Bootstrap Tests and Confidence Intervals for a Hazard Ratio When the Number of Observed Failures is Small, With Applications to Group Sequential Survival Studies

Abstract

We present a small sample approximation to the distribution of the efficient score statistic for testing the null hypothesis λ=λ ₀, where λ is the hazard ratio in a proportional hazards model. Here, “small sample” refers to a small number of failures in the observed data. Our basic approximation is to the conditional distribution of observations’ group memberships, given the observed number of failures and number of censored observations between successive failures; the implied distribution of the score statistic is found by simulation. Our method can be incorporated into sequential procedures for monitoring survival data and it successfully overcomes inaccuracies in the usual normal approximations that arise when only a few subjects have failed at early analyses.

A simulation study to assess the accuracy of our approximation required bootstrap simulation nested within experimental replications. Here, the computational demands were alleviated substantially by conducting the bootstrap tests themselves sequentially; using an innovative form of stochastic curtailment reduced the required computation by a factor of up to 25, 50 or even more.