Detecting qualitative interaction: a Bayesian approach.

Differences in treatment effects between centers in a multi-center trial may be important. These differences represent treatment by subgroup interaction. Peto defines qualitative interaction (QI) to occur when the simple treatment effect in one subgroup has a different sign than in another subgroup: this interaction is important. Interaction where the treatment effects are of the same sign in all subgroups is called quantitative and is often not important because the treatment recommendation is identical in all cases. A hierarchical model is used here with exchangeable mean responses to each treatment between subgroups. The posterior probability of QI and the corresponding Bayes factor are proposed as a diagnostic and as a test statistic. The model is motivated by two multi-center trials with binary responses. The frequentist power and size of the test using the Bayes factor are examined and compared with two other commonly used tests. The impact of imbalance between the sample sizes in each subgroup on power is examined, and the test based on the Bayes factor typically has better power for unbalanced designs, especially for small sample sizes. An exact test based on the Bayes factor is also suggested assuming the hierarchical model. The Bayes factor provides a concise summary of the evidence for or against QI. It is shown by example that it is easily adapted to summarize the evidence for 'clinically meaningful QI,' defined as the simple effects being of opposite signs and larger in absolute value than a minimal clinically meaningful effect.