Supremum Versions of the Log-Rank and Generalized Wilcoxon Statistics

Abstract Considerable progress has been made in generalizing to censored data hypothesis tests based on rank statistics. The most commonly used generalizations are the log-rank test statistic, which extends the Savage exponential scores statistic (Cox 1972; Mantel 1966) and the generalized Wilcoxon statistics (Gehan 1965; Peto and Peto 1972). Gill (1980) showed that in the two-sample problem the asymptotic efficiency properties of the Savage and Wilcoxon statistics are maintained in censored data by the log-rank and Peto and Peto Wilcoxon statistics, respectively, when censoring is the same in both samples. Heuristics and simulations have implied that some recently proposed supremum statistics, closely related to these two popular linear rank statistics, can be used to construct tests that provide good power against a much wider range of nonlocal alternatives (see, e.g., Fleming and Harrington 1981; Fleming, O'Brien, O'Fallon, and Harrington 1980; Gill 1980; Schumacher 1984). In this article we consider new and previously discussed classes of rank statistics that are related to the log-rank and generalized Wilcoxon tests. We focus on linear rank statistics and their Renyi-type supremum versions. These classes are either of the "asymptotically fully efficient" type or the "approximately distribution-free" type discussed by Leurgans (1983). The large sample distributions of these statistics are established for the general setting in which ties can be present in the data. By restricting attention to the general random censorship model, we obtain the asymptotic results by stating and applying a simplified version of Gill's (1980) limit theorems. We examine more thoroughly than in previous literature the relative operating characteristics of these newly and previously proposed censored data rank statistics. Important insights are obtained from simulations performed for small and moderate sample sizes, in equal and unequal sample sizes, and across a range of survival differences and degrees of censorship. For example, results reveal that supremum versions of the log-rank statistic are nearly as sensitive to proportional-hazards alternatives as the efficient log-rank test. In addition, these supremum versions provide greater sensitivity across a wide range of nonproportional-hazards configurations. This increase in power from the supremum statistics is less evident, however, when data are so heavily censored that one can only estimate early survival differences. Another important insight is obtained from the comparison in small samples of asymptotically fully efficient versus approximately distribution-free statistics. Although the former class has received much greater attention, the latter has the desirable property that the types of survival differences detected with highest power are less affected by degree of censorship. In certain circumstances, it would be appealing to apply simultaneously a linear rank statistic and its supremum version. Distributional results are obtained to show how this can be done appropriately and in a very straightforward manner. Key Words: Rank statisticsRenyi-type testProportional hazards