Accelerated Life Testing and ALTA - Revision history

Nicolette Young: /* Standardized vs. Fitted Values */

2015-12-16T22:37:24Z

Standardized vs. Fitted Values

← Older revision		Revision as of 22:37, 16 December 2015
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	A Standardized vs. Fitted Value plot helps to detect behavior not modeled in the underlying relationship. However, when heavy censoring is present, the plot is more difficult to interpret. In a Standardized vs. Fitted Value plot, the standardized residuals are plotted versus the scale parameter of the underlying life distribution (which is a function of stress) on log-linear paper (linear on the Y-axis). Therefore, the standardized residuals are plotted versus <math>\eta (V)\,\!</math> for the Weibull distribution, versus <math>{\mu }'(V)\,\!</math> for the lognormal distribution and versus <math>m(V)\,\!</math> for the exponential distribution.		A Standardized vs. Fitted Value plot helps to detect behavior not modeled in the underlying relationship. However, when heavy censoring is present, the plot is more difficult to interpret. In a Standardized vs. Fitted Value plot, the standardized residuals are plotted versus the scale parameter of the underlying life distribution (which is a function of stress) on log-linear paper (linear on the Y-axis). Therefore, the standardized residuals are plotted versus <math>\eta (V)\,\!</math> for the Weibull distribution, versus <math>{\mu }'(V)\,\!</math> for the lognormal distribution and versus <math>m(V)\,\!</math> for the exponential distribution.

	[[Image:standardized_fitted_value.png\|center\|~~400px~~\|Standardized Residuals vs. Fitted Value.]]		[[Image:standardized_fitted_value.png\|center\|500px\|Standardized Residuals vs. Fitted Value.]]

Nicolette Young: /* Cox-Snell Residuals */

2015-12-16T22:37:15Z

Cox-Snell Residuals

← Older revision		Revision as of 22:37, 16 December 2015
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	where <math>R({{T}_{i}})\,\!</math> is the calculated reliability value at failure time <math>{{T}_{i}}.\,\!</math> The Cox-Snell residuals are plotted on an exponential probability paper.		where <math>R({{T}_{i}})\,\!</math> is the calculated reliability value at failure time <math>{{T}_{i}}.\,\!</math> The Cox-Snell residuals are plotted on an exponential probability paper.

	[[Image:cox-snell_plot.png\|center\|~~400px~~\|Probability plot of the Cox-Snell residuals.]]		[[Image:cox-snell_plot.png\|center\|500px\|Probability plot of the Cox-Snell residuals.]]

	===Standardized vs. Fitted Values===		===Standardized vs. Fitted Values===

Nicolette Young: /* Standardized Residuals (SR) */

2015-12-16T22:37:02Z

Standardized Residuals (SR)

← Older revision		Revision as of 22:37, 16 December 2015
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	Then, under the assumed model, the standardized residuals should be normally distributed with a mean of 0 and a standard deviation of 1 ( <math>\tilde{\ }N(0,1)\,\!</math> ). Consequently, the standardized residuals for the lognormal distribution are commonly displayed on a normal probability plot.		Then, under the assumed model, the standardized residuals should be normally distributed with a mean of 0 and a standard deviation of 1 ( <math>\tilde{\ }N(0,1)\,\!</math> ). Consequently, the standardized residuals for the lognormal distribution are commonly displayed on a normal probability plot.

	[[Image:standardized_weibull.png\|center\|~~400px~~\|Probability plot of standardized residuals.]]		[[Image:standardized_weibull.png\|center\|500px\|Probability plot of standardized residuals.]]

