The size accuracy of combination tests

摘要

Adaptive clinical trials typically involve several independent stages. The P-values from each stage are synthesized through a so-called combination function which ensures that the overall test will be valid if the stagewise tests are valid. In practice however, approximate and possibly invalid stagewise tests are used. This paper studies how imperfections of the stagewise tests feed through into the combination test. Several general results are proven including some for discrete models. An approximation formula which directly links the combined size accuracy to the component size accuracy is given. In the wider context of adaptive clinical trials, the main conclusion is that the basic tests used should be size accurate at nominal sizes both much larger and also much smaller than nominal desired size. For binary outcomes, the implication is that the parametric bootstrap should be used.