10.2 The Weibull model: both AFT and proportional hazards
Theorem 10.2.1. A Weibull AFT model with scale \(\sigma \) and coefficients \(\beta \) is a proportional hazards model with hazard ratios \[ e^{\gamma _j} = e^{-\beta _j/\sigma } , \] so a Weibull regression can be reported either way.
Proof. With \(k = 1/\sigma \) and baseline \(S_0(t) = \exp \{-(t/\lambda )^k\}\), \[ S(t \mid x) = S_0\big (te^{-\beta ^{\mathsf T}x}\big ) = \exp \big \{-(t/\lambda )^ke^{-k\beta ^{\mathsf T}x}\big \} = S_0(t)^{\exp (-k\beta ^{\mathsf T}x)} . \] This is the form \(S(t\mid x) = S_0(t)^{\exp (\gamma ^{\mathsf T}x)}\) of a proportional hazards model (the section on the Cox model), with \(\gamma = -k\beta = -\beta /\sigma \). □
The Weibull is the only family with both properties; in particular, for the exponential (\(\sigma = 1\)) the hazard ratio is simply \(e^{-\beta _j}\), the reciprocal of the time ratio. The minus sign matters: a covariate that lengthens survival lowers the hazard.
Example 10.2.2. A Weibull AFT model for the lung data with age, sex and ECOG score gives \(\hat \sigma = 0.731\) and time ratios \(0.993\) per year of age, \(1.49\) for women against men and \(0.712\) per ECOG point (Software 10.5.1): women’s survival times are about one and a half times as long, and each ECOG point shortens them by \(29\%\). By Theorem 10.2.1, the hazard ratio for sex is \(e^{-0.4011/0.7311} = e^{-0.549} = 0.578\), against \(0.575\) from the Cox model of the section on the Cox model, and those for age and ECOG are \(1.010\) and \(1.591\), against \(1.011\) and \(1.590\). When the Weibull fits, the two models tell the same story in two languages.
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