10.1 The model

Let \(T_0\) be a survival time with survivor function \(S_0\), the baseline. The AFT model with covariates \(x\) and coefficients \(\beta \) says that \[ T = e^{\beta ^{\mathsf T}x}\,T_0 , \qquad \text {so}\qquad S(t \mid x) = S_0\big (te^{-\beta ^{\mathsf T}x}\big ) . \] The factor \(e^{\beta _j}\) is the time ratio (or acceleration factor) for a unit increase in \(x_j\): every quantile of \(T\) is multiplied by it. A time ratio above \(1\) means longer survival, the opposite direction to a hazard ratio. Differentiating, the hazard is \[ h(t \mid x) = e^{-\beta ^{\mathsf T}x}h_0\big (te^{-\beta ^{\mathsf T}x}\big ) , \] the baseline hazard read at a rescaled time and multiplied by the rate of rescaling, which is proportional to \(h_0(t)\) only for special \(h_0\). Taking logarithms gives the form in which AFT models are fitted, a linear model for log time, \[ \log T = \mu + \beta ^{\mathsf T}x + \sigma W , \] where \(W\) has a fixed standard distribution, \(\mu \) is an intercept and \(\sigma > 0\) a scale. The choice of \(W\) chooses the family (Table 1), and the parameters are estimated by maximum likelihood with censored observations contributing \(S(t_i \mid x_i)\), exactly as for the parametric models of the section on parametric models.


Distribution of \(W\)	Distribution of \(T\)	Shape of the hazard

standard extreme value (minimum)	Weibull, \(k = 1/\sigma \)	monotone
the same, with \(\sigma = 1\)	exponential	constant
standard normal	log-normal	rises, then falls
standard logistic	log-logistic, \(\alpha = 1/\sigma \)	falls (\(\sigma \ge 1\)), or rises then falls

Table 1: The common AFT families. The extreme value density is \(e^{w - e^w}\); it gives \(P(T > t) = \exp \{-(t/\lambda )^{1/\sigma }\}\) with \(\lambda = e^{\mu }\).

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