The article develops marginal models for multivariate longitudinal responses. Overall, the model consists of five regression submodels, one for the mean and four for the covariance matrix, with the latter resulting by considering various matrix decompositions. The decompositions that we employ are intuitive, easy to understand, and they do not rely on any assumptions such as the presence of an ordering among the multivariate responses. The regression submodels are semiparametric, with unknown functions represented by basis function expansions. We use spike-slap priors for the regression coefficients to achieve variable selection and function regularization, and to obtain parameter estimates that account for model uncertainty. An efficient Markov chain Monte Carlo algorithm for posterior sampling is developed. The simulation studies presented investigate the effects of priors on posteriors, the gains that one may have when considering multivariate longitudinal analyses instead of univariate ones, and whether these gains can counteract the negative effects of missing data. We apply the methods on a highly unbalanced longitudinal dataset with four responses observed over of period of 20 years
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