In this study, we are interested in comparing various computational approaches to Bayesian small area estimation of proportions in logistic regression models. The basic idea consists of incorporating into such a model nested random effects that reflect the complex structure of the data in a multistage sample design. As compared to the ordinary linear regression model, it is not feasible to obtain a closed form expression for the posterior distribution of the parameters. However, the proven optimality properties of empirical Bayes methods and their documented successful performance have made them popular (cf. Efron 1998). The EM algorithm has proven to be an extremely useful computational tool here for empirical Bayes estimation. The approximation often used in the M step is that proposed by Laird (1978), where the posterior is expressed as a multivariate normal distribution having its mean at the mode and covariance matrix equal to the inverse of the information matrix evaluated at the mode. Inspired by the
work of Zeger and Karim (1991), Wei and Tanner (1990), Gu and Li (1998) and Nielsen (2000) we also study a stochastic simulation method to approximate the posterior distribution. Alternatively, a hierarchical Bayes approach based on Gibbs sampling can also be employed. We present here the results of a Monte Carlo simulation study to compare point and interval estimates of small area proportions based on these three estimation methods. As the empirical Bayes estimators obtained are known to be biased, we use the bootstrap to correct for this.