A predictive density approach to predicting a future observable in multilevel models

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 densities, thereby extending results for the normal linear model (J. Amer. Statist. Assoc. 81 (1986) 196). 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 density 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 are simulated in order to assess the performance of these three methods. Both predictive intervals 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, somewhat surprising since the OLS and prior prediction methods are derived from the Kullback-Leibler divergence criterion. This suggests that the restricted class of predictive densities suggested by Levy and Perng for the normal linear model may need to be expanded for the multilevel model.