Likelihood Ratio Test Example

This example appears in the Accelerated life testing reference.

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:


 * $$\widehat{\beta }=\ 2.965820\,\!$$


 * $$\widehat{B}=\ 10,679.567542\,\!$$


 * $$\widehat{C}=\ 2.396615\cdot {{10}^{-9}}\,\!$$

The assumption of a common $$\beta \,\!$$ across the different stress levels can be visually assessed by using a probability plot. As you can see in the following plot, 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. For example, in the following figure, the $$\beta s\,\!$$ are compared for equality at the 10% level.



The LR test statistic, $$T\,\!$$, is calculated to be 0.481. Therefore, $$T=0.481\le 4.605={{\chi }^{2}}(0.9;2),\,\!$$ the $${\beta }'\,\!$$ s do not differ significantly at the 10% level. The individual likelihood values for each of the test stresses are shown next.