Arrhenius-Lognormal Model
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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