Difference in Restricted Mean Survival Time: Small Sample Distribution and Asymptotic Relative Efficiency

The difference between two arms in the restricted mean survival time is an alternative to the hazard ratio. It provides a more easily understood measure of the treatment effect of an intervention in a controlled clinical trial with a time to event endpoint. In this article, we consider the distribution of the estimate for small or moderate sample sizes or when the event rate is low and compare the asymptotic relative efficiency (ARE) and relative efficiency for practical sample sizes of the test to the standard logrank test for Weibull parent distributions. We find that for small sample sizes, the distribution can depart markedly from the normal distribution. We propose a method to approximate the correct null distribution using the Cornish–Fisher expansion and provide an example. For Weibull parent distributions, we consider two scenarios with either proportional hazards or nonproportional hazards where the hazard functions and survival distribution curves crossover. The ARE is compared for these two scenarios.