Parsimonious analysis of time-dependent effects in the Cox model

Cox's proportional hazards model can be extended to accommodate time-dependent effects of prognostic factors. We briefly review these extensions along with their varying degrees of freedom. Spending more degrees of freedom with conventional procedures (a priori defined interactions with simple functions of time, restricted natural splines, piecewise estimation for partitions of the time axis) allows the fitting of almost any shape of time dependence but at an increased risk of over-fit. This results in increased width of confidence intervals of time-dependent hazard ratios and in reduced power to confirm any time-dependent effect or even any effect of a prognostic factor. By means of comparative empirical studies the consequences of over-fitting time-dependent effects have been explored. We conclude that fractional polynomials, and similarly penalized likelihood approaches, today are the methods of choice, avoiding over-fit by parsimonious use of degrees of freedom but also permitting flexible modelling if time dependence of a usually a priori unknown shape is present in a data set. The paradigm of a parsimonious analysis of time-dependent effects is exemplified by means of a gastric cancer study.