Upload
voque
View
234
Download
0
Embed Size (px)
Citation preview
IntroductionThe linear model
An example
Asreml-R: an R package for mixed modelsusing residual maximum likelihood
David Butler1 Brian Cullis2 Arthur Gilmour3
1Queensland Department of Primary IndustriesToowoomba
2NSW Department of Primary IndustriesWagga Wagga Agricultural Institute
3NSW Department of Primary IndustriesOrange Agricultural Institute
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Outline
1 Introduction
2 The linear modelSpecifying the linear model in asreml-RThe asreml class
3 An exampleModels for a series of trials
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Introduction
ASReml: standalone program (Gilmour et al., 1999)
Designed to fit complex mixed models to large problems.Efficient computing strategies
Average Information algorithm (Gilmour et al., 1995)avoids forming expensive trace terms
Sparse matrix methodsavoid forming and storing zero cellsexploit variance structures with sparse inversesoptimize solution order
Direct product structures exploited(A ⊗ B)−1 = A−1 ⊗ B−1
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Introduction
ASReml: standalone program (Gilmour et al., 1999)
Designed to fit complex mixed models to large problems.Efficient computing strategies
Average Information algorithm (Gilmour et al., 1995)avoids forming expensive trace terms
Sparse matrix methodsavoid forming and storing zero cellsexploit variance structures with sparse inversesoptimize solution order
Direct product structures exploited(A ⊗ B)−1 = A−1 ⊗ B−1
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Introduction
ASReml: standalone program (Gilmour et al., 1999)
Designed to fit complex mixed models to large problems.Efficient computing strategies
Average Information algorithm (Gilmour et al., 1995)avoids forming expensive trace terms
Sparse matrix methodsavoid forming and storing zero cellsexploit variance structures with sparse inversesoptimize solution order
Direct product structures exploited(A ⊗ B)−1 = A−1 ⊗ B−1
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Introduction
ASReml: standalone program (Gilmour et al., 1999)
Designed to fit complex mixed models to large problems.Efficient computing strategies
Average Information algorithm (Gilmour et al., 1995)avoids forming expensive trace terms
Sparse matrix methodsavoid forming and storing zero cellsexploit variance structures with sparse inversesoptimize solution order
Direct product structures exploited(A ⊗ B)−1 = A−1 ⊗ B−1
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Asreml-R
asreml-R is the R interface to the ASReml fitting routines.
model specified as formula objects
initial values specified as list objectsasreml object
BLUPs of random effectsGLS estimates of fixed effectsREML estimates of variance componentspredictions from the linear model (if requested)
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Asreml-R
asreml-R is the R interface to the ASReml fitting routines.
model specified as formula objects
initial values specified as list objectsasreml object
BLUPs of random effectsGLS estimates of fixed effectsREML estimates of variance componentspredictions from the linear model (if requested)
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
The linear model
y = Xτ + Zu + e
y denotes the n × 1 vector of observationsτ is a p × 1 vector of fixed treatment effectsX is a n × p design matrixu is a q × 1 vector of random effectsZ is a n × q design matrixe is a n × 1 vector of residual errors
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
The linear model
[
ue
]
∼ N([
00
]
, θ
[
G(γ) 00 R(φ)
])
Where:G, R parameterized variance matrices
γ a vector of variance parameters relating to uφ a vector of variance parameters relating to eθ is a scale parameter
y ∼ N(Xτ , H)
H = R + ZGZ ′.
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
The linear model
[
ue
]
∼ N([
00
]
, θ
[
G(γ) 00 R(φ)
])
Where:G, R parameterized variance matrices
γ a vector of variance parameters relating to uφ a vector of variance parameters relating to eθ is a scale parameter
y ∼ N(Xτ , H)
H = R + ZGZ ′.
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
Variance structures for the errorsR structures
R may comprise t independent sections
R = ⊕tj=1Rj =
R1 0 . . . 00 R2 . . . 0...
