ASReml-S reference manual - VSN International

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8Examples8.1 IntroductionThis section considers the analysis of several examples to illustrate the capabilities ofasreml in the context of analysing real data-sets. We discuss some of the components returnedfrom asreml and indicate when potential problems may occur. Statistical conceptsand issues are discussed as necessary but we stress that the analyses are only illustrative.8.2 Split Plot DesignThe first example is the analysis of a split plot design originally presented by Yates(1935). The experiment was conducted to assess the effects on yield of three oat varieties(Golden Rain, Marvellous and Victory) with four levels of nitrogen application (0, 0.2, 0.4and 0.6 cwt/acre). The field layout consisted of six blocks (labelled I, II, III, IV, V and VI)with three whole-plots per block each split into four sub-plots. The three varieties wererandomly allocated to the three whole-plots while the four levels of nitrogen applicationwere randomly assigned to the four sub-plots within each whole-plot. The data is inTable 8.1.Table 8.1. A split-plot field trial of oat varieties and nitrogen applicationNitrogenBlock Variety 0.0cwt 0.2cwt 0.4cwt 0.6cwtGR 111 130 157 174I M 117 114 161 141V 105 140 118 156GR 61 91 97 100II M 70 108 126 149V 96 124 121 144GR 68 64 112 86III M 60 102 89 96V 89 129 132 124GR 74 89 81 122IV M 64 103 132 133V 70 89 104 117GR 62 90 100 116V M 80 82 94 126V 63 70 109 99GR 53 74 118 113VI M 89 82 86 104V 97 99 119 121A standard analysis of these data recognises the two basic elements inherent in theexperiment:
8.2 Split Plot Design 811. the stratification of the experiment units, that is the blocks, whole-plots and sub-plots,and2. the treatment structure that is superimposed on the experimental material.The latter is of prime interest in the presence of stratification. The aim of the analysis isto examine the importance of the treatment effects while accounting for the stratificationand restricted randomisation of the treatments to the experimental units.The function calls to initially create a data frame and perform the standard splitplotanalysis in asreml are given below. The variate/factor names are specified in theheader line of oats.txt, with factor names beginning with a capital letter. The functionasreml.read.table() recognises this convention and automatically creates the factorsin the data frame.> oats oats.asr names(oats)[1] "Blocks" "Nitrogen" "Subplots" "Variety" "Wplots" "yield"}The first five are factors describing the stratification, or experiment design, and appliedtreatments. The standard split plot analysis is achieved by fitting terms Blocks andBlocks:Wplots as random effects. It is not necessary to specify the residual term, whichis equivalent to Blocks:Wplots:Subplots, as the experimental units are uniquely definedby these three factors. The fixed effects include the main effects of both Variety andNitrogen and their interaction.The variance components are:> summary(oats.asr)$varcompgamma component std.error z.ratio constraintBlocks 1.2111646 214.4808 168.78653 1.270722 PositiveBlocks:Wplots 0.5989373 106.0637 67.87730 1.562579 PositiveR!variance 1.0000000 177.0864 37.33342 4.743375 PositiveFor simple variance component models such as the above, the default parameterisationfor the variance parameters is as the ratio to the residual variance. Thus asreml returnsthe gamma and component values for each term in the random model, which are thevariance ratio and component, respectively.The default synopsis for testing fixed effects in asreml is a table of incremental Wald tests(see Section 3.16):> wald(oats.asr)Wald tests for fixed effectsResponse: yieldTerms added sequentially; adjusted for those aboveDf Sum of Sq Wald statistic Pr(Chisq)(Intercept) 1 43419 245
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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
Page 42 and 43: 3.10 Conditional factors: the at()
Page 44 and 45: Table 3.2. Families and link functi
Page 46 and 47: 3.15 Multivariate analysis 34R!trai
Page 48 and 49: Source df F_inc F_con MSource df dd
Page 50 and 51: 4.1.1 Specifying variance models in
Page 52 and 53: 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 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
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References 130W. W. Stroup, P. S. B
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Index 132F statistics, 17fa(,k), 49
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