In analyzing data from multiple related studies, it often is of interest to borrow information across studies and to cluster similar studies. Although parametric hierarchical models are commonly used, of concern is sensitivity to the form chosen for the random-effects distribution. A Dirichlet process (DP) prior can allow the distribution to be unknown, while clustering studies; however, the DP does not allow local clustering of studies with respect to a subset of the coefficients without making independence assumptions. Motivated by this problem, we propose a matrix stick-breaking process (MSBP) as a prior for a matrix of random probability measures. Properties of the MSBP are considered, and methods are developed for posterior computation using Markov chain Monte Carlo. Using the MSBP as a prior for a matrix of study-specific regression coefficients, we demonstrate advantages over parametric modeling in simulated examples. The methods are further illustrated using a multinational uterotrophic bioassay study.