On the inverse of covariance matrices for unbalanced crossed designs

Abstract

This presentation addresses a long-standing open problem in the analysis of linear mixed models with crossed random effects under unbalanced designs: how to find an analytic expression for the inverse of $\bV$, the covariance matrix of the observed response.  The inverse matrix $\bV^{-1}$ is required for likelihood-based estimation and inference.  However, for unbalanced crossed designs, $\bV$ is dense and the lack of a closed-form representation for $\bV^{-1}$, until now, has made using likelihood-based methods computationally challenging and difficult to analyse mathematically. We use the Khatri--Rao product to represent $\bV$ and then to construct a modified covariance matrix whose inverse admits an exact spectral decomposition.  Building on this construction, we obtain an elegant and simple approximation to $\bV^{-1}$ for asymptotic unbalanced designs. 

For non-asymptotic settings, we derive an accurate and interpretable approximation under mildly unbalanced data and establish an exact inverse representation as a low-rank correction to this approximation, applicable to arbitrary degrees of unbalance. 

Simulation studies demonstrate the accuracy, stability, and computational tractability of the proposed framework.