ASReml-S reference manual - VSN International

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8.5 Balanced repeated measures 92Table 8.4. Summary of variance models fitted to the plant datanumber of REMLmodel parameters loglikelihood BICuniform 2 -196.88 401.95power 2 -182.98 374.15heterogeneous power 6 -171.50 367.57antedependence (order 1) 9 -160.37 357.51unstructured 15 -158.04 377.501. by fitting a random units term plus an independent residual using the multivariatedata frame,2. by specifying a cor() variance model for the R-structure, again using the multivariatedata frame,> grass1.asr summary(grass.asr)$loglik[1] -196.8768> summary(grass.asr)$varcompgamma component std.error z.ratio constraintunits 1.263422 159.8157 75.74879 2.109812 PositiveR!variance 1.000000 126.4943 25.82054 4.898979 Positive> summary(grass1.asr)$loglik[1] -196.8768> summary(grass1.asr)$varcompgamma component std.error z.ratio constraintR!variance 1.0000000 286.3098952 78.3448584 3.654482 PositiveR!trait.cor 0.5581911 0.5581911 0.1303821 4.281196 Unconstrained
8.5 Balanced repeated measures 93A more plausible model for repeated measures data would allow the correlations todecrease as the lag increases. The simplest model often used which accommodates thisis the first order autoregressive model, however since the heights are not measured atequally spaced time points we use the exp() power model. The correlation function isgiven by:ρ(u) = φ uwhere u is the time lag is weeks. The variance parameters from this model are:> summary(grass2.asr)$varcompgamma component std.error z.ratio constraintR!variance 1.0000000 301.3581311 96.67407884 3.117259 PositiveR!Time.pow 0.9190129 0.9190129 0.03117151 29.482466 UnconstrainedWhen fitting such models be careful to ensure the scale of the defining variate, here time,does not result in an estimate of φ too close to 1. For example, use of days in this examplewould result in an estimate for φ of about 0.993.> plot(grass2.asr, option = ’t’)creates a trend plot of residuals against the factors that index the experimental units.The first of these two plots, residuals against plant number for each time, is shown inFigure 8.4.Time: 10200-20-40Time: 5Time: 7Residuals200-20-40Time: 1 Time: 3200-20-402 4 6 8 10 12 14PlantFig. 8.4. residual ∼ Plant | Time for the exp() variance model for the plant dataThe residual plot suggests increasing variance over time. This can be modelled via theexph() variance function, which models Σ byΣ = D 0.5 CD 0.5
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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
Page 54 and 55: Σ = [σ ij ] :{σii = σ 2 , ∀i
Page 56 and 57: 4.4 Variance model functions 44init
Page 58 and 59: The Matérn class4.4 Variance model
Page 60 and 61: 4.4 Variance model functions 48Chol
Page 62 and 63: Required argumentsobj4.5 Rules for
Page 64 and 65: 4.7 Constraining variance parameter
Page 66 and 67: 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 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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