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Estimating heritabilities with large genomic models by established methods such as restricted maximum likelihood (REML) or Bayesian via Gibbs sampling is computationally expensive. Alternatively, heritability can be estimated indirectly by method R and by maximum predictivity, referred to as MaxPred here, at a much lower computing cost. By method R, the heritability used for predictions with whole and partial data is considered the best estimate when the predictions based on partial data are unbiased relative to those with the complete data. By MaxPred, the heritability estimate is the one that maximizes predictivity. This study compared heritability estimation with genomic information using average information REML (AI-REML), method R and MaxPred. A simulated population was generated with ten generations of 5000 animals each and an effective population size of 80. Each animal had one record for a trait with a heritability of 0.3, a phenotypic variance of 10.0 and was genotyped at 50 k SNP. In method R, the heritability estimate is found when the expectation of a regression coefficient is equal to one. The regression is the EBV of selection candidates calculated with the whole dataset regressed on the EBV of candidates calculated from a partial dataset. In this study, we used the GBLUP framework and therefore, GEBV was calculated. The partial dataset was created by removing the last generation of phenotypes. Predictivity was defined as the correlation between the adjusted phenotypes of the selection candidates and their GEBV calculated from the partial data. We estimated the heritability for populations that included between three and 10 generations. In every scenario, predictivity increased as more data was used and was the highest at the simulated heritability. However, the predictivity for all data subsets and all heritabilities compared did not differ more than 0.01, suggesting MaxPred is not the best indication for heritability estimation. For the whole dataset, the heritability was estimated as 0.30 ± 0.01, 0.26 ± 0.01 and 0.30 ± 0.04 for AI-REML without genomics, AI-REML with genomics and method R with genomics, respectively. Heritability estimation with genomics by method R reduced timing by 83%, implying a reduction in computing time from 9.5 to 1.6 h, on average, compared to AI-REML with genomics. Method R has the potential to estimate heritabilities with large genomic information at a low cost when many generations of animals are present; however, the standard error can be high when only a few iterations are used., (© 2024 The Authors. Journal of Animal Breeding and Genetics published by John Wiley & Sons Ltd.)

Increasing SNP density by incorporating sequence information only marginally increases prediction accuracies of breeding values in livestock. To find out why, we used statistical models and simulations to investigate the shape of distribution of estimated SNP effects (a profile) around quantitative trait nucleotides (QTNs) in populations with a small effective population size (Ne). A QTN profile created by averaging SNP effects around each QTN was similar to the shape of expected pairwise linkage disequilibrium (PLD) based on Ne and genetic distance between SNP, with a distinct peak for the QTN. Populations with smaller Ne showed lower but wider QTN profiles. However, adding more genotyped individuals with phenotypes dragged the profile closer to the QTN. The QTN profile was higher and narrower for populations with larger compared to smaller Ne. Assuming the PLD curve for the QTN profile, 80% of the additive genetic variance explained by each QTN was contained in ± 1/Ne Morgan interval around the QTN, corresponding to 2 Mb in cattle and 5 Mb in pigs and chickens. With such large intervals, identifying QTN is difficult even if all of them are in the data and the assumed genetic architecture is simplistic. Additional complexity in QTN detection arises from confounding of QTN profiles with signals due to relationships, overlapping profiles with closely spaced QTN, and spurious signals. However, small Ne allows for accurate predictions with large data even without QTN identification because QTNs are accounted for by QTN profiles if SNP density is sufficient to saturate the segments., Competing Interests: Conflicts of interest The authors declare no conflicts of interest., (© The Author(s) 2024. Published by Oxford University Press on behalf of The Genetics Society of America.)

Background: Validation by data truncation is a common practice in genetic evaluations because of the interest in predicting the genetic merit of a set of young selection candidates. Two of the most used validation methods in genetic evaluations use a single data partition: predictivity or predictive ability (correlation between pre-adjusted phenotypes and estimated breeding values (EBV) divided by the square root of the heritability) and the linear regression (LR) method (comparison of "early" and "late" EBV). Both methods compare predictions with the whole dataset and a partial dataset that is obtained by removing the information related to a set of validation individuals. EBV obtained with the partial dataset are compared against adjusted phenotypes for the predictivity or EBV obtained with the whole dataset in the LR method. Confidence intervals for predictivity and the LR method can be obtained by replicating the validation for different samples (or folds), or bootstrapping. Analytical confidence intervals would be beneficial to avoid running several validations and to test the quality of the bootstrap intervals. However, analytical confidence intervals are unavailable for predictivity and the LR method., Results: We derived standard errors and Wald confidence intervals for the predictivity and statistics included in the LR method (bias, dispersion, ratio of accuracies, and reliability). The confidence intervals for the bias, dispersion, and reliability depend on the relationships and prediction error variances and covariances across the individuals in the validation set. We developed approximations for large datasets that only need the reliabilities of the individuals in the validation set. The confidence intervals for the ratio of accuracies and predictivity were obtained through the Fisher transformation. We show the adequacy of both the analytical and approximated analytical confidence intervals and compare them versus bootstrap confidence intervals using two simulated examples. The analytical confidence intervals were closer to the simulated ones for both examples. Bootstrap confidence intervals tend to be narrower than the simulated ones. The approximated analytical confidence intervals were similar to those obtained by bootstrapping., Conclusions: Estimating the sampling variation of predictivity and the statistics in the LR method without replication or bootstrap is possible for any dataset with the formulas presented in this study., (© 2024. The Author(s).)

The exact accuracy of estimated breeding values can be calculated based on the prediction error variances obtained from the diagonal of the inverse of the left-hand side (LHS) of the mixed model equations (MME). However, inverting the LHS is not computationally feasible for large datasets, especially if genomic information is available. Thus, different algorithms have been proposed to approximate accuracies. This study aimed to: 1) compare the approximated accuracies from 2 algorithms implemented in the BLUPF90 suite of programs, 2) compare the approximated accuracies from the 2 algorithms against the exact accuracy based on the inversion of the LHS of MME, and 3) evaluate the impact of adding genotyped animals with and without phenotypes on the exact and approximated accuracies. Algorithm 1 approximates accuracies based on the diagonal of the genomic relationship matrix (G). In turn, algorithm 2 combines accuracies with and without genomic information through effective record contributions. The data were provided by the American Angus Association and included 3 datasets of growth, carcass, and marbling traits. The genotype file contained 1,235,930 animals, and the pedigree file contained 12,492,581 animals. For the genomic evaluation, a multi-trait model was applied to the datasets. To ensure the feasibility of inverting the LHS of the MME, a subset of data under single-trait models was used to compare approximated and exact accuracies. The correlations between exact and approximated accuracies from algorithms 1 and 2 of genotyped animals ranged from 0.87 to 0.90 and 0.98 to 0.99, respectively. The intercept and slope of the regression of exact on approximated accuracies from algorithm 2 ranged from 0.00 to 0.01 and 0.82 to 0.87, respectively. However, the intercept and the slope for algorithm 1 ranged from -0.10 to 0.05 and 0.98 to 1.10, respectively. In more than 80% of the traits, algorithm 2 exhibited a smaller mean square error than algorithm 1. The correlation between the approximated accuracies obtained from algorithms 1 and 2 ranged from 0.56 to 0.74, 0.38 to 0.71, and 0.71 to 0.97 in the groups of genotyped animals, genotyped animals without phenotype, and proven genotyped sires, respectively. The approximated accuracy from algorithm 2 showed a closer behavior to the exact accuracy when including genotyped animals in the analysis. According to the results, algorithm 2 is recommended for genetic evaluations since it proved more precise., (© The Author(s) 2024. Published by Oxford University Press on behalf of the American Society of Animal Science.)

Threshold models are often used in genetic analysis of categorical data, such as calving ease. Solutions in the liability scale are easily transformed into probabilities; therefore, estimated breeding values are published as the probability of expressing the category of main interest and are the industry's gold standard because they are easy to interpret and use for selection. However, because threshold models involve nonlinear equations and probability functions, implementing such a method is complex. Challenges include long computing time and convergence issues, intensified by including genomic data. Linear models are an alternative to overcome those challenges. Estimated breeding values computed using linear models are highly correlated (≥0.96) with those from threshold models; however, the lack of a transformation from the observed to the probability scale limits the use of linear models. The objective of this study was to propose transformations from observed to probability scale analogous to the transformation from liability to probability scale. We assessed computing time, peak memory use, correlations between estimated breeding values, and estimated genetic trends from linear and threshold models. With 11M animals in the pedigree and almost 965k genotyped animals, linear models were 5× faster to converge than threshold models (32 vs. 145 h), and peak memory use was the same (189 GB). The transformations proposed provided highly correlated probabilities from linear and threshold models. Correlations between direct (maternal) estimated breeding values from linear and threshold models and transformed to probabilities were ≥0.99 (0.97) for all animals in the pedigree, sires with/without progeny records, or animals with phenotypic records; therefore, estimated genetic trends were analogous, suggesting no loss of genetic progress in breeding programs that would adopt linear instead of threshold models. Furthermore, linear models reduced computing time by 5-fold compared to the threshold models; this enables weekly genetic evaluations and opens the possibility of using multi-trait models for categorical traits to improve selection effectiveness., (© The Author(s) 2024. Published by Oxford University Press on behalf of the American Society of Animal Science.)

Background: Identifying true positive variants in genome-wide associations (GWA) depends on several factors, including the number of genotyped individuals. The limited dimensionality of genomic information may give insights into the optimal number of individuals to be used in GWA. This study investigated different discovery set sizes based on the number of largest eigenvalues explaining a certain proportion of variance in the genomic relationship matrix (G). In addition, we investigated the impact on the prediction accuracy by adding variants, which were selected based on different set sizes, to the regular single nucleotide polymorphism (SNP) chips used for genomic prediction., Methods: We simulated sequence data that included 500k SNPs with 200 or 2000 quantitative trait nucleotides (QTN). A regular 50k panel included one in every ten simulated SNPs. Effective population size (Ne) was set to 20 or 200. GWA were performed using a number of genotyped animals equivalent to the number of largest eigenvalues of G (EIG) explaining 50, 60, 70, 80, 90, 95, 98, and 99% of the variance. In addition, the largest discovery set consisted of 30k genotyped animals. Limited or extensive phenotypic information was mimicked by changing the trait heritability. Significant and large-effect size SNPs were added to the 50k panel and used for single-step genomic best linear unbiased prediction (ssGBLUP)., Results: Using a number of genotyped animals corresponding to at least EIG98 allowed the identification of QTN with the largest effect sizes when Ne was large. Populations with smaller Ne required more than EIG98. Furthermore, including genotyped animals with a higher reliability (i.e., a higher trait heritability) improved the identification of the most informative QTN. Prediction accuracy was highest when the significant or the large-effect SNPs representing twice the number of simulated QTN were added to the 50k panel., Conclusions: Accurately identifying causative variants from sequence data depends on the effective population size and, therefore, on the dimensionality of genomic information. This dimensionality can help identify the most suitable sample size for GWA and could be considered for variant selection, especially when resources are restricted. Even when variants are accurately identified, their inclusion in prediction models has limited benefits., (© 2023. The Author(s).)

