ASReml-S reference manual - VSN International

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8.9 Balanced longitudinal data - Random coefficients and cubic smoothing splines 1140.5Regression residual0.0-0.5-2 -1 0 1 2 3Control BLUPFig. 8.10. Estimated deviations from regression of treated on control for each variety plottedagainst estimate for controlControl BLUP-Exposed BLUP10-1-2 -1 0 1 2 3Control BLUPFig. 8.11. Estimated difference between control and treated for each variety plotted againstestimate for control
8.9 Balanced longitudinal data - Random coefficients and cubic smoothing splines 115The implementation of cubic smoothing splines in asreml is based on the mixed modelformulation of Verbyla et al. [1999]. More recently the methodology has been extendedso that the user can specify knot points; in the original approach the knot points weretaken to be the ordered set of unique values of the explanatory variable. The specificationof knot points is particularly useful if the number of unique values in the explanatoryvariable is large, or if units are measured at different times.These data were originally reported by Draper and Smith [1998, ex24N, p559] and hverecently been reanalysed by Pinheiro and Bates [2000, p338]. The data are trunk circumferences(in millimetres) of each of 5 trees taken at 7 times (Figure 8.12). All treeswere measured at the same time so that the data are balanced. The aim of the studyis unclear, though both previous analyses involved modelling the overall growth curve,accounting for the obvious variation in both level and shape between trees.Pinheiro and Bates [2000] used a nonlinear mixed effects modelling approach, in whichthey modelled the growth curves by a three parameter logistic function of age:y =φ 11 + exp [−(x − φ 2 )/φ 3 ](8.13)where y is the trunk circumference, x is the tree age in days since December 31 1968, φ 1is the asymptotic height, φ 2 is the inflection point or the time at which the tree reaches0.5φ 1 , φ 3 is the time elapsed between trees reaching half and about 3/4 of φ 1 .200 400 600 800 1200 16004520015010050circ1 2320015010050200 400 600 800 1200 1600200 400 600 800 1200 1600ageFig. 8.12. Trellis plot of trunk circumference (mm) for each tree against age in days since 1December 1968.The data frame orange contains:
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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
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 122 and 123: 8.8 Paired Case-Control Study 110[
Page 124 and 125: 8.8 Paired Case-Control Study 112..
Page 128 and 129: 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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