On the inverse of covariance matrices for unbalanced crossed designs

TL;DR

This paper develops a novel analytical approach to invert covariance matrices in unbalanced crossed linear mixed models, enabling more efficient likelihood-based inference.

Contribution

It introduces a spectral decomposition method using the Khatri--Rao product to approximate and exactly represent the inverse of the covariance matrix in unbalanced designs.

Findings

Accurate approximation of the inverse covariance matrix for unbalanced designs.

Stable and computationally efficient method demonstrated through simulations.

Exact inverse representation as a low-rank correction for arbitrary unbalance levels.

Abstract

This paper 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 $\mathbf{V}$, the covariance matrix of the observed response. The inverse matrix $\mathbf{V}^{-1}$ is required for likelihood-based estimation and inference. However, for unbalanced crossed designs, $\mathbf{V}$ is dense and the lack of a closed-form representation for $\mathbf{V}^{-1}$, until now, has made using likelihood-based methods computationally challenging and difficult to analyse mathematically. We use the Khatri--Rao product to represent $\mathbf{V}$ 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 $\mathbf{V}^{-1}$ 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.