Rank tests for censored randomized block designs

Abstract

SUMMARY Rank tests are proposed and studied for the hypothesis of equal treatment effects in randomized block experiments. The procedures are applicable to samples containing arbitrarily right-censored observations as well as uncensored observations. In the case of logistic and double exponential rank scores, the tests are shown to be censored data extensions of the well-known Friedman (1937) and Brown & Mood (1951) procedures respectively. The application of the techniques to problems in survival analysis is also discussed. variate is subjected to arbitrary right censorship. In the context of biological assays, Sampford & Taylor (1959) have largely dealt with parametric normal theory approaches to both the estimation and testing of treatment effects. Patel (1975) has discussed a censored data modification of Friedman's (1937) statistic for testing the global hypothesis of equal treatment effects, while L. J. Wei has in unpublished work proposed a class of test procedures based on Gehan's (1965) scoring scheme. Downton (1976) and Morton (1978) have also discussed nonparametric tests for block designs. The latter develops a class of rank statistics arising by forming a linear combination of score components from Cox's (1975) partial likelihood. The present approach is to use the marginal probability of a generalized rank vector rather than a partial likelihood in the formation of a class of tests. The resulting rank tests differ from Morton's in several respects. For example, multiple distinct censored values between two consecutive uncensored observations would receive identical scores with the present scheme, distinct scores with Morton's. The specific objective of this paper is to propose a class of linear rank tests, derived under optimal local power considerations, for the global hypothesis of equal treatment effects. To develop this class of tests we exploit the concept of the accumulated rank vector probability of Prentice (1978). The principal theoretical results concerning the derivation of the test statistic and its null hypothesis permutation distribution follow in ? 2. The scores for the test statistic are given in explicit form for selected densities in ? 3. In ? 4 the test statistic is shown, when dealing with only uncensored data, to reduce to Friedman's (1937) statistic with logistic scores and to Brown & Mood's (1951) median test with double exponential scores. A discussion of the application of the techniques to both the log linear and proportional hazards models in survival analysis is the subject of ? 5.