Recursive combination tests

We present a method that extends the flexibility of adaptive designs to the number of interim analyses and to the choice of decision boundaries. At each stage of the trial, the design of the next stage can be determined using all of the information gathered so far. Additionally, one can specify the next stage as the final stage or can plan a further interim analysis. The method is based on a recursive application of the two-stage combination tests for p values. The crucial point is the appropriate definition of a p value function that combines p values from two separate stages to a single p value. Formally, we start with a two-stage combination test. However, the p value of the second stage can be replaced by the p value of a further combination test. This applies if the experimenter decides in the first interim analysis to per-form another interim analysis, thereby extending the design to at least three stages. Obviously, this can be also done in a recursive way in the following interim analyses. The test decision is based on an overall p value, which can be easily calculated by concatenation of the p value functions for the combination tests used. Under very general conditions, this procedure controls the level alpha. Recursive combination tests cover the classical group sequential and adaptive two-stage tests as special cases. By construction, they also allow for simple computation of confidence intervals and median unbiased point estimates.