Background: Reliabilities of best linear unbiased predictions (BLUP) of breeding values are defined as the squared correlation between true and estimated breeding values and are helpful in assessing risk and genetic gain. Reliabilities can be computed from the prediction error variances for models with a single base population but are undefined for models that include several base populations and when unknown parent groups are modeled as fixed effects. In such a case, the use of metafounders in principle enables reliabilities to be derived., Methods: We propose to compute the reliability of the contrast of an individual's estimated breeding value with that of a metafounder based on the prediction error variances of the individual and the metafounder, their prediction error covariance, and their genetic relationship. Computation of the required terms demands only little extra work once the sparse inverse of the mixed model equations is obtained, or they can be approximated. This also allows the reliabilities of the metafounders to be obtained. We studied the reliabilities for both BLUP and single-step genomic BLUP (ssGBLUP), using several definitions of reliability in a large dataset with 1,961,687 dairy sheep and rams, most of which had phenotypes and among which 27,000 rams were genotyped with a 50K single nucleotide polymorphism (SNP) chip. There were 23 metafounders with progeny sizes between 100,000 and 2000 individuals., Results: In models with metafounders, directly using the prediction error variance instead of the contrast with a metafounder leads to artificially low reliabilities because they refer to a population with maximum heterozygosity. When only one metafounder is fitted in the model, the reliability of the contrast is shown to be equivalent to the reliability of the individual in a model without metafounders. When there are several metafounders in the model, using a contrast with the oldest metafounder yields reliabilities that are on a meaningful scale and very close to reliabilities obtained from models without metafounders. The reliabilities using contrasts with ssGBLUP also resulted in meaningful values., Conclusions: This work provides a general method to obtain reliabilities for both BLUP and ssGBLUP when several base populations are included through metafounders., (© 2023. The Author(s).)

Historical data collection for genetic evaluation purposes is a common practice in animal populations; however, the larger the dataset, the higher the computing power needed to perform the analyses. Also, fitting the same model to historical and recent data may be inappropriate. Data truncation can reduce the number of equations to solve, consequently decreasing computing costs; however, the large volume of genotypes is responsible for most of the increase in computations. This study aimed to assess the impact of removing genotypes along with phenotypes and pedigree on the computing performance, reliability, and inflation of genomic predicted breeding value (GEBV) from single-step genomic best linear unbiased predictor for selection candidates. Data from two pig lines, a terminal sire (L1) and a maternal line (L2), were analyzed in this study. Four analyses were implemented: growth and "weaning to finish" mortality on L1, pre-weaning and reproductive traits on L2. Four genotype removal scenarios were proposed: removing genotyped animals without phenotypes and progeny (noInfo), removing genotyped animals based on birth year (Age), the combination of noInfo and Age scenarios (noInfo + Age), and no genotype removal (AllGen). In all scenarios, phenotypes were removed, based on birth year, and three pedigree depths were tested: two and three generations traced back and using the entire pedigree. The full dataset contained 1,452,257 phenotypes for growth traits, 324,397 for weaning to finish mortality, 517,446 for pre-weaning traits, and 7,853,629 for reproductive traits in pure and crossbred pigs. Pedigree files for lines L1 and L2 comprised 3,601,369 and 11,240,865 animals, of which 168,734 and 170,121 were genotyped, respectively. In each truncation scenario, the linear regression method was used to assess the reliability and dispersion of GEBV for genotyped parents (born after 2019). The number of years of data that could be removed without harming reliability depended on the number of records, type of analyses (multitrait vs. single trait), the heritability of the trait, and data structure. All scenarios had similar reliabilities, except for noInfo, which performed better in the growth analysis. Based on the data used in this study, considering the last ten years of phenotypes, tracing three generations back in the pedigree, and removing genotyped animals not contributing own or progeny phenotypes, increases computing efficiency with no change in the ability to predict breeding values., (© The Author(s) 2023. Published by Oxford University Press on behalf of the American Society of Animal Science.)

Genomic estimated breeding values (GEBV) of animals without phenotypes can be indirectly predicted using recursions on GEBV of a subset. To maximize predictive ability of indirect predictions (IP), the subset must represent the independent chromosome segments segregating in the population. We aimed to 1) determine the number of animals needed in recursions to maximize predictive ability, 2) evaluate equivalency IP-GEBV, and 3) investigate trends in predictive ability of IP derived from recent vs. distant generations or accumulating phenotypes from recent to past generations. Data comprised pedigree of 825K birds hatched over 12 overlapping generations, phenotypes for body weight (BW; 820K), residual feed intake (RF; 200K) and weight gain during a trial period (WG; 200K), and breast meat percent (BP; 43K). A total of 154K birds (last six generations) had genotypes. The number of animals that maximize predictive ability was assessed based on the number of largest eigenvalues explaining 99% of variation in the genomic relationship matrix (1Me = 7,131), twice (2Me), or a fraction of this number (i.e., 0.75, 0.50, or 0.25Me). Equivalency between IP and GEBV was measured by correlating these two sets of predictions. GEBV were obtained as if generation 12 (validation animals) was part of the evaluation. IP were derived from GEBV of animals from generations 8 to 11 or generations 11, 10, 9, or 8. IP predictive ability was defined as the correlation between IP and adjusted phenotypes. The IP predictive ability increased from 0.25Me to 1Me (11%, on average); the change from 1Me to 2Me was negligible (0.6%). The correlation IP-GEBV was the same when IP were derived from a subset of 1Me animals chosen randomly across generations (8 to 11) or from generation 11 (0.98 for BW, 0.99 for RF, WG, and BP). A marginal decline in the correlation was observed when IP were based on GEBV of animals from generation 8 (0.95 for BW, 0.98 for RF, WG, and BP). Predictive ability had a similar trend; from generation 11 to 8, it changed from 0.32 to 0.31 for BW, from 0.39 to 0.38 for BP, and was constant at 0.33(0.22) for RF(WG). Predictive ability had a slight to moderate increase accumulating up to four generations of phenotypes. 1Me animals provide accurate IP, equivalent to GEBV. A minimum decay in predictive ability is observed when IP are derived from GEBV of animals from four generations back, possibly because of strong selection or the model not being completely additive., (© The Author(s) 2023. Published by Oxford University Press on behalf of the American Society of Animal Science.)

Background: Although single-step GBLUP (ssGBLUP) is an animal model, SNP effects can be backsolved from genomic estimated breeding values (GEBV). Predicted SNP effects allow to compute indirect prediction (IP) per individual as the sum of the SNP effects multiplied by its gene content, which is helpful when the number of genotyped animals is large, for genotyped animals not in the official evaluations, and when interim evaluations are needed. Typically, IP are obtained for new batches of genotyped individuals, all of them young and without phenotypes. Individual (theoretical) accuracies for IP are rarely reported, but they are nevertheless of interest. Our first objective was to present equations to compute individual accuracy of IP, based on prediction error covariance (PEC) of SNP effects, and in turn, are obtained from PEC of GEBV in ssGBLUP. The second objective was to test the algorithm for proven and young (APY) in PEC computations. With large datasets, it is impossible to handle the full PEC matrix, thus the third objective was to examine the minimum number of genotyped animals needed in PEC computations to achieve IP accuracies that are equivalent to GEBV accuracies., Results: Correlations between GEBV and IP for the validation animals using SNP effects from ssGBLUP evaluations were ≥ 0.99. When all available genotyped animals were used for PEC computations, correlations between GEBV and IP accuracy were ≥ 0.99. In addition, IP accuracies were compatible with GEBV accuracies either with direct inversion of the genomic relationship matrix (G) or using the algorithm for proven and young (APY) to obtain the inverse of G. As the number of genotyped animals included in the PEC computations decreased from around 55,000 to 15,000, correlations were still ≥ 0.96, but IP accuracies were biased downwards., Conclusions: Theoretical accuracy of indirect prediction can be successfully obtained by computing SNP PEC out of GEBV PEC from ssGBLUP equations using direct or APY G inverse. It is possible to reduce the number of genotyped animals in PEC computations, but accuracies may be underestimated. Further research is needed to approximate SNP PEC from ssGBLUP to limit the computational requirements with many genotyped animals., (© 2022. The Author(s).)

Background: Single-step genomic predictions obtained from a breeding value model require calculating the inverse of the genomic relationship matrix [Formula: see text]. The Algorithm for Proven and Young (APY) creates a sparse representation of [Formula: see text] with a low computational cost. APY consists of selecting a group of core animals and expressing the breeding values of the remaining animals as a linear combination of those from the core animals plus an error term. The objectives of this study were to: (1) extend APY to marker effects models; (2) derive equations for marker effect estimates when APY is used for breeding value models, and (3) show the implication of selecting a specific group of core animals in terms of a marker effects model., Results: We derived a family of marker effects models called APY-SNP-BLUP. It differs from the classic marker effects model in that the row space of the genotype matrix is reduced and an error term is fitted for non-core animals. We derived formulas for marker effect estimates that take this error term in account. The prediction error variance (PEV) of the marker effect estimates depends on the PEV for core animals but not directly on the PEV of the non-core animals. We extended the APY-SNP-BLUP to include a residual polygenic effect and accommodate non-genotyped animals. We show that selecting a specific group of core animals is equivalent to select a subspace of the row space of the genotype matrix. As the number of core animals increases, subspaces corresponding to different sets of core animals tend to overlap, showing that random selection of core animals is algebraically justified., Conclusions: The APY-(ss)GBLUP models can be expressed in terms of marker effect models. When the number of core animals is equal to the rank of the genotype matrix, APY-SNP-BLUP is identical to the classic marker effects model. If the number of core animals is less than the rank of the genotype matrix, genotypes for non-core animals are imputed as a linear combination of the genotypes of the core animals. For estimating SNP effects, only relationships and estimated breeding values for core animals are needed., (© 2022. The Author(s).)

Official multibreed genomic evaluations for dairy cattle in the United States are based on multibreed BLUP evaluation followed by single-breed estimation of SNP effects. Single-step genomic BLUP (ssGBLUP) allows the straight computation of genomic (G)EBV in a multibreed context. This work aimed to develop ssGBLUP multibreed genomic predictions for US dairy cattle using the algorithm for proven and young (APY) to compute the inverse of the genomic relationship matrix. Only purebred Ayrshire (AY), Brown Swiss (BS), Guernsey (GU), Holstein (HO), and Jersey (JE) animals were considered. A 3-trait model with milk (MY), fat (FY), and protein (PY) yields was applied using about 45 million phenotypes recorded from January 2000 to June 2020. The whole data set included about 29.5 million animals, of which almost 4 million were genotyped. All the effects in the model were breed specific, and breed was also considered as fixed unknown parent groups. Evaluations were done for (1) each single breed separately (single); (2) HO and JE together (HO_JE); (3) AY, BS, and GU together (AY_BS_GU); (4) all the 5 breeds together (5_BREEDS). Initially, 15k core animals were used in APY for AY_BS_GU and 5_BREEDS, but larger core sets with more animals from the least represented breeds were also tested. The HO_JE evaluation had a fixed set of 30k core animals, with an equal representation of the 2 breeds, whereas HO and JE single-breed analysis involved 15k core animals. Validation for cows was based on correlations between adjusted phenotypes and (G)EBV, whereas for bulls on the regression of daughter yield deviations on (G)EBV. Because breed was correctly considered in the model, BLUP results for single and multibreed analyses were the same. Under ssGBLUP, predictability and reliability for AY, BS, and GU were on average 7% and 2% lower in 5_BREEDS compared with single-breed evaluations, respectively. However, validation parameters for these 3 breeds became better than in the single-breed evaluations when 45k animals were included in the core set for 5_BREEDS. Evaluations for Holsteins were more stable across scenarios because of the greatest number of genotyped animals and amount of data. Combining AY, BS, and GU into one evaluation resulted in predictions similar to the ones from single breed, especially when using about 30k core animals in APY. The results showed that single-step large-scale multibreed evaluations are computationally feasible, but fine tuning is needed to avoid a reduction in reliability when numerically dominant breeds are combined. Having evaluations for AY, BS, and GU separated from HO and JE may reduce inflation of GEBV for the first 3 breeds., (© 2022, The Authors. Published by Elsevier Inc. and Fass Inc. on behalf of the American Dairy Science Association®. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).)

