Common Shape Parameter Likelihood Ratio Test

In order to assess the assumption of a common shape parameter among the data obtained at various stress levels, the likelihood ratio (LR) test can be utilized [ 31]. This test applies to any distribution with a shape parameter. In the case of ALTA, it applies to the Weibull and lognormal distributions. When Weibull is used as the underlying life distribution, the shape parameter, β, is assumed to be constant across the different stress levels (i.e. stress independent). Similarly, the parameter , of the lognormal distribution is assumed to be constant across the different stress levels.

The likelihood ratio test is performed by first obtaining the LR test statistic, T. If the true shape parameters are equal, then the distribution of T is approximately chi-square with n - 1 degrees of freedom, where n is the number of test stress levels with two or more exact failure points. The LR test statistic, T, is calculated as follows:

are the likelihood values obtained by fitting a separate distribution to the data from each of the n test stress levels (with two or more exact failure times). The likelihood value , is obtained by fitting a model with a common shape parameter and a separate scale parameter for each of the n stress levels, using indicator variables.

Once the LR statistic has been calculated, then:

If T (α; n - 1), the n shape parameter estimates do not differ statistically significantly at the 100% level.
If T > (α; n - 1), the n shape parameter estimates differ statistically significantly at the 100α% level.
(α ;n - 1) is the 100(α) percentile of the chi-square distribution with n – 1 degrees of freedom.

Example

Consider the following times-to-failure data at three different stress levels.

The data set was analyzed using an Arrhenius-Weibull model. The analysis yields:

13.12.1.gif

The assumption of a common β across the different stress levels can be assessed visually using a probability plot.

Fig. 1: Probability plot of the three test stress levels.

In Figure 1, it can be seen that the plotted data from the different stress levels seem to be fairly parallel.

A better assessment can be made with the LR test, which can be performed using the Likelihood Ratio Test tool in ALTA 7. For example, in the following figure, the βs are compared for equality at the 10% level.

The individual likelihood values for each of the test stresses can be found in the Results tab of the Likelihood Ratio Test window.

The LR test statistic, T, is calculated to be 0.481. Therefore, T = 0.481 4.605 = (0.9; 2), the βs do not differ significantly at the 10% level.