Selects the best variancecovariance model for a set of environments (M.P. Boer, M. Malosetti, S.J. Welham & J.T.N.M. Thissen).
Options
PRINT = string tokens 
What to print (summary , best , model , components , effects , means , stratumvariances , monitoring , vcovariance , deviance , waldtests , missingvalues , covariancemodels ); default summ , best , comp , cova 

VCMODELS = string tokens 
Specifies the variancecovariance models that are to be compared for the set of environments (identity , diagonal , cs , hcs , outside , fa , fa2 , unstructured ); default iden , diag , cs , hcs , outs , fa , fa2 , unst 
CRITERION = string token 
Defines which criterion is used to compare the different covariance structures (aic , sic ); default sic 
FIXED = formula 
Defines extra fixed effects 
UNITFACTOR = factor 
Saves the units factor required to define the random model when UNITERROR is to be used 
MVINCLUDE = string tokens 
Whether to include units with missing values in the explanatory factors and variates and/or the yvariates (explanatory , yvariate ); default expl , yvar 
MAXCYCLE = scalar 
Limit on the number of iterations; default 100 
WORKSPACE = scalar 
Number of blocks of internal memory to be set up for use by the REML algorithm; default 100 
Parameters
TRAIT = variates 
Quantitative trait to be analysed; must be set 

GENOTYPES = factors 
Genotype factor; must be set 
ENVIRONMENTS = factors 
Environment factor; must be set 
UNITERROR = variate 
Uncertainty on trait means (derived from individual unit or plot error) to be included in QTL analysis; default * i.e. omitted 
SELECTEDMODEL = texts 
VCMODELS setting for the best variancecovariance model 
SAVE = REML save structures 
Save the details of each REML analysis for use in subsequent VDISPLAY and VKEEP directives 
Description
VGESELECT
selects the best covariance structure for genetic correlations between environments. The quantitative trait is specified by the TRAIT
parameter, and the environment and genotype factors are specified by the ENVIRONMENTS
and GENOTYPES
parameters respectively. The UNITERROR
parameter allows you to specify uncertainty on the trait means (derived from individual unit or plot error) to include in the random model; by default this is omitted. The UNITFACTOR
option allows you to save the factor that is needed to define the uniterror term (you would need this, for example, if you later wanted to save information about the term using VKEEP
).
The settings of the VCMODELS
option indicate which models to consider for the variancecovariance structure (see the Method Section for details). The CRITERION
option specifies whether to assess the different covariance structures by using the Bayesian Information Criterion (BIC), which is also known as the Schwarz Information Criterion (SIC), or by using Akaike’s Information Criterion (AIC). The default is to use the Schwarz (Bayesian) criterion. The SELECTEDMODEL
parameter can save the setting corresponding to the best covariance structure can be saved.
The PRINT
option controls the printed output. The summary
setting prints a summary of the analyses, and best
prints details of the best model. The other settings correspond to the settings of the PRINT
option of the REML
directive. The specified output is printed for each model specified by the MODELS
option.
The FIXED
option can be used to include extra fixed effects, e.g. selected QTLs (genetic predictors). There are also MVINCLUDE
, MAXCYCLE
and WORKSPACE
options which operate in the same way as these options in the REML
directive.
Options: PRINT
, VCMODELS
, CRITERION
, FIXED
, UNITFACTOR
, MVINCLUDE
, MAXCYCLE
, WORKSPACE
.
Parameters: TRAIT
, GENOTYPES
, ENVIRONMENTS
, UNITERROR
, SELECTEDMODEL
, SAVE
.
Method
The method selects the best variancecovariance matrix to model the genetic correlations between environments, based on the Schwarz (Bayesian) Information Criterion (BIC) or Akaike Information Criterion (AIC), as described by Malosetti et al. (2004) and Boer et al. (2007). The AIC and BIC are defined by:
AIC = deviance + 2 × p,
BIC (or SIC) = deviance + log(N) × p,
where N is the total number of observations, and p is the number of parameters in the variancecovariance matrix. The default is to use the Schwarz (Bayesian) criterion.
The variancecovariance models that can be specified by the VCMODELS
option to be compared are as follows:
Setting  Description  Variancecovariance matrix  Number of parameters 
identity 
Identity  I σ_{e}^{2}  1 
cs 
Compound symmetry  J σ_{g}^{2} + I σ_{e}^{2}  2 
diagonal 
Diagonal matrix (heteroscedastic)  D  n_{env} 
hcs 
Heterogeneous compound symmetry  J σ_{g}^{2} + D  n_{env} + 1 
outside 
Heterogeneity outside  √D K √D  n_{env} + 1 
fa 
First order factoranalytic model  λ λʹ T + D  2 × n_{env} 
fa2 
Second order factoranalytic model  3 × n_{env}  
unstructured 
√D K √D 
In this table n_{env} is the number of environments, σ2e and σ2g are scalars, and λ is a n_{env} dimensional vector. In addition, I is the n_{env} × n_{env} identity matrix, J is the n_{env} × n_{env} matrix with all values equal to one, K is the n_{env} × n_{env} matrix with one in its diagonal elements and θ in its offdiagonal elements, and D is a diagonal matrix containing the variances (σ_{ei}^{2}:i = 1…n_{env}).
The analyses are performed by the REML
directive, using the VSTRUCTURE
directive to specify the covariance models. The table below summarizes how the models are specified in Genstat notation.
Setting  VSTRUCTURE parameters 

Model  Heterogeneity  Order  Extra Random term  
identity 
identity 
None  
cs 
identity 
None  GENOTYPES 

diagonal 
diagonal 
None  
hcs 
diagonal 
None  GENOTYPES 

outside 
uniform 
Outside  
fa 
fa 
None  1  
fa2 
fa 
None  2  
unstructured 
unstructured 
None 
Action with RESTRICT
Restrictions are not allowed.
References
Boer, M.P., Wright, D,, Feng, L., Podlich, D.W., Luo, L., Cooper, M. & van Eeuwijk F.A. (2007). A mixedmodel quantitative trait loci (QTL) analysis for multipleenvironment trial data using environmental covariables for QTLbyenvironment interactions, with an example in maize. Genetics, 177, 18011813.
Malosetti, M., Voltas, J., Romagosa, I., Ullrich, S.E. & van Eeuwijk, F.A. (2004). Mixed models including environmental covariables for studying QTL by environment interaction. Euphytica, 137, 139145.
See also
Procedures: QMVAF
, QMBACKSELECT
, QMESTIMATE
, QMQTLSCAN
.
Commands for: REML analysis of linear mixed models, Statistical genetics and QTL estimation.
Example
CAPTION 'VGESELECT example'; STYLE=meta SPLOAD '%GENDIR%/Examples/F2maize_traits.gsh' VGESELECT [PRINT=summary,best,comp,covariance;\ VCMODELS=id,diag,cs,hcs,outside,fa,fa2,unstructured]\ TRAIT=yld; ENVIRONMENTS=E; GENOTYPES=G; SELECTEDMODEL=Bestmodel PRINT Bestmodel