Posterior F-Value In Bayesian Analysis Of Variance Using Winbugs

Abstract

Analysis of variance (ANOVA) is a standard method for describing and estimating heterogeneity among the means of a response variable across the levels of multiple categorical factors. In most experimental settings, ANOVA is used to test the presence of treatment effects. Frequentist approaches to making inferences about the variances of random cluster effects in hierarchical generalized linear models (HGLMs) have several limitations. These include reliance on asymptotic theory, questionable properties of classical likelihood ratio tests when pseudo-likelihood methods are used for estimation, and a failure to account for uncertainty in the estimation of features of prior distributions for model parameters. This paper compares and contrasts alternative approaches to making a specific type of inference about the variance components in an HGLM, focusing on the difference in the variance components. A Bayesian approach to making inferences about these types of differences is proposed that circumvents many of the problems associated with alternative frequentist approaches.Bayesian hypothesis testing literature on ANOVA is scant; the dominant treatment is still classical or frequentist. One impediment to adoption of Bayesian approach is lack of practical development, particularly a lack of ready-to-use formulas and algorithms. Markov Chain Monte Carlo (MCMC) and Gibbs sampling are used to obtain posterior point estimates from these posterior distributions. The 95% credible intervals (CI) were also obtained. Posterior F-values were obtained for the different priors and finally compared with that obtained using classical approach. The Bayesian test for ANOVA designs is useful to both researchers and students; both groups will get to appreciate the importance of Bayesian approach when applied to practical statistical problems.

Key Words: Bayesian Analysis of Variance, Variance Components, Hierarchical Generalized Linear Models, Posterior F-value, ANOVA.