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15 Examples 278(and in fact necessary and obtained by use of the !GU option) in this context sinceit should be considered as part of the variance structure for the combined varietymain effects and treatment by variety interactions. That is,[ ]σ2var (1 2 ⊗ u 1 + u 2 ) = 1 + σ2c 2 σ12σ12 σ1 2 + ⊗ I 44 (15.8)σ2 2tUsing the estimates from table 15.8 this structure is estimated as[ ]3.84 2.33⊗ I2.33 1.96 44Thus the variance of the variety effects in the control group (also known as thegenetic variance for this group) is 3.84. The genetic variance for the treated groupis much lower (1.96). The genetic correlation is 2.33/ √ 3.84 ∗ 1.96 = 0.85 whichis strong, supporting earlier indications of the dependence between the treatedand control root area (Figure 15.8).A multivariate approachIn this simple case in which the variance heterogeneity is associated with the twolevel factor tmt, the analysis is equivalent to a bivariate analysis in which the twotraits correspond to the two levels of tmt, namely sqrt(rootwt) for control andtreated. The model for each trait is given byy j = Xτ j + Z v u vj + Z r u rj + e j (j = c, t) (15.9)where y j is a vector of length n = 132 containing the sqrtroot values for variatej (j = c for control and j = t for treated), τ j corresponds to a constant termand u vj and u rj correspond to random variety and run effects. The design matricesare the same for both traits. The random effects and error are assumedto be independent ) Gaussian ) variables with zero means and variance structuresvar(u vj = σv 2 jI 44 , var(u rj = σr 2 jI 66 and var (e j ) = σj 2I 132. The bivariatemodel can be written as a direct extension of (15.9), namelyy = (I 2 ⊗ X) τ + (I 2 ⊗ Z v ) u v + (I 2 ⊗ Z r ) u r + e ∗ (15.10)where y = (y ′ c, y ′ t) ′ , u v = ( u ′ v c, u ′ v t) ′, u r = ( u ′ r c, u ′ r t) ′and e ∗ = (e ′ c, e ′ t) ′ .There is an equivalence between the effects in this bivariate model and the univariatemodel of (15.7). The variety effects for each trait (u v in the bivariatemodel) are partitioned in (15.7) into variety main effects and tmt.variety interactionsso that u v = 1 2 ⊗ u 1 + u 2 . There is a similar partitioning for the runeffects and the errors (see table 15.9).