Generalized Log-Rank Tests for Interval-Censored Failure Time Data JIANGUO SUN and QIANG ZHAO Department of Statistics, University of Missouri XINGQIU ZHAO Department of Mathematics and Statistics, McMaster University ABSTRACT. Several non-parametric test procedures have been proposed for incomplete survival data: interval-censored failure time data. However, most of them have unknown asymptotic properties with heuristically derived and/or complicated variance estimation. This article presents a class of generalized log-rank tests for this type of survival data and establishes their asymptotics. The methods are evaluated using simulation studies and illustrated by a set of real data from a cancer study. Key words: asymptotic distribution, clinical trials, interval-censoring, log-rank test, survival comparison 1. Introduction This paper discusses non-parametric comparison of survival functions based on incomplete survival data: interval-censored failure time data (cf. Li et al., 1997; Sun, 1998; Pan, 2000). By interval-censored data, we mean that the survival time of interest is observed only to belong to an interval instead of being exactly known or right-censored as usually assumed (cf. Li, 2003). One field in which interval-censored data often occur is observational or follow-up studies where patients are not continuously under observation. In this case, only the status about the occurrence of a certain event is observed at observation times, rather than the occurrence time of the event. One such example from a cancer study is provided in Finkelstein (1986) and will be discussed below in more details. Another field that commonly produces interval-censored failure time data is tumorgenicity experiments (cf. Lagakos & Louis, 1988). In this case, it is usually the case that the survival time of interest is either left-censored or right-censored, a special case of interval-censored data. Survival comparison is usually one of main goals in survival studies. For the problem, when right-censored failure time data are available, a number of well-established methods have been developed (cf. Fleming & Harrington, 1991; Kalbfleisch & Prentice, 2002). For the case of interval-censored failure time data, several authors have discussed the problem. For example, Peto & Peto (1972) considered the two-sample comparison problem under the Lehmann-type alternatives G 2 ðtÞ¼ G h 1 ðtÞ, where G 1 and G 2 are survival functions corresponding to the two different samples and h is a parameter. In this case, the comparison problem reduces to testing h ¼ 0 and they suggested using the score test, which they referred to as the log-rank test. Assuming the proportional hazards model, a special case of Lehmann-type alternatives, Finkelstein (1986) investigated the general k-sample comparison problem. For the problem, she also suggested applying the score test for testing regression parameters equal to zero. Following Finkelstein (1986), Sun (1996) studied the same problem without assuming the proportional hazards model and developed a non-parametric test using the idea behind the Ó Board of the Foundation of the Scandinavian Journal of Statistics 2005. Published by Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA Vol 32: 49–57, 2005