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2 Some theory 72.1 The linear mixed modelIntroductionIf y denotes the n × 1 vector of observations, the linear mixed model can bewritten asy = Xτ + Zu + e (2.1)where τ is the p × 1 vector of fixed effects, X is an n × p design matrix of fullcolumn rank which associates observations with the appropriate combination offixed effects, u is the q × 1 vector of random effects, Z is the n × q design matrixwhich associates observations with the appropriate combination of random effects,and e is the n × 1 vector of residual errors.The model (2.1) is called a linear mixed model or linear mixed effects model. Itis assumed [ ] ([ ] [ ])u 0 G(γ) 0∼ N , θ(2.2)e 0 0 R(φ)where the matrices G and R are functions of parameters γ and φ, respectively.The parameter θ is a variance parameter which we will refer to as the scaleparameter. In mixed effects models with more than one residual variance, arisingfor example in the analysis of data with more than one section (see below) orvariate, the parameter θ is fixed to one. In mixed effects models with a singleresidual variance then θ is equal to the residual variance (σ 2 ). In this case Rmust be a correlation matrix (see Table 2.1 for a discussion).Direct product structuresTo undertake variance modelling in ASReml you need to understand the formationof variance structures via direct products (⊗). The direct product of two matricesA (m×p) and B (n×q) is⎡⎤a 11B . . . a 1p B.⎢ . .. .⎥⎣a m1B . ⎦. . amp B.Direct products in R structuresConsider a vector of common errors associated with an experiment. The usualleast squares assumption (and the default in ASReml) is that these are independentlyand identically distributed (IID). However, if e was from a field experiment