ABSTRACT
In this chapter, linear models are extended to models with additional random components. We start with the general forms of the models and describe specific models as applications. Let y be an N -vector of responses, and X and Z be N × p and N × q model matrices for the fixed effect parameters β and random effect parameters v. The standard linear mixed model specifies
y = Xβ + Zv + e, (5.1)
where e ∼MVN(0,Σ), v ∼MVN(0, D), and v and e are independent. The variance matrices Σ and D are parameterized by an unknown variance component parameter τ , so random effect models are also known as variance component models. The random effect term v is often assumed to be MVN(0, σ2vIq), and the error term MVN(0, σ
2 eIN ), where
Ik is a k × k identity matrix, so the variance component parameter is τ = (σ2e , σ
If inferences are required about the fixed parameters only, they can be made from the implied multivariate normal model
y ∼MVN(Xβ, V ), where
V = ZDZ ′ +Σ.
