A class of weighted log-rank tests for survival data when the event is rare

In many epidemiological and medical follow-up studies, a majority of study subjects do not experience the event of interest during the follow-up period. An important example is the ongoing prostate, lung, colorectal, and ovarian cancer screening trial of the National Cancer Institute. In such a situation, the widely used G(rho) family of weighted log-rank statistics essentially reduces to the special case of the (unweighted) log-rank statistics. We propose a simple modification to the G(rho) family that adapts to survival data with rare events, a concept that we formulate in terms of a small number of events at the study endpoint relative to the sample size. The usual asymptotic properties, including convergence in distribution of the standardized statistics to the standard normal, are obtained under the rare event formulation. Semiparametric transformation models forming sequences of contiguous alternatives are considered and, for each rho, a specific such model is identified so that the corresponding modified G(rho) Statistic is asymptotically efficient. Simulation studies show that the proposed statistics do behave differently from the original G(rho) statistics when the event rate during the study period is low and the former could lead to a substantial efficiency gain over the latter. Extensions to the G(rho,gamma) family and to the regression problem are also given.