Hazard Ratio Estimation in Small Samples

Abstract

<p>When comparing survival times between groups in the setting of proportional hazards, the Cox model is typically used for estimation and inference, the latter based on large sample considerations. Mehrotra and Roth introduced a generalized log-rank (GLR) method for better statistical efficiency in estimating relative risk in small samples. In this article, we propose a refined GLR (RGLR) statistic by eliminating an unnecessary approximation in the development of the original GLR approach, and provide further insights into the performance of GLR and RGLR statistics. We also extend RGLR to allow for tied event times. We show across a variety of simulated scenarios that RGLR provides a smaller bias than commonly used Cox model, parametric models and GLR in small samples (up to 40 subjects per group), and has notably better efficiency relative to Cox and parametric models in terms of mean squared error. The RGLR method also consistently delivers adequate confidence interval coverage and Type I error control, while parametric methods and the Cox model tend to under-perform on that front in small samples. We further show that while the performance of the parametric model can be significantly influenced by misspecification of the true underlying survival distribution, the RGLR approach provides a consistently low bias and high relative efficiency. We apply all competing methods to data from two clinical trials. Supplementary materials for this article are available online.</p