Department of Statistics, UCLA

A predictive density function g' is obtained for the multilevel model which is optimal in minimizing a criterion based on Kullback-Leibler divergence for a restricted class of predictive densitites, thereby extending results for the normal linear model (levy & Perng 1986). Based upon this predictive density approach three prediction methods are examined: Multilevel, Prior, and OLS. The OLS prediction method corresponds to deriving a predictive density separately in each group, while the Prior prediction method corresponds to deriving a predictive denstiy for the entire model. The Multilevel prediction method merely adjusts the Prior prediction method by employing a well known shrinkage estimator from multilevel model estimation. Multilevel data is simulated in order to assess the preformance of these three methods. Both predictive inervals and predictive mean square error (PMSE) are used to assess the adequacy of prediction. The multilevel prediction method outperforms the OLS and prior prediction methods some what surprising since the OLS and Prior prediction methods are derived from the Kullback-Leibler divergence criterion. This suggests that the restricted class of predctive densities suggest by Levy & Perng for the normal linear model may need to be expanded for the multilevel model.