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Adaptive two-stage designs for clinical trials are well understood from a statistical perspective. However, there is still few research on how the stage-two sample size looks like when it is regarded as a function of the first-stage test statistic. In this paper, a formal proof on the concavity of the sample size function is provided if the design’s second stage is optimized such that it minimizes the expected sample size under the alternative under constraints on maximal type I error rate and minimal power.

A note on the shape of sample size functions of optimal adaptive two-stage designs