Variance model functions

Specifying variance structures in ASReml 4.

ASReml 4 includes the facility to associate a variance structure with a model factor through model term functions. Thus, the model term us(Trait) fits the model term Trait and associates the US variance structure with it.

All the variance structures that appear in Table 7.3 of the User Guide may be associated with model terms using functions names formed from the same letters but in lower case. For example CORUV becomes coruv().

The original more cumbersome syntax is still available, hopefully unaffected by the addition of the new syntax.

There are three parts to the new syntax:
1 including variance model functions in the linear model
2 specifying the R structure with a model equation
3 variance structure qualifiers.

The concepts are explained by reference to common types of models.

G structures

The general default for random terms in the model is the assumption the effects are IID (except for simple terms with an associated pedigree where the genetic relationships defined by the pedigree are recognised). This structure is formally specified by enclosing the term the idv() function.

Compound symmetry

A common model is mu site !r variety site.variety. This model can equivalently be fitted by dropping variety and associating a uniform variance/covariance matrix with site in the interaction term. That is mu site !r coruv(site).variety

Related models are

  ~ mu site !r coruh(site).variety # heterogeneous variance

  ~ mu site !r xfa1(site).variety # extended factor anaytic

  ~ mu site !r us(site).variety

R structures

Spatial Analysis is specified with the single RESIDUAL line

  residual ar1(row).ar1(column)

after the model line.

Multisite spatial analysis is specified with

 RESIDUAL at(site).ar1(row).ar1(column)

ASReml will sort the data records to conform to the specified structure.

When specifying R structures, it may be that the records are in a known standard order but not indexed by a coded factor. Then, the model term may have the number of levels as its argument. For example RESIDUAL ar(10).ar(15) defines a spatial correlation structure assuming 150 records mapping to a 10 by

Multivariate analysis.

ASReml now assumes unstructured variances in a multivariate analysis so that

 Y1 Y2 ~ Trait !r  Trait.group

is equivalent to

 Y1 Y2 ~ Trait !r us(Trait).group
 RESIDUAL units.us(Trait)

15 grid are sorted as 10 lots of 15.

Weighted analysis.

The !DISPERSION qualifier now applies to normal weighted analyses.

 yield !weight WT  !DISP  0.33 ~

scales the weights by a fixed variance of 0.33

 yield !weight WT  !DISP  ~

will estimate the variance parameter

 yield !weight WT   ~

implies no scaling is required.

Structure over several terms.

The syntax ![ idv(9) X1 X2 X3 X4 X5 X6 X7 X8 X9 !] defines an equal variance structure which is applied to the 9 covariates.

For random regression, use ![ us(2).Subject Subject age.Subject !]

Variance structure qualifiers.

The prefered option is to specify structure qualifiers in the variance function statement, after the name of the variable associated with the structure such as fun(term,quals) for example ar1(row,.3 !GF).ar1(col,0.7 !GF) would fir the model with the correlation parameters fixed at 0.3 for row and 0.7 for col.

The quals argument includes the qualifiers syntax is

 [!G...] [!=...]  [!INIT ...] [!USE ...]  [!COORD ...]

The !G... qualifier allows the setting of parameter space codes for each parameter.
The !=... qualifier allows the setting of EQUALITY constraints.
The !INIT qualifier enables to specification of initial values if it turns out that the default initial values are inadequate.

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