Bayesian analysis of three-way ANOVA model and a diagnostic test for the convergence of the estimation

The three-way analysis of variance (ANOVA) is one of the useful multilevel models in which a normality assumption is used for statistical inference. The main purpose of the paper is to evaluate some of the strengths of using Markov Chain Monte Carlo (MCMC) methods to this model. Experience shows that getting the mathematical expression for the joint posterior distribution can be quite difficult (see for details, Box and Tiao (1992)). But with the use of Bayesian software package WINBUGS, we can get the conditional distributions after specifying the model (the likelihood and the prior). While demonstrating these MCMC applications, a diagnostic procedure is shown after illustrating an example.