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.

Imprinted loci are those where the level of expression of an allele depends upon the allele's parent of origin. Imprinting is a widespread phenomenon and parent-of-origin effects have been reported for many qualitative and quantitative traits, in particular carcass traits. The effect of parent-of-origin effects on three quantitative traits - eye muscle area and fat depth at the P8 and 12/13th rib sites - measured on Angus and Hereford heifers and bull calves was examined. Parent-of-origin effects accounted for 12-45% of the total genetic variation for these traits.

This paper examines some of the implied effects commonly assumed when building relationship matrices. We propose the inclusion of an additional 'individual' in the genomic relationship matrix which models the mean of the founder population. It is shown that this resolves the problem of inconsistent prediction error variances due to alternate allele coding schemes.

Parent-of-origin effects arise when an individual's genes are modified during gametogenesis. Commonly known as imprinting, affected genes may be completely, or partially, suppressed. Individual loci in mice, human and sheep are known to be imprinted, and the quantitative effects of imprinted loci have been found for many carcass traits in cattle and pigs. Differentiating between five types of loci - direct additive loci and partially and completely imprinted loci by sires and dams - is not possible as their effects are confounded such that only three of seven parameters can be estimated. An analysis of Australian Hereford and Angus heifers and bulls for four ultrasonic measures of body composition - eye muscle area, rib fat, rump fat and intramuscular fat per cent - found parent-of-origin effects for both parents in most trait-gender data sets and that they were an average of 28% of the total genetic variance. No parent-of-origin effects were found for Hereford bull intramuscular fat per cent and the maternal parent-of-origin effects were not significant for Angus Heifer eye muscle area.

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.

A reparameterization of the multivariate linear mixed model in genetic evaluation to principal components is described. This yields an equivalent model with a sparser coefficient matrix in the mixed model equations and, thus, reduced computational requirements to solve them. It is especially advantageous for analyses incorporating genomic relationship information with many nonzero elements in the inverse of the relationship matrix. Moreover, the framework lends itself directly to dimension reduction and, thus, further computational savings by omitting genetic principal components with negligible eigenvalues. The potential impact on computational demands is illustrated for an application to single-step genomic evaluation of Australian sheep.

The impact of parameterising to genetic principal components and dimension reduction on computational requirements is examined for a subset of traits considered in single step evaluation of sheep in Australia. Together with judicious treatment of dense blocks due to genomic relationships in the mixed model equations, such models can reduce computational requirements many-fold.

A method for approximating prediction error variances and covariances among estimates of individual animals genetic effects for multiple-trait and random regression models is described. These approximations are used to calculate the prediction error variances of linear functions of the terms in the model. In the multiple-trait case these are indexes of estimated breeding values, and for random regression models these are estimated breeding values at individual points on the longitudinal scale. Approximate reliabilities for terms in the model and linear functions thereof are compared with corresponding reliabilities obtained from the inverse of the coefficient matrix in the mixed model equations. Results show good agreement between approximate and true values.

Background: A common measure employed to evaluate the efficacy of livestock improvement schemes is the genetic trend, which is calculated as the means of predicted breeding values for animals born in successive time periods. This implies that different cohorts refer to the same base population. For genetic evaluation schemes integrating genomic information with records for all animals, genotyped or not, this is often not the case: expected means for pedigree founders are zero whereas values for genotyped animals are expected to sum to zero at the (mean) time corresponding to the frequencies that are used to center marker allele counts when calculating genomic relationships. Methods: The paper examines estimates of genetic trends from single-step genomic evaluations. After a review of methods which propose to align pedigree-based and genomic relationship matrices, simulation is used to illustrate the effects of alignments and choice of assumed gene frequencies on trajectories of genetic trends. Results: The results show that methods available to alleviate differences between the founder populations implied by the two types of relationship matrices perform well; in particular, the meta-founder approach is advantageous. An application to data from routine genetic evaluation of Australian sheep is shown, confirming their effectiveness for practical data. Conclusions: Aligning pedigree and genomic relationship matrices for single step genetic evaluation for populations under selection is essential. Fitting meta-founders is an effective and simple method to avoid distortion of estimates of genetic trends.