Nonparametric linear tests with multiple events

A generalization of the nonparametric linear rank statistics is presented to handle the two-group comparison with multiple events. For a sample divided into two groups, in which each subject may experience at least two distinct failures, the logrank tests are extended to test the null hypothesis that the vector of the marginal survival distributions of the first group equals that of the second group. Two cases are distinguished depending on whether the null hypothesis does or does not imply the equality of the joint survival functions. In both cases, under the null hypothesis, the asymptotic joint distribution of the vector of the marginal statistics is shown to be Gaussian with covariance matrix consistently estimated using martingale properties. These theoretical results are illustrated by a simulation study and an application on the German Breast Cancer data. An extension to multiple hypotheses testing in multivariate proportional hazards models is also developed.