Precise Asymptotics for Linear Mixed Models with Crossed Random Effects
Jiming Jiang, Matt P. Wand, Swarnadip Ghosh
TL;DR
This work develops precise asymptotic normality results for maximum likelihood estimators in Gaussian linear mixed models that feature crossed random effects, unbalanced designs, and multivariate random slopes. The authors derive explicit leading-term asymptotic covariances for all parameter blocks, showing that fixed effects and covariance components exhibit near-nested-model-like behavior despite crossing. They provide practical statistical tools, including confidence intervals, Wald tests, and sample size calculations, with a detailed supplementary derivation of the Fisher information approximations. The results extend existing theory to broader, more realistic crossed designs and offer insights into the similarities and differences with nested random effects models, while outlining avenues for future non-Gaussian and sparse-data extensions.
Abstract
We obtain an asymptotic normality result that reveals the precise asymptotic behavior of the maximum likelihood estimators of parameters for a very general class of linear mixed models containing cross random effects. In achieving the result, we overcome theoretical difficulties that arise from random effects being crossed as opposed to the simpler nested random effects case. Our new theory is for a class of Gaussian response linear mixed models which includes crossed random slopes that partner arbitrary multivariate predictor effects and does not require the cell counts to be balanced. Statistical utilities include confidence interval construction, Wald hypothesis test and sample size calculations.