Background: The algorithm for proven and young (APY) has been suggested as a solution for recursively computing a sparse representation for the inverse of a large genomic relationship matrix (G). In APY, a subset of genotyped individuals is used as the core and the remaining genotyped individuals are used as noncore. Size and definition of the core are relevant research subjects for the application of APY, especially given the ever-increasing number of genotyped individuals., Methods: The aim of this study was to investigate several core definitions, including the most popular animals (MPA) (i.e., animals with high contributions to the genetic pool), the least popular males (LPM), the least popular females (LPF), a random set (Rnd), animals evenly distributed across genealogical paths (Ped), unrelated individuals (Unrel), or based on within-family selection (Fam), or on decomposition of the gene content matrix (QR). Each definition was evaluated for six core sizes based on prediction accuracy of single-step genomic best linear unbiased prediction (ssGBLUP) with APY. Prediction accuracy of ssGBLUP with the full inverse of G was used as the baseline. The dataset consisted of 357k pedigreed Duroc pigs with 111k pigs with genotypes and ~ 220k phenotypic records., Results: When the core size was equal to the number of largest eigenvalues explaining 50% of the variation of G (n = 160), MPA and Ped core definitions delivered the highest average prediction accuracies (~ 0.41-0.53). As the core size increased to the number of eigenvalues explaining 99% of the variation in G (n = 7320), prediction accuracy was nearly identical for all core types and correlations with genomic estimated breeding values (GEBV) from ssGBLUP with the full inversion of G were greater than 0.99 for all core definitions. Cores that represent all generations, such as Rnd, Ped, Fam, and Unrel, were grouped together in the hierarchical clustering of GEBV., Conclusions: For small core sizes, the definition of the core matters; however, as the size of the core reaches an optimal value equal to the number of largest eigenvalues explaining 99% of the variation of G, the definition of the core becomes arbitrary., (© 2022. The Author(s).)

Efficient computing techniques allow the estimation of variance components for virtually any traditional dataset. When genomic information is available, variance components can be estimated using genomic REML (GREML). If only a portion of the animals have genotypes, single-step GREML (ssGREML) is the method of choice. The genomic relationship matrix (G) used in both cases is dense, limiting computations depending on the number of genotyped animals. The algorithm for proven and young (APY) can be used to create a sparse inverse of G (GAPY~-1) with close to linear memory and computing requirements. In ssGREML, the inverse of the realized relationship matrix (H-1) also includes the inverse of the pedigree relationship matrix, which can be dense with a long pedigree, but sparser with short. The main purpose of this study was to investigate whether costs of ssGREML can be reduced using APY with truncated pedigree and phenotypes. We also investigated the impact of truncation on variance components estimation when different numbers of core animals are used in APY. Simulations included 150K animals from 10 generations, with selection. Phenotypes (h2 = 0.3) were available for all animals in generations 1-9. A total of 30K animals in generations 8 and 9, and 15K validation animals in generation 10 were genotyped for 52,890 SNP. Average information REML and ssGREML with G-1 and GAPY~-1 using 1K, 5K, 9K, and 14K core animals were compared. Variance components are impacted when the core group in APY represents the number of eigenvalues explaining a small fraction of the total variation in G. The most time-consuming operation was the inversion of G, with more than 50% of the total time. Next, numerical factorization consumed nearly 30% of the total computing time. On average, a 7% decrease in the computing time for ordering was observed by removing each generation of data. APY can be successfully applied to create the inverse of the genomic relationship matrix used in ssGREML for estimating variance components. To ensure reliable variance component estimation, it is important to use a core size that corresponds to the number of largest eigenvalues explaining around 98% of total variation in G. When APY is used, pedigrees can be truncated to increase the sparsity of H and slightly reduce computing time for ordering and symbolic factorization, with no impact on the estimates., (© The Author(s) 2022. Published by Oxford University Press on behalf of the American Society of Animal Science.)

Genomic prediction has become the new standard for genetic improvement programs, and currently, there is a desire to implement this technology for the evaluation of Angus cattle in Brazil. Thus, the main objective of this study was to assess the feasibility of evaluating young Brazilian Angus (BA) bulls and heifers for 12 routinely recorded traits using single-step genomic BLUP (ssGBLUP) with and without genotypes from American Angus (AA) sires. The second objective was to obtain estimates of effective population size (Ne) and linkage disequilibrium (LD) in the Brazilian Angus population. The dataset contained phenotypic information for up to 277,661 animals belonging to the Promebo breeding program, pedigree for 362,900, of which 1,386 were genotyped for 50k, 77k, and 150k single nucleotide polymorphism (SNP) panels. After imputation and quality control, 61,666 SNPs were available for the analyses. In addition, genotypes from 332 American Angus (AA) sires widely used in Brazil were retrieved from the AA Association database to be used for genomic predictions. Bivariate animal models were used to estimate variance components, traditional EBV, and genomic EBV (GEBV). Validation was carried out with the linear regression method (LR) using young-genotyped animals born between 2013 and 2015 without phenotypes in the reduced dataset and with records in the complete dataset. Validation animals were further split into progeny of BA and AA sires to evaluate if their progenies would benefit by including genotypes from AA sires. The Ne was 254 based on pedigree and 197 based on LD, and the average LD (±SD) and distance between adjacent single nucleotide polymorphisms (SNPs) across all chromosomes were 0.27 (±0.27) and 40743.68 bp, respectively. Prediction accuracies with ssGBLUP outperformed BLUP for all traits, improving accuracies by, on average, 16% for BA young bulls and heifers. The GEBV prediction accuracies ranged from 0.37 (total maternal for weaning weight and tick count) to 0.54 (yearling precocity) across all traits, and dispersion (LR coefficients) fluctuated between 0.92 and 1.06. Inclusion of genotyped sires from the AA improved GEBV accuracies by 2%, on average, compared to using only the BA reference population. Our study indicated that genomic information could help us to improve GEBV accuracies and hence genetic progress in the Brazilian Angus population. The inclusion of genotypes from American Angus sires heavily used in Brazil just marginally increased the GEBV accuracies for selection candidates., (© The Author(s) 2022. Published by Oxford University Press on behalf of the American Society of Animal Science.)

Single-step genomic BLUP (ssGBLUP) is a method for genomic prediction that integrates matrices of pedigree (A) and genomic (G) relationships into a single unified additive relationship matrix whose inverse is incorporated into a set of mixed model equations (MME) to compute genomic predictions. Pedigree information in dairy cattle is often incomplete. Missing pedigree potentially causes biases and inflation in genomic estimated breeding values (GEBV) obtained with ssGBLUP. Three major issues are associated with missing pedigree in ssGBLUP, namely biased predictions by selection, missing inbreeding in pedigree relationships, and incompatibility between G and A in level and scale. These issues can be solved using a proper model for unknown-parent groups (UPG). The theory behind the use of UPG is well established for pedigree BLUP, but not for ssGBLUP. This study reviews the development of the UPG model in pedigree BLUP, the properties of UPG models in ssGBLUP, and the effect of UPG on genetic trends and genomic predictions. Similarities and differences between UPG and metafounder (MF) models, a generalized UPG model, are also reviewed. A UPG model (QP) derived using a transformation of the MME has a good convergence behavior. However, with insufficient data, the QP model may yield biased genetic trends and may underestimate UPG. The QP model can be altered by removing the genomic relationships linking GEBV and UPG effects from MME. This altered QP model exhibits less bias in genetic trends and less inflation in genomic predictions than the QP model, especially with large data sets. Recently, a new model, which encapsulates the UPG equations into the pedigree relationships for genotyped animals, was proposed in simulated purebred populations. The MF model is a comprehensive solution to the missing pedigree issue. This model can be a choice for multibreed or crossbred evaluations if the data set allows the estimation of a reasonable relationship matrix for MF. Missing pedigree influences genetic trends, but its effect on the predictability of genetic merit for genotyped animals should be negligible when many proven bulls are genotyped. The SNP effects can be back-solved using GEBV from older genotyped animals, and these predicted SNP effects can be used to calculate GEBV for young-genotyped animals with missing parents., (The Authors. Published by Elsevier Inc. and Fass Inc. on behalf of the American Dairy Science Association®. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).)

The objectives of this study were to develop an efficient algorithm for calculating prediction error variances (PEVs) for genomic best linear unbiased prediction (GBLUP) models using the Algorithm for Proven and Young (APY), extend it to single-step GBLUP (ssGBLUP), and apply this algorithm for approximating the theoretical reliabilities for single- and multiple-trait models in ssGBLUP. The PEV with APY was calculated by block sparse inversion, efficiently exploiting the sparse structure of the inverse of the genomic relationship matrix with APY. Single-step GBLUP reliabilities were approximated by combining reliabilities with and without genomic information in terms of effective record contributions. Multi-trait reliabilities relied on single-trait results adjusted using the genetic and residual covariance matrices among traits. Tests involved two datasets provided by the American Angus Association. A small dataset (Data1) was used for comparing the approximated reliabilities with the reliabilities obtained by the inversion of the left-hand side of the mixed model equations. A large dataset (Data2) was used for evaluating the computational performance of the algorithm. Analyses with both datasets used single-trait and three-trait models. The number of animals in the pedigree ranged from 167,951 in Data1 to 10,213,401 in Data2, with 50,000 and 20,000 genotyped animals for single-trait and multiple-trait analysis, respectively, in Data1 and 335,325 in Data2. Correlations between estimated and exact reliabilities obtained by inversion ranged from 0.97 to 0.99, whereas the intercept and slope of the regression of the exact on the approximated reliabilities ranged from 0.00 to 0.04 and from 0.93 to 1.05, respectively. For the three-trait model with the largest dataset (Data2), the elapsed time for the reliability estimation was 11 min. The computational complexity of the proposed algorithm increased linearly with the number of genotyped animals and with the number of traits in the model. This algorithm can efficiently approximate the theoretical reliability of genomic estimated breeding values in ssGBLUP with APY for large numbers of genotyped animals at a low cost., (© The Author(s) 2021. Published by Oxford University Press on behalf of the American Society of Animal Science.)

