The genomic relationship matrix (GRM) can be inverted by the algorithm for proven and young (APY) based on recursion on a random subset of animals. While a regular inverse has a cubic cost, the cost of the APY inverse can be close to linear. Theory for the APY assumes that the optimal size of the subset (maximizing accuracy of genomic predictions) is due to a limited dimensionality of the GRM, which is a function of the effective population size (Ne). The objective of this study was to evaluate these assumptions by simulation. Six populations were simulated with approximate effective population size (Ne) from 20 to 200. Each population consisted of 10 nonoverlapping generations, with 25,000 animals per generation and phenotypes available for generations 1–9. The last 3 generations were fully genotyped assuming genome length L = 30. The GRM was constructed for each population and analyzed for distribution of eigenvalues. Genomic estimated breeding values (GEBV) were computed by single-step GBLUP, using either a direct or an APY inverse of GRM. The sizes of the subset in APY were set to the number of the largest eigenvalues explaining x% of variation (EIGx, x = 90, 95, 98, 99) in GRM. Accuracies of GEBV for the last generation with the APY inverse peaked at EIG98 and were slightly lower with EIG95, EIG99, or the direct inverse. Most information in the GRM is contained in ∼NeL largest eigenvalues, with no information beyond 4NeL. Genomic predictions with the APY inverse of the GRM are more accurate than by the regular inverse.

BackgroundA genomic relationship matrix (GRM) can be inverted efficiently with the Algorithm for Proven and Young (APY) through recursion on a small number of core animals. The number of core animals is theoretically linked to effective population size (N e). In a simulation study, the optimal number of core animals was equal to the number of largest eigenvalues of GRM that explained 98% of its variation. The purpose of this study was to find the optimal number of core animals and estimate N e for different species.MethodsDatasets included phenotypes, pedigrees, and genotypes for populations of Holstein, Jersey, and Angus cattle, pigs, and broiler chickens. The number of genotyped animals varied from 15,000 for broiler chickens to 77,000 for Holsteins, and the number of single-nucleotide polymorphisms used for genomic prediction varied from 37,000 to 61,000. Eigenvalue decomposition of the GRM for each population determined numbers of largest eigenvalues corresponding to 90, 95, 98, and 99% of variation.ResultsThe number of eigenvalues corresponding to 90% (98%) of variation was 4527 (14,026) for Holstein, 3325 (11,500) for Jersey, 3654 (10,605) for Angus, 1239 (4103) for pig, and 1655 (4171) for broiler chicken. Each trait in each species was analyzed using the APY inverse of the GRM with randomly selected core animals, and their number was equal to the number of largest eigenvalues. Realized accuracies peaked with the number of core animals corresponding to 98% of variation for Holstein and Jersey and closer to 99% for other breed/species. N e was estimated based on comparisons of eigenvalue decomposition in a simulation study. Assuming a genome length of 30 Morgan, N e was equal to 149 for Holsteins, 101 for Jerseys, 113 for Angus, 32 for pigs, and 44 for broilers.ConclusionsEigenvalue profiles of GRM for common species are similar to those in simulation studies although they are affected by number of genotyped animals and genotyping quality. For all investigated species, the APY required less than 15,000 core animals. Realized accuracies were equal or greater with the APY inverse than with regular inversion. Eigenvalue analysis of GRM can provide a realistic estimate of N e.

