Small area inference for binary variables in the National Health Interview Survey

The National Health Interview Survey is designed to produce precise estimates of finite population parameters for the entire United Stares but not for small geographical areas or subpopulations. Our investigation concerns estimates of proportions such as the probability of at least one visit to a doctor within the past 12 months. To include all sources of variation in the model, we carry out a Bayesian hierarchical analysis for the desired finite population quantities. First, for each cluster (county) a separate logistic regression relates the individual's probability of a doctor visit to his or her characteristics. Second, a multivariate linear regression links cluster regression parameters to covariates measured at the cluster level. We describe the numerical methods needed to obtain the desired posterior moments. Then we compare estimates produced using the exact numerical method with approximations. Finally, we compare the hierarchical Bayes estimates to empirical Bayes estimates and to standard methods, that is, synthetic estimates and estimates obtained from a conventional randomization-based approach. We use a cross-validation exercise to assess the quality of model fit. We also summarize the results of a separate study of the binary indicator of partial work limitation. Because we know the value of this variable for each respondent to the 1990 Census long form, we can compare estimates corresponding to alternative methods and models with very accurate estimates of the true values.