Multivariable Relationships: General Log-Linear and Proportional Hazards - Revision history

Lisa Hacker at 23:58, 23 June 2015

2015-06-23T23:58:29Z

← Older revision		Revision as of 23:58, 23 June 2015
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	The data can now be entered in ALTA and, with the assumption of an underlying life distribution and using MLE, the parameters of this model can be obtained.		The data can now be entered in ALTA and, with the assumption of an underlying life distribution and using MLE, the parameters of this model can be obtained.

	~~[[Category: Completed Theoretical Review]]~~

Chris Kahn at 17:18, 28 January 2013

2013-01-28T17:18:39Z

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Chris Kahn at 23:32, 25 January 2013

2013-01-25T23:32:36Z

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 =General Log-Linear Relationship=
-When a test involves multiple accelerating stresses or requires the inclusion of an engineering variable, a general multivariable relationship is needed. Such a relationship is the general log-linear relationship, which describes a life characteristic as a function of a vector of <math>n\,\!</math> stresses, or <math>\underline{X}=({{X}_{1}},{{X}_{2}}...{{X}_{n}}).\,\!</math> ALTA includes this relationship and allows up to eight stresses. Mathematically the relationship is given by:
+{{:General Log-Linear Relationship}}
 =Proportional Hazards Model=

Lisa Hacker: /* Indicator Variables */

2013-01-25T16:34:30Z

Indicator Variables

← Older revision		Revision as of 16:34, 25 January 2013
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	=Indicator Variables=		=Indicator Variables=
	Another advantage of the models ~~presented in this chapter~~ is that they allow for simultaneous analysis of continuous and categorical variables. Categorical variables are variables that take on discrete values such as the lot designation for products from different manufacturing lots. In this example, lot is a categorical variable, and it can be expressed in terms of indicator variables. Indicator variables only take a value of 1 or 0. For example, consider a sample of test units. A number of these units were obtained from Lot 1, others from Lot 2, and the rest from Lot 3. These three lots can be represented with the use of indicator variables, as follows:		Another advantage with both the General Log-Linear and Proportional Hazards models is that they allow for simultaneous analysis of continuous and categorical variables. Categorical variables are variables that take on discrete values such as the lot designation for products from different manufacturing lots. In this example, lot is a categorical variable, and it can be expressed in terms of indicator variables. Indicator variables only take a value of 1 or 0. For example, consider a sample of test units. A number of these units were obtained from Lot 1, others from Lot 2, and the rest from Lot 3. These three lots can be represented with the use of indicator variables, as follows:

	*Define two indicator variables, <math>{{X}_{1}}\,\!</math> and <math>{{X}_{2}}.\,\!</math>		*Define two indicator variables, <math>{{X}_{1}}\,\!</math> and <math>{{X}_{2}}.\,\!</math>

Richard House at 17:53, 24 September 2012

2012-09-24T17:53:31Z

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Richard House: /* Parametric Model Formulation */

2012-09-05T16:13:27Z

Parametric Model Formulation

← Older revision		Revision as of 16:13, 5 September 2012
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	\end{align}</math>		\end{align}</math>

	Solving for the parameters that maximize the log-likelihood function will yield the parameters for the PH-Weibull model. Note that for <math>\beta =1 \,\!</math>, the log-likelihood function becomes the log-likelihood function for the PH-exponential model, which is similar to the original form of the proportional hazards model proposed by Cox [[Appendix_E:_References\|[28]]].		Solving for the parameters that maximize the log-likelihood function will yield the parameters for the PH-Weibull model. Note that for <math>\beta =1 \,\!</math>, the log-likelihood function becomes the log-likelihood function for the PH-exponential model, which is similar to the original form of the proportional hazards model proposed by Cox and Oakes [[Appendix_E:_References\|[39]]].

	Note that the likelihood function of the GLL model is very similar to the likelihood function for the proportional hazards-Weibull model. In particular, the shape parameter of the Weibull distribution can be included in the regression coefficients as follows:		Note that the likelihood function of the GLL model is very similar to the likelihood function for the proportional hazards-Weibull model. In particular, the shape parameter of the Weibull distribution can be included in the regression coefficients as follows:

