrac{1}{5})^2[3(0-\tfrac{1}{5})^2+2(1-\tfrac{1}{5})^2]=0.288</math>
|-
| 2||17 ||F || 5|| <math>0.288+(\tfrac{1}{5})^2[(1-\tfrac{1}{5})^2+4(0-\tfrac{1}{5})^2]=0.320</math>
|-
| 1||17 ||S ||4 ||
|-
| 2||19 ||S ||3 ||
|-
| 3||20 ||F ||3 || <math>0.320+(\tfrac{1}{5})^2[(1-\tfrac{1}{5})^2+2(0-\tfrac{1}{5})^2]=0.394</math>
|-
| 5||22 ||F ||3 || <math>0.394+(\tfrac{1}{5})^2[2(0-\tfrac{1}{5})^2+4(1-\tfrac{1}{5})^2]=0.468</math>
|-
| 4||24 ||S ||2 ||
|-
| 3||25 || F||2 || <math>0.468+(\tfrac{1}{5})^2[(1-\tfrac{1}{5})^2+4(0-\tfrac{1}{5})^2]=0.593</math>
|-
| 5||25 || F||2 || <math>0.580+(\tfrac{1}{5})^2[(0-\tfrac{1}{5})^2+4(1-\tfrac{1}{5})^2]=0.718</math>
|-
| 3||26 ||S ||1 ||
|-
| 5||28 ||S ||0 ||
|}

Using Eqn.(MCF Bounds) and <math>{{K}_{5}}=1.644</math> for a 95% confidence level, the confidence bounds can be obtained as follows:

<center><math>\begin{matrix}
ID & Months & State & MC{{F}_{i}} & Va{{r}_{i}} & MC{{F}_{{{L}_{i}}}} & MC{{F}_{{{U}_{i}}}} \\
\text{1} & \text{5} & \text{F} & \text{0}\text{.20} & \text{0}\text{.032} & 0.0459 & 0.8709 \\
\text{2} & \text{6} & \text{F} & \text{0}\text{.40} & \text{0}\text{.064} & 0.1413 & 1.1320 \\
\text{1} & \text{10} & \text{F} & \text{0}\text{.60} & \text{0}\text{.096} & 0.2566 & 1.4029 \\
\text{3} & \text{12} & \text{F} & \text{0}\text{.80} & \text{0}\text{.128} & 0.3834 & 1.6694 \\
\text{2} & \text{13} & \text{F} & \text{1}\text{.00} & \text{0}\text{.160} & 0.5179 & 1.9308 \\
\text{4} & \text{13} & \text{F} & \text{1}\text{.20} & \text{0}\text{.192} & 0.6582 & 2.1879 \\
\text{1} & \text{15} & \text{F} & \text{1}\text{.40} & \text{0}\text{.224} & 0.8028 & 2.4413 \\
\text{4} & \text{15} & \text{F} & \text{1}\text{.60} & \text{0}\text{.256} & 0.9511 & 2.6916 \\
\text{5} & \text{16} & \text{F} & \text{1}\text{.80} & \text{0}\text{.288} & 1.1023 & 2.9393 \\
\text{2} & \text{17} & \text{F} & \text{2}\text{.0} & \text{0}\text{.320} & 1.2560 & 3.1848 \\
\text{1} & \text{17} & \text{S} & {} & {} & {} & {} \\
\text{2} & \text{19} & \text{S} & {} & {} & {} & {} \\
\text{3} & \text{20} & \text{F} & \text{2}\text{.33} & \text{0}\text{.394} & 1.4990 & 3.6321 \\
\text{5} & \text{22} & \text{F} & \text{2}\text{.66} & \text{0}\text{.468} & 1.7486 & 4.0668 \\
\text{4} & \text{24} & \text{S} & {} & {} & {} & {} \\
\text{3} & \text{25} & \text{F} & \text{3}\text{.16} & \text{0}\text{.593} & 2.1226 & 4.7243 \\
\text{5} & \text{25} & \text{F} & \text{3}\text{.66} & \text{0}\text{.718} & 2.5071 & 5.3626 \\
\text{3} & \text{26} & \text{S} & {} & {} & {} & {} \\
\text{5} & \text{28} & \text{S} & {} & {} & {} & {} \\
\end{matrix}</math></center>

The analysis presented in Examples 1 and 2 can be obtained automatically in Weibull ++ using the Non-Parametric RDA Specialized Folio, as shown next.

[[Image:RDAsystem1.png|thumb|center|400px| ]]

Note: In the above Folio, the <math>F</math> refers to failures and <math>E</math> refers to suspensions (or censoring ages).

The results with calculated MCF values and upper and lower 95% confidence limits are shown next along with the graphical plot.

[[Image:excelsheet.png|thumb|center|400px| ]]

[[Image:lda11.5.gif|thumb|center|400px| ]]

====Example 3====
The following table displays transmission repairs on a sample of 14 cars with manual transmission in a preproduction road test [31]. Here + denotes the censoring ages (how long a car has been observed).

<center><math>\begin{matrix}
Car ID & Mileage \\
\text{1} & \text{27099+} \\
\text{2} & \text{21999+} \\
\text{3} & \text{11891, 27583+} \\
\text{4} & \text{19966+} \\
\text{5} & \text{26146+} \\
\text{6} & \text{3648, 13957, 23193+} \\
\text{7} & \text{19823+} \\
\text{8} & \text{2890, 22707+} \\
\text{9} & \text{2714, 19275+} \\
\text{10} & \text{19803+} \\
\text{11} & \text{19630+} \\
\text{12} & \text{22056+} \\
\text{13} & \text{22940+} \\
\text{14} & \text{3240, 7690, 18965+} \\
\end{matrix}</math></center>

The car manufacturer seeks to estimate the mean cumulative number of repairs per car by 24,000 test miles (equivalently 5.5 x 24,000 = 132,000 customer miles) and to observe whether the population repair rate increases or decreases as a population ages.

====Solution to Example 3====
The data is entered into a Non-Parametric RDA Specialized Folio in Weibull++ as follows.

[[Image:systemRDA1.png|thumb|center|400px| ]]

The results are as follows,

[[Image:spreadsheetplain.png|thumb|center|400px| ]]

The results indicate that after 13,957 miles of testing, the estimated mean cumulative number of repairs per car is 0.5. Therefore, by 24,000 test miles, the estimated mean cumulative number of repairs per car is 0.5.

The MCF plot is shown next.

[[Image:lda11.8.gif|thumb|center|400px| ]]

A smooth curve through the MCF plot has a derivative that decreases as the population ages. That is, the repair rate decreases as each population ages. This is typical of products with manufacturing defects.

Template:Confidence limits for the MCF - Revision history

Lisa Hacker: Redirected page to Non-Parametric Recurrent Event Data Analysis

Kate Racaza: /* Confidence Limits for the MCF */

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

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

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

Harry Guo at 17:42, 22 February 2012

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

Nicolette Young at 21:01, 17 January 2012

Nicolette Young: Created page with '===Confidence Limits for the MCF=== Upper and lower conifidence limits for M({{t}_{i}}) are: ::\begin{align} & {{M}_{U}}({{t}_{i}})= & {{M}^{*}}({{t}_{i}}…'

Nicolette Young: Created page with '===Confidence Limits for the MCF=== Upper and lower conifidence limits for $M({{t}_{i}})$ are: :: $\begin{align} & {{M}_{U}}({{t}_{i}})= & {{M}^{*}}({{t}_{i}}…'$