G-structures

Introduction

Each random term in the linear model has default variance structure. In most cases, this is a scaled Identity. The exceptions are:
the giv() model term associates a GIV matrix with the term,
an A-inverse matrix is associated by default with !P pedigree factors unless it is expressed with the ide() model term.

A G-structure definition is required to assign some other variance structure to a model term. Usually a G structure is defined as a direct product of variance structures.

Functional specification

The functional specification of variance structures has been implemented in ASReml 4.

Each variance model structure has a name (VCODE): ID, AR1, AR1V, CORUH, DIAG, US and XFA1 are some commonly used structures. Under the functional specification, these names are converted to functions which are applied to the appropriate components of each model term, on the model line to produce a consolidated model term. For example
ar1(Row).ar1(Column)
diag(Site).id(Row) which is equivalent to diag(Site).Row
coruh(Site).Variety
xfa1(Site).Variety
us(Site).Variety

Initial parameter values and qualifiers may be specified as further arguments to the function. For example
xfa1(Site 0 0 .5 .5).Variety would force a correlation of 1 across sites.

Structural specification

In the original, structural specification, the model line was followed by a Variance Header Line. This specifies 3 things:

S, the number of sections of data (e.g. sites in a multisite analysis)

D, the layout dimensions of the data (e.g. 2 for a spatial analysis of plots in a grid)

G, the number of model terms which have an explicit variance structure specified in following lines.

The variance header is followed by SxD variance definitions for the residual, followed by the G structure definition lines.

Each G-structure definition consists of

G-structure header which identifies a model term and the number of components in the direct product structure,

variance structure definitions for each component.

A variance structure definition consists of
Size Sortkey VCODE [ qualifiers ] initial_values
Sortkey is usually 0 in G structures (but for spatial correlation models it points the the spatial coordinates).

Sample G-structure definitions

Multivariate sire model

A typical variance structure, assuming three traits, can be written as

 us(Trait).sire

where Trait.sire is the model term to which the structure applies and

US is the VCODE for an unstructured variance matrix. !GP is a qualifier specifying that the estimated unstructured variance matrix must be kept positive definite. Assuming 3 traits, the US structure has six parameters: 3 variances and 3 covariances in the order
V11
C21 V22
C31 C32 V22
(lower triangle rowwise). ASReml will obtain initial values as a proportion of the simple variances and covariances of the residual.

Multivariate animal model

A typical variance structure, assuming three traits, can be written as

us(Trait).nrm(animal)

where Trait.animal is the model term to which the structure applies.

US is the VCODE for an unstructured variance matrix. Assuming three traits, it has six parameterss: 3 variances and 3 covariances in the order V11 C21 V22 C31 C32 V22 (lower triangle rowwise). ASReml will obtain initial values as a proportion of the simple variances and covariances of the residual.

NRM is the VCODE for the numerator relationship matrix generated form the pedigree and associated with animal by the !P factor definition qualifier.

Genetic correlation across sites.

As a more complicated example, consider the analysis of say 50 variety trials where most varieties occur at most sites and all varieties occur at at least 2 sites.

Ultimately, we want to fit a factor analytic model but to get starting values for that, we first fit a uniform covariance model.

 coruv(site 0.1 1).variety

where site.variety is the model term to which the structure applies.

CORUV is the VCODE for a Uniform CORrelation matrix scaled by a single Varince. This is a simple model although it may take a while to run (equivalently, have two model terms variety site.variety and no explicit G-structure definition).

An extended factor analytic variance structure requires first that the xfak() variance function be used in the model. Assuming 1 factor, it can be written as

 xfa1(site).variety

The model term is actually a modified form of site.variety in that site has an extra level appended, to model the 'factor'

XFA1 is the VCODE for an extended factor analytic 1 variance matrix. It requires 100 initial values: 50 specific variances and 50 loadings.