ASReml-S reference manual - VSN International

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8.9 Balanced longitudinal data - Random coefficients and cubic smoothing splines 118var (vec(U)) = Σ ⊗ I 5where Σ is a 2 × 2 symmetric positive definite matrix. Non smooth variation can bemodelled at the overall mean (across trees) level and this is achieved by including thefactor dev(x) as a random term.The full model is:> orange1.asr orange1.asr summary(orange1.asr)$varcompgamma component std.error z.ratio constraintlink(~ x):Tree!Intercept.var 4.789034e+00 3.044970e+01 2.457813e+01 1.238894 Positivelink(~ x):Tree!x.var 9.392257e-05 5.971797e-04 4.240625e-04 1.408235 Positivespl(x) 1.004896e+02 6.389346e+02 4.131293e+02 1.546573 PositiveTree:spl(x) 1.116746e+00 7.100507e+00 4.935166e+00 1.438757 PositiveR!variance 1.000000e+00 6.358213e+00 3.652341e+00 1.740860 PositiveThe fitted curves from this model are shown in Figure 8.14. The fit is unacceptablebecause the spline has picked up too much curvature, suggesting there may be systematicnon-smooth variation at the overall level. This can be formally examined by includingthe dev(x) term as a random effect.
8.9 Balanced longitudinal data - Random coefficients and cubic smoothing splines 119200 400 600 800 1200 160045200150100Circumference2001 250315010050200 400 600 800 1200 1600200 400 600 800 1200 1600AgeFig. 8.14. Plot of fitted cubic smoothing spline for model 1Model 2 increased the REML log-likelihood by 3.70 (P < 0.05) with the spl(x) smoothingconstant approaching the boundary. The Season factor provides a possible explanation.When included in Model 3, Season has a Wald statistic of 107.3 (P < 0.01) and dev(x)becomes bounded. The spring measurements are lower than the autumn measurementsso growth is slower in winter. Models 4 and 5 successively examined each term, indicatingthat both smoothing constants are significant. Model 6 includes the covarianceparameter between the intercept and slope for each tree; this ensures that the model willbe translation invariant. This model requires care in the choice of starting values. Theasreml call, illustrating an alternative method for specifying initial values, and the fittedcomponents for model 6 are:> orange6.asr summary(orange6.asr)$varcompgamma component std.error z.ratio constraintlink(~ x):Tree!Intercept:Intercept 5.6011527956 31.7916580507 2.512945e+01 1.2651157 Unconstrainedlink(~ x):Tree!x:Intercept -0.0123766915 -0.0702490288 8.234755e-02 -0.8530798 Unconstrainedlink(~ x):Tree!x:x 0.0001080119 0.0006130664 4.344414e-04 1.4111604 Unconstrainedspl(x) 2.1666162231 12.2975259925 1.119901e+01 1.0980902 PositiveTree:spl(x) 1.3751301469 7.8051195890 5.222912e+00 1.4944001 PositiveR!variance 1.0000000000 5.6759133719 3.294276e+00 1.7229622 PositiveThe fitted values for individual trees (adjusted for Season) from model 6 together with amarginal prediction and approximate confidence intervals (2× standard error of prediction)are shown in Figure 8.15. The conclusions from this analysis are quite different from
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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
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6Prediction from the linear model6.
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Optional argumentslevels6.2 The pre
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6.2 The predict method 64The predic
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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 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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