ASReml-S reference manual - VSN International

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8.5 Balanced repeated measures 94where D is a diagonal matrix of variances and C is a correlation matrix with elementsgiven by c ij = φ |t i−t j | . Parameter estimates for the Heterogeneous power model are:> summary(grass3.asr)$varcompgamma component std.error z.ratio constraintR!variance 1.000000 1.000000 NA NA FixedR!Time.pow 0.906702 0.906702 0.04156582 21.813643 UnconstrainedR!Time.1 60.716256 60.716256 28.40226409 2.137726 PositiveR!Time.3 73.266300 73.266300 36.98538578 1.980953 PositiveR!Time.5 308.521040 308.521040 138.29720253 2.230855 PositiveR!Time.7 435.122455 435.122455 172.05226788 2.529013 PositiveR!Time.10 381.527027 381.527027 138.94461658 2.745893 PositiveNote that asreml fixes the scale parameter to 1 to ensure that the elements of D areidentifiable. The final two models considered are the antedependence model of order 1and the unstructured model. Both require as starting values the lower triangle of the fullvariance matrix. By default, asreml generates starting gammas of 0.15 for variances and0.10 for covariances and scales these by 1/2 of the simple variance of the response. Thisis adequate in many cases (including this example) but we would generally recommendusing the REML estimate of Σ from a previous model. For example, suitable startingvalues could be generated from the heterogeneous power model (grass3.asr) by:> r vcov i + q − 1. The antedependence model oforder 1 has 9 parameters for these data, 5 in D and 4 in U. The call using the defaultstarting values is shown above.The antedependence parameter estimates are given below and appear successively foreach time, that is, the element of D and then the row of U:> summary(grass4.asr)$varcompgamma component std.error z.ratio constraintR!variance 1.000000000 1.000000000 NA NA FixedR!trait.y1:y1 0.026862013 0.026862013 0.011023470 2.436802 UnconstrainedR!trait.y3:y1 -0.628357272 -0.628357272 0.246074475 -2.553525 UnconstrainedR!trait.y3:y3 0.037296080 0.037296080 0.015467150 2.411309 UnconstrainedR!trait.y5:y3 -1.490928182 -1.490928182 0.586337948 -2.542780 UnconstrainedR!trait.y5:y5 0.005994700 0.005994700 0.002468106 2.428867 UnconstrainedR!trait.y7:y5 -1.280440740 -1.280440740 0.206796644 -6.191787 UnconstrainedR!trait.y7:y7 0.007896965 0.007896965 0.003232797 2.442765 UnconstrainedR!trait.y10:y7 -0.967801877 -0.967801877 0.062806208 -15.409335 UnconstrainedR!trait.y10:y10 0.039063461 0.039063461 0.015947584 2.449491 UnconstrainedFinally, the estimated components for the unstructured model using default starting values.> summary(grass5.asr)$varcomp
8.6 Spatial analysis of a field experiment 95gamma component std.error z.ratio constraintR!variance 1.00000 1.00000 NA NA FixedR!trait.y1:y1 37.22619 37.22619 15.19745 2.449503 UnconstrainedR!trait.y3:y1 23.39345 23.39345 13.20621 1.771398 UnconstrainedR!trait.y3:y3 41.51952 41.51952 16.95023 2.449496 UnconstrainedR!trait.y5:y1 51.65238 51.65238 32.03335 1.612456 UnconstrainedR!trait.y5:y3 61.91690 61.91690 34.87103 1.775597 UnconstrainedR!trait.y5:y5 259.12143 259.12143 105.78295 2.449558 UnconstrainedR!trait.y7:y1 70.81131 70.81131 46.13709 1.534802 UnconstrainedR!trait.y7:y3 57.61452 57.61452 46.74100 1.232634 UnconstrainedR!trait.y7:y5 331.80679 331.80679 145.19689 2.285220 UnconstrainedR!trait.y7:y7 551.50690 551.50690 225.14337 2.449581 UnconstrainedR!trait.y10:y1 73.78571 73.78571 46.21175 1.596687 UnconstrainedR!trait.y10:y3 62.56905 62.56905 46.92608 1.333353 UnconstrainedR!trait.y10:y5 330.85060 330.85060 144.31864 2.292501 UnconstrainedR!trait.y10:y7 533.75583 533.75583 220.57976 2.419786 UnconstrainedR!trait.y10:y10 542.17548 542.17548 221.33410 2.449580 UnconstrainedTable 8.5. Summary of Wald statistics for fixed effects for variance models fitted to the plantdatamodelTmt (df=1) trait:Tmt (df=4)uniform 9.42 20.40power 6.85 24.53heterogeneous power 0.00 19.28antedependence (order 1) 4.19 15.63unstructured 1.72 17.86The antedependence model of order 1 is clearly the more parsimonious model (Table 8.4).There is a surprising level of discrepancy between models for the Wald tests (Table 8.5).The main effect of treatment is significant for the uniform, power and antedependencemodels.8.6 Spatial analysis of a field experimentThis section illustrates spatial and incomplete block analyses of a field experiment usingasreml . There has been a large amount of interest in developing techniques for the analysisof spatial data both in the context of field experiments and geostatistical data [Cullis andGleeson, 1991, Cressie, 1991, Gilmour et al., 1997, for example]. This example illustratesthe analysis of so-called regular spatial data, in which the data is observed on a lattice orregular grid. This is typical of most small plot designed field experiments. Spatial datais often irregularly spaced, either by design or because of the observational nature of thestudy. The techniques we present in the following can be extended for the analysis ofirregularly spaced spatial data, though, larger spatial data-sets may be computationallychallenging, depending on the degree of irregularity or models fitted.The data appears in Gilmour et al. [1995] and is from a field experiment designed to comparethe performance of 25 varieties of barley. The experiment was conducted at SlateHall Farm, UK, in 1976 and was designed as a balanced lattice square with 6 replicateslaid out in a 10 × 15 rectangular grid. Table 8.6 shows the layout of the experiment and
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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
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 104 and 105: 8.5 Balanced repeated measures 92Ta
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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