Penalized REML estimation can substantially reduce sampling variation in estimates of covariance matrices, and yield estimates of genetic parameters closer to population values than standard analyses. A number of suitable penalties based on prior distributions of correlation matrices from the Bayesian literature are described, and a simulation study is presented demonstrating their efficacy. Results show that reductions of 'loss' in estimates of the genetic covariance matrix, a conglomerate of sampling variance and bias, well over 50% are readily obtained for multivariate analyses of small samples. Default settings for a mild degree of penalization are proposed, which make such analyses suitable for routine use without increasing computational requirements.

Approximate lower bound sampling errors of maximum likelihood estimates of covariance components and their linear functions can be obtained from the inverse of the information matrix. For non-linear functions, sampling variances are commonly determined as the variance of their first order Taylor series expansions. This is used to obtain sampling errors for estimates of heritabilities and correlations, and these quantities can be computed with most software performing such analyses. In other instances, however, more complicated functions are of interest or the linear approximation is difficult or inadequate. A pragmatic alternative then is to evaluate sampling characteristics by repeated sampling of parameters from their asymptotic, multivariate normal distribution, calculating the function(s) of interest for each sample and inspecting the distribution across replicates. This paper demonstrates the use of this approach and examines the quality of approximation obtained.

Maximum likelihood estimation of genetic covariances subject to a penalty to reduce sampling variation has been shown to yield improved estimates, especially for analyses comprising many traits. However, this can increase computational requirements substantially. Similarly, penalties have been found to be beneficial in a maximum likelihood based approach for pooling results from analyses of subsets of traits. This paper examines the scope for using the latter method to apply penalties to results from multivariate analyses in a computationally undemanding post-estimation step. A simulation study is presented demonstrating that even slight changes to estimates in this way can result in 'regularized' values markedly closer to population values than standard, unpenalized estimates.

The era of genomic evaluation has brought the need to perform computations involving large, dense matrices. Particular tasks are the computation and inversion of the genomic relationship matrix. This paper investigates the suitability of Graphics Processing Units together with highly optimised software libraries for these computations, using blocked algorithms. It is shown that calculations are readily sped up by parallel processing, using freely available library routines, and that reductions in time by factors of 4 to 5 are achievable even for 'consumer' grade graphics cards.

This study assessed the impact of alternative parameterizations for the estimation of variance components on practical predictions of breeding values with MACE. Interbull MACE Holstein evaluations for somatic cell count (April 2009) and protein yield (August 2007) were considered. The MACE model was expressed in terms of a random regression model, which facilitates exploitation of principal component and factor analytic approaches. Both methods allow a reduction of the number of parameters to be estimated and benefit from the more parsimonious variance structure. Genetic parameters from different approaches were very similar, when the optimal fit was used. Over-fitting did not affect the estimates, but increased estimation time, whereas fitting too few parameters affected bull rankings in different countries.

Mixed model analyses via restricted maximum likelihood, fitting the so-called animal model, have become standard methodology for the estimation of genetic variances. Models involving multiple genetic variance components, due to different modes of gene action, are readily fitted. It is shown that likelihood-based calculations may provide insight into the quality of the resulting parameter estimates, and are directly applicable to the validation of experimental designs. This is illustrated for the example of a design suggested recently to estimate X-linked genetic variances. In particular, large sample variances and sampling correlations are demonstrated to provide an indication of 'problem' scenarios. Using simulation, it is shown that the profile likelihood function provides more appropriate estimates of confidence intervals than large sample variances. Examination of the likelihood function and its derivatives are recommended as part of the design stage of quantitative genetic experiments.

Penalized maximum likelihood estimation has been advocated for its capability to yield substantially improved estimates of covariance matrices, but so far only cases with equal numbers of records have been considered. We show that a generalization of the inverse Wishart distribution can be utilised to derive penalties which allow for differential penalization for different blocks of the matrices to be estimated. However, this requires multiple tuning factors to be determined and thus can increase computational requirements markedly. Simulation results are presented which indicate that the additional gains obtainable for estimates of genetic covariance components - over and above those from a simple, non-differential scheme - are moderate, even if numbers of records for different traits differ by orders of magnitude.