ASReml-S reference manual - VSN International

4.3 Types of variance models 41Table 4.1. Sequence of variance structures for the NIN field trialasreml call random term (G) residual error term (R)modelmodel1 2 1 21 yield ∼ Replicate + Variety - - - units idv() -1a1byield ∼ Variety,random = ∼ Replicateyield ∼ Variety,random = ∼ idv(Replicate),rcov = ∼ idv(units)Replicate idv() - units idv() -Replicate idv() - units idv() -2 yield ∼ Variety,rcov = ∼ Column:ar1(Row)- - - Column.Row id() ar1()2a2byield ∼ Variety,rcov = ∼ ar1(Column):ar1(Row)yield ∼ Variety,random = ∼ units,rcov = ∼ ar1(Column):ar1(Row)- - - Column.Row ar1() ar1()units idv() - Column.Row ar1() ar1()3 yield ∼ Variety,Column.Row ar1() ar1v units id() -random = ∼ ar1v(Column):ar1(Row)4.3 Types of variance modelsThere are three types of variance model that are used in fitting R and G structuresin asreml, namely, correlation models, homogeneous variance models and heterogeneousvariance models. These determine the form for each component of G and R. In thefollowing, we denote the variance matrix of any component relating to a term in randomor rcov by Σ.4.3.1 Correlation modelsIn correlation models all diagonal elements are identically equal to 1. Algebraically, ifΣ = [ρ ij ] , i, j = 1 . . . ω, denotes the correlation matrix for a particular model, then{ρii = 1, ∀iΣ = [ρ ij ] :ρ ij = ρ ji , |ρ ij | ≤ 1, i ≠ j.The simplest correlation model in asreml is the id() model, where Σ = I ω4.3.2 Homogeneous variance modelsIf the variance model is specified as a homogeneous variance model, the diagonal elementsall have the same positive value, σ 2 say. That is,