Background Metafounders are pseudo-individuals that encapsulate genetic heterozygosity and relationships within and across base pedigree populations, i.e. ancestral populations. This work addresses the estimation and usefulness of metafounder relationships in single-step genomic best linear unbiased prediction (ssGBLUP).ResultsWe show that ancestral relationship parameters are proportional to standardized covariances of base allelic frequencies across populations, such as fixation indexes. These covariances of base allelic frequencies can be estimated from marker genotypes of related recent individuals and pedigree. Simple methods for their estimation include naïve computation of allele frequencies from marker genotypes or a method of moments that equates average pedigree-based and marker-based relationships. Complex methods include generalized least squares (best linear unbiased estimator (BLUE)) or maximum likelihood based on pedigree relationships. To our knowledge, methods to infer coefficients from marker data have not been developed for related individuals. We derived a genomic relationship matrix, compatible with pedigree relationships, that is constructed as a cross-product of {−1,0,1} codes and that is equivalent (apart from scale factors) to an identity-by-state relationship matrix at genome-wide markers. Using a simulation with a single population under selection in which only males and youngest animals are genotyped, we observed that generalized least squares or maximum likelihood gave accurate and unbiased estimates of the ancestral relationship parameter (true value: 0.40) whereas the naïve method and the method of moments were biased (average estimates of 0.43 and 0.35). We also observed that genomic evaluation by ssGBLUP using metafounders was less biased in terms of estimates of genetic trend (bias of 0.01 instead of 0.12), resulted in less overdispersed (0.94 instead of 0.99) and as accurate (0.74) estimates of breeding values than ssGBLUP without metafounders and provided consistent estimates of heritability.ConclusionsEstimation of metafounder relationships can be achieved using BLUP-like methods with pedigree and markers. Inclusion of metafounder relationships reduces bias of genomic predictions with no loss in accuracy.

SummaryThe Algorithm for Proven and Young (APY) enables the implementation of single-step genomic BLUP (ssGBLUP) in large, genotyped populations by separating genotyped animals into core and non-core subsets and creating a computationally efficient inverse for the genomic relationship matrix (G). As APY became the choice for large-scale genomic evaluations in BLUP-based methods, a common question is how to choose the animals in the core subset. We compared several core definitions to answer this question. Simulations comprised a moderately heritable trait for 95,010 animals and 50,000 genotypes for animals across five generations. Genotypes consisted of 25,500 SNP distributed across 15 chromosomes.Genotyping errors and missing pedigree were also mimicked. Core animals were defined based on individual generations, equal representation across generations, and at random. For a sufficiently large core size, core definitions had the same accuracies and biases, even if the core animals had imperfect genotypes. When genotyped animals had unknown parents, accuracy and bias were significantly better (p ≤ .05) for random and across generation core definitions. K E Y W O R D S

BackgroundThe dimensionality of genomic information is limited by the number of independent chromosome segments (Me), which is a function of the effective population size. This dimensionality can be determined approximately by singular value decomposition of the gene content matrix, by eigenvalue decomposition of the genomic relationship matrix (GRM), or by the number of core animals in the algorithm for proven and young (APY) that maximizes the accuracy of genomic prediction. In the latter, core animals act as proxies to linear combinations of Me. Field studies indicate that a moderate accuracy of genomic selection is achieved with a small dataset, but that further improvement of the accuracy requires much more data. When only one quarter of the optimal number of core animals are used in the APY algorithm, the accuracy of genomic selection is only slightly below the optimal value. This suggests that genomic selection works on clusters of Me.ResultsThe simulation included datasets with different population sizes and amounts of phenotypic information. Computations were done by genomic best linear unbiased prediction (GBLUP) with selected eigenvalues and corresponding eigenvectors of the GRM set to zero. About four eigenvalues in the GRM explained 10% of the genomic variation, and less than 2% of the total eigenvalues explained 50% of the genomic variation. With limited phenotypic information, the accuracy of GBLUP was close to the peak where most of the smallest eigenvalues were set to zero. With a large amount of phenotypic information, accuracy increased as smaller eigenvalues were added.ConclusionsA small amount of phenotypic data is sufficient to estimate only the effects of the largest eigenvalues and the associated eigenvectors that contain a large fraction of the genomic information, and a very large amount of data is required to estimate the remaining eigenvalues that account for a limited amount of genomic information. Core animals in the APY algorithm act as proxies of almost the same number of eigenvalues. By using an eigenvalues-based approach, it was possible to explain why the moderate accuracy of genomic selection based on small datasets only increases slowly as more data are added.

