A Note on the Covariance Structure in a Linear Model

The covariance structure of a random or mixed linear model is sensitive to minor changes in the characterization of random effects, and what appear to be inconsequential differences in the description of random effects may imply nontrivial changes to the underlying linear model. When changes are made in the characterization of effects in a mixed model, the mean and covariance structure of both models must be considered to assure that the changes have resulted in a reparametrization of the original model. We consider this issue and the differences between reparametrizations for fixed effects and the covariance structure. Examples are presented to illustrate these differences. The examples could easily be presented or given as problems in regression courses, experimental design courses, or a first-year linear models course to augment students' understanding of linear models fundamentals.