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]\,\!\mu {}'[/math]:


[math]\,\!\mu {}'=\beta _{0}+\beta _{1}\times \frac{11605}{T}[/math]


where T is the temperature; [math]\,\!\beta _{1}[/math] is the activation energy; [math]\,\!11605[/math] is from the reciprocal of the Boltzmann constant. This function can be written in the following way:


[math]\,\!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]\,\!\alpha _{i}[/math].


In fact, the above model also can be expressed using the traditional Arrhenius model:


[math]\,\!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]\,\!\sigma =0.98[/math] , [math]\,\!\beta _{0}=-13.469[/math] , [math]\,\!\beta _{1}=0.6279[/math] (or [math]\,\!\alpha _{1}=7286.78[/math]).


  • The 95% confidence interval for [math]\,\!\sigma[/math] is [0.75, 1.28], for [math]\,\!\beta _{0}[/math] is [-19.1, -7.8] and for [math]\,\!\beta _{1}[/math] is [0.47, 0.79].


  • The variance/covariance matrix for [math]\,\!\sigma[/math] , [math]\,\!\beta _{0}[/math] and [math]\,\!\beta _{1}[/math] is:


[math]\,\!\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]\,\!\sigma[/math] , [math]\,\!\alpha _{0}[/math] and [math]\,\!\alpha _{1}[/math], the variance/covariance matrix is:


[math]\,\!\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:
Arrhenius Lognormal Analysis Summary.png


  • The 95% confidence intervals are:
Arrhenius Lognormal Parameter Bounds.png


The variance/covariance matrix for [math]\,\!\sigma[/math] , [math]\,\!\alpha _{0}[/math] and [math]\,\!\alpha _{1}[/math] is:

Arrhenius Lognormal Var Cov Results.png


  • 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.