Bayesian predictive inference for units with small sample sizes. The case of binary random variables

The National Health Interview Survey is designed to produce precise estimates for the entire United States but not for individual states. In this study, Bayesian predictive inference is used to provide point estimates and measures of variability for the desired finite population quantities. The investigation reported here concerns binary random variables such as the occurrence of at least one doctor visit within the past 12 months. The specification is hierarchic. First, for each cluster, there is a separate logistic regression relating a patient's probability of a doctor visit with his or her characteristics. Second, there is a multivariate linear regression linking the (cluster) regression parameters to covariates measured at the cluster level. A fully Bayesian analysis is carried out; this technique provides gains over synthetic estimation and conventional randomization-based analysis. The reported approach is potentially useful for any situation when the sample size associated with a unit of interest (e.g., a hospital or small geographic area) is too small to permit satisfactory inference using only the data from that unit.