Two-stage tests for studying monotherapy and combination therapy in two-by-two factorial trials

Two-stage testing involves a preliminary test of a nuisance parameter prior to testing a main hypothesis. In a two-by-two factorial trial, the treatment interaction is the nuisance to the inference about the efficacy of one of the treatments given alone. In comparing a combination therapy to both of its component therapies, the nuisance parameter is the difference in the component effects. When the preliminary test is an integral part of inference about the main parameter, the actual level of significance for the two-stage test procedure can be much higher than the desired nominal level. If one places no restriction on the value of the nuisance parameter, then any two-stage test with its significance level properly controlled has undesirable properties. This applies to comparative studies of combination agents relative to the component agents. When the interaction with an ineffective treatment is null, two-stage testing may have some power advantage for assessing monotherapy efficacy.