Template:PH Model - Revision history

Richard House: Redirected page to Multivariable Relationships: General Log-Linear and Proportional Hazards#Proportional Hazards Model

2012-08-15T22:46:28Z

Redirected page to Multivariable Relationships: General Log-Linear and Proportional Hazards#Proportional Hazards Model

Athanasios Gerokostopoulos: /* Parametric Model Formulation */

2012-02-27T22:58:40Z

Parametric Model Formulation

← Older revision		Revision as of 22:58, 27 February 2012
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	<br>		<br>
	Solving for the parameters that maximize the log-likelihood function will yield the parameters for the PH-Weibull model. Note that for <math>\beta </math> = 1, 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 [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 </math> = 1, 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]]].

	<br>		<br>

Athanasios Gerokostopoulos: /* Proportional Hazards Model */

2012-02-27T22:57:43Z

Proportional Hazards Model

← Older revision		Revision as of 22:57, 27 February 2012
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	<br>		<br>
	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 [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 [[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.
	<br>		<br>
	==Non-Parametric Model Formulation==		==Non-Parametric Model Formulation==

Athanasios Gerokostopoulos: /* Parametric Model Formulation */

2012-02-20T23:46:21Z

Parametric Model Formulation

← Older revision		Revision as of 23:46, 20 February 2012
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	<br>		<br>

	===Parametric Model Formulation===		==Parametric Model Formulation==
	<br>		<br>
	A parametric form of the proportional hazards model can be obtained by assuming an underlying distribution. In ALTA, the Weibull and exponential distributions are available. In this section we will consider the Weibull distribution to formulate the parametric proportional hazards model. In other words, it is assumed that the baseline failure rate is parametric and given by the Weibull distribution. In this case, the baseline failure rate is given by:		A parametric form of the proportional hazards model can be obtained by assuming an underlying distribution. In ALTA, the Weibull and exponential distributions are available. In this section we will consider the Weibull distribution to formulate the parametric proportional hazards model. In other words, it is assumed that the baseline failure rate is parametric and given by the Weibull distribution. In this case, the baseline failure rate is given by:

Athanasios Gerokostopoulos: /* Non-Parametric Model Formulation */

2012-02-20T23:45:42Z

Non-Parametric Model Formulation

← Older revision		Revision as of 23:45, 20 February 2012
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	The model has been widely used in the biomedical field [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 [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.
	<br>		<br>
	===Non-Parametric Model Formulation===		==Non-Parametric Model Formulation==
	<br>		<br>
	According to the PH model, the failure rate of a system is affected not only by its operation time, but also by the covariates under which it operates. For example, a unit may have been tested under a combination of different accelerated stresses such as humidity, temperature, voltage, etc. It is clear then that such factors affect the failure rate of a unit.		According to the PH model, the failure rate of a system is affected not only by its operation time, but also by the covariates under which it operates. For example, a unit may have been tested under a combination of different accelerated stresses such as humidity, temperature, voltage, etc. It is clear then that such factors affect the failure rate of a unit.

Athanasios Gerokostopoulos: /* Proportional Hazards Model */

2012-02-20T23:45:34Z

Proportional Hazards Model

← Older revision		Revision as of 23:45, 20 February 2012
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	==Proportional Hazards Model==		=Proportional Hazards Model=
	<br>		<br>
	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.

Athanasios Gerokostopoulos at 23:40, 20 February 2012

2012-02-20T23:40:32Z

← Older revision		Revision as of 23:40, 20 February 2012
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	<br>		<br>
	Solving for the parameters that maximize the log-likelihood function will yield the parameters for the PH-Weibull model. Note that for <math>\beta </math> = 1, 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 [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 </math> = 1, 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 [28].

	<br>		<br>
	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:

Athanasios Gerokostopoulos: /* Parametric Model Formulation */

2012-02-20T23:39:58Z

Parametric Model Formulation

← Older revision		Revision as of 23:39, 20 February 2012
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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left( \beta \cdot T_{i}^{\beta -1}{{e}^{-T_{i}^{\beta }\cdot {{e}^{\mathop{}_{j=0}^{m}{{a}_{j}}{{x}_{i,j}}}}}}{{e}^{\mathop{}_{j=0}^{m}{{a}_{j}}{{x}_{i,j}}}} \right) \\		\ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left( \beta \cdot T_{i}^{\beta -1}{{e}^{-T_{i}^{\beta }\cdot {{e}^{\mathop{\sum}_{j=0}^{m}{{a}_{j}}{{x}_{i,j}}}}}}{{e}^{\mathop{\sum}_{j=0}^{m}{{a}_{j}}{{x}_{i,j}}}} \right) -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( T_{i}^{\prime } \right)}^{\beta }}{{e}^{\mathop{\sum}_{j=0}^{m}{{a}_{j}}{{x}_{i,j}}}}+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
	~~& &~~ -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( T_{i}^{\prime } \right)}^{\beta }}{{e}^{\mathop{}_{j=0}^{m}{{a}_{j}}{{x}_{i,j}}}}+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
	\end{align}</math>		\end{align}</math>

