Likelihood Ratio Test Example: Difference between revisions

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<noinclude>{{Banner ALTA Examples}}
<noinclude>{{Banner ALTA Examples}}
''This example appears in the [[Additional_Tools|Accelerated Life Testing Data Analysis Reference]] book.''
''This example appears in the [https://help.reliasoft.com/reference/accelerated_life_testing_data_analysis Accelerated life testing reference].''


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


::<math>\widehat{\beta }=\ 2.965820\,\!</math>


::<math>\widehat{\beta }=\ 2.965820</math>
::<math>\widehat{B}=\ 10,679.567542\,\!</math>


::<math>\widehat{B}=\ 10,679.567542</math>
::<math>\widehat{C}=\ 2.396615\cdot {{10}^{-9}}\,\!</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.


[[Image:3linedplot.png|center|700px|Probability plot of the three test stress levels.]]


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.
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.   
 
 
[[Image:3linedplot.png|center|500px|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.   


[[Image:lkt.png|center|400px|]]
[[Image:lkt.png|center|400px|]]


 
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.
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 <math>10%</math>  level. The individual likelihood values for each of the test stresses are shown next.


[[Image:lktr.png|center|400px|]]
[[Image:lktr.png|center|400px|]]

Latest revision as of 19:04, 18 September 2023

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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.

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

Lkt.png

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.

Lktr.png