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Classical analysis of variance requires that model terms be labeled as fixedor random and typically culminate by comparing variability from each batch(factor) to variability from errors; without a standard methodology to assessthe magnitude of a batch's variability, to compare variability between batches,nor to consider the uncertainty in this assessment. In this paper we supportrecent work, placing ANOVA into a general multilevel framework, then refinethis through batch level model specifications, and develop it further byextension to the multivariate case. Adopting a Bayesian multilevel modelparametrization, with improper batch level prior densities, we derive a methodthat facilitates comparison across all sources of variability. Whereasclassical multivariate ANOVA often utilizes a single covariance criterion, e.g.determinant for Wilks' lambda distribution, the method allows arbitrarycovariance criteria to be employed. The proposed method also addressescomputation. By introducing implicit batch level constraints, which yieldimproper priors, the full posterior is efficiently factored, thus alleviatingcomputational demands. For a large class of models, the partitioning mitigates,or even obviates the need for methods such as MCMC. The method is illustratedwith simulated examples and an application focusing on climate projections withglobal climate models.

MMANOVA: A general multilevel framework for multivariate analysis of variance