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| In the book, the following results are provided: | | In the book, the following results are provided: |
| *ML estimations for the model parameters are: <math>\,\!</math> ,<math>\,\!</math> , <math>\,\!</math>(or <math>\,\!</math>). | | *ML estimations for the model parameters are: <math>\,\!\sigma =0.98</math> ,<math>\,\!\beta _{0}=-13.469</math> , <math>\,\!\beta _{1}=0.6279</math> (or <math>\,\!\alpha _{1}=7286.78</math>). |
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| *The 95% confidence interval for <math>\,\!</math> is [0.75, 1.28], for <math>\,\!</math> is [-19.1, -7.8] and for <math>\,\!</math> is [0.47, 0.79]. | | *The 95% confidence interval for <math>\,\!</math> is [0.75, 1.28], for <math>\,\!</math> is [-19.1, -7.8] and for <math>\,\!</math> is [0.47, 0.79]. |
Revision as of 18:49, 10 June 2014
ALTA_Reference_Examples_Banner.png
This example validates the calculation for the Arrhenius life stress relationship with a Lognormal distribution.
Reference Case
The data set is from Example 19.5 on page 498 in book Statistical Methods for Reliability Data by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998.
Data
Device-A was tested under several different temperature settings.
| Number in Group
|
State F/S
|
Time to State
|
Temperature (°F)
|
Subset ID
|
| 30 |
S |
5000 |
283.15 |
1
|
| 1 |
F |
1298 |
313.15 |
2
|
| 1 |
F |
1390 |
313.15 |
2
|
| 1 |
F |
3187 |
313.15 |
2
|
| 1 |
F |
3241 |
313.15 |
2
|
| 1 |
F |
3261 |
313.15 |
2
|
| 1 |
F |
3313 |
313.15 |
2
|
| 1 |
F |
4501 |
313.15 |
2
|
| 1 |
F |
4568 |
313.15 |
2
|
| 1 |
F |
4841 |
313.15 |
2
|
| 1 |
F |
4982 |
313.15 |
2
|
| 90 |
S |
5000 |
313.15 |
2
|
| 1 |
F |
581 |
333.15 |
3
|
| 1 |
F |
925 |
333.15 |
3
|
| 1 |
F |
1432 |
333.15 |
3
|
| 1 |
F |
1586 |
333.15 |
3
|
| 1 |
F |
2452 |
333.15 |
3
|
| 1 |
F |
2734 |
333.15 |
3
|
| 1 |
F |
2772 |
333.15 |
3
|
| 1 |
F |
4106 |
333.15 |
3
|
| 1 |
F |
4674 |
333.15 |
3
|
| 11 |
S |
5000 |
333.15 |
3
|
| 1 |
F |
283 |
353.15 |
4
|
| 1 |
F |
361 |
353.15 |
4
|
| 1 |
F |
515 |
353.15 |
4
|
| 1 |
F |
638 |
353.15 |
4
|
| 1 |
F |
854 |
353.15 |
4
|
| 1 |
F |
1024 |
353.15 |
4
|
| 1 |
F |
1030 |
353.15 |
4
|
| 1 |
F |
1045 |
353.15 |
4
|
| 1 |
F |
1767 |
353.15 |
4
|
| 1 |
F |
1777 |
353.15 |
4
|
| 1 |
F |
1856 |
353.15 |
4
|
| 1 |
F |
1951 |
353.15 |
4
|
| 1 |
F |
1964 |
353.15 |
4
|
| 1 |
F |
2884 |
353.15 |
4
|
| 1 |
S |
5000 |
353.15 |
4
|
Result
The following function is used for the Ln-Mean [math]\displaystyle{ \,\!\mu {}' }[/math] :
- [math]\displaystyle{ \,\!\mu {}'=\beta _{0}+\beta _{1}\times \frac{11605}{T} }[/math]
where T is the temperature; [math]\displaystyle{ \,\!\beta _{1} }[/math] is the activation energy; [math]\displaystyle{ \,\!11605 }[/math] is from reciprocal of the Boltzmann constant . This function can be written in the following way:
- [math]\displaystyle{ \,\!e^{{\mu }'}=e^{\alpha _{0}+\frac{\alpha _{1}}{T}} }[/math]
The above equation is the general log-linear model in ALTA. In ALTA, the coefficients are denoted by [math]\displaystyle{ \,\!\alpha _{i} }[/math] .
In fact, the above model also can be expressed using the traditional Arrhenius model:
- [math]\displaystyle{ \,\!e^{{\mu }'}=e^{\alpha _{0}+\frac{\alpha _{1}}{T}}=C\times e^{\frac{B}{T}} }[/math]
In the book, the following results are provided:
- ML estimations for the model parameters are: [math]\displaystyle{ \,\!\sigma =0.98 }[/math] ,[math]\displaystyle{ \,\!\beta _{0}=-13.469 }[/math] , [math]\displaystyle{ \,\!\beta _{1}=0.6279 }[/math] (or [math]\displaystyle{ \,\!\alpha _{1}=7286.78 }[/math]).
- The 95% confidence interval for [math]\displaystyle{ \,\! }[/math] is [0.75, 1.28], for [math]\displaystyle{ \,\! }[/math] is [-19.1, -7.8] and for [math]\displaystyle{ \,\! }[/math] is [0.47, 0.79].
- The variance/covariance matrix for , , and is
- [math]\displaystyle{ \,\!\begin{bmatrix}
0.0176 & -0.195 & 0.0059\\
-0.195 & 8.336 & -0.239\\
0.0059 & -.239& 0.0069
\end{bmatrix} }[/math]
In terms of [math]\displaystyle{ \,\! }[/math] , [math]\displaystyle{ \,\! }[/math] and [math]\displaystyle{ \,\! }[/math] , the variance/covariance matrix is
- [math]\displaystyle{ \,\! }[/math]
Results in ALTA