Beta-binomial/Poisson models have been used by many authors to model multivariate count data. Lora and Singer [Stat. Med. 27 (2008) 3366–3381] extended such models to accommodate repeated multivariate count data with overdipersion in the binomial component. To overcome some of the limitations of that model, we consider a beta-binomial/gamma-Poisson alternative that also allows for both overdispersion and different covariances between the Poisson counts. We obtain maximum likelihood estimates for the parameters using a Newton–Raphson algorithm and compare both models in a practical example.
"Beta-binomial/gamma-Poisson regression models for repeated counts with random parameters." Braz. J. Probab. Stat. 25 (2) 218 - 235, July 2011. https://doi.org/10.1214/10-BJPS118