On Variance Estimation for the One-Sample Log-Rank Test

The comparison of a time-to-event endpoint between a prospective study sample and a defined reference population is made with the one-sample log-rank test. This test is typically applied in epidemiologic studies as well as in specific phase II trial settings, e.g. in pediatric oncology. Its distributional properties are commonly derived in the large sample limit. It is however known from the literature, that the asymptotical approximations suffer when sample size is small. There have already been several attempts to address this problem. While some approaches do not allow easy power and sample size calculations, others lack a clear theoretical motivation and require further considerations. The problem itself can partly be attributed to the dependence of the compensated counting process and its variance estimator. For this purpose, we suggest a variance estimator which is uncorrelated to the compensated counting process. Moreover, this and other present approaches to variance estimation are covered as special cases by our general framework. For practical application, we provide power and sample size calculations for any approach fitting into this framework. Finally, we use simulations and real world data to study the empirical type I error and power performance of our method as compared to standard approaches.