Empirical Bayes estimators of small area proportions in multistage designs

The importance of small area estimation as a facet of survey sampling cannot be over-emphasized, Of late, there has been an increasing demand for small area statistics in both the public and private sectors. It is widely recognized that direct survey estimates for small areas are likely to be unstable due to the small sample sizes in the areas. This makes it necessary to ''borrow strength'' from related areas to obtain more accurate estimates. In this study, an empirical Bayes methodology for the estimation of small area proportions is proposed, implemented, and evaluated. The basic idea consists of incorporating into a logistic regression model, random effects that are nested in such a way as to reflect the complex structure of a multistage sample design. This yields both point estimates and associated naive measures of accuracy. The latter do not incorporate the uncertainty that arises from estimating the prior distribution of the random effects. We adjust these naively-estimated measures of uncertainty using the bootstrap techniques developed by Laird and Louis (1987). The proposed estimation approach is applied to data from a United States Census to predict local labour force participation rates. Results are compared with those obtained using unbiased and synthetic estimation, as well as a domain-adjusted synthetic estimation approach which incorporates predictor variables at both the individual and local area levels.