2004, Kirkpatrick, M, Meyer, K

Estimating the genetic and environmental variances for multivariate and function-valued phenotypes poses problems for estimation and interpretation. Even when the phenotype of interest has a large number of dimensions, most variation is typically associated with a small number of principal components (eigenvectors or eigenfunctions). We propose an approach that directly estimates these leading principal components; these then give estimates for the covariance matrices (or functions). Direct estimation of the principal components reduces the number of parameters to be estimated, uses the data efficiently, and provides the basis for new estimation algorithms. We develop these concepts for both multivariate and function-valued phenotypes and illustrate their application in the restricted maximum-likelihood framework.

2010, Meyer, Karin, Kirkpatrick, M

Knowledge of genetic parameters and variances is an essential pre-requisite for tasks such as the design of selection programmes or prediction of breeding values. Reliable estimation of these quantities is thus paramount. There is a growing trend to consider more and more complex phenotypes, necessitating multivariate analyses comprising numerous traits. Problems inherent in such analyses, arising from sampling variation and the resulting over-dispersion of sample eigenvalues, are well known. There has been longstanding interest in the 'regularization' of estimated covariance matrices. Generally, this involves a compromise between additional bias and reduced sampling variation of 'improved' estimators. Numerous simulation studies have demonstrated that this can improve the agreement between estimated and population covariance matrices; see Meyer and Kirkpatrick (2010) for a review. For instance, estimators of covariance matrices have been suggested which counter-act upwards bias of the largest and downwards bias of the smallest eigenvalues by shrinking them towards their mean. In quantitative genetic analyses, we attempt to partition covariances into their genetic and environmental components.

2005, Meyer, K, Kirkpatrick, M

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 moreparsimonious, 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'(2'k' − '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.

2005, Meyer, K, Kirkpatrick, M

'Repeated' measurements for a trait and individual, taken along some continuous scale such as time, can be thought of as representing points on a curve, where both means and covariances along the trajectory can change, gradually and continually. Such traits are commonly referred to as 'function-valued' (FV) traits. This review shows that standard quantitative genetic concepts extend readily to FV traits, with individual statistics, such as estimated breeding values and selection response, replaced by corresponding curves, modelled by respective functions. Covariance functions are introduced as the FV equivalent to matrices of covariances.Considering the class of functions represented by a regression on the continuous covariable, FV traits can be analysed within the linear mixed model framework commonly employed in quantitative genetics, giving rise to the so-called random regression model. Estimation of covariance functions, either indirectly from estimated covariances or directly from the data using restricted maximum likelihood or Bayesian analysis, is considered. It is shown that direct estimation of the leading principal components of covariance functions is feasible and advantageous. Extensions to multidimensional analyses are discussed.