ALTA ALTA Standard Folio Data PPH-Weibull: Difference between revisions
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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 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]. | ||
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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 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: | ||
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::<math>{{a}_{i,PH}}=-\beta \cdot {{a}_{i,GLL}}</math> | ::<math>{{a}_{i,PH}}=-\beta \cdot {{a}_{i,GLL}}</math> | ||
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:where: | :where: | ||
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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 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> | ||
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| | | valign="middle" | [http://reliawiki.com/index.php/Template:PH_Model PH Model] | ||
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Revision as of 22:10, 10 February 2012
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| Standard Folio Data PPH-Weibull |
| ALTA |
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Solving for the parameters that maximize Eqn. (PH LKV) will yield the parameters for the PH-Weibull model. Note that for [math]\displaystyle{ \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].
• [math]\displaystyle{ {{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]\displaystyle{ \beta }[/math] = 1 (exponential life distribution), Eqns. (PH LKV) and (GLL-LK) are identical, and [math]\displaystyle{ {{a}_{i,PH}}=-{{a}_{i,GLL}}. }[/math]
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| PH Model |
