This paper is concerned with the analysis of multivariate count data. A class of models is proposed, based on the work of Aitchison and Ho (1989), in which the correlation amongst the counts is represented by correlated, outcome-specific, latent effects. Several interesting special cases of the model are discussed and a tuned and efficient Markov chain Monte Carlo algorithm is developed to estimate the model. The ideas are illustrated with three real data examples of trivariate to sixteen variate correlated counts.