Predicting Random Effects From Finite Population Clustered Samples With Response Error
In many situations there is interest in parameters (e. g., mean) associated with the response distribution of individual clusters in a finite clustered population. We develop predictors of such parameters using a two-stage sampling probability model with response error. The probability model stems directly from finite population sampling without additional assumptions and thus is design-based. The predictors are closely related to best linear unbiased predictors (BLUP) that arise from common mixed-model methods, as well as to model-based predictors obtained via super population approaches for survey sampling. The context assumes clusters of equal size and equal size sampling of units within clusters. Target parameters may correspond to clusters realized in the sample, as well as nonrealized clusters. In either case, the predictors are linear and unbiased, and minimize the expected mean squared error. They correspond to the sum of predictors of responses for realized and nonrealized units in the cluster, accounting directly for the second-stage sampling fraction. In contrast, the BLUP commonly used in mixed models can be interpreted as predicting only the responses of second-stage units not observed for a cluster, not the cluster mean. The development reveals that two-stage sampling does not give rise to a more general variance structure often assumed in superpopulation models, even when variances within clusters are heterogeneous. With response error present, we predict target random variables defined as an expected (or average) response over units in a cluster.