Arrhenius-Lognormal Model

This example validates the calculation for the Arrhenius life stress relationship with a Lognormal distribution.

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.

Device-A was tested under several different temperature settings.

The following function is used for the Ln-Mean $$\,\!\mu {}'$$:


 * $$\,\!\mu {}'=\beta _{0}+\beta _{1}\times \frac{11605}{T}$$

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


 * $$\,\!e^{{\mu }'}=e^{\alpha _{0}+\frac{\alpha _{1}}{T}}$$

The above equation is the general log-linear model in ALTA. In ALTA, the coefficients are denoted by $$\,\!\alpha _{i}$$.

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


 * $$\,\!e^{{\mu }'}=e^{\alpha _{0}+\frac{\alpha _{1}}{T}}=C\times e^{\frac{B}{T}}$$

In the book, the following results are provided:


 * ML estimations for the model parameters are: $$\,\!\sigma =0.98$$, $$\,\!\beta _{0}=-13.469$$ , $$\,\!\beta _{1}=0.6279$$ (or $$\,\!\alpha _{1}=7286.78$$).


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


 * The variance/covariance matrix for $$\,\!\sigma$$, $$\,\!\beta _{0}$$ and $$\,\!\beta _{1}$$ is:


 * $$\,\!\begin{bmatrix}

0.0176 & -0.195 & 0.0059\\ -0.195 & 8.336 & -0.239\\ 0.0059 & -0.239 & 0.0069 \end{bmatrix}$$


 * In terms of $$\,\!\sigma$$, $$\,\!\alpha _{0}$$ and $$\,\!\alpha _{1}$$, the variance/covariance matrix is:


 * $$\,\!\begin{bmatrix}

0.0176 & -0.195 & 68.4695\\ -0.195 & 8.336 & -2773.5950\\ 68.4695 & -2773.5950 & 929264.5725 \end{bmatrix}$$


 * ML estimations for the model parameters are:




 * The 95% confidence intervals are:



The variance/covariance matrix for $$\,\!\sigma$$, $$\,\!\alpha _{0}$$ and $$\,\!\alpha _{1}$$ 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.