Testing for crossover of two hazard functions using Gail and Simon's method

Crossover of two hazard functions is sometimes called qualitative nonproportionality where the hazard ratio could be >1 in some time intervals but <1 in some other intervals. Investigators often wish to know whether a beneficial treatment effect exists over a long period of time (i.e., no crossover). This information is important for the management of safety and efficacy of a new treatment in long-term use. Also, if crossover occurs, the commonly used statistical methods, such as Cox proportional hazards model or linear rank tests, may not be appropriate. Graphical display may be used to visually examine whether there are crossovers in the observed hazard functions. A relevant question is whether the observed crossover of two hazard functions is due to chance variation. In this article, we propose a class of tests for crossover of two hazard functions. The study follow-up period is divided into nonoverlapping time intervals, and the weighted linear rank statistic, including logrank and generalized Wilcoxon statistics, can be calculated from each interval. These statistics are asymptotically independent and have normal distributions. Treating each interval as a "patient subset," qualitative tests of interactions between treatment and patient subsets can naturally apply. For our purpose, we consider the likelihood ratio test proposed by Gail and Simon. Two examples are used to illustrate this approach. The proposed test procedures are also studied through simulations.