Statistical inference using Kronecker structured covariance

Abstract

We present results for testing and estimation in the context of separable covariance models. We concentrate on two types of data: relational data and cross-classified data. Relational data is frequently represented by a square matrix and we are often interested in identifying patterns of similarity between entries in the matrix. Under the assumption of a separable covariance, a natural model for such data is based on the matrix-variate normal distribution. In the context of this model we develop a likelihood ratio test for testing for row and column dependence based on the observation of a single relational data matrix. We provide extensions of the test to accommodate common features of such data, such as undefined diagonal entries, a non-zero mean, multiple observations, and deviations from normality. We then develop an estimation procedure for mean and covariance parameters under this model. In the context of cross-classified data, the separable covariance structure plays a role in relating the different effects in an ANOVA decomposition. Specifically, for many types of categorical factors, it is plausible that levels of a factor that have similar main- effect coefficients may also have similar coefficients in higher-order interaction terms. We introduce a class of hierarchical prior distributions based on the array-variate normal that can adapt to such similarities and hence borrow information from main effects and lower- order interactions in order to improve estimation of higher-order interactions.