A Bayesian Comparison of Unit-Level and Fay–Herriot Models for Small Area Estimation

This paper compares two Bayesian small area estimation models, a weighted unit-level model and the Fay–Herriot model, under probability sampling. Posterior inference uses the multiplication rule, with noninformative priors for the unit-level model and a proper shrinkage prior for the Fay–Herriot model. The methodology is illustrated with body mass index data for eight California counties from the National Health and Nutrition Examination Survey and evaluated in two simulation studies without clustering or stratification. The two models produce similar posterior means, but the Fay–Herriot model exhibits larger posterior standard deviations, and the unit-level results are sensitive to the normalization of the survey weights. In the first simulation study, the unit-level credible intervals fall below nominal coverage, which a diagnostic attributes to using the weights as variance scalings when the errors have constant variance, whereas the Fay–Herriot intervals attain or exceed it. The second study, a calibrated Bayes check, shows approximately nominal coverage for both models under their prior predictive distributions, which does not imply nominal coverage at fixed parameter values. These results indicate that smaller posterior uncertainty is not by itself an advantage: while the Fay–Herriot model remains attractive when only area-level summaries are available, posterior uncertainty and interval coverage should be considered together, and the conclusions are conditional on the weighting and variance specifications used.