ASReml-S reference manual - VSN International

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8.8 Paired Case-Control Study 110[ ]3.84 2.33⊗ I2.33 1.96 44Thus the variance of the variety effects in the control group (also known as the geneticvariance for this group) is 3.84. The genetic variance for the treated group is much lower(1.96). The genetic correlation is 2.33/ √ 3.84 × 1.96 = 0.85 which is strong, supportingearlier indications of the dependence between the treated and control root area (Figure8.8).A multivariate approachIn this simple case in which the variance heterogeneity is associated with the two levelfactor Tmt, the analysis is equivalent to a bivariate analysis in which the two traitscorrespond to the two levels of Tmt, namely sqrtroot for control and treated. The modelfor each trait is given byy j = Xτ j + Z v u vj + Z r u rj + e j (j = c, t) (8.11)where y j is a vector of length n = 132 containing the sqrtroot values for variate j (j = cfor control and j = t for treated), τ j corresponds to a constant term and u vj and u rjcorrespond to random variety and run effects. The design matrices are the same for bothtraits. The random effects and error are assumed to be independent Gaussian variableswith zero means and variance structures var ( )u vj = σ2vjI 44 , var ( )u rj = σ2rjI 66 andvar (e j ) = σj 2I 132. The bivariate model can be written as a direct extension of (8.11),namelyy = (I 2 ⊗ X) τ + (I 2 ⊗ Z v ) u v + (I 2 ⊗ Z r ) u r + e ∗ (8.12)where y = (y ′ c, y ′ t) ′ , u v = ( u ′ v c, u ′ v t) ′,ur = ( 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 (8.9). The variety effects for each trait (u v in the bivariate model) are partitionedin (8.9) into variety main effects and tmt.variety interactions so that u v = 1 2 ⊗u 1 +u 2 .There is a similar partitioning for the run effects and the errors (Table 8.9).Table 8.9. Equivalence of random effects in bivariate and univariate analysesbivariate univariateeffects (model 8.12) (model 8.9)trait:Variety u v 1 2 ⊗ u 1 + u 2trait:Run u r 1 2 ⊗ u 3 + u 4Pair:trait e ∗ 1 2 ⊗ u 5 + eIn addition to the assumptions in the models for individual traits (8.11), the bivariateanalysis involves the assumptions cov (u vc ) u ′ v t= σ vct I 44 , cov (u rc ) u ′ r t= σ rct I 66 andcov (e c ) e ′ t = σ ct I 132 . Thus random effects and errors are correlated between traits. So,for example, the variance matrix for the variety effects for each trait is given by[ ]σ2var (u v ) = vcσ vctσ vct σv 2 ⊗ I 44t
8.8 Paired Case-Control Study 111This unstructured form for trait:Variety in the bivariate analysis is equivalent tothe Variety main effect plus heterogeneous Variety:Tmt interaction variance structure(8.10) in the univariate analysis. Similarly the unstructured form for trait:Run is equivalentto the Run main effect plus heterogeneous Run:Tmt interaction variance structure.The unstructured form for the errors (Pair:trait) in the bivariate analysis is equivalentto the Pair plus heterogeneous error (Pair:Tmt) variance in the univariate analysis.The asreml call is:> riceMV.asr summary(riceMV.asr)$loglik[1] -343.2199> summary(wormm.asr)$varcompgamma component std.error z.ratio constrainttrait:Variety!trait.syc:syc 3.8386404 3.8386404 1.1059311 3.4709 Unconstrainedtrait:Variety!trait.sye:syc 2.3332757 2.3332757 0.7755975 3.0083 Unconstrainedtrait:Variety!trait.sye:sye 1.9612537 1.9612537 0.7281807 2.6933 Unconstrainedtrait:Run!trait.syc:syc 1.7083269 1.7083269 0.6534826 2.6141 Unconstrainedtrait:Run!trait.sye:syc 0.3196590 0.3196590 0.5437117 0.5879 Unconstrainedtrait:Run!trait.sye:sye 2.5436703 2.5436703 0.7957811 3.1964 UnconstrainedR!variance 1.0000000 1.0000000 NA NA FixedR!trait.syc:syc2.1436774 2.1436774 0.4822556 4.4451 UnconstrainedR!trait.sye:syc0.9873042 0.9873042 0.3811844 2.5900 UnconstrainedR!trait.sye:sye2.3474190 2.3474190 0.5076600 4.6239 UnconstrainedThe resultant REML log-likelihood is identical to that of the heterogeneous univariateanalysis (column (b) of Table 8.8). The estimated variance parameters are summarisedin Table 8.10.Table 8.10. Estimated variance parameters from bivariate analysis of bloodworm datacontrol treatedsource variance variance covarianceus(trait):Variety 3.84 1.96 2.33us(trait):Run 1.71 2.54 0.32Pair:us(trait) 2.14 2.35 0.99Predicted variety means are obtained from:> riceMV.pv riceMV.pv$predictions$"trait:Variety":$"trait:Variety"$pvals:trait Variety predicted.value standard.error est.status1 syc AliCombo 14.953229 0.9180964 Estimable2 syc Amaroo 16.161198 0.9180985 Estimable3 syc Balilla 14.420236 0.9185701 Estimable4 syc Bluebelle 13.103132 0.9309747 Estimable5 syc Bogan 15.768223 0.9548522 Estimable
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Analysis of mixed modelsfor S langu
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D.G. ButlerQueensland Department of
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ContentsPreface . . . . . . . . . .
