Modeling common predictor effects on multivariate outcomes

Multivariate outcomes are commonly observed in medicine, public health, psychology and sociology. The typical (saturated) multivariate regression model has a separate set of regression coefficients for each outcome. However, multivariate outcomes are often quite similar and many outcomes can be expected to respond similarly to changes in covariate value. Given a set of outcomes likely to share common covariate effects, we propose a COmmon Predictor Effect (COPE) model, in which a single linear combination of the covariates predicts all outcomes and the magnitude of the effects is modified by an outcome-specific scale parameter. For continuous outcomes, we propose a two step iterative algorithm to fit the COPE model using available software for univariate data. Outcomes that share predictor effects need not be chosen a priori; we propose model selection tools to let the data select outcome clusters. We exhaust all possible outcome clustering and select best models by AIC or BIC for small number of outcomes. We develop a tree structured model selection method to select the best models efficiently when the number of outcomes is large. We apply the proposed methods to psychometric longitudinal data from adolescent children of HIV+ parents.;When outcomes are a mixture of continuous, binary and count variables, we propose a Bayesian multivariate COPE model and we use generalized linear models for the marginal distribution for each outcome. We explain how to set the prior distribution and we use Markov Chain Monte Carlo (MCMC) to estimate the posterior distribution. We propose two types of expanded models to diagnose whether each outcome indeed has predictor effects common with the other outcomes, and whether a particular predictor is commonly predictive for all outcomes. We determine a final model based on the diagnostic models. The method is applied to a psychometric data set of young people living with HIV.