Interim analysis based on the weighted log-rank test for delayed treatment effects under staggered patient entry

Fleming and Harrington's G(ρ,γ) class of weighted log-rank tests is appropriate for detecting delayed treatment effects such as those seen in cancer vaccines. A conditional power (CP) and an alpha spending function (ASF) approach are useful for interim analyses that are conducted with the aim of early termination due to futility and efficacy, respectively. However, calculation of the CP and the total Type I error probability are often not considered for delayed effects under the staggered patient entry. In this article, we first propose methods for calculating the CP analytically based on the weighted log-rank test. We compared the performances of the proposed methods with two other methods (i.e., usual log-rank test and optimal one) under the delayed alternatives. Our simulations demonstrated that the CP based on the weighted log-rank test was more powerful than that of the usual log-rank test and was comparable to the CP based on the optimal log-rank test. Second, we quantitatively evaluated the degree to which the Type I error probability was inflated when an ASF approach with forced independent increments assumption was applied to the weighted log-rank test. The proposed method will provide valuable tools in the decision-making stage of the interim analysis.