A flexible approach to the crossing hazards problem.

We propose a simple and flexible framework for the crossing hazards problem. The method is not confined to two-sample problems, but may also work with continuous exposure variables whose effect changes its sign at some time-point of the observed follow-up time. Penalized partial likelihood estimation relies upon the assumption of a smooth hazard ratio via low-rank basis splines with a conventional difference penalty to ensure smoothness, and additional ad hoc penalties to obtain restricted estimates useful in the context of crossing hazards. The framework naturally also leads to a statistical test that has good power for revealing a global effect under several alternatives, including crossing hazards. We provide the results from a real-data analysis and from some simulations to illustrate empirically the performance of the proposed approach as compared with the possible alternatives.