Predicting Random Effects from a Finite Population of Unequal Size Clusters based on Two-Stage Sampling

Prediction of random effects is an important problem with expanding applications. In the simplest context, the problem corresponds to prediction of the latent value (the mean) of a realized cluster selected via two-stage sampling. Best linear unbiased predictors developed from mixed models are widely used, but their development requires distributional assumptions or an infinite population framework. When the number and size of clusters is finite, super population models have been used to predict the contribution of the unobserved units to a realized cluster mean. Recently, predictors developed from a two-stage sampling model have been shown to out-perform the modelbased predictors. However, the random permutation model underlying these predictors is limited to settings where cluster sizes and sample sizes per cluster are equal. We present a new two-stage sampling model that can be used when cluster sizes and sample sizes per cluster differ. The model expands the set of random variables from the two-stage sampling model to accommodate the different size clusters. The resulting best linear unbiased predictor out performs any of the competing predictors, even when clusters and sample sizes per cluster are equal. The reduction in expected mean squared error relative to predictors developed under mixed model or super population model assumptions is likely due to the specificity of the expanded random permutation model to the problem. This suggests faithfully capturing the stochastic aspects of a problem is more important than simplifying assumptions in developing optimal solutions. Many other problems may be amenable to improved solutions based on extension of this approach. C07ed30.doc 11/29/2007 1:31 PM ii Contact Address: Edward J. Stanek III Department of Public Health 401 Arnold House University of Massachusetts at Amherst Amherst, Ma. 01002 Phone: 413-545-3812 Fax: 413-545-1645 Email: [email protected]