Adaptive controlled noninferiority group sequential trials

For studies comparing three independent arms: test group $T$, reference group $R$, and control group $C$, we consider the hierarchical testing of the a priori ordered hypotheses, that, in short, \begin{eqnarray*} \mbox{(I)}: &~& T > C, \\ \mbox{(II)}: &~& T > R - \Delta, \ \ \Delta > 0, \end{eqnarray*} in general adaptive group sequential designs. For normally distributed response variables with unknown variances, nested confidence intervals on the study parameters are derived at each stage of the trial, holding a predefined confidence level. During the course of the trial, the sample sizes can be calculated in a completely adaptive way based on the unblinded data of previous stages. Concrete formulae for sample size updating are provided in this paper. Moreover, in each interim analysis, it is possible to switch in the planning from showing noninferiority of $T$ in (II) to showing superiority of $T$, that is, $T > R$. A real data example is worked out in detail following an adaptive three-stage design of Pocock (1977) type. In the example, (I) is shown in the first stage and (II) in the second stage, so that the study stopped earlier at the second stage