Case studies in binary dispersion

It is common in biomedical studies with binary responses that variability in the observed number of events exceeds binomial variability, a phenomenon known as overdispersion. Failure to make an adjustment to the nominal standard errors can lead to seriously misleading inference for regression analysis. In this note, we examine a series of examples drawn from the literature to see which of two commonly used variance formulas is more adequate for describing overdispersion in applications. Two methods, residual analysis and formal comparison, are introduced. We recommend that both methods be employed in seeking an appropriate variance expression for binary responses. Each of the five data sets exhibits substantial overdispersion, one favoring the beta-binomial form, another favoring a constant overdispersion factor. The remaining three examples exhibit no preference.