Iterative Estimation of Variance Components for Non-Orthogonal Data

Abstract

Eisenhart [1947] introduced the term 'mixed model' to describe models useful in experiments where some effects, such as block or animal effects, can be thought of as random effects and other effects, for example treatments, are regarded as fixed. The estimation of variance components from these experiments is fully understood for various incomplete block arrangements with a high degree of symmetry (Nelder [1968]). For a general non-orthogonal design, however, difficulties arise and no simple known method is optimal under all conditions. Lack of balance is very common in records on animals, especially those arising from studies of quantitative genetics. Experimenters are fortunate if families of animals are all of the same size; even if an experiment begins with reasonable symmetry, accidental losses may introduce non-orthogonality. Cunningham and Henderson [1968] have proposed a general method of estimation, using iterative calculations. An algebraic oversight has corrupted their formulae, giving an iterative process that will usually fail to converge to anything reasonable.