	<br>		<br>
	:where:		where:

	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& R_{Li}^{\prime \prime }= & {{e}^{-T_{Li}^{\prime \prime \beta }{{e}^{\underset{j=0}{\mathop{\overset{n}{\mathop{\mathop{}_{}^{}}}\,}}\,{{\alpha }_{j}}{{x}_{j}}}}}} \\		& R_{Li}^{\prime \prime }= & {{e}^{-T_{Li}^{\prime \prime \beta }{{e}^{\underset{j=0}{\mathop{\overset{n}{\mathop{\mathop{\sum}_{}^{}}}\,}}\,{{\alpha }_{j}}{{x}_{j}}}}}} \\
	& R_{Ri}^{\prime \prime }= & {{e}^{-T_{Ri}^{\prime \prime \beta }{{e}^{\underset{j=0}{\mathop{\overset{n}{\mathop{\mathop{}_{}^{}}}\,}}\,{{\alpha }_{j}}{{x}_{j}}}}}}		& R_{Ri}^{\prime \prime }= & {{e}^{-T_{Ri}^{\prime \prime \beta }{{e}^{\underset{j=0}{\mathop{\overset{n}{\mathop{\mathop{\sum}_{}^{}}}\,}}\,{{\alpha }_{j}}{{x}_{j}}}}}}
	\end{align}</math>		\end{align}</math>

	<br>		<br>
	Solving for the parameters that maximize ~~Eqn. (PH LKV)~~ will yield the parameters for the PH-Weibull model. Note that for <math>\beta </math> = 1, ~~Eqn. (PH LKV)~~ becomes the likelihood function for the PH-exponential model, which is similar to the original form of the proportional hazards model proposed by Cox [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 </math> = 1, 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 [28].
	<br>		<br>
	Note that the likelihood function ~~given by Eqn. (~~GLL~~-LK)~~ is very similar to the likelihood function for the proportional hazards-Weibull model ~~given by Eqn. (PH LKV)~~. In particular, the shape parameter of the Weibull distribution can be included in the regression coefficients ~~of Eqn. (13)~~ 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:

	<br>		<br>
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	<br>		<br>
	:where:		where:
	<br>		<br>
	• <math>{{a}_{i,PH}}</math> are the parameters of the PH model.		• <math>{{a}_{i,PH}}</math> are the parameters of the PH model.
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	• <math>{{a}_{i,GLL}}</math> are the parameters of the general log-linear model.		• <math>{{a}_{i,GLL}}</math> are the parameters of the general log-linear model.

	In this case, the likelihood functions ~~given by Eqns. (PH LKV) and (GLL-LK)~~ are identical. Therefore, if no transformation on the covariates is performed, the parameter values that maximize ~~Eqn. (~~GLL~~-LK)~~ also maximize the likelihood function for the proportional hazards-Weibull (PHW) model ~~with parameters given by Eqn. (GLL Parameters)~~. Note that for <math>\beta </math> = 1 (exponential life distribution), ~~Eqns. (PH LKV) and (GLL-LK)~~ are identical, and <math>{{a}_{i,PH}}=-{{a}_{i,GLL}}.</math>		In this case, the likelihood functions are identical. Therefore, if no transformation on the covariates is performed, the parameter values that maximize the likelihood function of the GLL model also maximize the likelihood function for the proportional hazards-Weibull (PHW) model. Note that for <math>\beta </math> = 1 (exponential life distribution), the two likelihood functions are identical, and <math>{{a}_{i,PH}}=-{{a}_{i,GLL}}.</math>
	<br>		<br>

Athanasios Gerokostopoulos: /* Parametric Model Formulation */

2012-02-20T23:33:31Z

Parametric Model Formulation

← Older revision		Revision as of 23:33, 20 February 2012
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	===Parametric Model Formulation===		===Parametric Model Formulation===
	<br>		<br>
	A parametric form of the proportional hazards model can be obtained by assuming an underlying distribution. In ALTA ~~PRO~~, the Weibull and exponential distributions are available. In this section we will consider the Weibull distribution to formulate the parametric proportional hazards model. In other words, it is assumed that the baseline failure rate ~~in Eqn. (Prop. Failure Rate)~~ is parametric and given by the Weibull distribution. In this case, the baseline failure rate is given by:		A parametric form of the proportional hazards model can be obtained by assuming an underlying distribution. In ALTA, the Weibull and exponential distributions are available. In this section we will consider the Weibull distribution to formulate the parametric proportional hazards model. In other words, it is assumed that the baseline failure rate is parametric and given by the Weibull distribution. In this case, the baseline failure rate is given by:

