Author: Ryan Janicki (Center for Statistical Research and Methodology, U. S. Census Bureau)
Linear mixed effects models have been popular in small area estimation problems for modeling survey data when the sample size in one or more areas is too small for reliable inference. However, when the data are restricted to a bounded interval, the linear model may be inappropriate, particularly if the data are near the boundary. This paper investigates the use of a beta distribution as an alternative to the normal distribution as a sampling model for survey estimates of proportions which take values in (0; 1). Inference for small area proportions based on the posterior distribution of a beta regression model ensures that point estimates and credible intervals take values in (0; 1). Properties of a hierarchical Bayes small area model with a beta sampling distribution and logistic link function are presented and compared to those of the linear mixed effect model, and an MCMC algorithm for posterior inference is given. An example using 2010 Small Area Income and Poverty Estimates (SAIPE) data is given.