Joint modelling of cause-specific hazard functions with cubic splines: an application to a large series of breast cancer patients

The time of appearance of several kinds of relapses after a therapeutic intervention is of increasing interest in oncology. Typically, in breast cancer patients, events of clinical interest are intra-breast tumor recurrences and distant metastases, which act in a competitive way when considered as first failure. The evaluation of differential effects of clinical and biological variables on each event can improve the knowledge on the course of the disease and the targeting of future therapy. A simple tool for the joint smoothed estimation of cause-specific hazards functions and continuous covariate effects has been developed. Within the framework of generalized linear models with Poisson error, an extension of the piecewise exponential model is proposed, based on grouping follow-up times and continuous covariates. Interpolation of cause-specific hazards is obtained by resorting to cubic splines, which are piecewise polynomials of simple implementation with standard statistical software; their flexibility and smoothness are easily controlled by the number of knots and constraints on polynomial derivatives. The approach was applied to a data set of 2233 breast cancer patients treated with conservative surgery. It allowed modelling time-dependent and cause-specific effects of covariates on the hazard functions.