Background: Understanding whether genomic selection has been effective in livestock and when the results of genomic selection became visible are essential questions which we have addressed in this paper. Three criteria were used to identify practices of breeding programs over time: (1) the point of divergence of estimated genetic trends based on pedigree-based best linear unbiased prediction (BLUP) versus single-step genomic BLUP (ssGBLUP), (2) the point of divergence of realized Mendelian sampling (RMS) trends based on BLUP and ssGBLUP, and (3) the partition of genetic trends into that contributed by genotyped and non-genotyped individuals and by males and females., Methods: We used data on 282,035 animals from a commercial maternal line of pigs, of which 32,856 were genotyped for 36,612 single nucleotide polymorphisms (SNPs) after quality control. Phenotypic data included 228,427, 101,225, and 11,444 records for birth weight, average daily gain in the nursery, and feed intake, respectively. Breeding values were predicted in a multiple-trait framework using BLUP and ssGBLUP., Results: The points of divergence of the genetic and RMS trends estimated by BLUP and ssGBLUP indicated that genomic selection effectively started in 2019. Partitioning the overall genetic trends into that for genotyped and non-genotyped individuals revealed that the contribution of genotyped animals to the overall genetic trend increased rapidly from ~ 74% in 2016 to 90% in 2019. The contribution of the female pathway to the genetic trend also increased since genomic selection was implemented in this pig population, which reflects the changes in the genotyping strategy in recent years., Conclusions: Our results show that an assessment of breeding program practices can be done based on the point of divergence of genetic and RMS trends between BLUP and ssGBLUP and based on the partitioning of the genetic trend into contributions from different selection pathways. However, it should be noted that genetic trends can diverge before the onset of genomic selection if superior animals are genotyped retroactively. For the pig population example, the results showed that genomic selection was effective in this population., (© 2021. The Author(s).)

Accuracy of genomic predictions is an important component of the selection response. The objectives of this research were: 1) to investigate trends for prediction accuracies over time in a broiler population of accumulated phenotypes, genotypes, and pedigrees and 2) to test if data from distant generations are useful to maintain prediction accuracies in selection candidates. The data contained 820K phenotypes for a growth trait (GT), 200K for two feed efficiency traits (FE1 and FE2), and 42K for a carcass yield trait (CY). The pedigree included 1,252,619 birds hatched over 7 years, of which 154,318 from the last 4 years were genotyped. Training populations were constructed adding 1 year of data sequentially, persistency of accuracy over time was evaluated using predictions from birds hatched in the three generations following or in the years after the training populations. In the first generation, before genotypes became available for the training populations (first 3 years of data), accuracies remained almost stable with successive additions of phenotypes and pedigree to the accumulated dataset. The inclusion of 1 year of genotypes in addition to 4 years of phenotypes and pedigree in the training population led to increases in accuracy of 54% for GT, 76% for FE1, 110% for CY, and 38% for FE2; on average, 74% of the increase was due to genomics. Prediction accuracies declined faster without than with genomic information in the training populations. When genotypes were unavailable, the average decline in prediction accuracy across traits was 41% from the first to the second generation of validation, and 51% from the second to the third generation of validation. When genotypes were available, the average decline across traits was 14% from the first to the second generation of validation, and 3% from the second to the third generation of validation. Prediction accuracies in the last three generations were the same when the training population included 5 or 2 years of data, and a decrease of ~7% was observed when the training population included only 1 year of data. Training sets including genomic information provided an increase in accuracy and persistence of genomic predictions compared with training sets without genomic data. The two most recent years of pedigree, phenotypic, and genomic data were sufficient to maintain prediction accuracies in selection candidates. Similar conclusions were obtained using validation populations per year., (© The Author(s) 2021. Published by Oxford University Press on behalf of the American Society of Animal Science.)

Genomic selection has been adopted nationally and internationally in different livestock and plant species. However, understanding whether genomic selection has been effective or not is an essential question for both industry and academia. Once genomic evaluation started being used, estimation of breeding values with pedigree best linear unbiased prediction (BLUP) became biased because this method does not consider selection using genomic information. Hence, the effective starting point of genomic selection can be detected in two possible ways including the divergence of genetic trends and Realized Mendelian sampling (RMS) trends obtained with BLUP and single-step genomic BLUP (ssGBLUP). This study aimed to find the start date of genomic selection for a set of economically important traits in three livestock species by comparing trends obtained using BLUP and ssGBLUP. Three datasets were used for this purpose: 1) a pig dataset with 117k genotypes and 1.3M animals in pedigree, 2) an Angus cattle dataset consisted of ~842k genotypes and 11.5M animals in pedigree, and 3) a purebred broiler chicken dataset included ~154k genotypes and 1.3M birds in pedigree were used. The genetic trends for pigs diverged for the genotyped animals born in 2014 for average daily gain (ADG) and backfat (BF). In beef cattle, the trends started diverging in 2009 for weaning weight (WW) and in 2016 for postweaning gain (PWG), with little divergence for birth weight (BTW). In broiler chickens, the genetic trends estimated by ssGBLUP and BLUP diverged at breeding cycle 6 for two out of the three production traits. The RMS trends for the genotyped pigs diverged for animals born in 2014, more for ADG than for BF. In beef cattle, the RMS trends started diverging in 2009 for WW and in 2016 for PWG, with a trivial trend for BTW. In broiler chickens, the RMS trends from ssGBLUP and BLUP diverged strongly for two production traits at breeding cycle 6, with a slight divergence for another trait. Divergence of the genetic trends from ssGBLUP and BLUP indicates the onset of the genomic selection. The presence of trends for RMS indicates selective genotyping, with or without the genomic selection. The onset of genomic selection and genotyping strategies agrees with industry practices across the three species. In summary, the effective start of genomic selection can be detected by the divergence between genetic and RMS trends from BLUP and ssGBLUP., (© The Author(s) 2021. Published by Oxford University Press on behalf of the American Society of Animal Science.)

It is of interest to evaluate crossbred pigs for hot carcass weight (HCW) and birth weight (BW); however, obtaining a HCW record is dependent on livability (LIV) and retained tag (RT). The purpose of this study is to analyze how HCW evaluations are affected when herd removal and missing identification are included in the model and examine if accounting for the reasons for missing traits improves the accuracy of predicting breeding values. Pedigree information was available for 1,965,077 purebred and crossbred animals. Records for 503,716 commercial three-way crossbred terminal animals from 2014 to 2019 were provided by Smithfield Premium Genetics. Two pedigree-based models were compared; model 1 (M1) was a threshold-linear model with all four traits (BW, HCW, RT, and LIV), and model 2 (M2) was a linear model including only BW and HCW. The fixed effects used in the model were contemporary group, sex, age at harvest (for HCW only), and dam parity. The random effects included direct additive genetic and random litter effects. Accuracy, dispersion, bias, and Pearson correlations were estimated using the linear regression method. The heritabilities were 0.11, 0.07, 0.02, and 0.04 for BW, HCW, RT, and LIV, respectively, with standard errors less than 0.01. No difference was observed in heritabilities or accuracies for BW and HCW between M1 and M2. Accuracies were 0.33, 0.37, 0.19, and 0.23 for BW, HCW, RT, and LIV, respectively. The genetic correlation between BW and RT was 0.34 ± 0.03, and between BW and LIV was 0.56 ± 0.03. Similarly, the genetic correlation between HCW and RT was 0.26 ± 0.04, and between HCW and LIV was 0.09 ± 0.05, respectively. The positive and moderate genetic correlations between BW and other traits imply a heavier BW resulted in a higher probability of surviving to harvest. Genetic correlations between HCW and other traits were lower due to the large quantity of missing records. Despite the heritable and correlated aspects of RT and LIV, results imply no major differences between M1 and M2; hence, it is unnecessary to include these traits in classical models for BW and HCW., (© The Author(s) 2021. Published by Oxford University Press on behalf of the American Society of Animal Science.)

Genomic selection (GS) is now practiced successfully across many species. However, many questions remain, such as long-term effects, estimations of genomic parameters, robustness of genome-wide association study (GWAS) with small and large datasets, and stability of genomic predictions. This study summarizes presentations from the authors at the 2020 American Society of Animal Science (ASAS) symposium. The focus of many studies until now is on linkage disequilibrium between two loci. Ignoring higher-level equilibrium may lead to phantom dominance and epistasis. The Bulmer effect leads to a reduction of the additive variance; however, the selection for increased recombination rate can release anew genetic variance. With genomic information, estimates of genetic parameters may be biased by genomic preselection, but costs of estimation can increase drastically due to the dense form of the genomic information. To make the computation of estimates feasible, genotypes could be retained only for the most important animals, and methods of estimation should use algorithms that can recognize dense blocks in sparse matrices. GWASs using small genomic datasets frequently find many marker-trait associations, whereas studies using much bigger datasets find only a few. Most of the current tools use very simple models for GWAS, possibly causing artifacts. These models are adequate for large datasets where pseudo-phenotypes such as deregressed proofs indirectly account for important effects for traits of interest. Artifacts arising in GWAS with small datasets can be minimized by using data from all animals (whether genotyped or not), realistic models, and methods that account for population structure. Recent developments permit the computation of P-values from genomic best linear unbiased prediction (GBLUP), where models can be arbitrarily complex but restricted to genotyped animals only, and single-step GBLUP that also uses phenotypes from ungenotyped animals. Stability was an important part of nongenomic evaluations, where genetic predictions were stable in the absence of new data even with low prediction accuracies. Unfortunately, genomic evaluations for such animals change because all animals with genotypes are connected. A top-ranked animal can easily drop in the next evaluation, causing a crisis of confidence in genomic evaluations. While correlations between consecutive genomic evaluations are high, outliers can have differences as high as 1 SD. A solution to fluctuating genomic evaluations is to base selection decisions on groups of animals. Although many issues in GS have been solved, many new issues that require additional research continue to surface., (© The Author(s) 2021. Published by Oxford University Press on behalf of the American Society of Animal Science.)

