Adaptive designs with arbitrary dependence structure.

Adaptive designs were originally developed for independent and uniformly distributed p-values. There are trial settings where independence is not satisfied or where it may not be possible to check whether it is satisfied. In these cases, the test statistics and p-values of each stage may be dependent. Since the probability of a type I error for a fixed adaptive design depends on the true dependence structure between the p-values of the stages, control of the type I error rate might be endangered if the dependence structure is not taken into account adequately. In this paper, we address the problem of controlling the type I error rate in two-stage adaptive designs if any dependence structure between the test statistics of the stages is admitted (worst case scenario). For this purpose, we pursue a copula approach to adaptive designs. For two-stage adaptive designs without futility stop, we derive the probability of a type I error in the worst case, that is for the most adverse dependence structure between the p-values of the stages. Explicit analytical considerations are performed for the class of inverse normal designs. A comparison with the significance level for independent and uniformly distributed p-values is performed. For inverse normal designs without futility stop and equally weighted stages, it turns out that correcting for the worst case is too conservative as compared to a simple Bonferroni design.