Bayesian analysis of a three-component hierarchical design model

Inferences about the parameters in the three-component hierarchical design model yijk=μ+ai+bij+eijk are considered from a Bayesian viewpoint. Under the usual normality and independence assumptions and adopting a non-informative reference prior distribution, various features of the posterior distribution of the variance components σ12= var (eijk), σ22 = var (bij) and σ32 = var (ai) are discussed, including inferences about a variance ratio, the relative contributions of the components and the magnitude of the individual components. A scaled χ2 approximation technique is developed for the marginal distributions of the components which can be applied to general q-component model. In addition, a Bayesian solution to the problem of pooling variance estimates is given.