.... . .
...0 0 . . . Rt
Each section may be the direct product of two or moredimensions
R i = R i1 ⊗ R i2 ⊗ . . .
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
Variance structures for the errorsR structures
R may comprise t independent sections
R = ⊕tj=1Rj =
R1 0 . . . 00 R2 . . . 0...
.... . .
...0 0 . . . Rt
Each section may be the direct product of two or moredimensions
R i = R i1 ⊗ R i2 ⊗ . . .
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
Variance of the random effectsG structures
The vector of random effects is often composed of bsubvectors
u = [u′
1 u′
2 . . . u′
b]′
The u i are assumed N(0, θGi).
As for R
G = ⊕bi=1Gi =
G1 0 . . . 00 G2 . . . 0...
.... . .
...0 0 . . . Gb
Assuming separability Gi = Gi1 ⊗ Gi2 ⊗ . . . ⊗ Gif
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
Variance of the random effectsG structures
The vector of random effects is often composed of bsubvectors
u = [u′
1 u′
2 . . . u′
b]′
The u i are assumed N(0, θGi).
As for R
G = ⊕bi=1Gi =
G1 0 . . . 00 G2 . . . 0...
.... . .
...0 0 . . . Gb
Assuming separability Gi = Gi1 ⊗ Gi2 ⊗ . . . ⊗ Gif
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
Specifying the linear model
fit.asr < − asreml (fixed=, random=, rcov=, data= )
Fixed effectsfixed = y ∼ model formula
Random effects (G structures)random = ∼ model formula
Error model (R structures)rcov = ∼ model formula
Sparse fixedsparse = ∼ model formulaVariance matrix for solutions not available
Factors crossed or nested - determined by coding.
y may be a matrix
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
Specifying the linear model
fit.asr < − asreml (fixed=, random=, rcov=, data= )
Fixed effectsfixed = y ∼ model formula
Random effects (G structures)random = ∼ model formula
Error model (R structures)rcov = ∼ model formula
Sparse fixedsparse = ∼ model formulaVariance matrix for solutions not available
Factors crossed or nested - determined by coding.
y may be a matrix
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
Specifying the linear model
fit.asr < − asreml (fixed=, random=, rcov=, data= )
Fixed effectsfixed = y ∼ model formula
Random effects (G structures)random = ∼ model formula
Error model (R structures)rcov = ∼ model formula
Sparse fixedsparse = ∼ model formulaVariance matrix for solutions not available
Factors crossed or nested - determined by coding.
y may be a matrix
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
Specifying variance models
G structures
The default variance model is (scaled) identity.
Variance models for random terms are specified usingspecial functions.
For example
random = ∼ diag(A):B
specifies a diagonal variance structure of orderlength(levels(A)) for A and a (default) identity for B.
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
Specifying variance models
G structures
The default variance model is (scaled) identity.
Variance models for random terms are specified usingspecial functions.
For example
random = ∼ diag(A):B
specifies a diagonal variance structure of orderlength(levels(A)) for A and a (default) identity for B.
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
Specifying variance models
R structures
Default σ2In, where n < − nrow(data)
Specified using special functions.
Example: a series of t independent experiments indexedby the factor Trial,
rcov = ∼ at(Trial):ar1(A):ar1(B)
specifies separable autoregressive processes across Aand B at each level of Trial
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
Specifying variance models
R structures
Default σ2In, where n < − nrow(data)
Specified using special functions.
Example: a series of t independent experiments indexedby the factor Trial,
rcov = ∼ at(Trial):ar1(A):ar1(B)
specifies separable autoregressive processes across Aand B at each level of Trial
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
Special FunctionsModel functionslin(obj=x) Includes the named factor as a variate.spl(obj=x) Spline random factor.pol(obj=x , t) Orthogonal polynomials of order |t |.
