Tests against qualitative interaction: exact critical values and robust tests

Consider a study to evaluate treatment A with a placebo in two or more groups of patients. If treatment A is beneficial to one group of patients and harmful to another, then we say that there is qualitative interaction or crossover interaction between patient groups and the treatments. Gail and Simon (1985, Biometrics 41, 361-372) developed a large-sample procedure for this testing problem. Their test has received favorable coverage in the literature. In this article, we obtain corresponding exact finite sample results for normal error distribution and provide a table of critical values. The test statistic is similar to the familiar F-ratio, and its p-value is equal to a weighted sum of tail areas of F-distributions. The computations to implement this are simple. A simulation study shows that the exact critical values provided here for normal error distribution are preferable to the asymptotic critical values for a wide range of error distributions. We also develop tests that are power robust against long-tailed error distributions. Our robust test uses M-estimators instead of the least squares estimators. We show that the efficiency robustness of the M-estimator translates to power robustness of the corresponding test. Therefore, our robust tests are better if outliers are expected. A simulation study illustrates the substantial power advantages of our robust tests.