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Open Access Methodology

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

Karin Meyer* and Mark Kirkpatrick

Author Affiliations

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

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Genetics Selection Evolution 2005, 37:1-30  doi:10.1186/1297-9686-37-1-1


The electronic version of this article is the complete one and can be found online at: http://www.gsejournal.org/content/37/1/1


Received:1 April 2004
Accepted:9 August 2004
Published:15 January 2005

© 2004 INRA, EDP Sciences

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.

Keywords:
covariances; principal components; restricted maximum likelihood; reduced rank

Methodology

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