BAYESIAN-ANALYSIS OF BINARY AND POLYCHOTOMOUS RESPONSE DATA

A vast literature in statistics, biometrics, and econometrics is conce rned with the analysis of binary and polychotomous response data. The classical approach fits a categorical response regression model using maximum likelihood, and inferences about the model are based on the as sociated asymptotic theory. The accuracy of classical confidence state ments is questionable for small sample sizes. In this article, exact B ayesian methods for modeling categorical response data are developed u sing the idea of data augmentation. The general approach can be summar ized as follows. The probit regression model for binary outcomes is se en to have an underlying normal regression structure on latent continu ous data. Values of the latent data can be simulated from suitable tru ncated normal distributions. If the latent data are known, then the po sterior distribution of the parameters can be computed using standard results for normal linear models. Draws from this posterior are used t o sample new latent data, and the process is iterated with Gibbs sampl ing. This data augmentation approach provides a general framework for analyzing binary regression models. It leads to the same simplificatio n achieved earlier for censored regression models. Under the proposed framework, the class of probit regression models can be enlarged by us ing mixtures of normal distributions to model the latent data. In this normal mixture class, one can investigate the sensitivity of the para meter estimates to the choice of ''link function,'' which relates the linear regression estimate to the fitted probabilities. In addition, t his approach allows one to easily fit Bayesian hierarchical models. On e specific model considered here reflects the belief that the vector o f regression coefficients lies on a smaller dimension linear subspace. The methods can also be generalized to multinomial response models wi th J > 2 categories. In the ordered multinomial model, the J categorie s are ordered and a model is written linking the cumulative response p robabilities with the linear regression structure. In the unordered mu ltinomial model, the latent variables have a multivariate normal distr ibution with unknown variance-covariance matrix. For both multinomial models, the data augmentation method combined with Gibbs sampling is o utlined. This approach is especially attractive for the multivariate p robit model, where calculating the likelihood can be difficult.