APPLICATIONS OF BAYESIAN METHODS IN ANALYSIS OF VARIANCE

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. 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 lack of ready-to-use formulae and algorithms. The aim of this research was to construct a Bayesian hierarchical model for hypothesis test in ANOVA designs using non-informative priors, conditionally conjugate priors as well as the Zellner-g priors. First, the posterior distributions were obtained. Then the effects of various hyper parameters on variance parameters in ANOVA were illustrated. Markov Chain Monte Carlo (MCMC) and Gibbs sampling were then used to obtain posterior point estimates from these posterior distributions. The 95% credible intervals were also obtained and then used to draw inferences. Posterior F-values were obtained for the different priors and finally compared with those obtained using classical approach. Conditional conjugate Normal posterior distribution for means was obtained while conditional conjugate inverse gamma posterior distributions for the variances were also obtained. An F-Value of 4.598 was obtained using the classical approach while posterior F-value of 4.56 was obtained for normal priors for means and conjugate inverse Gamma for the variances. Posterior F-value of 4.62 was obtained using Zellner-g prior (g=n=30) whereas Posterior F-value of 4.52 was obtained using Zellner-g prior (g=k2=30).The results indicated that the F-Values obtained using the classical and the Bayesian approach are similar. 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. v