The Beta-Binomial Multivariate Model for Correlated Categorical Data

Abstract

Over the past year, a significant amount of research has explored the logistic regression models for
analyzing correlated categorical data. In these models, it is assumed that the data occur in clusters,
where individuals within each cluster are correlated, but individuals from different clusters are
assumed independent. A commonly used in modeling correlated categorical univariate data is to
assume that individual counts are generated from a Binomial distribution, with probabilities vary
between individuals according to a Beta distribution. The marginal distribution of the counts is then
Beta-Binomial. In this paper, a generalization of the model is made allowing the number of
respondent m, to be random. Thus both the number units m, and the underlying probability vector
are allowed to vary. We proposed the model for correlated categorical data, which is generalized to
account for extra variation by allowing the vectors of proportions to vary according to a Dirichlet
distribution. The model is a mixture distribution of multinomial and Dirichlet distribution, and we
call the model as the beta-binomial multivariate model.