Describing extra-binomial variation with partially correlated models

Data in the form of proportions are often analyzed under a binomial model. However, because genuine random sampling is often infeasible, the subjects in the sample may be collected in clumps and the variances of the observed proportions may be considerably larger than those corresponding to the binomial model. A set of data from a study of the proportion of subjects testing positive to the disease toxoplasmosis is used in this article to motivate partially correlated binomial models capable of describing data observed in practical situations where clumped sampling is likely to appear, According to these models, the extra-binomial variance of the observed frequencies may range from a linear to a quadratic function of the sample size. An efficient algorithm for the evaluation of the resulting probability mass function is given.