	===Cox-Snell Residuals===		===Cox-Snell Residuals===

Nicolette Young: /* Acceleration Factor Plots */

2015-12-16T22:36:48Z

Acceleration Factor Plots

← Older revision		Revision as of 22:36, 16 December 2015
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	The Acceleration Factor vs. Stress plot is generated using the equation above at a constant use stress level and at a varying accelerated stress. In the below figure, the Acceleration Factor vs. Stress was plotted for a constant use level of 300K. Since <math>{{L}_{A}}={{L}_{u}}\,\!</math>, the value of the acceleration factor at 300K is equal to 1. The acceleration factor for a temperature of 450K is approximately 8. This means that the life at the use level of 300K is eight times higher than the life at 450K.		The Acceleration Factor vs. Stress plot is generated using the equation above at a constant use stress level and at a varying accelerated stress. In the below figure, the Acceleration Factor vs. Stress was plotted for a constant use level of 300K. Since <math>{{L}_{A}}={{L}_{u}}\,\!</math>, the value of the acceleration factor at 300K is equal to 1. The acceleration factor for a temperature of 450K is approximately 8. This means that the life at the use level of 300K is eight times higher than the life at 450K.

	[[Image:accfactplot.png\|center\|~~400px~~\|Acceleration Factor vs. Stress Plot.]]		[[Image:accfactplot.png\|center\|500px\|Acceleration Factor vs. Stress Plot.]]

	==Residual Plots==<!-- THIS SECTION HEADER IS LINKED TO EXTERNAL SEARCH IN THE W++/A HELP FILE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK. -->		==Residual Plots==<!-- THIS SECTION HEADER IS LINKED TO EXTERNAL SEARCH IN THE W++/A HELP FILE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK. -->

Nicolette Young: /* Standard Deviation Plots */

2015-12-16T22:21:31Z

Standard Deviation Plots

← Older revision		Revision as of 22:21, 16 December 2015
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	Standard Deviation vs. Stress is a useful plot in accelerated life testing analysis and provides information about the spread of the data at each stress level.		Standard Deviation vs. Stress is a useful plot in accelerated life testing analysis and provides information about the spread of the data at each stress level.

	[[Image:stdevplot.png\|center\|~~400px~~\|Standard Deviation vs. Stress plot.]]		[[Image:stdevplot.png\|center\|500px\|Standard Deviation vs. Stress plot.]]

	==Acceleration Factor Plots==		==Acceleration Factor Plots==

David J. Groebel: /* Standardized Residuals (SR) */

2014-10-13T22:54:14Z

Standardized Residuals (SR)

← Older revision		Revision as of 22:54, 13 October 2014
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	'''SR for the Lognormal Distribution'''		'''SR for the Lognormal Distribution'''

	Once the parameters have been estimated ~~using rank regression~~, the fitted or calculated responses can be calculated by:		Once the parameters have been estimated, the fitted or calculated responses can be calculated by:

	::<math>{{\widehat{e}}_{i}}=\frac{\ln ({{T}_{i}})-{{\widehat{\mu }}^{\prime }}}{{{\widehat{\sigma }}_{{{T}'}}}}\,\!</math>		::<math>{{\widehat{e}}_{i}}=\frac{\ln ({{T}_{i}})-{{\widehat{\mu }}^{\prime }}}{{{\widehat{\sigma }}_{{{T}'}}}}\,\!</math>

Richard House at 21:42, 21 September 2012

2012-09-21T21:42:28Z

Richard House at 02:28, 30 August 2012

2012-08-30T02:28:16Z

← Older revision		Revision as of 02:28, 30 August 2012
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	==Pdf Plots==		==Pdf Plots==
	The ~~<math>~~pdf~~</math>~~ is a function of time and stress. For this reason, a 2-dimensional plot of the ~~<math>~~pdf~~</math>~~ vs. Time at a given stress and a 3-dimensional plot of the ~~<math>~~pdf~~</math>~~ vs. Time and Stress can be obtained in ALTA.		The ''pdf'' is a function of time and stress. For this reason, a 2-dimensional plot of the ''pdf'' vs. Time at a given stress and a 3-dimensional plot of the ''pdf'' vs. Time and Stress can be obtained in ALTA.

