On the inverse of covariance matrices for unbalanced crossed designs

2025/12/10 by Lyu, Ziyang, Sisson, S. A., Welsh, A. H.
#FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2512.09273

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 V, the covariance matrix of the observed response. The inverse matrix V^-1 is required for likelihood-based estimation and inference. However, for unbalanced crossed designs, V is dense and the lack of a closed-form representation for 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 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 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.

Citations

• Asymptotics for EBLUPs within crossed mixed effect models
• Backfitting for large scale crossed random effects regressions

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