ASReml-S reference manual - VSN International

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8.8 Paired Case-Control Study 106The multivariate data frame> riceMV names(riceMV)[1] "Pair" "Run" "Variety" "yc" "ye" "syc" "sye"contains factors Variety and Run and variates for root weight and square root of rootweight for both the control and exposed treatments (yc, ye, syc, sye respectively).A plot of the treated vs the control root area (on the square root scale) for each variety isshown in Figure 8.8. There is a strong dependence between the treated and control rootarea, which is not surprising. The aim of the experiment was to determine the toleranceof varieties to bloodworms and identify the most tolerant varieties. The definition oftolerance should allow for the fact that varieties differ in their inherent seedling vigour(Figure 8.8). The initial approach was to regress the treated root area against the controlroot area and define the index of vigour as the residual from this regression. This is clearlyinefficient since there is error in both variables. We seek to determine an index of tolerancefrom the joint analysis of treated and control root area.141210sye864210 15 20sycFig. 8.8. Rice bloodworm data: Plot of square root of root weight for treated versus controlStandard analysisPreliminary analyses indicated variance heterogeneity so that subsequent analyses wereconducted on the square root scale. The allocation of bloodworm treatments withinvarieties and varieties within runs defines a nested block structure of the formRun/Variety/Tmt = Run + Run:Variety + Run:Variety:Tmt( = Run + Pair + Pair:Tmt )( = Run + Run:Variety + units )There is an additional blocking term, however, due to the fact that the bloodworms withina run are derived from the same batch of larvae whereas between runs the bloodwormscome from different sources. This defines a block structure of the form
Run/Tmt/Variety = Run + Run:Tmt + Run:Tmt:Variety( = Run + Run:Tmt + Pair:Tmt )Combining the two provides the full block structure for the design:Run + Run:Variety + Run:Tmt + Run:Tmt:Variety= Run + Run:Variety + Run:Tmt + units= Run + Pair + Run:Tmt + Pair:Tmt8.8 Paired Case-Control Study 107In line with the aims of the experiment the treatment structure comprises variety andtreatment main effects and treatment by variety interactions.In the traditional approach the terms in the block structure are regarded as randomand the treatment terms as fixed. The choice of treatment terms as fixed or randomdepends largely on the aims of the experiment. The aim of this example is to selectthe best varieties. The definition of best is somewhat more complex since it does notinvolve the single trait sqrt(rootwt) but rather two traits, namely sqrt(rootwt) inthe presence/absence of bloodworms. To minimize selection bias the variety main effectsand Tmt:Variety interactions are taken as random. The main effect of treatment is fittedas fixed to allow for the likely scenario that rather than a single population of treatmentby variety effects there are in fact two populations (control and treated) with a differentmean for each. There is evidence of this prior to analysis with the large difference in meansqrt(rootwt) for the two groups (14.93 and 8.23 for control and treated respectively).The inclusion of Tmt as a fixed effect ensures that BLUPs of Tmt:Variety effects areshrunk to the correct mean (treatment means rather than an overall mean).The model for the data is given byy = Xτ + Z 1 u 1 + Z 2 u 2 + Z 3 u 3 + Z 4 u 4 + Z 5 u 5 + e (8.9)where y is a vector of length n = 264 containing the sqrtroot values, τ corresponds toa constant term and the fixed treatment contrast and u 1 . . . u 5 correspond to randomVariety, Tmt:Variety, Run, Tmt:Run and Variety:Run effects. The random effects anderror are assumed to be independent Gaussian variables with zero means and variancestructures var (u i ) = σ 2 i I b i(where b i is the length of u i , i = 1 . . . 5) and var (e) = σ 2 I n .The asreml call is:> rice1.asr summary(rice1.asr)$loglik[1] -345.2559> summary(rice1.asr)$varcompgamma component std.error z.ratio constraintVariety 1.8085661 2.3782170 0.7914703 3.004809 PositiveVariety:Tmt 0.3743885 0.4923110 0.2764182 1.781037 PositiveRun 0.2444393 0.3214312 0.5482284 0.586309 PositivePair 0.7421389 0.9758932 0.3883409 2.512981 PositiveTmt:Run 1.3291572 1.7478068 0.4793480 3.646217 PositiveR!variance 1.0000000 1.3149738 0.2974417 4.420946 Positive> wald(rice1.asr)Wald tests for fixed effectsResponse: sqrtroot
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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
Page 68 and 69: 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 120 and 121: Terms added sequentially; adjusted
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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