A Two-Stage Procedure For Comparing Hazard Rate Functions

Comparison of two hazard rates is important in applications related to times to occurrence of a specific event. Conventional comparison procedures, such as the logrank, Gehan-Wilcoxon, and Peto-Peto tests, are powerful only when the two hazard rates do not cross each other. Because crossing hazard rates are common in practice, a number of procedures have been proposed in the literature for comparing such rates. However, most of these procedures only consider the alternative hypothesis with crossing hazard rates; many other realistic cases, including those when the two hazard rates run parallel to each other, are excluded from consideration. In this paper, we propose a two-stage procedure that considers all possible alternatives, including ones with crossing or running parallel hazard rates. To define its significance level and p-value properly, a new procedure for handling the crossing hazard rates problem is suggested, which has the property that its test statistic is asymptotically independent of the test statistic of the logrank test. We show that the two-stage procedure, with the logrank test and the suggested procedure for handling the crossing hazard rates problem used in its two stages, performs well in applications in comparing two hazard rates.