Time series type modelsar1(), ar2() Autoregressivema1(), ma2() Moving average
Metric based models in < or <2
exp(), gau() One dimensionalaexp(), agau() Anisotropic 2Dmtrn() Matérn class
General structure modelscor(), corb(), corg() Correlationdiag(), us(), ante(), chol() Variancefa(obj=x, q) Factor Analytic with q factors
Known structuresped(), giv() Use known inverse matrices.
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
Special FunctionsModel functionslin(obj=x) Includes the named factor as a variate.spl(obj=x) Spline random factor.pol(obj=x , t) Orthogonal polynomials of order |t |.
Time series type modelsar1(), ar2() Autoregressivema1(), ma2() Moving average
Metric based models in < or <2
exp(), gau() One dimensionalaexp(), agau() Anisotropic 2Dmtrn() Matérn class
General structure modelscor(), corb(), corg() Correlationdiag(), us(), ante(), chol() Variancefa(obj=x, q) Factor Analytic with q factors
Known structuresped(), giv() Use known inverse matrices.
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
Special FunctionsModel functionslin(obj=x) Includes the named factor as a variate.spl(obj=x) Spline random factor.pol(obj=x , t) Orthogonal polynomials of order |t |.
Time series type modelsar1(), ar2() Autoregressivema1(), ma2() Moving average
Metric based models in < or <2
exp(), gau() One dimensionalaexp(), agau() Anisotropic 2Dmtrn() Matérn class
General structure modelscor(), corb(), corg() Correlationdiag(), us(), ante(), chol() Variancefa(obj=x, q) Factor Analytic with q factors
Known structuresped(), giv() Use known inverse matrices.
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
Special FunctionsModel functionslin(obj=x) Includes the named factor as a variate.spl(obj=x) Spline random factor.pol(obj=x , t) Orthogonal polynomials of order |t |.
Time series type modelsar1(), ar2() Autoregressivema1(), ma2() Moving average
Metric based models in < or <2
exp(), gau() One dimensionalaexp(), agau() Anisotropic 2Dmtrn() Matérn class
General structure modelscor(), corb(), corg() Correlationdiag(), us(), ante(), chol() Variancefa(obj=x, q) Factor Analytic with q factors
Known structuresped(), giv() Use known inverse matrices.
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
Special FunctionsModel functionslin(obj=x) Includes the named factor as a variate.spl(obj=x) Spline random factor.pol(obj=x , t) Orthogonal polynomials of order |t |.
Time series type modelsar1(), ar2() Autoregressivema1(), ma2() Moving average
Metric based models in < or <2
exp(), gau() One dimensionalaexp(), agau() Anisotropic 2Dmtrn() Matérn class
General structure modelscor(), corb(), corg() Correlationdiag(), us(), ante(), chol() Variancefa(obj=x, q) Factor Analytic with q factors
Known structuresped(), giv() Use known inverse matrices.
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
The asreml class
Component Descriptionloglik log likelihood at terminationgammas vector of variance parameter estimatescoefficients list of fixed, random and sparse coefficientsvcoeff variance of the coefficientsfitted.values fitted valuesresiduals residualssigma2 residual variancepredictions list of predictions if specifiedG.param list object of variance models for random termsR.param list object of variance models for error term
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An example
Specifying the linear model in asreml-RThe asreml class
asreml methods
coef() List with components fixed, random and sparse.
resid() Vector of residuals.
fitted() Vector of fitted values.
summary() List including the asreml() call, REMLlog-likelihood, variance parameters, coefficients,residuals and components of C−1 if requested.
wald() A table of Wald tests for each fixed term.
plot() Residual plots including the sample variogram,distribution, fitted values and trend plots.
predict() Predictictions from the linear model (eg, tables ofadjusted means). See Gilmour et al. (2004) andWelham et al. (2004).
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An exampleModels for a series of trials
Example: Multi-environment trials
In the context of a plant genetic improvement program,
It is important to know how genotype performance varieswith a change in environment, that is, to investigate (G×E)interaction.
