Testing Against Ordered Alternatives for Censored Survival Data

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Abstract
In animal carcinogenesis experiments or comparative clinical trials where the r groups correspond to increasing dosages of an agent or an increasing number of additional modalities and the primary outcome is time to a certain event (e.g., tumor occurrence, death) T, ordered alternatives of the form H1 : S1 ≤ S2 ≤ · · · ≤ Sr , where Si(t) = Pr (Ti > t) is the survival function for group are of special interest. Using the counting process formulation of the problem and generalizing earlier results, we show that the class of all two-sample weighted logrank statistics can be extended to test specifically for ordered alternatives. Ordered versions of the Gehan-Wilcoxon, unweighted logrank, and Peto–Prentice–Wilcoxon tests are investigated in detail. Asymptotic relative efficiencies under Lehmann and scale alternatives are evaluated analytically and numerically among these three tests and against Tarone's trend test (1975) for Weibull, gamma, and lognormal survival distributions. Effects of censoring and unequal group sizes are also explored. This class of tests has the advantage of having an asymptotic X2 (1) distribution, as opposed to the D distribution for the test proposed by Mau (1988). They can be easily calculated from modifying existing programs for two-sample weighted logrank tests. The ordered versions of the logrank test and the Peto–Prentice–Wilcoxon test are particularly recommended for their versatility under a variety of situations. The use of the tests is illustrated with data from a carcinogenesis experiment and from a colon cancer trial.