ASReml-S reference manual - VSN International

2.1 The linear mixed model 8Direct products in R structuresConsider a vector of common errors associated with an experiment. The usual leastsquares assumption (and the default in asreml) is that these are independently and identicallydistributed (IID). However, if the data was from a field experiment laid out ina rectangular array of r rows by c columns, say, we could arrange the residuals e as amatrix and potentially consider that they were autocorrelated within rows and columns.Writing the residuals as a vector in field order, that is, by sorting the residuals rowswithin columns (plots within blocks) the variance of the residuals might then beσ 2 e Σ c (ρ c ) ⊗ Σ r (ρ r )where Σ c (ρ c ) and Σ r (ρ r ) are correlation matrices for the row model (order r, autocorrelationparameter ρ r ) and column model (order c, autocorrelation parameter ρ c )respectively. More specifically, a two-dimensional separable autoregressive spatial structure(AR1 ⊗ AR1) is sometimes assumed for the common errors in a field trial analysis(see Gogel (1997) and Cullis et al. (1998) for examples). In this case⎡1ρ r 1Σ r =ρ 2 r ρ r 1⎢ . . .⎣ . . .ρ r−1rρ r−2r. ..ρ r−3r . . . 1⎤ ⎡1ρ c 1and Σ c =ρ 2 c ρ c 1⎥ ⎢⎦ ⎣...ρ c−1cρ c−2c. ..ρ c−3c . . . 1⎤.⎥⎦See 3.15Alternatively, the residuals might relate to a multivariate analysis with n t traits and nunits and be ordered traits within units. In this case an appropriate variance structuremight beI n ⊗ Σwhere Σ (n t×n t ) is a variance matrix.Direct products in G structuresLikewise, the random terms in u in the model may have a direct product variance structure.For example, for a field trial with s sites, g varieties and the effects ordered varietieswithin sites, the model term site.variety may have the variance structureΣ ⊗ I gwhere Σ is the variance matrix for sites. This would imply that the varieties are independentrandom effects within each site, have different variances at each site, and arecorrelated across sites. Important Whenever a random term is formed as the interactionof two factors you should consider whether the IID assumption is sufficient or if a directproduct structure might be more appropriate.2.1.3 Variance structures for the errors: R structuresThe vector e will in some situations be a series of vectors indexed by a factor or factors.The convention we adopt is to refer to these as sections. Thus e = [e ′ 1, e ′ 2, . . . , e ′ s] ′ and thee j represent the errors of sections of the data. For example, these sections may represent