Precise Asymptotics for Linear Mixed Models with Crossed Random Effects

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.

💡 Research Summary

This paper establishes a rigorous asymptotic normality theory for maximum‑likelihood estimators (MLEs) in a very general class of Gaussian linear mixed models that contain crossed random effects. While mixed‑effects models with nested random factors have been extensively studied, the crossed‑effects case poses substantial technical challenges because the marginal covariance matrix of the response lacks a block‑diagonal structure, preventing the straightforward application of classical matrix‑algebraic arguments.

The authors consider data indexed by two crossing factors, i = 1,…,m and i′ = 1,…,m′, with n_{ii′} observations in each cell (i,i′). The model is
Y_{ii′} | U_i , U_{i′}, X_A^{ii′}, X_B^{ii′}  ∼  N\big( X_A^{ii′}(β_A + U_i + U_{i′}) + X_B^{ii′}β_B,; σ^2 I_{n_{ii′}} \big),
with random effects U_i ∼ N(0, Σ) and U_{i′} ∼ N(0, Σ′). The dimensions of the fixed‑effect design matrices are d_A and d_B, respectively. By stacking all cells, the authors write the full conditional distribution as
Y | X_A,X_B ∼ N\big( X_Aβ_A + X_Bβ_B,; V(Σ,Σ′,σ^2) \big),
where V is a sum of three components: the contribution of the two crossed random‑effect covariance matrices and the residual variance σ^2 I.

Maximum‑likelihood estimation proceeds by maximizing the log‑likelihood ℓ(β_A,β_B,Σ,Σ′,σ^2). The paper’s central result (Result 1) shows that, under a set of regularity conditions—(A1) both m and m′ grow to infinity at comparable rates, (A2) the within‑cell sample sizes n_{ii′} diverge and are uniformly bounded away from zero relative to the overall average n, and (A3) the rows of X_A and X_B are non‑degenerate with finite second moments—the appropriately scaled estimator vector converges in distribution to a standard multivariate normal distribution.

Specifically, the following scaling matrices appear:

• For the crossed‑random‑effects fixed‑effect vector β_A, the scaling is (Σ/m + Σ′/m′)^{‑1/2}.
• For the ordinary fixed‑effect vector β_B, the scaling is (σ^2 C_{β_B} mm′ n)^{‑1/2}, where C_{β_B}=E