Estimating a smooth baseline hazard function for the Cox model

The aim of fitting a Cox model to time-to-event data is to estimate the effect of covariates on the baseline hazard function. The baseline hazard function, not itself estimated within the model, is the hazard function obtained when all covariate are set to zero. In several applications, it is important to have an explicit, preferably smooth, estimate of the baseline hazard function, or more generally the baseline distribution function. A key application is model-based prediction of survival probabilities in independent data. In particular, external validation of a Cox model should include an assessment of the calibration of prediction, and this requires an explicit baseline. We propose using simple parametric models (fractional polynomials and restricted cubic splines) to approximate the baseline log cumulative hazard function to a sufficient degree of accuracy. Other functions, including hazard and survival, are easily derived from the log cumulative hazard. We derive and compare estimated functions in two real datasets, one of prognostic factors in breast cancer, the other a randomized controlled trial in oesophageal cancer.