Likelihood Ratio Test Example
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This example appears in the Accelerated life testing reference.
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]\displaystyle{ \widehat{\beta }=\ 2.965820\,\! }[/math]
- [math]\displaystyle{ \widehat{B}=\ 10,679.567542\,\! }[/math]
- [math]\displaystyle{ \widehat{C}=\ 2.396615\cdot {{10}^{-9}}\,\! }[/math]
The assumption of a common [math]\displaystyle{ \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.
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]\displaystyle{ \beta s\,\! }[/math] are compared for equality at the 10% level.
The LR test statistic, [math]\displaystyle{ T\,\! }[/math], is calculated to be 0.481. Therefore, [math]\displaystyle{ T=0.481\le 4.605={{\chi }^{2}}(0.9;2),\,\! }[/math] the [math]\displaystyle{ {\beta }'\,\! }[/math] s do not differ significantly at the 10% level. The individual likelihood values for each of the test stresses are shown next.
