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Studies often estimate associations between an outcome and multiple variates. For example, studies of diagnostic test accuracy estimate sensitivity and specificity, and studies of predictive and prognostic factors typically estimate associations for multiple factors. Meta-analysis is a family of statistical methods for synthesizing estimates across multiple studies. Multivariate models exist that account for within-study correlations and between-study heterogeneity. The number of parameters that must be estimated in existing models is quadratic in the number of variates (e.g., risk factors). This means they may not be usable if data are sparse with many variates and few studies. We propose a new model that addresses this problem by approximating a variance-covariance matrix that models within-study correlation and between-study heterogeneity in a low-dimensional space using random projection. The number of parameters that must be estimated in this model scales linearly in the number of variates and quadratically in the dimension of the approximating space, making estimation more tractable. We performed a simulation study to compare coverage, bias, and precision of estimates made using the proposed model to those from univariate meta-analyses. We demonstrate the method using data from an ongoing systematic review on predictors of pain and function after total knee arthroplasty. Finally, we suggest a decision tool to help analysts choose among available models.