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.

Multilevel modelling is often used in the social sciences for analyzing data that has a hiearchial structure, e.g.. students nesteded within schools. In an earlier study, we investigated the performance of various prediction rules for predicting a future observable within a hierachial data set (Afshartous & de Leeuw 2002). We apply the multilevel prediction approach to the NELS:88 educational data in order to asses the improvement in prediction; the goal is to develop model selction criteria (multilevel cross-validation and multilevel bootstrap) that assess the predictive ability of multilevel models. We also introduce two plots that 1) aid in the visualization the amount to which the multilevel model predictions are "shrunk" or translated from the OLS predictions, 2) help identify if certain groups exist for which the predictions are pariculary good or bad.

We present a decomposition of prediction error for the multilevel model in the context of predicting a future observable y,j in the jth group of a hierarchial dataset. The multilevel prediction rule is used for prediction and the components of prediction error are estimated via a simulation study that spans the various combinations of level-1 (individual) and level-2 (group) sample sizes and different intraclass correlation values. These components of prediction error provide information with respect to the cost of aparameter estimation versus data inputation for predicting future values in a hierarchial data set. Specifically, the cost of parameter estimation is very small compared to data imputation.

Multilevel modeling is an increasingly popular technique for analyzing hierarchial data. We consider the problem of predicting a future observable y,j in the Jth group of a hierarchial dataset. Three prediction rules are presented and assessed via a Monte Carlo study that extensively covers both the sample size and parameter space. Specifically, the sample size space concerns the various combinations of level-1 and level-2 sample sizes, while the parameter space concerns different intraclass correlation values. The three prediction rules employ OLS, Prior, and Multilevel estimators for the level-1 coefficients Bj. The multilevel prediction rule performs the best across all design conditions, and the prior prediction rule degrades as the number of groups J increases.