Abstract
BACKGROUND: Standard futility analyses designed for a proportional hazards setting may have serious drawbacks when non-proportional hazards are present. One important type of non-proportional hazards occurs when the treatment effect is delayed. That is, there is little or no early treatment effect but a substantial later effect. METHODS: We define optimality criteria for futility analyses in this setting and propose simple search procedures for deriving such rules in practice. RESULTS: We demonstrate the advantages of the optimal rules over commonly used rules in reducing the average number of events, the average sample size, or the average study duration under the null hypothesis with minimal power loss under the alternative hypothesis. CONCLUSION: Optimal futility rules can be derived for a non-proportional hazards setting that control the loss of power under the alternative hypothesis while maximizing the gain in early stopping under the null hypothesis.