ASReml-S reference manual - VSN International

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8.9 Balanced longitudinal data - Random coefficients and cubic smoothing splines 116> orange names(orange)[1] "Tree" "x" "circ" "Season"where Tree is a factor with 5 levels, x is tree age in days since 31 December 1968, circis the trunk circumference and Season is a factor with two levels, Spring and Autumn.The factor Season was included after noting that tree age spans several years and ifconverted to day of year, measurements were taken in either April/May (Spring) orSeptember/October (Autumn).Initially we restrict the dataset to tree 1 to demonstrate fitting cubic splines in asreml.The model includes the intercept and linear regression of trunk circumference on x andan additional random term spl(x) which includes a random term with a special designmatrix with 7−2 = 5 columns which relate to the vector, δ whose elements δ i , i = 2, . . . , 6are the second differentials of the cubic spline at the knot points. The second differentialsof a natural cubic spline are zero at the first and last knot points [Green and Silverman,1994].> orange.asr summary(orange.asr)$varcompgamma component std.error z.ratio constraintspl(x) 0.07876884 3.954159 9.950608 0.3973786 PositiveR!variance 1.00000000 50.199529 37.886791 1.3249876 Positive> wald(orange.asr, denDF="default")$WaldDf denDF F inc Pr(Intercept) 1 3.5 1382.0 3.124431e-06x 1 3.5 217.5 1.229875e-04$stratumVariancesdf Variance spl(x) R!variancespl(x) 1.488944 97.64263 11.99606 1R!variance 3.511056 50.20223 0.00000 1Predicted values of the spline curve at nominated points can be obtained by:> orange.pv
8.9 Balanced longitudinal data - Random coefficients and cubic smoothing splines 117140120Circumference100806040200 400 600 800 1000 1200 1400 1600AgeFig. 8.13. Fitted cubic smoothing spline for tree 1Table 8.11. ANOVA decomposition for the orange datastratum decomposition type df or ne(Intercept) 1 f 1Agex f 1spl(x) r 5residual r 7TreeTree rc 5Age:Treex:Tree rc 5spl(x):Tree r 25errorrAn overall spline is included as well as tree deviation splines. We note that the interceptand slope for the tree deviation splines are assumed to be random effects. This isconsistent with Verbyla et al. [1999]. In this sense the tree deviation splines play a rolein modelling the conditional curves for each tree and variance modelling. The interceptand slope for each tree are included as random coefficients (denoted by rc in Table 8.11).Thus, if U 5×2 is the matrix of intercepts (column 1) and slopes (column 2) for each tree,then we assume that
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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
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 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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