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ContentsviiB Available variance mod
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List of Figures1.1 Weekly body weig
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1.3 Data sets used 21.2.2 Help and
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1.3 Data sets used 41.3.2 Repeated
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1.3 Data sets used 6101 Sire 1 0102
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2.1 The linear mixed model 8Direct
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2.2 Estimation 10There is a corresp
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2.2 Estimation 12where H ij = ∂ 2
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2.5 Inference for random effects 14
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2.6 Inference for fixed effects 16T
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2.6 Inference for fixed effects 18t
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3Fitting the mixed model3.1 Introdu
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3.3 Introducing the asreml function
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3.6 Getting help3.7 The asreml func
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3.8 Fixed terms 26na.method.Xkeep.o
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3.8 Fixed terms 28Summary of reserv
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3.10 Conditional factors: the at()
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Table 3.2. Families and link functi
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3.15 Multivariate analysis 34R!trai
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Source df F_inc F_con MSource df dd
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4.1.1 Specifying variance models in
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4.2 A sequence of structures for th
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Σ = [σ ij ] :{σii = σ 2 , ∀i
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4.4 Variance model functions 44init
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The Matérn class4.4 Variance model
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4.4 Variance model functions 48Chol
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Required argumentsobj4.5 Rules for
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4.7 Constraining variance parameter
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5Genetic analysis5.1 IntroductionIn
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5.2 Pedigree, G-inverse objects and
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5.4 Using Pedigree and G-inverse ob
Page 72 and 73: 6Prediction from the linear model6.
Page 74 and 75: Optional argumentslevels6.2 The pre
Page 76 and 77: 6.2 The predict method 64The predic
Page 78 and 79: 6.4 Complicated weighting 666.4 Com
Page 80 and 81: 7The asreml class and related metho
Page 82 and 83: maxiter maximum number of iteration
Page 84 and 85: 7.3 Methods and related functions 7
Page 88 and 89: pedigree7.3 Methods and related fun
Page 92 and 93: 8Examples8.1 IntroductionThis secti
Page 94 and 95: 8.2 Split Plot Design 82In this exa
Page 96 and 97: 8.3 Unbalanced nested design 84[4,]
Page 98 and 99: 8.3 Unbalanced nested design 86adju
Page 100 and 101: 8.4 Sources of variability in unbal
Page 102 and 103: 8.5 Balanced repeated measures 902
Page 104 and 105: 8.5 Balanced repeated measures 92Ta
Page 106 and 107: 8.5 Balanced repeated measures 94wh
Page 108 and 109: 8.6 Spatial analysis of a field exp
Page 110 and 111: 8.6 Spatial analysis of a field exp
Page 112 and 113: Variety predicted.value standard.er
Page 114 and 115: 8.7 Unreplicated early generation v
Page 116 and 117: 8.7 Unreplicated early generation v
Page 118 and 119: 8.8 Paired Case-Control Study 106Th
Page 120 and 121: Terms added sequentially; adjusted
Page 124 and 125: 8.8 Paired Case-Control Study 112..
Page 126 and 127: 8.9 Balanced longitudinal data - Ra
Page 134 and 135: A.3 Aliasing and singularities 122W
Page 136 and 137: Table B.1: Details of the available
Page 138 and 139: B Available variance models 126Deta
Page 140 and 141: ReferencesG. E. P. Box. Analysis of
Page 142 and 143: References 130W. W. Stroup, P. S. B
Page 144 and 145: Index 132F statistics, 17fa(,k), 49
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