Testing for the Proportionality of Hazards in Two Samples Against the Increasing Cumulative Hazard Ratio Alternative

A number of tests of the proportional hazards hypothesis have been proposed in the past. In recent years, researchers have proposed tests geared specially for the alternative hypothesis of "increasing hazard ratio", keeping in mind the case of crossing hazards. This alternative may be too restrictive in many situations. In this paper we develop a test of the proportional hazards model for the weaker "increasing cumulative hazard ratio" alternative. The work is motivated by a data analytic example given by Gill & Schumacher (1987) where their test fails to reject the null hypothesis even though the faster ageing of one group is quite apparent from a plot. The normalized test statistic proposed here has an asymptotically normal distribution under either hypothesis. We also present two graphical methods related to our analytical test.