A hierarchical Bayesian model for binary data incorporating selection bias

We consider a Bayesian nonignorable model to accommodate a nonignorable selection mechanism for predicting small area proportions. Our main objective is to extend a model on selection bias in a previously published paper, coauthored by four authors, to accommodate small areas. These authors assume that the survey weights (or their reciprocals that we also call selection probabilities) are available, but there is no simple relation between the binary responses and the selection probabilities. To capture the nonignorable selection bias within each area, they assume that the binary responses and the selection probabilities are correlated. To accommodate the small areas, we extend their model to a hierarchical Bayesian nonignorable model and we use Markov chain Monte Carlo methods to fit it. We illustrate our methodology using a numerical example obtained from data on activity limitation in the U.S. National Health Interview Survey. We also perform a simulation study to assess the effect of the correlation between the binary responses and the selection probabilities.