Testing Variance Components in Mixed Linear Models

Introduction

Consider the mixed model

$$Y = X\beta + Z\gamma + \varepsilon,$$

where Y is an n ×1 observable random vector, X is an n ×p known matrix, β is a p ×1 vector of unknown parameters, Z is another known n ×m matrix, γ is an m ×1 unobservable random vector such that γ ∼ N _m (0, θ ₁ I), θ ₁ ≥ 0, and ε is another unobservable n ×1 random vector such that ε ∼ N _n (0, θ ₀ I), θ ₀ > 0. It is also assumed that γ and ε are independent and that n > rank(X, Z) > rank(X). Therefore, we have

$$Y \sim {N}_{n}(X\beta,{\theta }_{0}I + {\theta }_{1}ZZ^{\prime}).$$

(2)

Model (1) can be generalized to more than two variance components and has proven useful to practitioners in a variety of fields such as genetics, biology, psychology, and agriculture, where it is usually of interest to test the null hypothesis θ ₁ = 0 against the alternative θ ₁ > 0, or equivalently,

$${H}_{0} : \rho = 0,\qquad \mbox{ vs}\qquad {H}_{1} : \rho> 0,$$

where ρ = θ ₁ ∕ θ ₀.

Wald Test

Wald (1947) proposed an exact...