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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.\nThe *Cox proportional hazards model* is the following [[statistical model|K3AMMSY8DD]].\nThe conditional [[hazard function|Y959NKQ3XW]] of $T$ given $x$ is\n$$h(t \\mid x) = h_0(t) \\, e^{\\beta^T x}, \\qquad t \\ge 0,$$\nwhere $\\beta \\in \\R^p$ is an unknown parameter and the baseline hazard $h_0 \\colon [0, \\infty) \\to [0, \\infty)$ is an unknown function.\n\n## Remarks\n\nFor two covariate vectors $x$ and $x'$, the hazard ratio $h(t \\mid x) / h(t \\mid x') = e^{\\beta^T (x - x')}$ does not depend on $t$; this is the proportional hazards property.\nEquivalently, the survival functions satisfy $S(t \\mid x) = S_0(t)^{e^{\\beta^T x}}$, where $S_0$ is the baseline survival function.\n\nGiven right-[[censored|MY8ZZP84NP]] data $(\\tilde t_i, \\delta_i, x_i)$ without tied times, $\\beta$ is estimated by maximising the partial [[likelihood|0VW9E4QXR6]]\n$$L(\\beta) = \\prod_{i \\,:\\, \\delta_i = 1} \\frac{e^{\\beta^T x_i}}{\\sum_{j \\,:\\, \\tilde t_j \\ge \\tilde t_i} e^{\\beta^T x_j}}.$$\n\nExample: the Weibull model with shape $k \u003e 0$ and hazard $h(t \\mid x) = k t^{k-1} \\lambda^{-k} e^{\\beta^T x}$ is a proportional hazards model with $h_0(t) = k t^{k-1} \\lambda^{-k}$.\n\nNon-example: if a treatment has hazard $h_1(t) = 2t$ and the control has $h_0(t) = 1$, the ratio $2t$ changes with time and the curves cross at $t = 1/2$.\nNo constant $\\beta$ satisfies $h_1 = h_0 e^{\\beta}$, so these two groups do not follow a proportional hazards model.\n",
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