Methods for predicting specific college course grades, based on small numbers of observations, were investigated. These methods use collateral information across potentially diverse institutions to obtain refined within-group parameter estimates. One method, referred to as pooled least squares with adjusted intercepts, assumes that slopes and residual variances are homogeneous across selected colleges. The second method, referred to as Bayesian m-group regression, allows estimates of slopes and residual variances to vary across colleges, without ignoring available collateral information. These central prediction models were compared with the more usual procedure of deriving regression equations within each college considered in isolation. Data were obtained from colleges participating in the American College Testing standard research services program during the 1983-84 academic year and during at least one of 1984-85, 1985-86, or 1986-87 academic years. Six analysis groups were used, ranging in size from 11 to 17 colleges, each with at least 20 and less than 100 observations. It was found that for both models using collateral information, a sample size of 20 resulted in a level of cross-validated prediction accuracy comparable to that obtained using the within-college least squares procedure at colleges with 50 or more observations. The Bayesian approach outperformed the pooled least squares approach. It is noted that the Bayesian approach is highly adaptive to different structures and can thus be expected to outperform the other two procedures across most situations. Three tables in the text and 12 in Appendix A provide study data. Appendix B contains six graphs illustrating the models. (Author/SLD)