General Log-Linear (GLL)-Weibull Model: Difference between revisions

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{{Reference Example|ALTA_Reference_Examples_Banner.png|ALTA_Reference_Examples}}
{{Reference Example|{{Banner ALTA Reference Examples}}}}
This example validates the calculation of GLL relationship for Weibull distribution.
 
This example validates the results for the GLL life-stress relationship with a Weibull distribution in ALTA standard folios.




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{{Reference_Example_Heading2}}


The data is given below.
The following table shows the data.
{| {{table}}
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!State F/S
!State F/S
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::<math>\,\!ln\left ( \eta  \right )=\alpha _{0}+\alpha _{1}\frac{1}{T}</math>  
::<math>\,\!ln\left ( \eta  \right )=\alpha _{0}+\alpha _{1}\frac{1}{T}</math>  


The book has the following results:
The book has the following results:
*The model parameters are: <math>\,\!\alpha _{0}=-3.156</math> , <math>\,\!\alpha _{1}=4390</math> and <math>\,\!\beta =2.27</math>.
*The model parameters are: <math>\,\!\alpha _{0}=-3.156</math> , <math>\,\!\alpha _{1}=4390</math> and <math>\,\!\beta =2.27</math>.


*The variance of each parameter is: <math>\,\!Var\left ( \alpha _{0} \right )=3.08</math> , <math>\,\!Var\left ( \alpha _{1} \right )=484,819.5</math> and <math>\,\!Var\left ( \beta\right )=0.1396</math> .


*The two-sided 90% confidence intervals for the model parameters are:  <math>\,\!\left [ \alpha _{0,L},\alpha _{0,U}  \right ]=\left [ -6.044,-0.269 \right ]</math>  , <math>\,\!\left [ \alpha _{1,L},\alpha _{1,U}  \right ]=\left [ 3244.8,5535.3 \right ]</math> and <math>\,\!\left [ \beta _{1,L},\beta _{1,U}  \right ]=\left [ 1.73,2.97 \right ]</math> .  
*The variance of each parameter is: <math>\,\!Var\left ( \alpha _{0} \right )=3.08</math> , <math>\,\!Var\left ( \alpha _{1} \right )=484819.5</math> and <math>\,\!Var\left ( \beta\right )=0.1396</math> .
 
 
*The two-sided 90% confidence intervals for the model parameters are:  <math>\,\!\left [ \alpha _{0,L},\alpha _{0,U}  \right ]=\left [ -6.044, -0.269 \right ]</math>  , <math>\,\!\left [ \alpha _{1,L},\alpha _{1,U}  \right ]=\left [ 3244.8, 5535.3 \right ]</math> and <math>\,\!\left [ \beta _{1,L},\beta _{1,U}  \right ]=\left [ 1.73, 2.97 \right ]</math> .
 
 
*The estimated B10 life at temperature of 35°C is 24,286 hours. The two-sided 90% confidence interval is [10371, 56867].


*The estimated B10 life at temperature of 35°C is 24,286 hours. The two-sided 90% confidence interval is [10,371, 56,867].
    
    
*The estimated reliability at 35°C and 10,000 hours is <math>\,\!R\left ( 10,000 \right )=0.9860</math> . The two-sided 90% confidence interval is [0.892, 0.998].
*The estimated reliability at 35°C and 10,000 hours is <math>\,\!R\left ( 10000 \right )=0.9860</math> . The two-sided 90% confidence interval is [0.892, 0.998].




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General Log-Linear (GLL)-Weibull Model

This example validates the results for the GLL life-stress relationship with a Weibull distribution in ALTA standard folios.


Reference Case

The data set is from Example 7.14 on page 297 in book Life Cycle Reliability Engineering by Dr. Guangbin Yang, John Wiley & Sons, 2007.


Data

The following table shows the data.

State F/S Time to State (Hr) Temperature (°C) Group ID
F 1138 100 1
F 1944 100 1
F 2764 100 1
F 2846 100 1
F 3246 100 1
F 3803 100 1
F 5046 100 1
F 5139 100 1
S 5500 100 1
S 5500 100 1
S 5500 100 1
S 5500 100 1
F 1121 120 2
F 1572 120 2
F 2329 120 2
F 2573 120 2
F 2702 120 2
F 3702 120 2
F 4277 120 2
S 4500 120 2
F 420 150 3
F 650 150 3
F 703 150 3
F 838 150 3
F 1086 150 3
F 1125 150 3
F 1387 150 3
F 1673 150 3
F 1896 150 3
F 2037 150 3


Result

The model used in the book is:

[math]\displaystyle{ \,\!ln\left ( \eta \right )=\alpha _{0}+\alpha _{1}\frac{1}{T} }[/math]


The book has the following results:

  • The model parameters are: [math]\displaystyle{ \,\!\alpha _{0}=-3.156 }[/math] , [math]\displaystyle{ \,\!\alpha _{1}=4390 }[/math] and [math]\displaystyle{ \,\!\beta =2.27 }[/math].


  • The variance of each parameter is: [math]\displaystyle{ \,\!Var\left ( \alpha _{0} \right )=3.08 }[/math] , [math]\displaystyle{ \,\!Var\left ( \alpha _{1} \right )=484819.5 }[/math] and [math]\displaystyle{ \,\!Var\left ( \beta\right )=0.1396 }[/math] .


  • The two-sided 90% confidence intervals for the model parameters are: [math]\displaystyle{ \,\!\left [ \alpha _{0,L},\alpha _{0,U} \right ]=\left [ -6.044, -0.269 \right ] }[/math] , [math]\displaystyle{ \,\!\left [ \alpha _{1,L},\alpha _{1,U} \right ]=\left [ 3244.8, 5535.3 \right ] }[/math] and [math]\displaystyle{ \,\!\left [ \beta _{1,L},\beta _{1,U} \right ]=\left [ 1.73, 2.97 \right ] }[/math] .


  • The estimated B10 life at temperature of 35°C is 24,286 hours. The two-sided 90% confidence interval is [10371, 56867].


  • The estimated reliability at 35°C and 10,000 hours is [math]\displaystyle{ \,\!R\left ( 10000 \right )=0.9860 }[/math] . The two-sided 90% confidence interval is [0.892, 0.998].


Results in ALTA

In ALTA, the GLL model with Weibull distribution is used. Since temperature is the stress, the reciprocal transform is used. The results are:

  • The model parameters are:
Temperature GLL Weibull Analysis Summary.png


  • The variances of the parameters are:
Temperature GLL Weibull Var Cov Results.png


  • The two-sided 90% confidence intervals for the model parameters are:
Temperature GLL Weibull Parameter Bounds.png


  • The estimated B10 life and its two-sided 90% confidence intervals are:
Temperature GLL Weibull QPC B10 Life.png


  • The estimated reliability with its two-sided 90% confidence interval at 35°C and 10,000 hours are:
Temperature GLL Weibull QPC Reliability.png
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