Likelihood Ratio Test Example

From ReliaWiki
Jump to navigation Jump to search
ALTA Examples Banner.png

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:

[math]\widehat{\beta }=\ 2.965820\,\![/math]
[math]\widehat{B}=\ 10,679.567542\,\![/math]
[math]\widehat{C}=\ 2.396615\cdot {{10}^{-9}}\,\![/math]

The assumption of a common [math]\beta \,\![/math] 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.

Probability plot of the three test stress levels.

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 [math]\beta s\,\![/math] are compared for equality at the 10% level.


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