Correlated frailty model for multivariate longitudinal count data

A joint model of a sequence of different longitudinal counts using a correlated Poisson-gamma model was extended to allow frailties to be marginally gamma distributed with different variance parameters. In this case the frailty factor is split into the sum of two components: one is shared by entire group and the other is an individual term. That is, a time varying frailty term is included in the model to handle both between heterogeneity and within-subject serial correlation. Because of the intractability to obtain the full likelihood, we use the composite likelihood procedure based on all bivariate contributions of a multivariate longitudinal vector. We illustrate the methodology by modeling the number of mild and moderat/severe episodes of diarrhea occurred between two following treatments in a placebo-controlled trial conducted in Brazil. The proposed model can be reduced to that proposed by Henderson an Shimakura (2003) when there is evidence to support the hypothesis of frailty variances being equal. A limitation of this model is that it assumes that there is always a positive correlation between frailties.