A comparison of computational approaches to Bayesian small area estimation of proportions in hierarchical logistic models

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 e ects that
re ect the complex structure of the data in a multistage sample design. As com-
pared 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 algo-
rithm 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 distri-
bution 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 dis-
tribution. 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.
Keywords and phrases: Logistic Regression, Generalized Linear Models, Overdis-
persion, Random E ects, Stochastic Simulation, EM algorithm, Gibbs Sampling.