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

Revision as of 20:30, 10 June 2014

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ALTA_Reference_Examples

This example validates the calculation of GLL relationship for Weibull distribution.

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 data is given below.

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 )=484,819.5 }[/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 [10,371, 56,867].

The estimated reliability at 35°C and 10,000 hours is [math]\displaystyle{ \,\!R\left ( 10,000 \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 estimated reliability with its two-sided 90% confidence interval at 35°C and 10,000 hours are:

@@ Line 85: / Line 85: @@
 ::<math>\,\!ln\left ( \eta  \right )=\alpha _{0}+\alpha _{1}\frac{1}{T}</math>
 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 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 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].