On variance estimation for the one-sample log-rank test

TL;DR

This paper introduces a new variance estimator for the one-sample log-rank test in survival analysis, improving accuracy in small samples and providing a unified framework for various methods.

Contribution

It proposes a novel variance estimator uncorrelated with the counting process and offers a comprehensive framework for sample size and power calculations.

Findings

The new estimator reduces type I error in small samples.

Simulation studies show improved power over standard methods.

Real data application confirms practical utility.

Abstract

Time-to-event endpoints show an increasing popularity in phase II cancer trials. The standard statistical tool for such one-armed survival trials is the one-sample log-rank test. 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 sample size and power 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 methodology as compared to standard approaches.