The objective of this study was to assess the reliability and bias of estimated breeding values (EBV) from traditional BLUP with unknown parent groups (UPG), genomic EBV (GEBV) from single-step genomic BLUP (ssGBLUP) with UPG for the pedigree relationship matrix (A) only (SS_UPG), and GEBV from ssGBLUP with UPG for both A and the relationship matrix among genotyped animals (A 22 ; SS_UPG2) using 6 large phenotype-pedigree truncated Holstein data sets. The complete data included 80 million records for milk, fat, and protein yields from 31 million cows recorded since 1980. Phenotype-pedigree truncation scenarios included truncation of phenotypes for cows recorded before 1990 and 2000 combined with truncation of pedigree information after 2 or 3 ancestral generations. A total of 861,525 genotyped bulls with progeny and cows with phenotypic records were used in the analyses. Reliability and bias (inflation/deflation) of GEBV were obtained for 2,710 bulls based on deregressed proofs, and on 381,779 cows born after 2014 based on predictivity (adjusted cow phenotypes). The BLUP reliabilities for young bulls varied from 0.29 to 0.30 across traits and were unaffected by data truncation and number of generations in the pedigree. Reliabilities ranged from 0.54 to 0.69 for SS_UPG and were slightly affected by phenotype-pedigree truncation. Reliabilities ranged from 0.69 to 0.73 for SS_UPG2 and were unaffected by phenotype-pedigree truncation. The regression coefficient of bull deregressed proofs on (G)EBV (i.e., GEBV and EBV) ranged from 0.86 to 0.90 for BLUP, from 0.77 to 0.94 for SS_UPG, and was 1.00 ± 0.03 for SS_UPG2. Cow predictivity ranged from 0.22 to 0.28 for BLUP, 0.48 to 0.51 for SS_UPG, and 0.51 to 0.54 for SS_UPG2. The highest cow predictivities for BLUP were obtained with the most extreme truncation, whereas for SS_UPG2, cow predictivities were also unaffected by phenotype-pedigree truncations. The regression coefficient of cow predictivities on (G)EBV was 1.02 ± 0.02 for SS_UPG2 with the most extreme truncation, which indicated the least biased predictions. Computations with the complete data set took 17 h with BLUP, 58 h with SS_UPG, and 23 h with SS_UPG2. The same computations with the most extreme phenotype-pedigree truncation took 7, 36, and 15 h, respectively. The SS_UPG2 converged in fewer rounds than BLUP, whereas SS_UPG took up to twice as many rounds. Thus, the ssGBLUP with UPG assigned to both A and A 22 provided accurate and unbiased evaluations, regardless of phenotype-pedigree truncation scenario. Old phenotypes (before 2000 in this data set) did not affect the reliability of predictions for young selection candidates, especially in SS_UPG2., (The Authors. Published by Elsevier Inc. and Fass Inc. on behalf of the American Dairy Science Association®. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).)

The introduction of animals from a different environment or population is a common practice in commercial livestock populations. In this study, we modeled the inclusion of a group of external birds into a local broiler chicken population for the purpose of genomic evaluations. The pedigree was composed of 242,413 birds and genotypes were available for 107,216 birds. A five-trait model that included one growth, two yield, and two efficiency traits was used for the analyses. The strategies to model the introduction of external birds were to include a fixed effect representing the origin of parents and to use unknown parent groups (UPG) or metafounders (MF). Genomic estimated breeding values (GEBV) were obtained with single-step GBLUP using the Algorithm for Proven and Young. Bias, dispersion, and accuracy of GEBV for the validation birds, that is, from the most recent generation, were computed. The bias and dispersion were estimated with the linear regression (LR) method,whereas accuracy was estimated by the LR method and predictive ability. When fixed UPG were fit without estimated inbreeding, the model did not converge. In contrast, models with fixed UPG and estimated inbreeding or random UPG converged and resulted in similar GEBV. The inclusion of an extra fixed effect in the model made the GEBV unbiased and reduced the inflation. Genomic predictions with MF were slightly biased and inflated due to the unbalanced number of observations assigned to each metafounder. When combining local and external populations, the greatest accuracy can be obtained by adding an extra fixed effect to account for the origin of parents plus UPG with estimated inbreeding or random UPG. To estimate the accuracy, the LR method is more consistent among scenarios, whereas the predictive ability greatly depends on the model specification., (© The Author(s) 2021. Published by Oxford University Press on behalf of the American Society of Animal Science.)

Genomic information has a limited dimensionality (number of independent chromosome segments [Me]) related to the effective population size. Under the additive model, the persistence of genomic accuracies over generations should be high when the nongenomic information (pedigree and phenotypes) is equivalent to Me animals with high accuracy. The objective of this study was to evaluate the decay in accuracy over time and to compare the magnitude of decay with varying quantities of data and with traits of low and moderate heritability. The dataset included 161,897 phenotypic records for a growth trait (GT) and 27,669 phenotypic records for a fitness trait (FT) related to prolificacy in a population with dimensionality around 5,000. The pedigree included 404,979 animals from 2008 to 2020, of which 55,118 were genotyped. Two single-trait models were used with all ancestral data and sliding subsets of 3-, 2-, and 1-generation intervals. Single-step genomic best linear unbiased prediction (ssGBLUP) was used to compute genomic estimated breeding values (GEBV). Estimated accuracies were calculated by the linear regression (LR) method. The validation population consisted of single generations succeeding the training population and continued forward for all generations available. The average accuracy for the first generation after training with all ancestral data was 0.69 and 0.46 for GT and FT, respectively. The average decay in accuracy from the first generation after training to generation 9 was -0.13 and -0.19 for GT and FT, respectively. The persistence of accuracy improves with more data. Old data have a limited impact on the predictions for young animals for a trait with a large amount of information but a bigger impact for a trait with less information., (© The Author(s) 2021. Published by Oxford University Press on behalf of the American Society of Animal Science.)

The stability of genomic evaluations depends on the amount of data and population parameters. When the dataset is large enough to estimate the value of nearly all independent chromosome segments (~10K in American Angus cattle), the accuracy and persistency of breeding values will be high. The objective of this study was to investigate changes in estimated breeding values (EBV) and genomic EBV (GEBV) across monthly evaluations for 1 yr in a large genotyped population of beef cattle. The American Angus data used included 8.2 million records for birth weight, 8.9 for weaning weight, and 4.4 for postweaning gain. A total of 10.1 million animals born until December 2017 had pedigree information, and 484,074 were genotyped. A truncated dataset included animals born until December 2016. To mimic a scenario with monthly evaluations, 2017 data were added 1 mo at a time to estimate EBV using best linear unbiased prediction (BLUP) and GEBV using single-step genomic BLUP with the algorithm for proven and young (APY) with core group fixed for 1 yr or updated monthly. Predictions from monthly evaluations in 2017 were contrasted with the predictions of the evaluation in December 2016 or the previous month for all genotyped animals born until December 2016 with or without their own phenotypes or progeny phenotypes. Changes in EBV and GEBV were similar across traits, and only results for weaning weight are presented. Correlations between evaluations from December 2016 and the 12 consecutive evaluations were ≥0.97 for EBV and ≥0.99 for GEBV. Average absolute changes for EBV were about two times smaller than for GEBV, except for animals with new progeny phenotypes (≤0.12 and ≤0.11 additive genetic SD [SDa] for EBV and GEBV). The maximum absolute changes for EBV (≤2.95 SDa) were greater than for GEBV (≤1.59 SDa). The average(maximum) absolute GEBV changes for young animals from December 2016 to January and December 2017 ranged from 0.05(0.25) to 0.10(0.53) SDa. Corresponding ranges for animals with new progeny phenotypes were from 0.05(0.88) to 0.11(1.59) SDa for GEBV changes. The average absolute change in EBV(GEBV) from December 2016 to December 2017 for sires with ≤50 progeny phenotypes was 0.26(0.14) and for sires with >50 progeny phenotypes was 0.25(0.16) SDa. Updating the core group in APY without adding data created an average absolute change of 0.07 SDa in GEBV. Genomic evaluations in large genotyped populations are as stable and persistent as the traditional genetic evaluations, with less extreme changes., (© The Author(s) 2021. Published by Oxford University Press on behalf of the American Society of Animal Science.)

Pedigree information is often missing for some animals in a breeding program. Unknown-parent groups (UPGs) are assigned to the missing parents to avoid biased genetic evaluations. Although the use of UPGs is well established for the pedigree model, it is unclear how UPGs are integrated into the inverse of the unified relationship matrix (H-inverse) required for single-step genomic best linear unbiased prediction. A generalization of the UPG model is the metafounder (MF) model. The objectives of this study were to derive 3 H-inverses and to compare genetic trends among models with UPG and MF H-inverses using a simulated purebred population. All inverses were derived using the joint density function of the random breeding values and genetic groups. The breeding values of genotyped animals (u2) were assumed to be adjusted for UPG effects (g) using matrix Q2 as u2∗=u2+Q2g before incorporating genomic information. The Quaas-Pollak-transformed (QP) H-inverse was derived using a joint density function of u2∗ and g updated with genomic information and assuming nonzero cov(u2∗,g'). The modified QP (altered) H-inverse also assumes that the genomic information updates u2∗ and g, but cov(u2∗,g')=0. The UPG-encapsulated (EUPG) H-inverse assumed genomic information updates the distribution of u2∗. The EUPG H-inverse had the same structure as the MF H-inverse. Fifty percent of the genotyped females in the simulation had a missing dam, and missing parents were replaced with UPGs by generation. The simulation study indicated that u2∗ and g in models using the QP and altered H-inverses may be inseparable leading to potential biases in genetic trends. Models using the EUPG and MF H-inverses showed no genetic trend biases. These 2 H-inverses yielded the same genomic EBV (GEBV). The predictive ability and inflation of GEBVs from young genotyped animals were nearly identical among models using the QP, altered, EUPG, and MF H-inverses. Although the choice of H-inverse in real applications with enough data may not result in biased genetic trends, the EUPG and MF H-inverses are to be preferred because of theoretical justification and possibility to reduce biases., (© The Author(s) 2021. Published by Oxford University Press on behalf of the American Society of Animal Science. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com.)

Single-step genomic BLUP (ssGBLUP) requires compatibility between genomic and pedigree relationships for unbiased and accurate predictions. Scaling the genomic relationship matrix (G) to have the same averages as the pedigree relationship matrix (i.e., scaling by averages) is one way to ensure compatibility. This requires computing both relationship matrices, calculating averages, and changing G, whereas only the inverses of those matrices are needed in the mixed model equations. Therefore, the compatibility process can add extra computing burden. In the single-step Bayesian regression, the scaling is done by including a mean (μ g ) as a fixed effect in the model. The parameter μ g can be interpreted as the average of the breeding values of the genotyped animals. In this study, such scaling, called automatic, was implemented in ssGBLUP via Quaas-Pollak transformation of the inverse of the relationship matrix used in ssGBLUP (H), which combines the inverses of the pedigree and genomic relationship matrices. Comparisons involved a simulated data set, and the genomic relationship matrix was computed using different allele frequencies either from the current population (i.e., realized allele frequencies), equal among all the loci, or from the base population. For all of the scenarios, we computed bias [defined as the average difference between true breeding values (TBV) and genomic estimated breeding values (GEBV)], accuracy (defined as the correlation between TBV and GEBV), and dispersion (defined as the regression coefficient of GEBV on TBV). With no scaling, the bias expressed in terms of genetic standard deviations was 0.86, 0.64, and 0.58 with realized, equal, and base population allele frequencies, respectively. With scaling by averages, which is currently used in ssGBLUP, bias was 0.07, 0.08, and 0.03, respectively. With automatic scaling, bias was 0.18 regardless of allele frequencies. Accuracies were similar among scaling methods, but about 0.1 lower in the scenario without scaling. The GEBV were more inflated without any scaling, whereas the automatic scaling performed similarly to the scaling by averages. The average dispersion for those methods was 0.94. When μ g was treated as random, with the variance equal to differences between pedigree and genomic relationships, the bias was the same as with the scaling by averages. The automatic scaling is biased, especially when μ g is treated as a fixed effect. The bias may be small in real data with fewer generations, when traits are undergoing weak selection, or when the number of genotyped animals is large., (Copyright © 2021 American Dairy Science Association. Published by Elsevier Inc. All rights reserved.)

