ASReml-S reference manual - VSN International

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The Matérn class4.4 Variance model functions 46asreml uses an extended Matérn class which accomodates geometric anisotropy and achoice of metrics for random fields observed in two dimensions. This extension, describedin detail in Haskard [2006], is given byρ(h; φ) = ρ M (d(h; δ, α, λ); φ, ν)where h = (h x , h y ) T is the spatial separation vector, (δ, α) governs geometric anisotropy,(λ) specifies the choice of metric and (φ, ν) are the parameters of the Matérn correlationfunction. The function isρ M (d; φ, ν) = { 2 ν−1 Γ(ν) } ( ) ν (−1 d dK ν , (4.1)φ φ)estimationwhere φ > 0 is a range parameter, ν > 0 is a smoothness parameter, Γ(·) is the gammafunction, K ν (.) is the modified Bessel function of the third kind of order ν (Abramowitzand Stegun, 1965, section 9.6) and d is the distance defined in terms of X and Y axes:h x = x i − x j ; h y = y i − y j ; s x = cos(α)h x + sin(α)h y ; s y = cos(α)h x − sin(α)h y ;d = (δ|s x | λ + |s y | λ /δ) 1/λ .For a given ν, the range parameter φ affects the rate of decay of ρ(·) with increasing d.The parameter ν > 0 controls the analytic smoothness of the underlying process u s , theprocess being ⌈ν⌉ − 1 times mean-square differentiable, where ⌈ν⌉ is the smallest integergreater than or equal to ν (Stein, 1999, page 31). Larger ν correspond to smootherprocesses. asreml uses numerical derivatives for ν when its current value is outside theinterval [0.2,5].When ν = m + 1 2 with m a non-negative integer, ρ M (·) is the product of exp(−d/φ) anda polynomial of degree m in d. Thus ν = 1 2yields the exponential correlation function,ρ M (d; φ, 1 2) = exp(−d/φ), and ν = 1 yields Whittle’s elementary correlation function,ρ M (d; φ, 1) = (d/φ)K 1 (d/φ) (Webster and Oliver, 2001).When ν = 1.5 thenρ M (d; φ, 1.5) = exp(−d/φ)(1 + d/φ)which is the correlation function of a random field which is continuous and once differentiable.This has been used recently by Kammann and Wand [2003]. As ν → ∞ thenρ M (·) tends to the gaussian correlation function.The metric parameter λ is not estimated by asreml; it is usually set to 2 for Euclideandistance. Setting λ = 1 provides the cityblock metric, which together with ν = 0.5 modelsa separable AR1timesAR1 process. Cityblock metric may be appropriate when the dominantspatial processes are alighned with rows/columns as occurs in field experiments.Geometric anisotropy is discussed in most geostatistical books [Webster and Oliver, 2001,Diggle et al., 2003] but rarely are the anisotropy angle or ratio estimated from the data.Similarly the smoothness parameter ν is often set a-priori [Kammann and Wand, 2003,Diggle et al., 2003]. However Stein [1999] and Haskard et al. [2005] demonstrate that νcan be reliably estimated even for modest sized data-sets, subject to caveats regardingthe sampling design.EstimationThe order of the parameters in mtrn(), with their defaults, is (φ, ν = 0.5, δ = 1, α =0, λ = 2). Parameters are fixed or estimated depending on the data type (numeric orcharacter) of the argument to the respective parameter.• If an argument is numeric, it is treated as a starting value for estimation and giventhe constraint code P (positive).
4.4 Variance model functions 47• This behaviour can be altered by concatenating the numeric value followed by theconstraint code (P, U or F) into a character string.• If an argument is absent from the call, the corresponding parameter is held fixed atits default value.For example, to fit a Matérn model with only φ estimated and the other parameters set attheir defaults then we could use mtrn(phi = 0.1) where the starting value for estimationis given as 0.1.To fix ν some value other than the default and estimate φ, the fixed value and constraintcode are given as a single string to the nu argument. That is mtrn(phi = 0.1, nu = ”1.0F”)The parameters φ and ν are highly correlated so it may be better to <strong>manual</strong>ly cover agrid of ν values.We note that there is non-uniqueness in the anisotropy parameters of this metric d(·)since inverting δ and adding π 2to α gives the same distance. This non-uniqueness can beremoved by constraining 0 ≤ α < π 2and δ > 0, or by constraining 0 ≤ α < π and either0 < δ ≤ 1 or δ ≥ 1. With λ = 2, isotropy occurs when δ = 1, and then the rotation angleα is irrelevant: correlation contours are circles, compared with ellipses in general. Withλ = 1, correlation contours are diamonds.4.4.4 General structure modelscor(obj, init=NA)corb(obj, k=1, init=NA)corg(obj, init=NA)diag(obj, init=NA)us(obj, init=NA)chol(obj, k=1, init=NA)cholc(obj, k=1, init=NA)ante(obj, k=1, init=NA)fa(obj, k=1, init=NA)DescriptionThe class of general variance models includes the simple, banded and general correlationmodels (cor, corb, corg), the diagonal, unstructured, Cholesky and antedependencevariance models (diag, us, chol, cholc, ante) and the factor analytic structure (fa).Required argumentsobja factor in the data frame.Optional argumentsinita vector of initial parameter values. This vector can have an optional namesattribute to set the boundary constraint for each parameter. In this case, thename of each element may be one of ”P”, ”U” or ”F” for positive, unconstrainedor fixed, respectively.
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Analysis of mixed modelsfor S langu
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D.G. ButlerQueensland Department of
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ContentsPreface . . . . . . . . . .
Page 10 and 11: ContentsviiB Available variance mod
Page 12 and 13: List of Figures1.1 Weekly body weig
Page 14 and 15: 1.3 Data sets used 21.2.2 Help and
Page 16 and 17: 1.3 Data sets used 41.3.2 Repeated
Page 18 and 19: 1.3 Data sets used 6101 Sire 1 0102
Page 20 and 21: 2.1 The linear mixed model 8Direct
Page 22 and 23: 2.2 Estimation 10There is a corresp
Page 24 and 25: 2.2 Estimation 12where H ij = ∂ 2
Page 26 and 27: 2.5 Inference for random effects 14
Page 28 and 29: 2.6 Inference for fixed effects 16T
Page 30 and 31: 2.6 Inference for fixed effects 18t
Page 32 and 33: 3Fitting the mixed model3.1 Introdu
Page 34 and 35: 3.3 Introducing the asreml function
Page 36 and 37: 3.6 Getting help3.7 The asreml func
Page 38 and 39: 3.8 Fixed terms 26na.method.Xkeep.o
Page 40 and 41: 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 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 106 and 107: 8.5 Balanced repeated measures 94wh
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8.6 Spatial analysis of a field exp
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8.6 Spatial analysis of a field exp
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Variety predicted.value standard.er
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8.7 Unreplicated early generation v
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8.7 Unreplicated early generation v
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8.8 Paired Case-Control Study 106Th
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Terms added sequentially; adjusted
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8.8 Paired Case-Control Study 110[
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8.8 Paired Case-Control Study 112..
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8.9 Balanced longitudinal data - Ra
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A.3 Aliasing and singularities 122W
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Table B.1: Details of the available
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B Available variance models 126Deta
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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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