ASReml-S reference manual - VSN International

4.7 Constraining variance parameters 51See Sections 2.1and B• asreml does not have an implicit scale parameter when G structures are defined inthe random model formula. For this reason one, and only one, of the models in theG structure term must be a variance function; an initial value must be supplied forthe associated scale parameter, this is discussed under Initial values and constraints forvariance parameters on page 29.• When the G structure involves more than one variance model, one must be either anhomogeneous or a heterogeneous variance model and the rest should be correlationmodels; if more than one are non-correlation models then constraints should be usedto avoid identifiability problems, that is, to prevent attempts to estimate confoundedparameters.4.6 G structures involving more than one random termlink()It is usually the case that a variance structure applies to a particular term in the linearmodel and that there is no covariance between random terms. The link() function, withsome restrictions, can be used to estimate a covariance between random terms in certaincases.For example, in random regressions we would generally wish to estimate a covariancebetween intercept and slope. The syntax for a longitudinal analysis of animal liveweightover time, say, is> asreml(. . . , random = us(link(∼ time)):Animal, . . . )Ignoring the variance model, the syntax link(∼ time):Animal generates Animal+time:Animalterms and ensures they remain adjacent in the design matrix. The us() variance modelspecifies a variance parameter for the intercept (Animal), a variance parameter for theslopes (time:Animal) and a covariance between them.Note that in the current version this syntax is restricted to simple cases such as this andthe factor defining the subjects must appear on the right hand side of the : operator.The above is equivalent, but not identical because of scaling differences, to> asreml(. . . , random = us(pol(time)):Animal, . . . )4.7 Constraining variance parametersEquality and more general relationships among variance parameters are specified in asremlwith a simple linear model. Let κ be the n κ vector of unconstrained variance parametersand T be a n κ × n c matrix imposing the linear constraints T ′ κ = s. This isequivalent toκ = Mθ + Eswhere θ is the n c vector of constrained parameters. Constraints are specified in asremlby the matrix M which must have a dimnames attribute with the names of κ as its rownames.Note that in asreml n κ need only encompass the subset of variance parameters amongwhich constraints will be applied, rather than the entire set.The function asreml.constraints() generates the matrix M from a linear model formulaand a data frame in which to1. resolve the terms named in that formula, and