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 perio d. An important example is the ongoing prostate, lung, colorectal, and ovar ian cancer screening trial of the National Cancer Institute. In such a situ ation, the widely used G(rho) family of weighted log-rank statistics essent ially reduces to the special case of the (unweighted) log-rank statistics. We propose a simple modification to the G(rho) family that adapts to surviv al data with rare events, a concept that we formulate in terms of a small n umber of events at the study endpoint relative to the sample size. The usua l asymptotic properties, including convergence in distribution of the stand ardized statistics to the standard normal, are obtained under the rare even t formulation. Semiparametric transformation models forming sequences of co ntiguous alternatives are considered and, for each rho, a specific such mod el is identified so that the corresponding modified G(rho) Statistic is asy mptotically efficient. Simulation studies show that the proposed statistics do behave differently from the original G(rho) statistics when the event r ate during the study period is low and the former could lead to a substanti al efficiency gain over the latter. Extensions to the G(rho,gamma) family a nd to the regression problem are also given.