  ANOVA decompositions are a standard method for describing and estimating
heterogeneity among the means of a response variable across levels of multiple
categorical factors. In such a decomposition, the complete set of main effects
and interaction terms can be viewed as a collection of vectors, matrices and
arrays that share various index sets defined by the factor levels. For many
types of categorical factors, it is plausible that an ANOVA decomposition
exhibits some consistency across orders of effects, in that the levels of a
factor that have similar main-effect coefficients may also have similar
coefficients in higher-order interaction terms. In such a case, estimation of
the higher-order interactions should be improved by borrowing information from
the main effects and lower-order interactions. To take advantage of such
patterns, this article introduces a class of hierarchical prior distributions
for collections of interaction arrays that can adapt to the presence of such
interactions. These prior distributions are based on a type of array-variate
normal distribution, for which a covariance matrix for each factor is
estimated. This prior is able to adapt to potential similarities among the
levels of a factor, and incorporate any such information into the estimation of
the effects in which the factor appears. In the presence of such similarities,
this prior is able to borrow information from well-estimated main effects and
lower-order interactions to assist in the estimation of higher-order terms for
which data information is limited.
