Reparameterization and invariant covariance matrices of factors in linear models

Abstract

Let the vector ζ consist of sampled random elements of factors in a linear mixed model. Let P be a permutation matrix. The covariance matrix D(ζ) is called P-invariant if D(ζ)=D(Pζ). It will be demonstrated that there is a strong correspondence between the spectrum of D(ζ) and certain reparametrization conditions on the factors. In particular, the classical reparametrization condition ∑ζ_i=0 has a clear presentation through the eigenvalues of D(ζ). This correspondence is useful for modelling data.