Identify genotypes with broad or specific adaptation.
G×E is assessed in a series of designed experiments in arange of environments (METs)
Environments may be geographic locations and/or years
Smith et al. (2005) present a useful review.
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An exampleModels for a series of trials
Example: Multi-environment trials
In the context of a plant genetic improvement program,
It is important to know how genotype performance varieswith a change in environment, that is, to investigate (G×E)interaction.
Identify genotypes with broad or specific adaptation.
G×E is assessed in a series of designed experiments in arange of environments (METs)
Environments may be geographic locations and/or years
Smith et al. (2005) present a useful review.
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An exampleModels for a series of trials
Example: Mixed model for MET data
y = Xτ + Z gug + Z ouo + e
Assume var(
ug)
= Gg = Ge ⊗ Ig
Ge is the genetic variance matrix:
Ge =
σ2g1
σg12 σg13 · · · σg1t
σ2g2
σg23 · · · σg2t
σ2g3
· · · σg3t
. . .σ2
gt
Allow separate spatial covariance structures for the errorsfor each trial
R j = σ2j Rcj (φcj
) ⊗ Rrj (φrj)
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An exampleModels for a series of trials
Example: METs in ASReml-R
asreml(yield ∼ trial + . . . ,random = ∼ us(trial):genotype + . . . ,rcov = ∼ at(trial):ar1(column):ar1(row), . . . )
trial is a (fixed) factor with t levels
genotype is a (random) factor with g levels
us(trial):genotype models genotype effects in each trialwith variance Ge ⊗ Ig where Ge is an unstructured form
at(trial):ar1(column):ar1(row) models the residual effectsfor each trial with an AR1×AR1 correlation structure.
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An exampleModels for a series of trials
Example: METs in ASReml-R
asreml(yield ∼ trial + . . . ,random = ∼ us(trial):genotype + . . . ,rcov = ∼ at(trial):ar1(column):ar1(row), . . . )
trial is a (fixed) factor with t levels
genotype is a (random) factor with g levels
us(trial):genotype models genotype effects in each trialwith variance Ge ⊗ Ig where Ge is an unstructured form
at(trial):ar1(column):ar1(row) models the residual effectsfor each trial with an AR1×AR1 correlation structure.
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An exampleModels for a series of trials
Example: METs in ASReml-R
asreml(yield ∼ trial + . . . ,random = ∼ us(trial):genotype + . . . ,rcov = ∼ at(trial):ar1(column):ar1(row), . . . )
trial is a (fixed) factor with t levels
genotype is a (random) factor with g levels
us(trial):genotype models genotype effects in each trialwith variance Ge ⊗ Ig where Ge is an unstructured form
at(trial):ar1(column):ar1(row) models the residual effectsfor each trial with an AR1×AR1 correlation structure.
