ASReml-S reference manual - VSN International

4.4 Variance model functions 49the initial values are given asc(σ 11 , σ 21 , σ 22 , . . . , σ 44 )antedependenceThe k-factor antedependence ante(,k) structure models Σ ω×ω asΣ −1 = UDU ′where U ω×ω is a unit upper triangular matrix with elements {u ij } whereand D = diag(d 1 , . . . , d ω ).u ii = 1u ij = 0, i > ju ij = u ij , 1 ≤ i − j ≤ kConsidering the above example for a factor Site with 4 levels,asreml(. . . ,ante(Site,1). . . )generates Σ −1 = UDU ′ where⎡⎤ ⎡⎤1 u 12 0 0 d 1 0 0 0⎢ 0 1 uU =23 0 ⎥ ⎢ 0 d⎣⎦ D =2 0 0 ⎥⎣⎦0 0 1 u 34 0 0 d 3 00 0 0 10 0 0 d 4factor analyticIn asreml the parameters for ante are given as for us, chol and cholc as the lower trianglerow-wise of an unstructured matrix.In a factor analytic model of order k (fa(,k)), the variance matrix Σ ω×ω is modelled asΣ = ΓΓ ′ + Ψwhere Γ (ω×k) is a matrix of loadings and Ψ ω×ω is a diagonal matrix whose elements arereferred to as specific variances.For example, if Site is a factor with 4 levels, the component matrices forasreml(. . . ,fa(Site,1). . . )are⎡ ⎤ ⎡⎤l 1 ψ 1 0 0 0⎢ lΓ = 2 ⎥ ⎢ 0 ψ⎣ ⎦ Ψ =2 0 0 ⎥⎣⎦l 3 0 0 ψ 3 0l 4 0 0 0 ψ 4where the parameters are given in the order c(diag(Ψ), vec(Γ)).A key issue with factor analytic models is that it is possible for the REML estimatesof the specific variances to be zero. The fa() algorithm [Thompson et al., 2003] used inasreml permits some (or all) specific variances to be set to zero.If k > 1, constraints on the elements of Γ are required for identifiability. These constraintsare chosen to be: l 12 = 0 for k = 1; l 13 = l 23 = 0 for k = 3;, etc. asreml sets these elementsto zero and their corresponding boundary constraints to ”F”.4.4.5 Known relationship structuresgiv(obj, var=FALSE, init=NA)ped(obj, var=FALSE, init=NA)