Modelling multivariate, overdispersed binomial data with additive and multiplicative random effects

When modelling multivariate binomial data, it often occurs that it is necessary to take into consideration both clustering and overdispersion, the former arising from the dependence between data, and the latter due to the additional variability in the data not prescribed by the distribution. If interest lies in accommodating both phenomena at the same time, we can use separate sets of random effects that capture the within-cluster association and the extra variability. In particular, the random effects for overdispersion can be included in the model either additively or multiplicatively. For this purpose, we propose a series of Bayesian hierarchical models that deal simultaneously with both phenomena. The proposed models are applied to bivariate repeated prevalence data for hepatitis C virus (HCV) and human immunodeficiency virus (HIV) infection in injecting drug users in Italy from 1998 to 2007.