To have your steak and eat it: Genetic principal component analysis for beef cattle data

Quantitative genetic analyses usually deal with several, if not many, correlated traits or effects. Generally, the matrices of covariances among these effects are considered to be 'unstructured', i.e. for k traits we have k(k + 1)/2 distinct (co)variance components, and restrictions on estimates are imposed only to ensure that estimated matrices are positive semi-definite, i.e. do not have negative eigenvalues. In contrast, in other areas of statistics covariance matrices are often assumed to be structured. Parametric forms, such as compound symmetry or auto-regressive covariances (e.g. Jennrich and Schluchter 1986) are common assumptions for longitudinal or spatial data. Alternative parameterisations are based on the eigen-vectors and -values of the covariances matrices concerned. In particular, principal component (PC) analysis is widely utilised to summarise multivariate information and as a dimension reduction technique. So far, PC analyses (PCA) for genetic (or other random) effects have by and large been carried out in 2 steps, first obtaining full rank estimates of covariance matrices, and then performing an eigen-decomposition of the estimates. A better approach is to estimate the PCs directly and, at the same time, to restrict estimation to the most important components only (Kirkpatrick and Meyer 2004). This is readily accommodated within the usual linear, mixed model framework, requiring only a simple reparameterisation. This paper reviews direct estimation of PCs, and presents an application to an analysis of carcass traits of beef cattle.