Generation of over-dispersed and under-dispersed binomial variates

This article proposes an algorithm for generating over-dispersed and under-dispersed binomial variates with specified mean and variance. The over-dispersed/under-dispersed distributions are derived from correlated binary variables with an underlying continuous multivariate distribution. Different multivariate distributions or different correlation matrices result in different over-dispersed (or under-dispersed) distributions. The overdispersed binomial distributions that are generated from three different correlation matrices of a multivariate normal are compared with the beta-binomial distribution for various mean and over-dispersion parameters by quantile-quantile (Q-Q) plots. The two distributions appear to be similar. The under-dispersed binomial distribution is simulated to model an example data set that exhibits under-dispersed binomial variation.