Journal of Official Statistics, Vol.6, No.2, 1990. pp. 133–155
A Bayesian Loglinear Model Analysis of Categorical Data
Robert M. Leighty and William J. Johnson
Abstract:We illustrate a two-staged Bayesian strategy for making inferences about the loglinear model parameters that summarize the interaction structure in a multi-dimensional contingency table. In the first stage, we locate full and reduced uniformly ordered models whose parameter vectors enclose all important parameters. In the second-stage, posterior regions are used to identify important loglinear model terms. Parameters in these terms are estimated by Bayesian posterior means that compromise full and reduced model maximum likelihood estimates (MLE's). The likelihood is summarized by the approximate normal distribution of the sufficient full-model MLE. A hierarchical normal-Cauchy-tail prior, centered at the reduced-model MLE, is assumed. A relative precision hyper-parameter measures our belief in the reduced model. We illustrate three methods of approximating the posterior moments: the Laplace method, an empirical Bayes method, and the diagonalized covariance method. Our Bayesian strategy is then used to reanalyze the Ries-Smith detergent preference data.
Keywords:Hierarchical Bayes; Laplace approximation; credible regions.
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