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.

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