Some exact tests for comparing survival curves in the presence of unequal right censoring

Abstract

SUMMARY It is shown that through the use of conditional tests after artificial censoring one may produce analogues of standard nonparametric tests, such as the log rank and generalized Wilcoxon, that are exact even in the presence of unequal right censoring where standard permutation tests fail. The artificial censoring leads to a loss in power, but in important special cases, as for example in the study of rare diseases, this can be minor. The power loss is investigated through a study of the asymptotic efficiency of the proposed tests. An application is made to test for a suspected carcinogen. Two standard nonparametric tests for comparing survival curves in the presence of right censoring are the log rank test of Mantel (1966) and Cox (1972), and the generalized Wilcoxon test of Gehan (1965) and Breslow (1970). The asymptotic validity of these tests holds even in the presence of unequal censoring. When samples are small, or more generally when the number of failures is small, it is common practice to replace asymptotic tests by tests based on computing, or simulating, the permutation distributions of the score statistics associated with the asymptotic tests. This produces exact tests provided the censoring mechanisms are the same in the groups being compared. Since this is seldom true, permutation tests are, in general, only approximate. Jennrich (1983) considered the asymptotic behaviour of the log rank permutation test in the presence of unequal censoring. When censoring intensities are roughly equal and sample sizes are similar, the permutation test offers a reasonable, though possibly conservative, alternative to the asymptotic test. In general, however, the permutation test can be either conservative or nonconservative, sometimes to a disturbing degree. Here random artificial censoring and conditioning are used to make standard tests exact even in the presence of unequal censoring. As will be shown, and was observed by Efron (1967) who considered an alternative form, artificial censoring generally leads to a loss of power. While this can be large, there are important problems where the loss is minor using the methods proposed here. These are problems in which most of the subjects under study are censored before they fail. As with all randomized tests, the results depend not only on the basic data, but also on the randomizations introduced.