Genet. Sel. Evol. 37 (2005) 1-30
DOI: 10.1051/gse:2004034

Restricted maximum likelihood estimation of genetic principal components and smoothed covariance matrices

Karin Meyer and Mark Kirkpatrick

Animal Genetics and Breeding Unit (A joint venture with NSW Agriculture.) , University of New England, Armidale NSW 2351, Australia Section of Integrative Biology, 1 University Station C-0930, University of Texas, Austin, Texas 78712, USA

(Received 1 April 2004; accepted 9 August 2004 )

Abstract - Principal component analysis is a widely used `dimension reduction' technique, albeit generally at a phenotypic level. It is shown that we can estimate genetic principal components directly through a simple reparameterisation of the usual linear, mixed model. This is applicable to any analysis fitting multiple, correlated genetic effects, whether effects for individual traits or sets of random regression coefficients to model trajectories. Depending on the magnitude of genetic correlation, a subset of the principal component generally suffices to capture the bulk of genetic variation. Corresponding estimates of genetic covariance matrices are more parsimonious, have reduced rank and are smoothed, with the number of parameters required to model the dispersion structure reduced from k(k+1)/2 to m(2k-m+1)/2 for k effects and m principal components. Estimation of these parameters, the largest eigenvalues and pertaining eigenvectors of the genetic covariance matrix, via restricted maximum likelihood using derivatives of the likelihood, is described. It is shown that reduced rank estimation can reduce computational requirements of multivariate analyses substantially. An application to the analysis of eight traits recorded via live ultrasound scanning of beef cattle is given.

Key words: covariances / principal components / restricted maximum likelihood / reduced rank

Correspondence and reprints: kmeyer@didgeridoo.une.edu.au

© INRA, EDP Sciences 2004