Richard House: /* Proportional Hazards Model */

2012-09-05T15:52:41Z

Proportional Hazards Model

← Older revision		Revision as of 15:52, 5 September 2012
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	=Proportional Hazards Model=		=Proportional Hazards Model=
	Introduced by D. R. Cox, the Proportional Hazards (PH) model was developed in order to estimate the effects of different covariates influencing the times-to-failure of a system.		Introduced by D. R. Cox, the Proportional Hazards (PH) model was developed in order to estimate the effects of different covariates influencing the times-to-failure of a system.
	The model has been widely used in the biomedical field, as discussed in [[Appendix_E:_References\|[22]]], and recently there has been an increasing interest in its application in reliability engineering. In its original form, the model is non-parametric, (i.e., no assumptions are made about the nature or shape of the underlying failure distribution). In this reference, the original non-parametric formulation as well as a parametric form of the model will be considered utilizing a Weibull life distribution. In ALTA, the proportional hazards model is included in its parametric form and can be used to analyze data with up to eight variables. The GLL-Weibull and GLL-exponential models are actually special cases of the proportional hazards model. However, when using the proportional hazards in ALTA, no transformation on the covariates (or stresses) can be performed.		The model has been widely used in the biomedical field, as discussed in Leemis [[Appendix_E:_References\|[22]]], and recently there has been an increasing interest in its application in reliability engineering. In its original form, the model is non-parametric, (i.e., no assumptions are made about the nature or shape of the underlying failure distribution). In this reference, the original non-parametric formulation as well as a parametric form of the model will be considered utilizing a Weibull life distribution. In ALTA, the proportional hazards model is included in its parametric form and can be used to analyze data with up to eight variables. The GLL-Weibull and GLL-exponential models are actually special cases of the proportional hazards model. However, when using the proportional hazards in ALTA, no transformation on the covariates (or stresses) can be performed.

	==Non-Parametric Model Formulation==		==Non-Parametric Model Formulation==

Richard House: /* Parametric Model Formulation */

2012-09-04T20:37:07Z

Parametric Model Formulation

← Older revision		Revision as of 20:37, 4 September 2012
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	\end{align}</math>		\end{align}</math>

	Solving for the parameters that maximize the log-likelihood function will yield the parameters for the PH-Weibull model. Note that for <math>\beta =1 \,\!</math>, the log-likelihood function becomes the log-likelihood function for the PH-exponential model, which is similar to the original form of the proportional hazards model proposed by Cox [[~~Reference Appendix D~~: ~~References~~\|[28]]].		Solving for the parameters that maximize the log-likelihood function will yield the parameters for the PH-Weibull model. Note that for <math>\beta =1 \,\!</math>, the log-likelihood function becomes the log-likelihood function for the PH-exponential model, which is similar to the original form of the proportional hazards model proposed by Cox [[Appendix_E:_References\|[28]]].

	Note that the likelihood function of the GLL model is very similar to the likelihood function for the proportional hazards-Weibull model. In particular, the shape parameter of the Weibull distribution can be included in the regression coefficients as follows:		Note that the likelihood function of the GLL model is very similar to the likelihood function for the proportional hazards-Weibull model. In particular, the shape parameter of the Weibull distribution can be included in the regression coefficients as follows:

Richard House: /* Proportional Hazards Model */

2012-09-04T20:36:41Z

Proportional Hazards Model

← Older revision		Revision as of 20:36, 4 September 2012
Line 112:		Line 112:
	=Proportional Hazards Model=		=Proportional Hazards Model=
	Introduced by D. R. Cox, the Proportional Hazards (PH) model was developed in order to estimate the effects of different covariates influencing the times-to-failure of a system.		Introduced by D. R. Cox, the Proportional Hazards (PH) model was developed in order to estimate the effects of different covariates influencing the times-to-failure of a system.
	The model has been widely used in the biomedical field [[~~Reference Appendix D~~: ~~References~~\|[22]]], and recently there has been an increasing interest in its application in reliability engineering. In its original form, the model is non-parametric, (i.e., no assumptions are made about the nature or shape of the underlying failure distribution). In this reference, the original non-parametric formulation as well as a parametric form of the model will be considered utilizing a Weibull life distribution. In ALTA, the proportional hazards model is included in its parametric form and can be used to analyze data with up to eight variables. The GLL-Weibull and GLL-exponential models are actually special cases of the proportional hazards model. However, when using the proportional hazards in ALTA, no transformation on the covariates (or stresses) can be performed.		The model has been widely used in the biomedical field, as discussed in [[Appendix_E:_References\|[22]]], and recently there has been an increasing interest in its application in reliability engineering. In its original form, the model is non-parametric, (i.e., no assumptions are made about the nature or shape of the underlying failure distribution). In this reference, the original non-parametric formulation as well as a parametric form of the model will be considered utilizing a Weibull life distribution. In ALTA, the proportional hazards model is included in its parametric form and can be used to analyze data with up to eight variables. The GLL-Weibull and GLL-exponential models are actually special cases of the proportional hazards model. However, when using the proportional hazards in ALTA, no transformation on the covariates (or stresses) can be performed.