In the pig industry, purebred animals are raised in nucleus herds and selected to produce crossbred progeny to perform in commercial environments. Crossbred and purebred performances are different, correlated traits. All purebreds in a pen have their performance assessed together at the end of a performance test. However, only selected crossbreds are removed (based on visual inspection) and measured at different times creating many small contemporary groups (CGs). This may reduce estimated breeding value (EBV) prediction accuracies. Considering this sequential recording of crossbreds, the objective was to investigate the impact of different CG definitions on genetic parameters and EBV prediction accuracy for crossbred traits. Growth rate (GP) and ultrasound backfat (BFP) records were available for purebreds. Lifetime growth (GX) and backfat (BFX) were recorded on crossbreds. Different CGs were tested: CG_all included farm, sex, birth year, and birth week; CG_week added slaughter week; and CG_day used slaughter day instead of week. Data of 124,709 crossbreds were used. The purebred phenotypes (62,274 animals) included three generations of purebred ancestors of these crossbreds and their CG mates. Variance components for four-trait models with different CG definitions were estimated with average information restricted maximum likelihood. Purebred traits' variance components remained stable across CG definitions and varied slightly for BFX. Additive genetic variances (and heritabilities) for GX fluctuated more: 812 ± 36 (0.28 ± 0.01), 257 ± 15 (0.17 ± 0.01), and 204 ± 13 (0.15 ± 0.01) for CG_all, CG_week, and CG_day, respectively. Age at slaughter (AAS) and hot carcass weight (HCW) adjusted for age were investigated as alternatives for GX. Both have potential for selection but lower heritabilities compared with GX: 0.21 ± 0.01 (0.18 ± 0.01), 0.16 ± 0.02 (0.16 + 0.01), and 0.10 ± 0.01 (0.14 ± 0.01) for AAS (HCW) using CG_all, CG_week, and CG_day, respectively. The predictive ability, linear regression (LR) accuracy, bias, and dispersion of crossbred traits in crossbreds favored CG_day, but correlations with unadjusted phenotypes favored CG_all. In purebreds, CG_all showed the best LR accuracy, while showing small relative differences in bias and dispersion. Different CG scenarios showed no relevant impact on BFX EBV. This study shows that different CG definitions may affect evaluation stability and animal ranking. Results suggest that ignoring slaughter dates in CG is more appropriate for estimating crossbred trait EBV for purebred animals., (© The Author(s) 2020. Published by Oxford University Press on behalf of the American Society of Animal Science.)

Single-step genomic best linear unbiased prediction with the Algorithm for Proven and Young (APY) is a popular method for large-scale genomic evaluations. With the APY algorithm, animals are designated as core or noncore, and the computing resources to create the inverse of the genomic relationship matrix (GRM) are reduced by inverting only a portion of that matrix for core animals. However, using different core sets of the same size causes fluctuations in genomic estimated breeding values (GEBVs) up to one additive standard deviation without affecting prediction accuracy. About 2% of the variation in the GRM is noise. In the recursion formula for APY, the error term modeling the noise is different for every set of core animals, creating changes in breeding values. While average changes are small, and correlations between breeding values estimated with different core animals are close to 1.0, based on the normal distribution theory, outliers can be several times bigger than the average. Tests included commercial datasets from beef and dairy cattle and from pigs. Beyond a certain number of core animals, the prediction accuracy did not improve, but fluctuations decreased with more animals. Fluctuations were much smaller than the possible changes based on prediction error variance. GEBVs change over time even for animals with no new data as genomic relationships ties all the genotyped animals, causing reranking of top animals. In contrast, changes in nongenomic models without new data are small. Also, GEBV can change due to details in the model, such as redefinition of contemporary groups or unknown parent groups. In particular, increasing the fraction of blending of the GRM with a pedigree relationship matrix from 5% to 20% caused changes in GEBV up to 0.45 SD, with a correlation of GEBV > 0.99. Fluctuations in genomic predictions are part of genomic evaluation models and are also present without the APY algorithm when genomic evaluations are computed with updated data. The best approach to reduce the impact of fluctuations in genomic evaluations is to make selection decisions not on individual animals with limited individual accuracy but on groups of animals with high average accuracy., (© The Author(s) 2020. Published by Oxford University Press on behalf of the American Society of Animal Science.)

Reliable single-nucleotide polymorphisms (SNP) effects from genomic best linear unbiased prediction BLUP (GBLUP) and single-step GBLUP (ssGBLUP) are needed to calculate indirect predictions (IP) for young genotyped animals and animals not included in official evaluations. Obtaining reliable SNP effects and IP requires a minimum number of animals and when a large number of genotyped animals are available, the algorithm for proven and young (APY) may be needed. Thus, the objectives of this study were to evaluate IP with an increasingly larger number of genotyped animals and to determine the minimum number of animals needed to compute reliable SNP effects and IP. Genotypes and phenotypes for birth weight, weaning weight, and postweaning gain were provided by the American Angus Association. The number of animals with phenotypes was more than 3.8 million. Genotyped animals were assigned to three cumulative year-classes: born until 2013 (N = 114,937), born until 2014 (N = 183,847), and born until 2015 (N = 280,506). A three-trait model was fitted using the APY algorithm with 19,021 core animals under two scenarios: 1) core 2013 (random sample of animals born until 2013) used for all year-classes and 2) core 2014 (random sample of animals born until 2014) used for year-class 2014 and core 2015 (random sample of animals born until 2015) used for year-class 2015. GBLUP used phenotypes from genotyped animals only, whereas ssGBLUP used all available phenotypes. SNP effects were predicted using genomic estimated breeding values (GEBV) from either all genotyped animals or only core animals. The correlations between GEBV from GBLUP and IP obtained using SNP effects from core 2013 were ≥0.99 for animals born in 2013 but as low as 0.07 for animals born in 2014 and 2015. Conversely, the correlations between GEBV from ssGBLUP and IP were ≥0.99 for animals born in all years. IP predictive abilities computed with GEBV from ssGBLUP and SNP predictions based on only core animals were as high as those based on all genotyped animals. The correlations between GEBV and IP from ssGBLUP were ≥0.76, ≥0.90, and ≥0.98 when SNP effects were computed using 2k, 5k, and 15k core animals. Suitable IP based on GEBV from GBLUP can be obtained when SNP predictions are based on an appropriate number of core animals, but a considerable decline in IP accuracy can occur in subsequent years. Conversely, IP from ssGBLUP based on large numbers of phenotypes from non-genotyped animals have persistent accuracy over time., (© The Author(s) 2020. Published by Oxford University Press on behalf of the American Society of Animal Science.)

Early application of genomic selection relied on SNP estimation with phenotypes or de-regressed proofs (DRP). Chips of 50k SNP seemed sufficient for an accurate estimation of SNP effects. Genomic estimated breeding values (GEBV) were composed of an index with parent average, direct genomic value, and deduction of a parental index to eliminate double counting. Use of SNP selection or weighting increased accuracy with small data sets but had minimal to no impact with large data sets. Efforts to include potentially causative SNP derived from sequence data or high-density chips showed limited or no gain in accuracy. After the implementation of genomic selection, EBV by BLUP became biased because of genomic preselection and DRP computed based on EBV required adjustments, and the creation of DRP for females is hard and subject to double counting. Genomic selection was greatly simplified by single-step genomic BLUP (ssGBLUP). This method based on combining genomic and pedigree relationships automatically creates an index with all sources of information, can use any combination of male and female genotypes, and accounts for preselection. To avoid biases, especially under strong selection, ssGBLUP requires that pedigree and genomic relationships are compatible. Because the inversion of the genomic relationship matrix (G) becomes costly with more than 100k genotyped animals, large data computations in ssGBLUP were solved by exploiting limited dimensionality of genomic data due to limited effective population size. With such dimensionality ranging from 4k in chickens to about 15k in cattle, the inverse of G can be created directly (e.g., by the algorithm for proven and young) at a linear cost. Due to its simplicity and accuracy, ssGBLUP is routinely used for genomic selection by the major chicken, pig, and beef industries. Single step can be used to derive SNP effects for indirect prediction and for genome-wide association studies, including computations of the P-values. Alternative single-step formulations exist that use SNP effects for genotyped or for all animals. Although genomics is the new standard in breeding and genetics, there are still some problems that need to be solved. This involves new validation procedures that are unaffected by selection, parameter estimation that accounts for all the genomic data used in selection, and strategies to address reduction in genetic variances after genomic selection was implemented., (© The Author(s) 2020. Published by Oxford University Press on behalf of the American Society of Animal Science.)

Causal variants inferred from sequence data analysis are expected to increase accuracy of genomic selection. In this work we evaluated the gain in reliability of genomic predictions, for stature in US Holsteins, when adding selected sequence variants to a pre-existent SNP chip. Two prediction methods were tested: de-regressed proofs assuming heterogeneous (genomic BLUP; GBLUP) residual variances and by single-step GBLUP (ssGBLUP) using actual phenotypes. Phenotypic data included 3,999,631 records for stature on 3,027,304 Holstein cows. Genotypes on 54,087 SNP markers (54k) were available for 26,877 bulls. Additionally, 16,648 selected sequence variants were combined with the 54k markers, for a total of 70,735 (70k) markers. In all methods, SNP in the genomic relationship matrix (G) were unweighted or weighted iteratively, with weights derived either by SNP effects squared or by a nonlinear method that resembles BayesA (nonlinear A). Reliability of genomic predictions were obtained by cross validation. With unweighted G derived from 54k markers, the reliabilities (× 100) were 72.4 for GBLUP and 75.3 for ssGBLUP. With unweighted G derived from 70k markers, the reliabilities were 73.4 and 76.0, respectively. Weighting by nonlinear A changed reliabilities to 73.3, and 75.9, respectively. Addition of selected sequence variants had a small effect on reliabilities. Weighting by quadratic functions reduced reliabilities. Weighting by nonlinear A increased reliabilities for GBLUP but had only a small effect in ssGBLUP. Reliabilities for direct genomic values extracted from ssGBLUP using unweighted G with 54k were higher than reliabilities by any GBLUP. Thus, ssGBLUP seems to capture more information than GBLUP and there is less room for extra reliability. Improvements in GBLUP may be because the weights in G change the covariance structure, which can explain a proportion of the variance that is accounted for when a heterogeneous residual variance is assumed by considering a different number of daughters per bull., (Copyright © 2019 American Dairy Science Association. Published by Elsevier Inc. All rights reserved.)

