Arrhenius-Lognormal Model
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Arrhenius-Lognormal Model |
This example validates the results for the Arrhenius life stress relationship with a Lognormal distribution in ALTA standard folios.
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. The following table shows the data.
| 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 the 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{ \,\!\sigma }[/math] is [0.75, 1.28], for [math]\displaystyle{ \,\!\beta _{0} }[/math] is [-19.1, -7.8] and for [math]\displaystyle{ \,\!\beta _{1} }[/math] is [0.47, 0.79].
- The variance/covariance matrix for [math]\displaystyle{ \,\!\sigma }[/math] , [math]\displaystyle{ \,\!\beta _{0} }[/math] and [math]\displaystyle{ \,\!\beta _{1} }[/math] is:
- [math]\displaystyle{ \,\!\begin{bmatrix} 0.0176 & -0.195 & 0.0059\\ -0.195 & 8.336 & -0.239\\ 0.0059 & -0.239 & 0.0069 \end{bmatrix} }[/math]
- In terms of [math]\displaystyle{ \,\!\sigma }[/math] , [math]\displaystyle{ \,\!\alpha _{0} }[/math] and [math]\displaystyle{ \,\!\alpha _{1} }[/math], the variance/covariance matrix is:
- [math]\displaystyle{ \,\!\begin{bmatrix} 0.0176 & -0.195 & 68.4695\\ -0.195 & 8.336 & -2773.5950\\ 68.4695 & -2773.5950 & 929264.5725 \end{bmatrix} }[/math]
- The log-likelihood value is -321.7.
Results in ALTA
- ML estimations for the model parameters are:
- The 95% confidence intervals are:
The variance/covariance matrix for [math]\displaystyle{ \,\!\sigma }[/math] , [math]\displaystyle{ \,\!\alpha _{0} }[/math] and [math]\displaystyle{ \,\!\alpha _{1} }[/math] is:
- The log-likelihood value is -321.7.
It can be seen that all the results in ALTA are very close to the results in the book.