Modelling multivariate outcomes in hierarchical data, with application to cluster randomised trials

In the cluster randomised study design, the data collected have a hierarchical structure and often include multivariate outcomes. We present a flexible modelling strategy that permits several normally distributed outcomes to be analysed simultaneously, in which intervention effects as well as individual-level and cluster-level between-outcome correlations are estimated. This is implemented in a Bayesian framework which has several advantages over a classical approach, for example in providing credible intervals for functions of model parameters and in allowing informative priors for the intracluster correlation coefficients. In order to declare such informative prior distributions, and fit models in which the between-outcome covariance matrices are constrained, priors on parameters within the covariance matrices are required. Careful specification is necessary however, in order to maintain non-negative definiteness and symmetry between the different outcomes. We propose a novel solution in the case of three multivariate outcomes, and present a modified existing approach and novel alternative for four or more outcomes. The methods are applied to an example of a cluster randomised trial in the prevention of coronary heart disease. The modelling strategy presented would also be useful in other situations involving hierarchical multivariate outcomes.