Prediction with measurement errors in finite populations.

We address the problem of selecting the best linear unbiased predictor (BLUP) of the latent value (e.g., serum glucose fasting level) of sample subjects with heteroskedastic measurement errors. Using a simple example, we compare the usual mixed model BLUP to a similar predictor based on a mixed model framed in a finite population (FPMM) setup with two sources of variability, the first of which corresponds to simple random sampling and the second, to heteroskedastic measurement errors. Under this last approach, we show that when measurement errors are subject-specific, the BLUP shrinkage constants are based on a pooled measurement error variance as opposed to the individual ones generally considered for the usual mixed model BLUP. In contrast, when the heteroskedastic measurement errors are measurement condition-specific, the FPMM BLUP involves different shrinkage constants. We also show that in this setup, when measurement errors are subject-specific, the usual mixed model predictor is biased but has a smaller mean squared error than the FPMM BLUP which point to some difficulties in the interpretation of such predictors.

Performance of Balanced Two-Stage Empirical Predictors of Realized Cluster Latent Values from Finite Populations: A Simulation Study.

Predictors of random effects are usually based on the popular mixed effects model developed under the assumption that the sample is obtained from a conceptual infinite population; such predictors are employed even when the actual population is finite. Two alternatives that incorporate the finite nature of the population are obtained from the superpopulation model proposed by Scott and Smith (1969, JASA, 64: 830-840) or from the finite population mixed model recently proposed by Stanek and Singer (2004, JASA, 99:1119-1130). Predictors derived under the latter model with the additional assumptions that all variance components are known and that within-cluster variances are equal have smaller mean squared error than the competitors based on either the mixed effects or Scott and Smith's models. As population variances are rarely known, we propose method of moment estimators to obtain empirical predictors and conduct a simulation study to evaluate their performance. The results suggest that the finite population mixed model empirical predictor is more stable than its competitors since, in terms of mean squared error, it is either the best or the second best and when second best, its performance lies within acceptable limits. When both cluster and unit intra-class correlation coefficients are very high (e.g., 0.95 or more), the performance of the empirical predictors derived under the three models is similar.