7.4 The baseline hazard and predicted survival

Although the partial likelihood does not need \(h_0\), predictions do. Given \(\hat \beta \), the Breslow estimator of the baseline cumulative hazard is \[ \hat H_0(t) = \sum _{t_{(j)} \leq t}\frac {d_j}{\sum _{l \in R_j}e^{\hat \beta ^{\mathsf T}x_l}} , \] the Nelson–Aalen estimator with each subject at risk counted with weight \(e^{\hat \beta ^{\mathsf T}x_l}\). The predicted survival of a subject with covariates \(x\) is \(\hat S(t \mid x) = \exp \bigl (-\hat H_0(t)e^{\hat \beta ^{\mathsf T}x}\bigr )\). In R, survfit applied to a fitted Cox model with a data frame of covariate values gives these curves, which the next section uses for adjusted survival curves.

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