The objective of this study was to compare genetic trends from single-step genomic BLUP (ssGBLUP) and traditional BLUP models for milk production traits of US Holsteins. Phenotypes were 305-d milk, fat, and protein yields from 21,527,040 cows recorded between January 1990 and August 2015. The pedigree file included 29,651,623 animals and was limited to 3 generations back from recorded or genotyped animals. Genotypes for 764,029 animals were used, and analyses were by a 3-trait repeatability model as used in the US official genetic evaluation. Unknown-parent groups were incorporated into the inverse of a relationship matrix (H -1 in ssGBLUP and A -1 in BLUP) with the QP transformation. For ssGBLUP, 18,359 genotyped animals were randomly chosen as core animals to calculate the inverse of the genomic relationship matrix with the APY algorithm. Computations took 6.5 h and 1.4 GB of memory for BLUP, and 13 h and 115 GB of memory for ssGBLUP. For genotyped sires with at least 10 daughters, the average genetic levels for predicted transmitting ability (PTA) and genomic PTA were similar up to 2008, with a higher level for ssGBLUP later (approximately by 36 kg for milk, 2.1 kg for fat, and 1.1 kg for protein for bulls born in 2010). For genotyped cows, the average genetic levels were similar up to 2006, with a higher level for ssGBLUP (approximately by 91 kg for milk, 3.6 kg for fat, and 2.7 kg for protein for cows born in 2012). For all cows, the average levels were slightly higher for ssGBLUP, with much smaller differences than for genotyped cows. Trends for BLUP indicate bias due to genomic preselection for genotyped sires and cows. For official evaluations released in December 2016, traditional PTA had the same trend as multiple-step genomic PTA for both genotyped bulls and cows except for the youngest bulls, who had traditional PTA slightly lower than genomic PTA. For genotyped bulls born in recent years, genetic gain for official traditional and genomic evaluations was similar in contrast to ssGBLUP and BLUP differences. Official PTA for cows were adjusted so that the Mendelian sampling variance was comparable with that for bulls, and those adjustments likely removed bias due to genomic preselection from traditional PTA, especially for genotyped cows. The ssGBLUP method seems to account partially for that bias and is computationally suitable for national evaluations., (The Authors. Published by FASS Inc. and Elsevier Inc. on behalf of the American Dairy Science Association®. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).)

When the environment on which the animals are raised is very diverse, selecting the best sires for different environments may require the use of models that account for genotype by environment interaction (G × E). The main objective of this study was to evaluate the existence of G × E for yearling weight (YW) in Nellore cattle using reaction norm models with only pedigree and pedigree combined with genomic relationships. Additionally, genomic regions associated with each environment gradient were identified. A total of 67,996 YW records were used in reaction norm models to calculate EBV and genomic EBV. The method of choice for genomic evaluations was single-step genomic BLUP (ssGBLUP). Traditional and genomic models were tested on the ability to predict future animal performance. Genetic parameters for YW were obtained with the average information restricted maximum likelihood method, with and without adding genomic information for 5,091 animals. Additive genetic variances explained by windows of 200 adjacent SNP were used to identify genomic regions associated with the environmental gradient. Estimated variance components for the intercept and the slope in traditional and genomic models were similar. In both models, the observed changes in heritabilities and genetic correlations for YW across environments indicate the occurrence of genotype by environment interactions. Both traditional and genomic models were capable of identifying the genotype by environment interaction; however, the inclusion of genomic information in reaction norm models improved the ability to predict animals' future performance by 7.9% on average. The proportion of genetic variance explained by the top SNP window was 0.77% for the regression intercept (BTA5) and 0.82% for the slope (BTA14). Single-step GBLUP seems to be a suitable model to predict genetic values for YW in different production environments.

We investigated the importance of SNP weighting in populations with 2,000 to 25,000 genotyped animals. Populations were simulated with two effective sizes (20 or 100) and three numbers of QTL (10, 50 or 500). Pedigree information was available for six generations; phenotypes were recorded for the four middle generations. Animals from the last three generations were genotyped for 45,000 SNP. Single-step genomic BLUP (ssGBLUP) and weighted ssGBLUP (WssGBLUP) were used to estimate genomic EBV using a genomic relationship matrix (G). The WssGBLUP performed better in small genotyped populations; however, any advantage for WssGBLUP was reduced or eliminated when more animals were genotyped. WssGBLUP had greater resolution for genome-wide association (GWA) as did increasing the number of genotyped animals. For few QTL, accuracy was greater for WssGBLUP than ssGBLUP; however, for many QTL, accuracy was the same for both methods. The largest genotyped set was used to assess the dimensionality of genomic information (number of effective SNP). The number of effective SNP was considerably less in weighted G than in unweighted G. Once the number of independent SNP is well represented in the genotyped population, the impact of SNP weighting becomes less important., (© 2017 Blackwell Verlag GmbH.)

Background: Much effort is put into identifying causative quantitative trait nucleotides (QTN) in animal breeding, empowered by the availability of dense single nucleotide polymorphism (SNP) information. Genomic selection using traditional SNP information is easily implemented for any number of genotyped individuals using single-step genomic best linear unbiased predictor (ssGBLUP) with the algorithm for proven and young (APY). Our aim was to investigate whether ssGBLUP is useful for genomic prediction when some or all QTN are known., Methods: Simulations included 180,000 animals across 11 generations. Phenotypes were available for all animals in generations 6 to 10. Genotypes for 60,000 SNPs across 10 chromosomes were available for 29,000 individuals. The genetic variance was fully accounted for by 100 or 1000 biallelic QTN. Raw genomic relationship matrices (GRM) were computed from (a) unweighted SNPs, (b) unweighted SNPs and causative QTN, (c) SNPs and causative QTN weighted with results obtained with genome-wide association studies, (d) unweighted SNPs and causative QTN with simulated weights, (e) only unweighted causative QTN, (f-h) as in (b-d) but using only the top 10% causative QTN, and (i) using only causative QTN with simulated weight. Predictions were computed by pedigree-based BLUP (PBLUP) and ssGBLUP. Raw GRM were blended with 1 or 5% of the numerator relationship matrix, or 1% of the identity matrix. Inverses of GRM were obtained directly or with APY., Results: Accuracy of breeding values for 5000 genotyped animals in the last generation with PBLUP was 0.32, and for ssGBLUP it increased to 0.49 with an unweighted GRM, 0.53 after adding unweighted QTN, 0.63 when QTN weights were estimated, and 0.89 when QTN weights were based on true effects known from the simulation. When the GRM was constructed from causative QTN only, accuracy was 0.95 and 0.99 with blending at 5 and 1%, respectively. Accuracies simulating 1000 QTN were generally lower, with a similar trend. Accuracies using the APY inverse were equal or higher than those with a regular inverse., Conclusions: Single-step GBLUP can account for causative QTN via a weighted GRM. Accuracy gains are maximum when variances of causative QTN are known and blending is at 1%.

This paper evaluates an efficient implementation to multiply the inverse of a numerator relationship matrix for genotyped animals () by a vector (). The computation is required for solving mixed model equations in single-step genomic BLUP (ssGBLUP) with the preconditioned conjugate gradient (PCG). The inverse can be decomposed into sparse matrices that are blocks of the sparse inverse of a numerator relationship matrix () including genotyped animals and their ancestors. The elements of were rapidly calculated with the Henderson's rule and stored as sparse matrices in memory. Implementation of was by a series of sparse matrix-vector multiplications. Diagonal elements of , which were required as preconditioners in PCG, were approximated with a Monte Carlo method using 1,000 samples. The efficient implementation of was compared with explicit inversion of with 3 data sets including about 15,000, 81,000, and 570,000 genotyped animals selected from populations with 213,000, 8.2 million, and 10.7 million pedigree animals, respectively. The explicit inversion required 1.8 GB, 49 GB, and 2,415 GB (estimated) of memory, respectively, and 42 s, 56 min, and 13.5 d (estimated), respectively, for the computations. The efficient implementation required <1 MB, 2.9 GB, and 2.3 GB of memory, respectively, and <1 sec, 3 min, and 5 min, respectively, for setting up. Only <1 sec was required for the multiplication in each PCG iteration for any data sets. When the equations in ssGBLUP are solved with the PCG algorithm, is no longer a limiting factor in the computations.

Animal feeding is the most important economic component of beef production systems. Selection for feed efficiency has not been effective mainly due to difficult and high costs to obtain the phenotypes. The application of genomic selection using SNP can decrease the cost of animal evaluation as well as the generation interval. The objective of this study was to compare methods for genomic evaluation of feed efficiency traits using different cross-validation layouts in an experimental beef cattle population genotyped for a high-density SNP panel (BovineHD BeadChip assay 700k, Illumina Inc., San Diego, CA). After quality control, a total of 437,197 SNP genotypes were available for 761 Nelore animals from the Institute of Animal Science, Sertãozinho, São Paulo, Brazil. The studied traits were residual feed intake, feed conversion ratio, ADG, and DMI. Methods of analysis were traditional BLUP, single-step genomic BLUP (ssGBLUP), genomic BLUP (GBLUP), and a Bayesian regression method (BayesCπ). Direct genomic values (DGV) from the last 2 methods were compared directly or in an index that combines DGV with parent average. Three cross-validation approaches were used to validate the models: 1) YOUNG, in which the partition into training and testing sets was based on year of birth and testing animals were born after 2010; 2) UNREL, in which the data set was split into 3 less related subsets and the validation was done in each subset a time; and 3) RANDOM, in which the data set was randomly divided into 4 subsets (considering the contemporary groups) and the validation was done in each subset at a time. On average, the RANDOM design provided the most accurate predictions. Average accuracies ranged from 0.10 to 0.58 using BLUP, from 0.09 to 0.48 using GBLUP, from 0.06 to 0.49 using BayesCπ, and from 0.22 to 0.49 using ssGBLUP. The most accurate and consistent predictions were obtained using ssGBLUP for all analyzed traits. The ssGBLUP seems to be more suitable to obtain genomic predictions for feed efficiency traits on an experimental population of genotyped animals.

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., (Copyright © 2016 by the Genetics Society of America.)

The objectives of this study were to develop and evaluate an efficient implementation in the computation of the inverse of genomic relationship matrix with the recursion algorithm, called the algorithm for proven and young (APY), in single-step genomic BLUP. We validated genomic predictions for young bulls with more than 500,000 genotyped animals in final score for US Holsteins. Phenotypic data included 11,626,576 final scores on 7,093,380 US Holstein cows, and genotypes were available for 569,404 animals. Daughter deviations for young bulls with no classified daughters in 2009, but at least 30 classified daughters in 2014 were computed using all the phenotypic data. Genomic predictions for the same bulls were calculated with single-step genomic BLUP using phenotypes up to 2009. We calculated the inverse of the genomic relationship matrix GAPY(-1) based on a direct inversion of genomic relationship matrix on a small subset of genotyped animals (core animals) and extended that information to noncore animals by recursion. We tested several sets of core animals including 9,406 bulls with at least 1 classified daughter, 9,406 bulls and 1,052 classified dams of bulls, 9,406 bulls and 7,422 classified cows, and random samples of 5,000 to 30,000 animals. Validation reliability was assessed by the coefficient of determination from regression of daughter deviation on genomic predictions for the predicted young bulls. The reliabilities were 0.39 with 5,000 randomly chosen core animals, 0.45 with the 9,406 bulls, and 7,422 cows as core animals, and 0.44 with the remaining sets. With phenotypes truncated in 2009 and the preconditioned conjugate gradient to solve mixed model equations, the number of rounds to convergence for core animals defined by bulls was 1,343; defined by bulls and cows, 2,066; and defined by 10,000 random animals, at most 1,629. With complete phenotype data, the number of rounds decreased to 858, 1,299, and at most 1,092, respectively. Setting up GAPY(-1) for 569,404 genotyped animals with 10,000 core animals took 1.3h and 57 GB of memory. The validation reliability with APY reaches a plateau when the number of core animals is at least 10,000. Predictions with APY have little differences in reliability among definitions of core animals. Single-step genomic BLUP with APY is applicable to millions of genotyped animals., (Copyright © 2016 American Dairy Science Association. Published by Elsevier Inc. All rights reserved.)