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An exampleModels for a series of trials
MET data set
Stage 2 trials taken from the Qld barley program (Kellyet al., 2007)
14 environments over 2 years of trialling: 2003/41255 unique genotypes tested
698 in 2003720 in 2004163 genotypes common across years
Partially replicated designs (Cullis et al., 2006)
Response variate is grain yield
Pedigrees traced back four generations
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An exampleModels for a series of trials
Analysis strategy
1 Initial spatial model for each experimentanalyse each trial separately, orjoint analysis with a diagonal variance model
qb.asr1 <- asreml(yield ∼ Site,random = ∼ diag(Site):Genotype,rcov = ∼ at(Site):ar1(Column):ar1(Row),data = qb)
17,663 equations56 variance parameters
2 Model global trend and extraneous effects.3 Model G×E.4 Predict genotype effects
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An exampleModels for a series of trials
Analysis strategy
1 Initial spatial model for each experimentanalyse each trial separately, orjoint analysis with a diagonal variance model
qb.asr1 <- asreml(yield ∼ Site,random = ∼ diag(Site):Genotype,rcov = ∼ at(Site):ar1(Column):ar1(Row),data = qb)
17,663 equations56 variance parameters
2 Model global trend and extraneous effects.3 Model G×E.4 Predict genotype effects
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An exampleModels for a series of trials
Analysis strategy
1 Initial spatial model for each experimentanalyse each trial separately, orjoint analysis with a diagonal variance model
qb.asr1 <- asreml(yield ∼ Site,random = ∼ diag(Site):Genotype,rcov = ∼ at(Site):ar1(Column):ar1(Row),data = qb)
17,663 equations56 variance parameters
2 Model global trend and extraneous effects.3 Model G×E.4 Predict genotype effects
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An exampleModels for a series of trials
plot(qb.asr1,option=’v’)
510
1520
2040
6080
1000.00.20.40.60.81.0
Column
Row
2003Biloela
510
1520
2040
6080
1000.00.20.40.60.81.0
Column
Row
2003Breeza
510
1520
2040
6080
1000.00.20.40.60.81.0
Column
Row
2003Brookstead
510
1520
2040
6080
1000.00.20.40.60.81.0
Column
Row
2003Clifton
510
1520
2040
6080
1000.00.20.40.60.81.0
Column
Row
2003Kurumbul
510
1520
2040
6080
1000.00.20.40.60.81.0
Column
Row
2003Narrabri
510
1520
2040
6080
1000.00.20.40.60.81.0
Column
Row
2003Tamworth
510
1520
2040
6080
1000.00.20.40.60.81.0
Column
Row
2004BillaBilla
510
1520
2040
6080
1000.00.20.40.60.81.0
Column
Row
2004Biloela
510
1520
2040
6080
1000.00.20.40.60.81.0
Column
Row
2004Breeza
510
1520
2040
6080
1000.00.20.40.60.81.0
Column
Row
2004Brookstead
510
1520
2040
6080
1000.00.20.40.60.81.0
Column
Row
2004Gilgandra
510
1520
2040
6080
1000.00.20.40.60.81.0
Column
Row
2004Narrabri
510
1520
2040
6080
1000.00.20.40.60.81.0
Column
Row
2004Walgett
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An exampleModels for a series of trials
Models for G×E
Diagonal variance structure analagous to individualanalyses.
Assumes that the genetic effects in different environmentsare un-correlated. Unlikely to be sensible.The us() model is the most general form for Ge. Difficulties:
With many environments, the number of parameters is largeDifficult to fit REML estimate of matrix can be singular - notfull rank
Factor Analytic (FA) variance model a good approximationto US and handles not full rank
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An exampleModels for a series of trials
Models for G×E
Diagonal variance structure analagous to individualanalyses.
Assumes that the genetic effects in different environmentsare un-correlated. Unlikely to be sensible.The us() model is the most general form for Ge. Difficulties:
With many environments, the number of parameters is largeDifficult to fit REML estimate of matrix can be singular - notfull rank
Factor Analytic (FA) variance model a good approximationto US and handles not full rank
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An exampleModels for a series of trials
Models for G×E
Diagonal variance structure analagous to individualanalyses.
