MMANOVA: A general multilevel framework for multivariate analysis of variance

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