	[[Image:pdf2d.png\|center\|400px\| 2D ''pdf'' plot.]]		[[Image:pdf2d.png\|center\|400px\| 2D ''pdf'' plot.]]
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	[[Image:pdf3d.gif\|center\|400px\| 3D ''pdf'' plot.]]		[[Image:pdf3d.gif\|center\|400px\| 3D ''pdf'' plot.]]

	A ~~<math>~~pdf~~</math>~~ plot represents the relative frequency of failures as a function of time and stress. Although the ~~<math>~~pdf~~</math>~~ plot is less important in most reliability applications than the other plots available in ALTA, it provides a good way of visualizing the distribution and its characteristics such as its shape, skewness, mode, etc.		A ''pdf'' plot represents the relative frequency of failures as a function of time and stress. Although the ''pdf'' plot is less important in most reliability applications than the other plots available in ALTA, it provides a good way of visualizing the distribution and its characteristics such as its shape, skewness, mode, etc.

	==Life-Stress Plots==		==Life-Stress Plots==
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	[[Image:ch4-7.png\|center\|400px\|Typical Arrhenius Life vs. Stress plot.]]		[[Image:ch4-7.png\|center\|400px\|Typical Arrhenius Life vs. Stress plot.]]

	Each line in the figure above represents the path for extrapolating a life measure, such as a percentile, from one stress level to another. The slope and intercept of those lines are the parameters of the life-stress relationship (whenever the relationship can be linearized). The imposed ~~<math>~~pdfs~~</math>~~ represent the distribution of the data at each stress level.		Each line in the figure above represents the path for extrapolating a life measure, such as a percentile, from one stress level to another. The slope and intercept of those lines are the parameters of the life-stress relationship (whenever the relationship can be linearized). The imposed ''pdfs'' represent the distribution of the data at each stress level.

	==Standard Deviation Plots==		==Standard Deviation Plots==

Richard House: /* Life-Stress Plots */

2012-08-21T23:26:49Z

Life-Stress Plots

← Older revision		Revision as of 23:26, 21 August 2012
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	[[Image:ch4-7.png\|center\|400px\|Typical Arrhenius Life vs. Stress plot.]]		[[Image:ch4-7.png\|center\|400px\|Typical Arrhenius Life vs. Stress plot.]]

	Each line in the figure above represents the path for extrapolating a life measure, such as a percentile, from one stress level to another. The slope and intercept of those lines are the parameters of the life-stress relationship (whenever the relationship can be linearized). The imposed <math>~~pdf~~</math>s represent the distribution of the data at each stress level.		Each line in the figure above represents the path for extrapolating a life measure, such as a percentile, from one stress level to another. The slope and intercept of those lines are the parameters of the life-stress relationship (whenever the relationship can be linearized). The imposed <math>pdfs</math> represent the distribution of the data at each stress level.

	==Standard Deviation Plots==		==Standard Deviation Plots==

Richard House: /* Standardized vs. Fitted Values */

2012-08-21T23:26:03Z

Standardized vs. Fitted Values

← Older revision		Revision as of 23:26, 21 August 2012
Line 113:		Line 113:

	===Standardized vs. Fitted Values===		===Standardized vs. Fitted Values===
	A Standardized vs. Fitted Value plot helps to detect behavior not modeled in the underlying relationship. However, when heavy censoring is present, the plot is more difficult to interpret. In a Standardized vs. Fitted Value plot, the standardized residuals are plotted versus the scale parameter of the underlying life distribution (which is a function of stress) on log-linear paper (linear on the Y-axis). Therefore, the standardized residuals are plotted versus <math>\eta (V)</math> for the Weibull distribution, versus <math>{\mu }'(V)</math> for the lognormal distribution and versus <math>m(V)</math> for the exponential distribution.		A Standardized vs. Fitted Value plot helps to detect behavior not modeled in the underlying relationship. However, when heavy censoring is present, the plot is more difficult to interpret. In a Standardized vs. Fitted Value plot, the standardized residuals are plotted versus the scale parameter of the underlying life distribution (which is a function of stress) on log-linear paper (linear on the Y-axis). Therefore, the standardized residuals are plotted versus <math>\eta (V)\,\!</math> for the Weibull distribution, versus <math>{\mu }'(V)\,\!</math> for the lognormal distribution and versus <math>m(V)\,\!</math> for the exponential distribution.

