Interval Estimation for Small Area Proportions with Small True Proportions from Stratified Random Sampling Survey Data∗†

Consider interval estimation of m small area proportions Pi (i = 1, · · · ,m), where we assume a stratified random sampling design with equal number of observations n in each stratum, and where the domains of interest are the strata. A 100(1 − α)% confidence interval for Pi that has appeared repeatedly in the literature and is used in application is given by P̂ i ± zα/2 √ msei, where P̂ i and msei are an empirical Bayes estimator of Pi and an associated second-order unbiased mean squared error estimator (i = 1, · · · ,m). In the case where no covariates are available, the underlying model is pi|Pi ind. ∼ N (Pi, ψi), Pi ind. ∼ N (μ,A), where pi is the sample proportion for domain i (i = 1, . . . ,m); ψi are known sampling variances; and μ and A are unknown hyperparameters. We refer to models that use the normality assumption on both levels of the hierarchy as “NormalNormal Models.” The well-documented problems of the normal approximation to the binomial raise questions about the accuracy of confidence intervals based on the Normal-Normal model above when the domain sample sizes are small or when the true domain proportions are close to 0 or 1. We argue that a more reasonable model in this setting is a beta-binomial model in which the sampled stratum counts have binomial distributions and the prior distribution of the true stratum proportions follows a beta distribution. The Beta-Binomial Model has also previously appeared in the literature as a candidate for modelling small area proportions. We examine a new empirical Bayes confidence interval based on this model. We perform simulation studies under the Beta-Binomial Model that compare the peformance of this CI and an alternative CI constructed using the Normal-Normal Model.