Assumes that the genetic effects in different environmentsare un-correlated. Unlikely to be sensible.The us() model is the most general form for Ge. Difficulties:
With many environments, the number of parameters is largeDifficult to fit REML estimate of matrix can be singular - notfull rank
Factor Analytic (FA) variance model a good approximationto US and handles not full rank
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An exampleModels for a series of trials
Known genetic effects
A better genetic variance model most likely achieved bypartitioning genetic effects into additive and non-additive.If ug = ag + ig , then
Assume ag ∼ N(0, σ2aA)
Assume ig ∼ N(0, σ2i I)
var(
ug)
= Gae ⊗ A + Gie ⊗ I
Asreml-R1 ainv < − asreml.Ainverse(pedigree)$ginv2 asreml( . . . , ped(genotype), . . . + . . . , ide(genotype), . . . ,
ginverse=list(genotype=ainv), . . . )
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An exampleModels for a series of trials
Known genetic effects
A better genetic variance model most likely achieved bypartitioning genetic effects into additive and non-additive.If ug = ag + ig , then
Assume ag ∼ N(0, σ2aA)
Assume ig ∼ N(0, σ2i I)
var(
ug)
= Gae ⊗ A + Gie ⊗ I
Asreml-R1 ainv < − asreml.Ainverse(pedigree)$ginv2 asreml( . . . , ped(genotype), . . . + . . . , ide(genotype), . . . ,
ginverse=list(genotype=ainv), . . . )
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An exampleModels for a series of trials
Known genetic effects
A better genetic variance model most likely achieved bypartitioning genetic effects into additive and non-additive.If ug = ag + ig , then
Assume ag ∼ N(0, σ2aA)
Assume ig ∼ N(0, σ2i I)
var(
ug)
= Gae ⊗ A + Gie ⊗ I
Asreml-R1 ainv < − asreml.Ainverse(pedigree)$ginv2 asreml( . . . , ped(genotype), . . . + . . . , ide(genotype), . . . ,
ginverse=list(genotype=ainv), . . . )
Butler, Cullis and Gilmour asreml-R
IntroductionThe linear model
An exampleModels for a series of trials
The final model
asreml(yield ∼ Site+at(Site,c(3,6,8,13)):lincol + at(Site,c(3,8,10,11)):linrow +at(Site,3):lincol:linrow + at(Site,4):fx4 + at(Site,6):fx6,
random = ∼ fa(Site,3):ped(Genotype) + fa(Site):ide(Genotype) +at(Site,c(2,4,5,7,9,11,12)):Column + at(Site,c(2)):Row,
rcov = ∼ at(Site):ar1(Column):ar1(Row),ginverse = list(Genotype=ainv), data=qb)
50,115 equations
134 parameters
Butler, Cullis and Gilmour asreml-R
ReferencesMore about asreml-R and ASReml
References
Cullis, B., Smith, A., and Coombes, N. (2006). On the design of early generationvariety trials with correlated data. Journal of Agricultural, Biological andEnvironmental Statistics, (in press).
Gilmour, A. R., Thompson, R., and Cullis, B. R. (1995). Average information REML: Anefficient algorithm for variance parameter estimation in linear mixed models.Biometrics, 51, 1440–1450.
Gilmour, A. R., Cullis, B. R., Welham, S. J., and Thompson, R. (1999). ASREML,reference manual. Biometric bulletin, no 3, NSW Agriculture, Orange AgriculturalInstitute, Forest Road, Orange 2800 NSW Australia.
Gilmour, A. R., Cullis, B. R., Welham, S. J., Gogel, B. J., and Thompson, R. (2004). Anefficient computing strategy for prediction in mixed linear models. ComputationalStatistics and Data Analysis, 44, 571–586.
Kelly, A., Cullis, B. R., Gilmour, A., Smith, A. B., Eccleston, J. A., and Thompson, R.(2007). Estimation in a multiplicative mixed model involving a genetic relationshipmatrix. In preparation.
Smith, A., Cullis, B., and Thompson, R. (2005). The analysis of crop cultivar breedingand evaluation trials: An overview of current mixed model approaches. Journal ofAgricultural Science, Cambridge, 143, 1–14.
Welham, S. J., Cullis, B. R., Gogel, B. J., Gilmour, A. R., and Thompson, R. (2004).Prediction in linear mixed models. Australian and New Zealand Journal of Statistics,46, 325–347.
Butler, Cullis and Gilmour asreml-R
ReferencesMore about asreml-R and ASReml
More about asreml-R and ASReml
Visit:
www.vsni.co.uk
VSN International Ltd.5 The WaterhouseWaterhouse StreetHemel HempsteadHerts HP1 1ES
United Kingdom
Butler, Cullis and Gilmour asreml-R