	[[Image:standardized_fitted_value.png\|center\|400px\|Standardized Residuals vs. Fitted Value.]]		[[Image:standardized_fitted_value.png\|center\|400px\|Standardized Residuals vs. Fitted Value.]]

← Older revision		Revision as of 21:42, 21 September 2012
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	The acceleration factor is a unitless number that relates a product's life at an accelerated stress level to the life at the use stress level. It is defined by:		The acceleration factor is a unitless number that relates a product's life at an accelerated stress level to the life at the use stress level. It is defined by:

	::<math>{{A}_{F}}=\frac{{{L}_{u}}}{{{L}_{A}}}</math>		::<math>{{A}_{F}}=\frac{{{L}_{u}}}{{{L}_{A}}}\,\!</math>

	where:		where:
	::*<math>{{L}_{u}}\,\!</math> is the life at the use stress level.		::*<math>{{L}_{u}}\,\!</math> is the life at the use stress level.
	::*<math>{{L}_{A}}\,\!</math> is the life at the accelerated level.		::*<math>{{L}_{A}}\,\!</math> is the life at the accelerated level.

	As it can be seen, the acceleration factor depends on the life-stress relationship (i.e., Arrhenius, Eyring, etc.) and is thus a function of stress.		As it can be seen, the acceleration factor depends on the life-stress relationship (i.e., Arrhenius, Eyring, etc.) and is thus a function of stress.

	The Acceleration Factor vs. Stress plot is generated using the equation above at a constant use stress level and at a varying accelerated stress. In the below figure, the Acceleration Factor vs. Stress was plotted for a constant use level of 300K. Since <math>{{L}_{A}}={{L}_{u}}\,\!</math> , the value of the acceleration factor at 300K is equal to 1. The acceleration factor for a temperature of 450K is approximately 8. This means that the life at the use level of 300K is eight times higher than the life at 450K.		The Acceleration Factor vs. Stress plot is generated using the equation above at a constant use stress level and at a varying accelerated stress. In the below figure, the Acceleration Factor vs. Stress was plotted for a constant use level of 300K. Since <math>{{L}_{A}}={{L}_{u}}\,\!</math>, the value of the acceleration factor at 300K is equal to 1. The acceleration factor for a temperature of 450K is approximately 8. This means that the life at the use level of 300K is eight times higher than the life at 450K.

	[[Image:accfactplot.png\|center\|400px\|Acceleration Factor vs. Stress Plot.]]		[[Image:accfactplot.png\|center\|400px\|Acceleration Factor vs. Stress Plot.]]
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	Once the parameters have been estimated, the standardized residuals for the Weibull distribution can be calculated by:		Once the parameters have been estimated, the standardized residuals for the Weibull distribution can be calculated by:

	::<math>{{\widehat{e}}_{i}}=\widehat{\beta }\left[ \ln ({{T}_{i}})-\ln (\widehat{\eta }(V)) \right]</math>		::<math>{{\widehat{e}}_{i}}=\widehat{\beta }\left[ \ln ({{T}_{i}})-\ln (\widehat{\eta }(V)) \right]\,\!</math>

	Then, under the assumed model, these residuals should look like a sample from an extreme value distribution with a mean of 0. For the Weibull distribution the standardized residuals are plotted on a smallest extreme value probability paper. If the Weibull distribution adequately describes the data, then the standardized residuals should appear to follow a straight line on such a probability plot. Note that when an observation is censored (suspended), the corresponding residual is also censored.		Then, under the assumed model, these residuals should look like a sample from an extreme value distribution with a mean of 0. For the Weibull distribution the standardized residuals are plotted on a smallest extreme value probability paper. If the Weibull distribution adequately describes the data, then the standardized residuals should appear to follow a straight line on such a probability plot. Note that when an observation is censored (suspended), the corresponding residual is also censored.
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	Once the parameters have been estimated using rank regression, the fitted or calculated responses can be calculated by:		Once the parameters have been estimated using rank regression, the fitted or calculated responses can be calculated by:

