7.1 The model

For a subject with covariates \(x = (x_1, \ldots , x_p)\), the proportional hazards model is \[ h(t \mid x) = h_0(t)\exp (\beta _1x_1 + \cdots + \beta _px_p) = h_0(t)e^{\beta ^{\mathsf T}x} , \] where \(h_0(t)\), the baseline hazard, is the hazard of a subject with all covariates zero, and is left unspecified. For two subjects with covariates \(x\) and \(x'\), \[ \frac {h(t \mid x)}{h(t \mid x')} = e^{\beta ^{\mathsf T}(x - x')} , \] which does not depend on \(t\): the hazards are proportional. This is the model’s one strong assumption, checked in a later section. Its consequences for interpretation are simple:

• for a binary covariate (say \(1\) for treated, \(0\) for control), \(e^{\beta }\) is the hazard ratio (HR) of treated to control, at every time;
• for a continuous covariate, \(e^{\beta }\) is the hazard ratio for a one-unit increase, and \(e^{c\beta }\) for an increase of \(c\) units;
• \(\beta > 0\) (HR \(> 1\)) means higher risk and shorter survival; \(\beta < 0\) (HR \(< 1\)) lower risk; \(\beta = 0\) no effect.

In terms of survival, \(S(t \mid x) = S_0(t)^{\exp (\beta ^{\mathsf T}x)}\), since \(H(t \mid x) = H_0(t)e^{\beta ^{\mathsf T}x}\): a hazard ratio of \(2\) squares the survival probability at every time (\(0.8\) becomes \(0.64\)). A hazard ratio is not a ratio of probabilities of dying, nor of median survival times, though for small risks it is close to the first.

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