A1826
Title: On the inverse of covariance matrices for unbalanced crossed designs
Authors: Ziyang Lyu - University of New South Wales (Australia) [presenting]
Scott Sisson - University of New South Wales (Austria)
Alan Welsh - the Australian National University (Australia)
Abstract: A long-standing open problem in the analysis of linear mixed models with crossed random effects under unbalanced designs concerns finding an analytic expression for the inverse of the covariance matrix of the observed response. The inverse matrix is required for likelihood-based estimation and inference. However, for unbalanced crossed designs, the covariance matrix is dense and the lack of a closed-form representation for the inverse has made using likelihood-based methods computationally challenging and difficult to analyze mathematically. The Khatri--Rao product is used to represent the covariance matrix and to construct a modified covariance matrix whose inverse admits an exact spectral decomposition. Building on this construction, an elegant and simple approximation to the inverse of the covariance matrix for asymptotic unbalanced designs is obtained. For non-asymptotic settings, an accurate and interpretable approximation is derived under mildly unbalanced data and an exact inverse representation is established 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.