	<br>		<br>
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	<br>		<br>
	::<math>\lambda (t,\underline{X})=\frac{\beta }{\eta }{{\left( \frac{t}{\eta } \right)}^{\beta -1}}\cdot {{e}^{\mathop{}_{j=1}^{m}{{a}_{j}}{{x}_{j}}}}</math>		::<math>\lambda (t,\underline{X})=\frac{\beta }{\eta }{{\left( \frac{t}{\eta } \right)}^{\beta -1}}\cdot {{e}^{\mathop{\sum}_{j=1}^{m}{{a}_{j}}{{x}_{j}}}}</math>

	<br>		<br>
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	<br>		<br>
	::<math>\lambda (t,\underline{X})=\beta \cdot {{t}^{\beta -1}}\cdot {{e}^{\mathop{}_{j=0}^{m}{{a}_{j}}{{x}_{j}}}}</math>		::<math>\lambda (t,\underline{X})=\beta \cdot {{t}^{\beta -1}}\cdot {{e}^{\mathop{\sum}_{j=0}^{m}{{a}_{j}}{{x}_{j}}}}</math>

	<br>		<br>
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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& R(t,\underline{X})= & {{e}^{-\~~mathop{}_~~{0}^{t}\lambda (u)du}} \\		R(t,\underline{X})=\ {{e}^{-\int_{0}^{t}\lambda (u)du}} =\ {{e}^{-\int_{0}^{t}\lambda (u,\underline{X})du}} =\ {{e}^{-{{t}^{\beta }}\cdot {{e}^{\mathop{\sum}_{j=0}^{m}{{a}_{j}}{{x}_{j}}}}}}
	~~& = &~~ {{e}^{-\~~mathop{}_~~{0}^{t}\lambda (u,\underline{X})du}} \\
	~~& = &~~ {{e}^{-{{t}^{\beta }}\cdot {{e}^{\mathop{}_{j=0}^{m}{{a}_{j}}{{x}_{j}}}}}}
	\end{align}</math>		\end{align}</math>

	<br>		<br>
	The <math>pdf</math> can be obtained by taking the partial derivative of the reliability function ~~given by Eqn. (PH Rel)~~ with respect to time. The PH <math>pdf</math> is:		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:

	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& f(t,\underline{X})= & \lambda (t,\underline{X})\cdot R(t,\underline{X}) \\		f(t,\underline{X})= & \lambda (t,\underline{X})\cdot R(t,\underline{X}) =\ \beta \cdot {{t}^{\beta -1}}{{e}^{\left[ \mathop{\sum}_{j=0}^{m}{{a}_{j}}{{x}_{j}}-{{t}^{\beta }}\cdot {{e}^{\mathop{\sum}_{j=0}^{m}{{a}_{j}}{{x}_{j}}}} \right]}}
	~~& = &~~ \beta \cdot {{t}^{\beta -1}}{{e}^{\left[ \mathop{}_{j=0}^{m}{{a}_{j}}{{x}_{j}}-{{t}^{\beta }}\cdot {{e}^{\mathop{}_{j=0}^{m}{{a}_{j}}{{x}_{j}}}} \right]}}
	\end{align}</math>		\end{align}</math>

Athanasios Gerokostopoulos: /* Non-Parametric Model Formulation */

2012-02-20T23:28:23Z

Non-Parametric Model Formulation

← Older revision		Revision as of 23:28, 20 February 2012
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	<br>		<br>
	According to the PH model, the failure rate of a system is affected not only by its operation time, but also by the covariates under which it operates. For example, a unit may have been tested under a combination of different accelerated stresses such as humidity, temperature, voltage, etc. It is clear then that such factors affect the failure rate of a unit.		According to the PH model, the failure rate of a system is affected not only by its operation time, but also by the covariates under which it operates. For example, a unit may have been tested under a combination of different accelerated stresses such as humidity, temperature, voltage, etc. It is clear then that such factors affect the failure rate of a unit.

			<br>
	The instantaneous failure rate (or hazard rate) of a unit is given by:		The instantaneous failure rate (or hazard rate) of a unit is given by:

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	<br>		<br>
	• a positive function <math>g(x,\underline{A})</math> , independent of time, which incorporates the effects of a number of covariates such as humidity, temperature, pressure, voltage, etc.		• a positive function <math>g(x,\underline{A})</math> , independent of time, which incorporates the effects of a number of covariates such as humidity, temperature, pressure, voltage, etc.

	<br>		<br>
	The failure rate of a unit is then given by:		The failure rate of a unit is then given by:
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	<br>		<br>
	:where:		where:
	<br>		<br>
	::<math>\underline{X}</math> is a row vector consisting of the covariates:		::<math>\underline{X}</math> is a row vector consisting of the covariates:
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	::<math>\underline{A}={{({{a}_{1}},{{a}_{2}},...{{a}_{m}})}^{T}}</math>		::<math>\underline{A}={{({{a}_{1}},{{a}_{2}},...{{a}_{m}})}^{T}}</math>

	:where:		<br>
			where: