Analysis of survival data with nonproportional hazard functions

The log-rank test or the proportional hazard model is a valuable, widely accepted method for analyzing time-to-response data from comparative clinical trials. When the hazard ratio is constant in time, this procedure is optimal. Indiscriminate or unthinking use of this approach results in problems in the determination of treatment differences. For example, when the true survival curves intersect, the hazard ratio cannot be constant, i.e., the hazard functions are not proportional. It is shown that by considering time-by-treatment interactions we gain flexibility in describing the relationships among hazard functions. In this paper we demonstrate with the results of a clinical trial how available methodology can be used to permit tests for the appropriateness of the model and to enable informative analysis of such data.