In this paper, we propose a class of multivariate random effects models allowing for the inclusion of study-level covariates to carry out meta-analyses. As existing algorithms for computing maximum likelihood estimates often converge poorly or may not converge at all when the random effects are multi-dimensional, we develop an efficient expectation–maximization algorithm for fitting multi-dimensional random effects regression models. In addition, we also develop a new methodology for carrying out variable selection with study-level covariates. We examine the performance of the proposed methodology via a simulation study. We apply the proposed methodology to analyze metadata from 26 studies involving statins as a monotherapy and in combination with ezetimibe. In particular, we compare the low-density lipoprotein cholesterol-lowering efficacy of monotherapy and combination therapy on two patient populations (naïve and non-naïve patients to statin monotherapy at baseline), controlling for aggregate covariates. The proposed methodology is quite general and can be applied in any meta-analysis setting for a wide range of scientific applications and therefore offers new analytic methods of clinical importance. Copyright © 2012 John Wiley & Sons, Ltd.