Accelerated failure time model

Remarks

Equivalently, $T = e^{\beta^T x} T_0$, where $T_0$ has survival function $S_0$, so that $\log T = \beta^T x + \log T_0$. Parametric versions fix the law of $\log T_0$, for example normal (log-normal model), logistic (log-logistic model) or extreme value (Weibull model). If $S_0$ has hazard function $h_0$, then $h(t \mid x) = e^{-\beta^T x} h_0(t e^{-\beta^T x})$.

Example: if $T_0$ is exponential with rate $1$, then $S(t \mid x) = \exp(-t e^{-\beta^T x})$, so $T$ given $x$ is exponential with rate $e^{-\beta^T x}$. Its median $e^{\beta^T x} \log 2$ is the baseline median times $e^{\beta^T x}$, as the model requires.

Non-example: the family $S(t \mid x) = \exp(-t^{k(x)})$, with a shape $k(x) > 0$ that varies with $x$, is not an accelerated failure time model. If two members with shapes $k_1 \ne k_2$ were related by a change of time scale, then $\exp(-t^{k_2}) = \exp(-(ct)^{k_1})$ for all $t > 0$ and some $c > 0$. This is impossible, since $t^{k_2} / t^{k_1}$ is not constant in $t$.