ABSTRACT
Section 3.4.4 introduced the binomial distribution (Equation 3.7) underlying inferences concerning a simple proportion π, which is normally estimated by p = r/n. The Poisson distribution is a closely related model, widely used to represent data on counts of events. It represents the limiting case of the binomial distribution as the group size n → ∞ while the expected number of events ER = nπ = μ is held constant at a finite value and π = μ/n → 0. Many applications in epidemiology involve a large population size with a low incidence or prevalence of disease, leading to a small to moderate number of cases positive for the disease. Under these circumstances, the Poisson distribution serves as a simplified proxy for the binomial. The fact that the Poisson distribution has a single parameter, μ was a great advantage when users relied on extensive tables to obtain confidence limits. For example, Documenta Geigy Scientific Tables (Diem and Lentner 1970) used 19 pages to list Clopper-Pearson intervals for π, but only two pages for exact intervals for μ. Correspondingly, it is simpler to evaluate coverage properties and so forth in the Poisson case.
