Median Rank for Multiple Censored Data: Difference between revisions

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{{Reference Example}}
{{Reference Example}}


This example compares the median rank calculation for multiple censored data.
This example validates the median rank calculation for multiple censored data in Weibull++ standard folios.




{{Reference Example Heading1}}
{{Reference Example Heading1}}
Table 3.1 on page 78 in book “''Reliability & Life Testing Handbook Vol 2''” by Dr. Kececioglu, Prentice-Hall, 1994.
Table 3.1 on page 78 in the book ''Reliability & Life Testing Handbook Vol 2'' by Dr. Kececioglu, Prentice-Hall, 1994.




{{Reference Example Heading2}}
{{Reference Example Heading2}}
{| border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
{| {{table}}
|-
|- align="center"
!Num. In Stage
!Num. In Stage
!State F or S
!State F or S
!Time to Failure
!Time to Failure
|- align="center"
|-
|1 ||F |||5100
|1 ||F |||5100
|- align="center"
|-
|1 ||S |||9500
|1 ||S |||9500
|- align="center"
|-
|1 ||F |||15000
|1 ||F |||15000
|- align="center"
|-
|1 ||S |||22000
|1 ||S |||22000
|- align="center"
|-
|1 ||F |||40000
|1 ||F |||40000
|}  
|}  
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{{Reference Example Heading3}}
{{Reference Example Heading3}}
{| border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
{| {{table}}
|-
|- align="center"
!Num. In Stage
!Num. In Stage
!State F or S
!State F or S
!Time to Failure
!Time to Failure
!Median Rank (%)
!Median Rank (%)
|- align="center"
|-
|1 ||F |||5100 |||12.94
|1 ||F |||5100 |||12.94
|- align="center"
|-  
|1 ||S |||9500 |||
|1 ||S |||9500 |||
|- align="center"
|-  
|1 ||F |||15000 |||36.1
|1 ||F |||15000 |||36.1
|- align="center"
|-  
|1 ||S |||22000 |||
|1 ||S |||22000 |||
|- align="center"
|-  
|1 ||F |||40000 |||70.84
|1 ||F |||40000 |||70.84
|}  
|}  
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The differences between the results in Weibull++ and the book are due to the method of calculating the median ranks (MR). In the book, the following approximation method is used.
The differences between the results in Weibull++ and the book are due to the method of calculating the median ranks (MR). In the book, the following approximation method is used:


::<math>MR_{i}\approx \frac{MON_{i}-0.3}{N+0.4}</math>
::<math>MR_{i}\approx \frac{MON_{i}-0.3}{N+0.4}</math>
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In Weibull++, the following exact method is used.
In Weibull++, the following exact method is used:


::<math>MR_{i}= \frac{1}{1+\frac{N-MON_{i}+1}{MON_{i}}F_{0.5,m,n}}</math>
::<math>MR_{i}= \frac{1}{1+\frac{N-MON_{i}+1}{MON_{i}}F_{0.5,m,n}}</math>


   
   
where <math>m=2(N-MON_{i}+1), n=2xMON_{i}\cdot F_{0.5,m,n}\,\!</math> is the 50 percentile of a F distribution with degree of freedom of ''m'' and ''n''.
where <math>m=2(N-MON_{i}+1), n=2\times MON_{i}\cdot F_{0.5,m,n}\,\!</math> is the 50 percentile of a F distribution with degree of freedom of ''m'' and ''n''.

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Median Rank for Multiple Censored Data

This example validates the median rank calculation for multiple censored data in Weibull++ standard folios.


Reference Case

Table 3.1 on page 78 in the book Reliability & Life Testing Handbook Vol 2 by Dr. Kececioglu, Prentice-Hall, 1994.


Data

Num. In Stage State F or S Time to Failure
1 F 5100
1 S 9500
1 F 15000
1 S 22000
1 F 40000


Result

Num. In Stage State F or S Time to Failure Median Rank (%)
1 F 5100 12.94
1 S 9500
1 F 15000 36.1
1 S 22000
1 F 40000 70.84


Results in Weibull++

The coordinates of each point in the following plot shows the failure time and the corresponding median rank.


Median Ranks Multiple Censored Data.png


The differences between the results in Weibull++ and the book are due to the method of calculating the median ranks (MR). In the book, the following approximation method is used:

[math]\displaystyle{ MR_{i}\approx \frac{MON_{i}-0.3}{N+0.4} }[/math]


where [math]\displaystyle{ MR_{i}\,\! }[/math] is the median rank at the [math]\displaystyle{ ith\,\! }[/math] failure time; [math]\displaystyle{ MON_{i}\,\! }[/math] is the mean order number; [math]\displaystyle{ N\,\! }[/math] is the total samples. For the step by step calculation of mean order number (MON), please refer to the book “Reliability & Life Testing Handbook Vol 2” by Dr. Kececioglu, Prentice-Hall, 1994.


In Weibull++, the following exact method is used:

[math]\displaystyle{ MR_{i}= \frac{1}{1+\frac{N-MON_{i}+1}{MON_{i}}F_{0.5,m,n}} }[/math]


where [math]\displaystyle{ m=2(N-MON_{i}+1), n=2\times MON_{i}\cdot F_{0.5,m,n}\,\! }[/math] is the 50 percentile of a F distribution with degree of freedom of m and n.

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