Hierarchical Bayesian Nonresponse Models for Binary Data From Small Areas With Uncertainty About Ignorability

Abstract

In the National Crime Survey (NCS), data on victimization can be poststratified into domains determined by urbanization, type of place, and poverty level. There is much difficulty in the analysis of binary data with substantial nonresponse. We consider three Bayesian hierarchical models for binary nonresponse data, like those from the NCS, which are clustered within a number of domains or areas. As in small area estimation, one key feature is that each model “borrows strength” across the areas through the selection approach to nonresponse. This is necessary to estimate the parameters with the least association to the observed data (i.e., weakly identified parameters). The first model assumes that the nonresponse mechanism is ignorable, and the second model assumes that it is nonignorable. We argue that a discrete model expansion (a probabilistic mixture) may be inappropriate for modeling uncertainty about ignorability. Therefore, we propose a third model through a continuous model expansion on an odds ratio for each area. When the odds ratio is 1, we have the ignorable model; otherwise, the model is nonignorable. One important feature is that uncertainty about ignorability is incorporated by “centering” on the ignorable model. We analyze the poststratified data from the NCS to reveal latent features associated with nonresponse. The complexity of the posterior distributions of the parameters forces us to implement the methodology using Markov chain Monte Carlo methods. When the proportion of households with a characteristic (i.e., victimization in the NCS) and the response probability of a household in the population are estimated, we find that the nonignorable model and the expansion model are similar but that they differ from the ignorable model. Although considerable prior information about the nonresponse mechanism is needed, the expansion model indicates that nonresponse for most of the areas is nonignorable. An analysis shows that inference is not very sensitive to an important distribution assumption, and a simulation exercise shows that the expansion model works very well.