ABSTRACT

Mixed-effects models include both fixed effects and random effects, where random effects are usually introduced to model correlation within a cluster and/or spatial correlations. They provide flexible tools to model both the mean and the covariance structures simultaneously. The simplest mixed-effects model is perhaps the classical two-way

mixed model. Suppose A is a fixed factor with a levels, B is a random factor with b levels, and the design is balanced. The two-way mixed model assumes that

yijk = µ+ αi + βj + (αβ)ij + ǫijk,

i = 1, . . . , a; j = 1, . . . , b; k = 1, . . . ,m, (9.1)

where yijk is the kth observation at level i of factor A and level j of factor B, µ is the overall mean, αi and βj are main effects, (αβ)ij is the interaction, and ǫijk are random errors. Since factor B is random, βj

and (αβ)ij are random effects. It is usually assumed that βj iid∼ N(0, σ2b ),

(αβ)ij iid∼ N(0, σ2ab), ǫijk iid∼ N(0, σ2), and they are mutually independent.