ALTA ALTA Standard Folio Data PPH-Weibull

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Standard Folio Data PPH-Weibull
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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]. 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: [math]\displaystyle{ {{a}_{i,PH}}=-\beta \cdot {{a}_{i,GLL}} }[/math] where: • [math]\displaystyle{ {{a}_{i,PH}} }[/math] are the parameters of the PH model. • [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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