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Biometrika 1967 54(1-2):109-125; doi:10.1093/biomet/54.1-2.109
© 1967 by Biometrika Trust
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Bayesian analysis of a three-component hierarchical design model

G. C. TIAO and G. E. P. BOX

University of Wisconsin

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 {sigma}12= var (eijk), {sigma}22 = var (bij) and {sigma}32 = 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 {chi}2 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.


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