Bayesian analysis of nonignorable missing categorical data: An application to bone mineral density and family income

We consider a problem in which an analysis is needed for categorical data from a single two-way table with partial classification (i.e., both item and unit nonresponses). We assume that this is the only information available. A Bayesian methodology permits modeling different patterns of missingness under ignorability and nonignorability assumptions. We construct a nonignorable nonresponse model which is obtained from the ignorable nonresponse model via a model expansion using a data-dependent prior; the nonignorable nonresponse model robustifies the ignorable nonresponse model. A multinomial-Dirichlet model, adjusted for the nonresponse, is used to estimate the cell probabilities, and a Bayes factor is used to test for association. We illustrate our methodology using data on bone mineral density and family income. A sensitivity analysis is used to assess the effects of the data-dependent prior. The ignorable and nonignorable nonresponse models are compared using a simulation study, and there are subtle differences between these models.