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.

Estimates of large genetic covariance matrices are commonly obtained by pooling results from a series of analyses of small subsets of traits. Procedures available to pool the part-estimates differ in their efficacy in accounting for unequal accuracies of estimates and sampling correlations, and ensuring that pooled matrices are within the parameter space. We propose a maximum likelihood (ML) approach to combine estimates, treating sets from individual part-analyses as matrices of mean squares and cross-products from independent families. This facilitates simultaneous pooling of estimates for all sources of variation considered, readily allows for weighted estimation or a given structure of the pooled matrices, and provides a framework for regularized estimation by penalizing the likelihood. A simulation study is presented, comparing the quality of combined estimates for several procedures, including truncation or shrinkage of either canonical or individual matrix eigen-values, iterative summation of expanded part matrices, and the ML approach, considering a range of penalties. Shrinking eigen-values of individual matrices towards their mean reduced losses in the pooled estimates, but substantially increased proportional losses in their phenotypic counterparts and thus yielded estimates differing most from corresponding full multivariate analyses of all traits. Assuming a simple pseudo-pedigree structure when combining estimates for all random effects simultaneously using ML allowed sampling correlations between estimates of different components from the same part-analysis to be approximated sufficiently to yield pooled matrices closest to full multivariate results, with little change in phenotypic components. Imposing a mild penalty to shrink matrices for random effects towards their sum proved highly advantageous, markedly reducing losses in estimates and more than compensating for the reduction in efficiency of using the data inherent in analyses by parts. Penalized ML provides a flexible alternative to current methods for pooling estimates from part-analyses with good sampling properties, and should be adopted more widely.

Estimates of covariance functions for weights of Angus cattle from birth to 3000 days of age were obtained using Bayesian analysis. Data consisted of records in 69 herds with at least 50 mature cow weights, and records in 6 additional herds with 60% or more animals having at least four weights, 551,259 records on 197,915 animals in total. The model of analysis fitted contemporary groups and cubic regressions on orthoganal polynomials of age nested within sex, birth type, dam age class and lactation status as fixed effects. Random effects fitted were cubic and quartic regressions on orthogonal polynomials of age for animals' direct genetic and permanent environmental effects, and quadratic regressions, restricted to 0 to 600 days of age, for maternal genetic and environmental effects. Measurement error variances were modelled through a step function with 32 classes, yielding 69 covariance components to be estimated.

Genomic information can accurately specify relationships among animals, including between those without known common ancestors. Genetic variances estimated with genomic data relate to unknown, more distant, founder populations than those defined by the pedigree. Starting from different sets of assumptions, the properties of some alternative genomic relationship matrices (G) are explored. Although the assumptions and matrices differ, the resulting sets of estimated breeding values predict the differences between animals identically, despite obtaining different estimates of the additive genetic variance - showing that there are many ways of building G that provide identical results. For some methods integer and logic, rather than floating point, operations will expedite building G many-fold.

A strategy to reduce computational demands of genome-wide association studies fitting a mixed model is presented. Improvements are achieved by utilizing a large proportion of calculations that remain constant across the multiple analyses for individual markers involved, with estimates obtained without inverting large matrices.

A simulation study examining the effects of 'regularization' on estimates of genetic covariance matrices for small samples is presented. This is achieved by penalizing the likelihood, and three types of penalties are examined. It is shown that regularized estimation can substantially enhance the accuracy of estimates of genetic parameters. Penalties shrinking estimates of genetic covariances or correlations towards their phenotypic counterparts acted somewhat differently to those aimed reducing the spread of sample eigenvalues. While improvements of estimates were found to be comparable overall, shrinkage of genetic towards phenotypic correlations resulted in least bias.

Estimates of direct and maternal variance and heritability for weights at each week (up to 280 days of age) and month of age (up to 600 days of age) in Zebu cattle are presented. More than one million records on 200 000 animals, weighed every 90 days from birth to 2 years of age, were available. Data were split according to week (data sets 1) or month (data sets 2) of age at recording, creating 54 and 21 data sets, respectively. The model of analysis included contemporary groups as fixed effects, and age of dam (linear and quadratic) and age of calf (linear) effects as covariables. Random effects fitted were additive direct and maternal genetic effects, and maternal permanent environmental effect. Direct heritability estimates decreased from 0.28 at birth, to 0.12-0.13 at about 150 days of age, stayed more or less constant at 0.14-0.16 until 270 days of age and increased with age after that, up to 0.25-0.26. Maternal heritability estimates increased from birth (0.01) to a peak of 0.14 for data sets 1 and 0.07-0.08 for data sets 2 at about 180-210 days of age, before decreasing slowly to 0.07 and 0.05, respectively, at 300 days, and then rapidly diminished after 300 days of age. Permanent environmental effects were 1.5 to four times higher than genetic maternal effects and showed a similar trend.

Estimates of covariance functions and genetic parameters were obtained for growth of Angus cattle from birth to 820 days of age. Data comprised 84 533 records on 20 731 animals in 43 herds, with a high proportion of animals with 4 or more weights recorded. Changes in weights were modelled through random regression on orthogonal polynomials of age at recording. A total of 11 combinations of quadratic, cubic, quartic and quintic polynomials to model direct and maternal genetic effects and permanent environmental effects were considered. Results showed good agreement for all models at ages with many records, but differed at the highest ages and at very early ages with few weights available. Cubic polynomials appeared to be most problematic. The order of polynomial fit for permanent environmental effects of the animal dominated estimates of phenotypic variances and mean squares for residual errors. A model fitting a quartic polynomial for these effects and quadratic polynomials for the other random effects, appeared to be the best compromise between detailedness of the model which could be supported by the data, plausibility of results, and fit, measured as mean square error.

We propose a new version of functional data model for analyzing familial related individuals, where the within-subject correlation depends smoothly on a covariate such as age and the between-subject correlation follows family-wise genetic association. Our motivating example concerns measurements of weight as a function of age in sibling cows from independent families. Observations are sparsely sampled from trajectories of a phenotype contaminated with measurement error, where the phenotypic trajectory consists of a genetic component and an environmental component. By combining information across individuals, the genetic and environmental covariance are estimated via smoothing techniques. We study the genetic and environmental effects using principal component analysis, taking into account the genetic correlation to enhance the subject-level signal extraction. We show via the real data and simulations that incorporating the correlation structure improves predictions of individual phenotypic trajectories.

Background: Analysis of data on genotypes with different expression in different environments is a classic problem in quantitative genetics. A review of models for data with genotype × environment interactions and related problems is given, linking early, analysis of variance based formulations to their modern, mixed model counterparts. Results: It is shown that models developed for the analysis of multi-environment trials in plant breeding are directly applicable in animal breeding. In particular, the 'additive main effect, multiplicative interaction' models accommodate heterogeneity of variance and are characterised by a factor-analytic covariance structure. While this can be implemented in mixed models by imposing such structure on the genetic covariance matrix in a standard, multi-trait model, an equivalent model is obtained by fitting the common and specific factors genetic separately. Properties of the mixed model equations for alternative implementations of factor-analytic models are discussed, and extensions to structured modelling of covariance matrices for multi-trait, multi-environment scenarios are described. Conclusion: Factor analytic models provide a natural framework for modelling genotype × environment interaction type problems. Mixed model analyses fitting such models are likely to see increasing use due to the parsimonious description of covariance structures available, the scope for direct interpretation of factors as well as computational advantages.