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.

Under the new (ASReml 4 variance wrapper function) syntax the primary rule is that variances smust not be overspecified. Thus the model term ar1v(row).ar1v(column) will fail because there are two, confounded variances specified. ar1(row).ar1(column) will fail as a term in the linear model because no variance term is specified. However, ASReml will usually supply a variance parameter when this is specified as a residual.

Generally, ASReml under the new syntax will have the correct order of effects. However, when a structure is applied over several model terms, the user must check the structure matches the effects. Thus for the random regression case

!{ animal animal.age us(2).animal !}

is correct because us(2).animal assumes 2 sets of animal effects. However

!{ animal animal.age animal.us(2) !}

is wrong.

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 AR1V 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 σ²(I+Σ_C ⊗ λ Σ_R).)
Under the functional syntax, we would write

 yield ~ mu variety !r repl ar1(column 0.4).ar1v(row  0.3 0.1)

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_iitial_vlues

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.

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