ASReml-S reference manual - VSN International

4Specifying the variance structures4.1 IntroductionThis chapter introduces variance model specification in asreml, a complex aspect of themodelling process. The key concepts are:• The mixed linear modelhas a residual termand random effectswhere in the most complex formsand eachy = Xτ + Zu + ee ∼ N(0, θR)u ∼ N(0, θG)R = ⊕ i R iG = ⊕ j G jR i = R i (φ i )G j = G j (δ j )where φ i and δ j parameterise the respective variance models.• We use the terms R structure and G structure to refer to the matrices R i and G jabove in a syntactic manner, respectively,• R and G structures are typically formed as a direct product of particular variancemodels,• The order of terms in a direct product must agree with the order of effects in thecorresponding model term,• Variance models may be correlation matrices or variance matrices with equal or unequalvariances on the diagonal. A model for a correlation matrix (eg. ar1) can beconverted to an equal variance form (eg. ar1v) and to a heterogeneous variance form(eg. ar1h),• Variances are sometimes estimated as variance component ratios (relative to the overallscale parameter, θ).Chapter 2 gives theoretical details. We begin this chapter by considering an ordered sequenceof variance structures for the NIN variety trial (see Section 4.2) as an introductionto variance modelling in practice; we then consider the topics in detail.