Likelihood Ratio Test Example - Revision history

Lisa Hacker at 19:04, 18 September 2023

2023-09-18T19:04:14Z

← Older revision		Revision as of 19:04, 18 September 2023
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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].''

	</noinclude>		</noinclude>

Nicolette Young at 17:18, 21 December 2015

2015-12-21T17:18:39Z

← Older revision		Revision as of 17:18, 21 December 2015
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	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.		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\|~~600px~~\|Probability plot of the three test stress levels.]]		[[Image:3linedplot.png\|center\|700px\|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.		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.

Nicolette Young at 17:18, 21 December 2015

2015-12-21T17:18:16Z

← Older revision		Revision as of 17:18, 21 December 2015
Line 35:		Line 35:
	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.		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\|~~500px~~\|Probability plot of the three test stress levels.]]		[[Image:3linedplot.png\|center\|600px\|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.		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.

Richard House at 22:46, 21 September 2012

2012-09-21T22:46:37Z

← Older revision		Revision as of 22:46, 21 September 2012
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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.		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\|500px\|Probability plot of the three test stress levels.]]		[[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.		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 10% level. The individual likelihood values for each of the test stresses are shown next.

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

Richard House at 04:39, 22 August 2012

2012-08-22T04:39:16Z

← Older revision		Revision as of 04:39, 22 August 2012
Line 26:		Line 26:

	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>
Line 34:		Line 33:
	::<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.
	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\|500px\|Probability plot of the three test stress levels.]]		[[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.
	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\|]]

Kate Racaza: pulled content from Additional_Tools page

2012-08-14T02:45:48Z

pulled content from Additional_Tools page

New page

<noinclude>{{Banner ALTA Examples}}
''This example appears in the [[Additional_Tools|Accelerated Life Testing Data Analysis Reference]] book.''

</noinclude>
Consider the following times-to-failure data at three different stress levels.

{||border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
!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.

[[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|]]

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|]]