## Normal linear mixed models

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.