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  "body": "\n## Definition\n\nLet $T$ be an event time with values in $(0, \\infty)$ and let $x \\in \\R^p$ be a vector of covariates.\nAn *accelerated failure time model* is the following [[statistical model|K3AMMSY8DD]].\nThe conditional [[survival function|QTM1HZM53Q]] of $T$ given $x$ is\n$$S(t \\mid x) = S_0\\!\\left(t \\, e^{-\\beta^T x}\\right), \\qquad t \\ge 0,$$\nwhere $\\beta \\in \\R^p$ is a parameter and $S_0$ is a baseline survival function.\n\n## Remarks\n\nEquivalently, $T = e^{\\beta^T x} T_0$, where $T_0$ has survival function $S_0$, so that $\\log T = \\beta^T x + \\log T_0$.\nParametric versions fix the law of $\\log T_0$, for example normal (log-normal model), logistic (log-logistic model) or extreme value (Weibull model).\nIf $S_0$ has [[hazard function|Y959NKQ3XW]] $h_0$, then $h(t \\mid x) = e^{-\\beta^T x} h_0(t e^{-\\beta^T x})$.\n\nExample: 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 $e^{-\\beta^T x}$.\nIts median $e^{\\beta^T x} \\log 2$ is the baseline median times $e^{\\beta^T x}$, as the model requires.\n\nNon-example: the family $S(t \\mid x) = \\exp(-t^{k(x)})$, with a shape $k(x) \u003e 0$ that varies with $x$, is not an accelerated failure time model.\nIf 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 $t \u003e 0$ and some $c \u003e 0$.\nThis is impossible, since $t^{k_2} / t^{k_1}$ is not constant in $t$.\n",
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