ASReml-S reference manual - VSN International

2.2 Estimation 10There is a corresponding partition in Z, Z = [Z 1 Z 2 . . . Z b ]. As before each submatrix,G i , is assumed to be the kronecker product of one, two or three component matrices.These matrices are indexed for each of the factors constituting the term in the linearmodel. For example, the term site:genotype has two factors and so the matrix G i iscomprised of two component matrices defining the variance structure for each factor inthe term.Models for the component matrices G i include the standard model for which G i = γ i I qias well as direct product models for correlated random factors given byG i = G i1 ⊗ G i2 ⊗ G i3for three component factors. The vector u i is therefore assumed to be the vector representationof a 3-way array. For two factors the vector u i is simply the vec of a matrixwith rows and columns indexed by the component factors in the term, where vec of amatrix is a function which stacks the columns of its matrix argument below each other.A range of models are available for the components of both R and G. They includecorrelation (C) models (that is, where the diagonals are 1), or covariance (V ) models andare discussed in detail in Chapter 4 (see Section 4.3). Some correlation models include• autoregressive (order 1 or 2)• moving average (order 1 or 2)• ARMA(1,1)• uniform• banded• general correlation.Some of the covariance models include• diagonal (that is, independent with heterogeneous variances)• antedependence• unstructured• factor analytic.There is the facility within asreml to allow for a nonzero covariance between the subvectorsof u, for example in random regression models . In this setting the intercept and say theslope for each unit are assumed to be correlated and it is more natural to consider thethe two component terms as a single term, which gives rise to a single G structure. Thisconcept is discussed later.2.2 EstimationEstimation involves two processes that are very strongly linked. One process involvesestimation of τ and predic- tion of u (although the latter may not always be of interest)for given θ, φ and γ. The other process involves estimation of these variance parameters.Note that in the following sections we have set θ = 1 to simplify the presentation ofresults.