Download pdf guide - VSN International

2 Some theory 8laid out in a rectangular array of r rows by c columns, we could arrange the residualsas a matrix and might consider that they were autocorrelated within rowsand columns. Writing the residuals as a vector in field order, that is, by sortingthe residuals rows within columns (plots within blocks) the variance of theresiduals 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 parameterρ c ) respectively. More specifically, a two-dimensional separable autoregressivespatial structure (AR1 ⊗ AR1) is sometimes assumed for the common errors ina field trial analysis (see Gogel (1997) and Cullis et al. (1998) for examples). Inthis case⎡⎤⎡⎤11ρ r 1ρρ 2 c 1r ρ r 1⎥ and Σ c = ⎢ ρ 2 c ρ c 1⎥ .Σ r =⎢⎣ .ρ r−1r.ρ r−2r. . .. ⎥⎦ρ r−3r . . . 1⎢⎣ .ρ c−1c.ρ c−2c. . .. ⎥⎦ρ c−3c . . . 1See Chapter 8for further detailsAlternatively, the residuals might relate to a multivariate analysis with n t traitsand n units and be ordered traits within units. In this case an appropriatevariance structure might beI n ⊗ Σwhere Σ (nt×nt) is a general or unstructured variance matrix.Direct products in G structuresLikewise, the random terms in u in the model may have a direct product variancestructure. For example, for a field trial with s sites, g varieties and the effectsordered varieties within sites, the model term site.variety may have the variancestructureΣ ⊗ I gwhere Σ is the variance matrix for sites. This would imply that the varieties areindependent random effects within each site, have different variances at each site,and are correlated across sites. Important Whenever a random term is formedas the interaction of two factors you should consider whether the IID assumptionis sufficient or if a direct product structure might be more appropriate.