# G-structures

## Principles of combining variance models

Variance structures are sometimes defined as the direct product of
variance models. For example, a two factor interaction may involve two
variance models, one for each of the two factors in the
interaction. Some of the rules for combining variance models differ
for R structures and G structures.
Briefly, the rules are:
when combining variance models in both R
and G structures, the resulting direct product structure must match
the ordered effects with the outer factor first, for example, the G structure in

yield ~ mu variety !r repl column.row
0 0 1
column.row 2
column 0 AR1 0.4
row 0 ARV1 0.3 0.1

is for column.row which tells
ASReml that the direct product structure matches the effects ordered
rows ` within` columns. (The variance model can be written as
`sigma` ^2(**I**+`Sigma_C` .dp. `lambda `**Sigma_R**).)
Thus the G structure
definition line for column is specified first,
ASReml automatically includes and
estimates an error
variance parameter for each section of an R structure. The
variance structures defined by the user should therefore normally be correlation
matrices. A variance model can be specified but the !S2==1
qualifier would then be required to fix the error variance at
1 and prevent ASReml trying to estimate two confounded parameters
(error variance and the parameter corresponding to the
variance model specified.
ASReml does not have an implicit scale
parameter for G structures that are defined explicitly. For
this reason the model supplied when the G
structure involves just one variance model must be a variance model;
an initial value must be supplied for
this associated scale parameter; this is discussed
under ` additional_initial_values`
when the G
structure involves more than one variance model, one must be either a
homogeneous or a heterogeneous variance model and the rest should be
correlation models; if more than one are non-correlation models then the !GF
qualifier should be used to avoid identifiability problems, that is,
ASReml trying to estimate both parameters when they are confounded.
**Return to start**