	==Non-Parametric Model Formulation==		==Non-Parametric Model Formulation==

Richard House at 02:58, 30 August 2012

2012-08-30T02:58:09Z

← Older revision		Revision as of 02:58, 30 August 2012
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	===GLL Exponential===		===GLL Exponential===
	The GLL-exponential model can be derived by setting <math>m=L(\underline{X})\,\!</math> in the exponential ~~<math>~~pdf~~</math>~~, yielding the following GLL-exponential ~~<math>~~pdf~~</math>~~ :		The GLL-exponential model can be derived by setting <math>m=L(\underline{X})\,\!</math> in the exponential ''pdf'', yielding the following GLL-exponential ''pdf'':

	<math>f(t,\underline{X})={{e}^{-\left( {{\alpha }_{0}}+\underset{j=1}{\overset{n}{\mathop{\sum }}}\,{{\alpha }_{j}}{{X}_{j}} \right)}}{{e}^{-\left( {{\alpha }_{0}}+\underset{j=1}{\overset{n}{\mathop{\sum }}}\,{{\alpha }_{j}}{{X}_{j}} \right)\cdot t}}</math>		<math>f(t,\underline{X})={{e}^{-\left( {{\alpha }_{0}}+\underset{j=1}{\overset{n}{\mathop{\sum }}}\,{{\alpha }_{j}}{{X}_{j}} \right)}}{{e}^{-\left( {{\alpha }_{0}}+\underset{j=1}{\overset{n}{\mathop{\sum }}}\,{{\alpha }_{j}}{{X}_{j}} \right)\cdot t}}</math>
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	===GLL Weibull===		===GLL Weibull===
	The GLL-Weibull model can be derived by setting <math>\eta =L(\underline{X})\,\!</math> in Weibull ~~<math>~~pdf~~</math>~~, yielding the following GLL-Weibull ~~<math>~~pdf~~</math>~~ :		The GLL-Weibull model can be derived by setting <math>\eta =L(\underline{X})\,\!</math> in Weibull ''pdf'', yielding the following GLL-Weibull ''pdf'':

	<math>f(t,\underline{X})=\beta \cdot {{t}^{\beta -1}}{{e}^{-\beta \left( {{\alpha }_{0}}+\underset{j=1}{\overset{n}{\mathop{\sum }}}\,{{\alpha }_{j}}{{X}_{j}} \right)}}{{e}^{-{{t}^{\beta }}{{e}^{-\beta \left( {{\alpha }_{0}}+\underset{j=1}{\overset{n}{\mathop{\sum }}}\,{{\alpha }_{j}}{{X}_{j}} \right)}}}}</math>		<math>f(t,\underline{X})=\beta \cdot {{t}^{\beta -1}}{{e}^{-\beta \left( {{\alpha }_{0}}+\underset{j=1}{\overset{n}{\mathop{\sum }}}\,{{\alpha }_{j}}{{X}_{j}} \right)}}{{e}^{-{{t}^{\beta }}{{e}^{-\beta \left( {{\alpha }_{0}}+\underset{j=1}{\overset{n}{\mathop{\sum }}}\,{{\alpha }_{j}}{{X}_{j}} \right)}}}}</math>
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	===GLL Lognormal===		===GLL Lognormal===
	The GLL-lognormal model can be derived by setting <math>\breve{T}=L(\underline{X})\,\!</math>		The GLL-lognormal model can be derived by setting <math>\breve{T}=L(\underline{X})\,\!</math>
	in the lognormal ~~<math>~~pdf~~</math>~~, yielding the following GLL-lognormal ~~<math>~~pdf~~</math>~~ :		in the lognormal ''pdf'', yielding the following GLL-lognormal ''pdf'':

	<math>f(t,\underline{X})=\frac{1}{t\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-{{\alpha }_{0}}-\underset{j=1}{\overset{n}{\mathop{\sum }}}\,{{\alpha }_{j}}{{X}_{j}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>		<math>f(t,\underline{X})=\frac{1}{t\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-{{\alpha }_{0}}-\underset{j=1}{\overset{n}{\mathop{\sum }}}\,{{\alpha }_{j}}{{X}_{j}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
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	\end{align}</math>		\end{align}</math>

	The ~~<math>~~pdf~~</math>~~ can be obtained by taking the partial derivative of the reliability function with respect to time. The PH ~~<math>~~pdf~~</math>~~ is:		The ''pdf'' can be obtained by taking the partial derivative of the reliability function with respect to time. The PH ''pdf'' is:

	::<math>\begin{align}		::<math>\begin{align}