ASReml-S reference manual - VSN International

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Terms added sequentially; adjusted for those aboveDf Sum of Sq Wald statistic Pr(Chisq)(Intercept) 1 1953.17 1485.33 < 2.2e-16 ***Tmt 1 617.16 469.33 < 2.2e-16 ***residual (MS) 1.318.8 Paired Case-Control Study 108The estimated variance components from this analysis also appear in column (a) of Table8.8. The variance component for the Variety main effects is large. There is evidence ofVariety:Tmt interactions so we may expect some discrimination between varieties interms of tolerance to bloodworms.Given the large difference (p < 0.001) between Tmt means we may wish to allow for heterogeneityof variance associated with Tmt. Thus we fit a separate Variety:Tmt variancefor each level of Tmt so that instead of assuming var (u 2 ) = σ 2 2I 88 we assume[ ]σ2var (u 2 ) = 2c 00 σ2t2 ⊗ I 44where σ 2 2c and σ 2 2t are the Variety:Tmt interaction variances for control and treated respectively.This model can be fitted using a diagonal variance structure for the treatmentpart of the interaction. We also fit a separate Run:Tmt variance for each level of Tmt andheterogeneity at the residual level, by including an extra at(Tmt,2):units term. Wehave chosen level 2 of Tmt as we expect more variation for the exposed treatment andthus the extra variance component for this term should be positive.By default, asreml sets the parameter constraint for variance components to Positive. Toallow for negative components, which may have meaning in this particular example, wemust set the parameter constraints to Unconstrained. The following sequence of calls• creates default R and G parameter list objects (start.values=T) in temp• opens the default text editor where the parameter constraints can be changed to Uand the result saved to RG.rice• fits the model using the G level parameter settings in RG.rice through the G.paramargument.> temp rice2.asr summary(rice2.asr)$loglik[1] -343.2199> summary(worm2.asr)$varcompgamma component std.error z.ratio constraintVariety 2.018113 2.333895 0.776098 3.007 PositiveVariety:Tmt!Tmt.Control.var 1.302060 1.505799 0.665177 2.263 UnconstrainedVariety:Tmt!Tmt.Exposed.var -0.321861 -0.372224 0.456151 -0.816 UnconstrainedRun 0.276148 0.319358 0.543446 0.587 Positive
8.8 Paired Case-Control Study 109Pair 0.853857 0.987463 0.381194 2.590 PositiveTmt:Run!Tmt.Control.var 1.200871 1.388777 0.635834 2.184 UnconstrainedTmt:Run!Tmt.Exposed.var 1.923409 2.224372 0.723924 3.072 Unconstrainedat(Tmt, Exposed):units 0.176145 0.203707 0.631838 0.322 PositiveR!variance 1.000000 1.156474 0.417382 2.770 Positive> wald(rice2.asr)Wald tests for fixed effectsResponse: sqrtrootTerms added sequentially; adjusted for those aboveDf Sum of Sq Wald statistic Pr(Chisq)(Intercept) 1 1476.86 1277.03 < 2.2e-16 ***Tmt 1 519.01 448.78 < 2.2e-16 ***residual (MS) 1.16Table 8.8. Estimated variance components from univariate analyses of bloodworm data. (a)Model with homogeneous variance for all terms and (b) model with heterogeneous variance forinteractions involving tmtsource(a)(b)homogeneous heterogeneouscontrol treatedVariety 2.378 2.333Variety:Tmt 0.492 1.505 -0.372Run 0.321 0.319Run:Tmt 1.748 1.389 2.224Variety:Run (Pair) 0.976 0.987Tmt:Pair 1.315 1.156 1.360REML log-likelihood -345.256 -343.22The estimated variance components from this analysis are given in column (b) of Table8.8. There is no significant variance heterogeneity at the residual or Run:Tmt level. Thisindicates that the square root transformation of the data has successfully stabilised theerror variance. There is, however, significant variance heterogeneity for Variety:Tmtinteractions with the variance being much greater for the control group. This reflectsthe fact that in the absence of bloodworms the potential maximum root area is greater.Note that the Variety:Tmt interaction variance for the treated group is negative. Thenegative component is meaningful (and in fact necessary and obtained by changing theconstraint codes for variance parameters to U as described above) in this context sinceit should be considered as part of the variance structure for the combined variety maineffects and treatment by variety interactions. That is,[ ]σ2var (1 2 ⊗ u 1 + u 2 ) = 1 + σ2c 2 σ12σ12 σ1 2 + σ2t2 ⊗ I 44 (8.10)Using the estimates from Table 8.8 this structure is estimated as
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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
Page 70 and 71: 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 122 and 123: 8.8 Paired Case-Control Study 110[
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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ASReml-S reference manual - VSN International

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