Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease Incidence.

We discuss the use of the beta-binomial distribution for the description of plant disease incidence data, collected on the basis of scoring plants as either "diseased" or "healthy". The beta-binomial is a discrete probability distribution derived by regarding the probability of a plant being diseased (a constant in the binomial distribution) as a beta-distributed variable. An important characteristic of the beta-binomial is that its variance is larger than that of the binomial distribution with the same mean. The beta-binomial distribution, therefore, may serve to describe aggregated disease incidence data. Using maximum likelihood, we estimated beta-binomial parameters p (mean disease incidence) and θ (an index of aggregation) for four previously published sets of disease incidence data in which there were some indications of aggregation. Goodness-of-fit tests showed that, in all these cases, the beta-binomial provided a good description of the observed data and resulted in a better fit than did the binomial distribution. The relationship between the parameters of the beta-binomial distribution and those of variance-mean relationships for aggregated disease-incidence data is shown.