Bayesian Analysis of a Sensitive Proportion for a Small Area

Without accounting for sensitive items in sample surveys, sampled units may not respond (nonignorable nonresponse) or they respond untruthfully. There are several survey designs that address this problem and we will review some of them. In our study, we have binary data from clusters within small areas, obtained from a version of the unrelated‐question design, and the sensitive proportion is of interest for each area. A hierarchical Bayesian model is used to capture the variation in the observed binomial counts from the clusters within the small areas and to estimate the sensitive proportions for all areas. Both our example on college cheating and a simulation study show significant reductions in the posterior standard deviations of the sensitive proportions under the small‐area model as compared with an analogous individual‐area model. The simulation study also demonstrates that the estimates under the small‐area model are closer to the truth than for the corresponding estimates under the individual‐area model. Finally, for small areas, we discuss many extensions to accommodate covariates, finite population sampling, multiple sensitive items and optional designs.