Genetic variance is a central parameter in quantitative genetics and breeding. Assessing changes in genetic variance over time as well as the genome is therefore of high interest. Here, we extend a previously proposed framework for temporal analysis of genetic variance using the pedigree-based model, to a new framework for temporal and genomic analysis of genetic variance using marker-based models. To this end, we describe the theory of partitioning genetic variance into genic variance and within-chromosome and between-chromosome linkage-disequilibrium, and how to estimate these variance components from a marker-based model fitted to observed phenotype and marker data. The new framework involves three steps: (i) fitting a marker-based model to data, (ii) sampling realisations of marker effects from the fitted model and for each sample calculating realisations of genetic values and (iii) calculating the variance of sampled genetic values by time and genome partitions. Analysing time partitions indicates breeding programme sustainability, while analysing genome partitions indicates contributions from chromosomes and chromosome pairs and linkage-disequilibrium. We demonstrate the framework with a simulated breeding programme involving a complex trait. Results show good concordance between simulated and estimated variances, provided that the fitted model is capturing genetic complexity of a trait. We observe a reduction of genetic variance due to selection and drift changing allele frequencies, and due to selection inducing negative linkage-disequilibrium.

Genetically linked small and large dairy cattle populations were simulated to test the effect of different sources of information from foreign populations on the accuracy of predicting breeding values for young animals in a small population. A large dairy cattle population (P) with >20 generations was simulated, and a small subpopulation (P) with 3 generations was formed as a related population, including phenotypes and genomic information. Predicted breeding values for young animals in the small population were calculated using BLUP and single-step genomic BLUP (ssGBLUP) in 4 different scenarios: (S1) 3,166 phenotypes, 22,855 pedigree animals, and 1,000 to 6,000 genotypes for P; (S2) S1 plus genomic estimated breeding value (GEBV) for 4,475 sires from P as external information; (S3) S1 plus 221,580 phenotypes, 402,829 pedigree animals, and 53,558 genotypes for P; and (S4) single nucleotide polymorphism (SNP) effects calculated based on P data. The ability to predict true breeding value was assessed in the youngest third of the genotyped animals in the small population. When data only from the small population were used and 1,000 animals were genotyped, the accuracy of GEBV was only 1 point greater than the estimated breeding value accuracy (0.32 vs. 0.31). Adding external GEBV for sires from P did not considerably increase accuracy (0.33 vs. 0.32 in S1). Combining phenotypes, pedigree, and genotypes for P and P was beneficial for predicting accuracy of GEBV in the small population, and the prediction accuracy of GEBV in this scenario was 0.38 compared with 0.31 from estimated breeding values. When SNP effects from P were used to predict GEBV for young genotyped animals from P, accuracy was greatest (0.56). With 6,000 genotyped animal in P, accuracy was greatest (0.61) with the combined populations. In a small population with few genotypes, the highest accuracy of evaluation may be obtained by using SNP effects derived from a related large population.

We investigated the effects of different strategies for genotyping populations on variance components and heritabilities estimated with an animal model under restricted maximum likelihood (REML), genomic REML (GREML), and single‐step GREML (ssGREML). A population with 10 generations was simulated. Animals from the last one, two or three generations were genotyped with 45,116 SNP evenly distributed on 27 chromosomes. Animals to be genotyped were chosen randomly or based on EBV. Each scenario was replicated five times. A single trait was simulated with three heritability levels (low, moderate, high). Phenotypes were simulated for only females to mimic dairy sheep and also for both sexes to mimic meat sheep. Variance component estimates from genomic data and phenotypes for one or two generations were more biased than from three generations. Estimates in the scenario without selection were the most accurate across heritability levels and methods. When selection was present in the simulations, the best option was to use genotypes of randomly selected animals. For selective genotyping, heritabilities from GREML were more biased compared to those estimated by ssGREML, because ssGREML was less affected by selective or limited genotyping.