Optimal monotone conditional error functions

Abstract

This paper presents a general method that provides optimal monotone conditional error functions for confirmatory adaptive two-stage designs with conditional power based sample size recalculations. The presented method builds on a previously developed general theory for optimal adaptive two-stage designs where sample sizes are reassessed for a specific conditional power and the goal is to minimize the expected sample size. The previous theory can easily lead to a non-monotonous conditional error function, which is highly undesirable for logical reasons and, as we show, can harm type I error rate control for composite null hypotheses. We also show that type I error control is generally guaranteed with a conditional error function (CEF) that is non-increasing in the first stage p-value. We present a method that extends the existing theory by introducing an intermediate monotonising steps that can easily be implemented and provides a non-increasing conditional error function. We show mathematically that the monotonising step provides the optimal non-increasing conditional error function. We illustrate the method with several examples using optconerrf, an R package implemented for this paper.