ASReml-S reference manual - VSN International

2Some theory2.1 The linear mixed model2.1.1 IntroductionIf y denotes the n × 1 vector of observations, the linear mixed model can be written asy = Xτ + Zu + e (2.1)where τ is the p×1 vector of fixed effects, X is an n×p design matrix of full column rankwhich associates observations with the appropriate combination of fixed effects, u is theq × 1 vector of random effects, Z is the n × q design matrix which associates observationswith the appropriate combination of random effects, and e is the n × 1 vector of residualerrors.The model (2.1) is called a linear mixed modelassumed [ ] ([ ]u 0∼ N , θe 0or linear mixed effects model. It is])(2.2)[G(γ) 00 R(φ)where the matrices G and R are functions of parameters γ and φ, respectively. Theparameter θ is a variance parameter which we will refer to as the scale parameter. Inmixed effects models with more than one residual variance, arising for example in theanalysis of data with more than one section (see below) or variate, the parameter θ isfixed to one. In mixed effects models with a single residual variance then θ is equal tothe residual variance (σ 2 ). In this case R must be correlation matrix (see Table 2.1 for adiscussion).2.1.2 Direct product structuresTo undertake variance modelling in asreml it is important to understand the formation ofvariance structures via direct products (⊗). The direct product of two matrices A (m×p)and B (n×q) is⎡⎤a 11B . . . a 1p B.⎢ . .. .⎥⎣a m1B . ⎦. . amp B.