Qualitative interactions in multifactor studies

In clinical trials qualitative interaction or crossover interaction is said to occur when one treatment is superior for some sets of patients and the alternative treatment is superior for other subsets. Here we propose a definition of no qualitative interaction with respect to a single continuous covariate which implies that one treatment is superior to the other treatment over a prespecified range of the covariate. Further, in studies involving patients cross-classified by two or more prognostic factors, we define a marginal qualitative interaction with respect to a treatment factor and a prognostic factor by averaging in some sense over the remaining prognostic factors. In some situations, the methodology of Gail and Simon (1985, Biometrics 41, 361-372) is shown to be appropriate. Their procedure assumes independent estimates of treatment effect for each subset of patients and this may not be appropriate for the generalizations we propose. Therefore, we generalize the procedure of Gail and Simon to the case of two correlated estimates of treatment effect, providing a table of critical values. Results for one-sided tests for the case of J correlated estimates are also obtained. We also present an example illustrating our procedure.