# Likelihood Ratio Test Example

This example appears in the Accelerated Life Testing Data Analysis Reference book.

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

Stress 406 K 416 K 426 K
Time Failed (hrs) 248 164 92
456 176 105
528 289 155
731 319 184
813 340 219
543 235

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.