A conditional error function approach for adaptive enrichment designs with continuous endpoints

Adaptive enrichment designs offer an efficient and flexible way to demonstrate the efficacy of a treatment in a clinically defined full population or in, eg, biomarker-defined subpopulations while controlling the family-wise Type I error rate in the strong sense. Frequently used testing strategies in designs with two or more stages include the combination test and the conditional error function approach. Here, we focus on the latter and present some extensions. In contrast to previous work, we allow for multiple subgroups rather than one subgroup only. For nested as well as nonoverlapping subgroups with normally distributed endpoints, we explore the effect of estimating the variances in the subpopulations. Instead of using a normal approximation, we derive new t-distribution-based methods for two different scenarios. First, in the case of equal variances across the subpopulations, we present exact results using a multivariate t-distribution. Second, in the case of potentially varying variances across subgroups, we provide some improved approximations compared to the normal approximation. The performance of the proposed conditional error function approaches is assessed and compared to the combination test in a simulation study. The proposed methods are motivated by an example in pulmonary arterial hypertension.