	::<math>{{\widehat{e}}_{i}}=\frac{\ln ({{T}_{i}})-{{\widehat{\mu }}^{\prime }}}{{{\widehat{\sigma }}_{{{T}'}}}}</math>		::<math>{{\widehat{e}}_{i}}=\frac{\ln ({{T}_{i}})-{{\widehat{\mu }}^{\prime }}}{{{\widehat{\sigma }}_{{{T}'}}}}\,\!</math>

	Then, under the assumed model, the standardized residuals should be normally distributed with a mean of 0 and a standard deviation of 1 ( <math>\tilde{\ }N(0,1)</math> ). Consequently, the standardized residuals for the lognormal distribution are commonly displayed on a normal probability plot.		Then, under the assumed model, the standardized residuals should be normally distributed with a mean of 0 and a standard deviation of 1 ( <math>\tilde{\ }N(0,1)\,\!</math> ). Consequently, the standardized residuals for the lognormal distribution are commonly displayed on a normal probability plot.

	[[Image:standardized_weibull.png\|center\|400px\|Probability plot of standardized residuals.]]		[[Image:standardized_weibull.png\|center\|400px\|Probability plot of standardized residuals.]]
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	The Cox-Snell residuals are given by:		The Cox-Snell residuals are given by:

	::<math>{{\widehat{e}}_{i}}=-\ln [R({{T}_{i}})]</math>		::<math>{{\widehat{e}}_{i}}=-\ln [R({{T}_{i}})]\,\!</math>

	where <math>R({{T}_{i}})\,\!</math> is the calculated reliability value at failure time <math>{{T}_{i}}.\,\!</math> The Cox-Snell residuals are plotted on an exponential probability paper.		where <math>R({{T}_{i}})\,\!</math> is the calculated reliability value at failure time <math>{{T}_{i}}.\,\!</math> The Cox-Snell residuals are plotted on an exponential probability paper.

	~~::<math></math>~~

	[[Image:cox-snell_plot.png\|center\|400px\|Probability plot of the Cox-Snell residuals.]]		[[Image:cox-snell_plot.png\|center\|400px\|Probability plot of the Cox-Snell residuals.]]

	===Standardized vs. Fitted Values===		===Standardized vs. Fitted Values===
	A Standardized vs. Fitted Value plot helps to detect behavior not modeled in the underlying relationship. However, when heavy censoring is present, the plot is more difficult to interpret. In a Standardized vs. Fitted Value plot, the standardized residuals are plotted versus the scale parameter of the underlying life distribution (which is a function of stress) on log-linear paper (linear on the Y-axis). Therefore, the standardized residuals are plotted versus <math>\eta (V)\,\!</math> for the Weibull distribution, versus <math>{\mu }'(V)\,\!</math> for the lognormal distribution and versus <math>m(V)\,\!</math> for the exponential distribution.		A Standardized vs. Fitted Value plot helps to detect behavior not modeled in the underlying relationship. However, when heavy censoring is present, the plot is more difficult to interpret. In a Standardized vs. Fitted Value plot, the standardized residuals are plotted versus the scale parameter of the underlying life distribution (which is a function of stress) on log-linear paper (linear on the Y-axis). Therefore, the standardized residuals are plotted versus <math>\eta (V)\,\!</math> for the Weibull distribution, versus <math>{\mu }'(V)\,\!</math> for the lognormal distribution and versus <math>m(V)\,\!</math> for the exponential distribution.

	[[Image:standardized_fitted_value.png\|center\|400px\|Standardized Residuals vs. Fitted Value.]]		[[Image:standardized_fitted_value.png\|center\|400px\|Standardized Residuals vs. Fitted Value.]]