title: Adaptive designs with arbitrary dependence structure.
abstract: 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.
authors.affiliation: Institute of Biostatistics and Clinical Research, University of Münster, Schmeddingstraße 56, 48149, Münster, Germany.
authors.forename: Rene
authors.initials: R
authors.lastname: Schmidt
authors.forename: Andreas
authors.initials: A
authors.lastname: Faldum
authors.forename: Olaf
authors.initials: O
authors.lastname: Witt
authors.forename: Joachim
authors.initials: J
authors.lastname: Gerss
mesh_terms: D001699:Biometry / Q000379:methods*; D002986:Clinical Trials as Topic / Q000379:methods*; D006801:Humans
keywords: Adaptive designs; Copulas; Dependent test statistics; Inflation of type I error rate; Inverse normal method