In this chapter linear models are extended to models with additional random components. We start with the general form of the model 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 sometimes assumed to be MVN(0, σ2vIq), and the error term MVN(0, σ

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. For known variance components, the MLE

βˆ = (XtV −1X)−1XtV −1y (5.2)

is the BLUE and BUE. When the variance components are unknown, we plug in the variance component estimators, resulting in a non-linear estimator for the mean parameters.