Combining purebreed and crossbreed information is beneficial for genetic evaluation of some livestock species. Genetic evaluations can use relationships based on genomic information, relying on allele frequencies that are breed specific. Single-step genomic BLUP (ssGBLUP) does not account for different allele frequencies, which could limit the genetic gain in crossbreed evaluations. In this study, we tested the performance of different breed-specific genomic relationship matrices () in ssGBLUP for crossbreed evaluations; we also tested the importance of genotyping crossbred animals. Genotypes were available for purebreeds (AA and BB) and crossbreeds (F) in simulated and real pig populations. The number of genotyped animals was, on average, 4,315 for the simulated population and 15,798 for the real population. Cross-validation was performed on 1,200 and 3,117 F animals in the simulated and real populations, respectively. Simulated scenarios were under no artificial selection, mass selection, or BLUP selection. Two genomic relationship matrices were constructed based on breed-specific allele frequencies: 1) , a genomic relationship matrix centered by breed-specific allele frequencies, and 2) , a genomic relationship matrix centered and scaled by breed-specific allele frequencies. All (the across-breed genomic relationship matrix), , and were also tuned to account for selective genotyping. Using breed-specific allele frequencies reduced the number of negative relationships between 2 purebreeds, pulling the average closer to 0, as in the pedigree-based relationship matrix. For simulated populations that included mass selection, genomic EBV (GEBV) in F, when using and , were, on average, 10% more accurate than ; however, after tuning to account for selective genotyping, provided the same accuracy as for breed-specific genomic relationship matrices. For the real population, accuracies for litter size in F were 0.62 for , , and , and tuning had no impact on accuracy, except for , which was 1 percentage point less accurate. Accuracy of GEBV for number of stillborns in F1 was 0.5 for all tested genomic relationship matrices with no changes after tuning. We observed that genotyping F increased accuracies of GEBV for the same animals by up to 39% compared with having genotypes for only AA and BB. In crossbreed evaluations, accounting for breed-specific allele frequencies promoted changes in G that were not influential enough to improve accuracy of GEBV. Therefore, the best performance of ssGBLUP for crossbreed evaluations requires genotypes for pure- and crossbreeds and no breed-specific adjustments in the realized relationship matrix.

Many computations with SNP data including genomic evaluation, parameter estimation, and genome-wide association studies use an inverse of the genomic relationship matrix. The cost of a regular inversion is cubic and is prohibitively expensive for large matrices. Recent studies in cattle demonstrated that the inverse can be computed in almost linear time by recursion on any subset of ∼10,000 individuals. The purpose of this study is to present a theory of why such a recursion works and its implication for other populations. Assume that, because of a small effective population size, the additive information in a genotyped population has a small dimensionality, even with a very large number of SNP markers. That dimensionality is visible as a limited number of effective SNP effects, independent chromosome segments, or the rank of the genomic relationship matrix. Decompose a population arbitrarily into core and noncore individuals, with the number of core individuals equal to that dimensionality. Then, breeding values of noncore individuals can be derived by recursions on breeding values of core individuals, with coefficients of the recursion computed from the genomic relationship matrix. A resulting algorithm for the inversion called "algorithm for proven and young" (APY) has a linear computing and memory cost for noncore animals. Noninfinitesimal genetic architecture can be accommodated through a trait-specific genomic relationship matrix, possibly derived from Bayesian regressions. For populations with small effective population size, the inverse of the genomic relationship matrix can be computed inexpensively for a very large number of genotyped individuals., (Copyright © 2016 by the Genetics Society of America.)

Background: Most developments in quantitative genetics theory focus on the study of intra-breed/line concepts. With the availability of massive genomic information, it becomes necessary to revisit the theory for crossbred populations. We propose methods to construct genomic covariances with additive and non-additive (dominance) inheritance in the case of pure lines and crossbred populations., Results: We describe substitution effects and dominant deviations across two pure parental populations and the crossbred population. Gene effects are assumed to be independent of the origin of alleles and allelic frequencies can differ between parental populations. Based on these assumptions, the theoretical variance components (additive and dominant) are obtained as a function of marker effects and allelic frequencies. The additive genetic variance in the crossbred population includes the biological additive and dominant effects of a gene and a covariance term. Dominance variance in the crossbred population is proportional to the product of the heterozygosity coefficients of both parental populations. A genomic BLUP (best linear unbiased prediction) equivalent model is presented. We illustrate this approach by using pig data (two pure lines and their cross, including 8265 phenotyped and genotyped sows). For the total number of piglets born, the dominance variance in the crossbred population represented about 13 % of the total genetic variance. Dominance variation is only marginally important for litter size in the crossbred population., Conclusions: We present a coherent marker-based model that includes purebred and crossbred data and additive and dominant actions. Using this model, it is possible to estimate breeding values, dominant deviations and variance components in a dataset that comprises data on purebred and crossbred individuals. These methods can be exploited to plan assortative mating in pig, maize or other species, in order to generate superior crossbred individuals in terms of performance.

The purpose of this study was to examine accuracy of genomic selection via single-step genomic BLUP (ssGBLUP) when the direct inverse of the genomic relationship matrix (G) is replaced by an approximation of G(-1) based on recursions for young genotyped animals conditioned on a subset of proven animals, termed algorithm for proven and young animals (APY). With the efficient implementation, this algorithm has a cubic cost with proven animals and linear with young animals. Ten duplicate data sets mimicking a dairy cattle population were simulated. In a first scenario, genomic information for 20k genotyped bulls, divided in 7k proven and 13k young bulls, was generated for each replicate. In a second scenario, 5k genotyped cows with phenotypes were included in the analysis as young animals. Accuracies (average for the 10 replicates) in regular EBV were 0.72 and 0.34 for proven and young animals, respectively. When genomic information was included, they increased to 0.75 and 0.50. No differences between genomic EBV (GEBV) obtained with the regular G(-1) and the approximated G(-1) via the recursive method were observed. In the second scenario, accuracies in GEBV (0.76, 0.51 and 0.59 for proven bulls, young males and young females, respectively) were also higher than those in EBV (0.72, 0.35 and 0.49). Again, no differences between GEBV with regular G(-1) and with recursions were observed. With the recursive algorithm, the number of iterations to achieve convergence was reduced from 227 to 206 in the first scenario and from 232 to 209 in the second scenario. Cows can be treated as young animals in APY without reducing the accuracy. The proposed algorithm can be implemented to reduce computing costs and to overcome current limitations on the number of genotyped animals in the ssGBLUP method., (© 2015 Blackwell Verlag GmbH.)

The objective of this study was to remove bottlenecks generally found in a computer program for average-information REML. The refinements included improvements to setting-up mixed-model equations on a hash table with a faster hash function as sparse matrix storage, changing sparse structures in calculation of traces, and replacing a sparse matrix package using traditional methods (FSPAK) with a new package using supernodal methods (YAMS); the latter package quickly processed sparse matrices containing large, dense blocks. Comparisons included 23 models with data sets from broiler, swine, beef, and dairy cattle. Models included single-trait, multiple-trait, maternal, and random regression models with phenotypic data; selected models used genomic information in a single-step approach. Setting-up mixed model equations was completed without abnormal termination in all analyses. Calculations in traces were accelerated with a hash format, especially for models with a genomic relationship matrix, and the maximum speed was 67 times faster. Computations with YAMS were, on average, more than 10 times faster than with FSPAK and had greater advantages for large data and more complicated models including multiple traits, random regressions, and genomic effects. These refinements can be applied to general average-information REML programs.

Recent use of genomic (marker-based) relationships shows that relationships exist within and across base population (breeds or lines). However, current treatment of pedigree relationships is unable to consider relationships within or across base populations, although such relationships must exist due to finite size of the ancestral population and connections between populations. This complicates the conciliation of both approaches and, in particular, combining pedigree with genomic relationships. We present a coherent theoretical framework to consider base population in pedigree relationships. We suggest a conceptual framework that considers each ancestral population as a finite-sized pool of gametes. This generates across-individual relationships and contrasts with the classical view which each population is considered as an infinite, unrelated pool. Several ancestral populations may be connected and therefore related. Each ancestral population can be represented as a "metafounder," a pseudo-individual included as founder of the pedigree and similar to an "unknown parent group." Metafounders have self- and across relationships according to a set of parameters, which measure ancestral relationships, i.e., homozygozities within populations and relationships across populations. These parameters can be estimated from existing pedigree and marker genotypes using maximum likelihood or a method based on summary statistics, for arbitrarily complex pedigrees. Equivalences of genetic variance and variance components between the classical and this new parameterization are shown. Segregation variance on crosses of populations is modeled. Efficient algorithms for computation of relationship matrices, their inverses, and inbreeding coefficients are presented. Use of metafounders leads to compatibility of genomic and pedigree relationship matrices and to simple computing algorithms. Examples and code are given., (Copyright © 2015 by the Genetics Society of America.)

The purpose of this study was to estimate correlations between purebred and F1 crossbred performance to verify the appropriateness of current models used in multibreed selection. Records on birth weight (WB) and weaning weight (WW) from purebred Limousins (LIM) and Limousin × Angus progeny (F1) were used to estimate genetic parameters using a multiple-trait (purebred and F1 weights were different traits) approach. For WB, there were 148,647 records for LIM and 17,981 for F1, and for WW, there were 81,585 records for LIM and 21,778 for F1. The fixed effect in models for LIM and F1 animals was contemporary group. Random effects for LIM animals were direct genetic, maternal genetic, and maternal permanent environment effects. Random effects for F1 were sire and dam. The pedigree for Angus dams used for crossing was unavailable and therefore these dams were assumed unrelated. The direct h2 estimates (SE) for purebred animals were 0.41 (0.05) and 0.24 (0.02) for WB and WW, respectively. For F1, the same estimates were 0.22 (0.09) and 0.32 (0.05). Genetic correlations estimates between purebreds and crossbreds were 0.84 (0.07) and 0.64 (0.18) for WB and WW, respectively. The genetic correlation for WW estimated in this study suggests that F1 and purebred information for this trait should not be treated, genetically, as the same trait due to different genetic effects molding it. However, the genetic correlation for WB was much higher, indicating that this trait in purebreds and F1 is essentially the same trait.

Quality control filtering of single-nucleotide polymorphisms (SNPs) is a key step when analyzing genomic data. Here we present a practical method to identify low-quality SNPs, meaning markers whose genotypes are wrongly assigned for a large proportion of individuals, by estimating the heritability of gene content at each marker, where gene content is the number of copies of a particular reference allele in a genotype of an animal (0, 1, or 2). If there is no mutation at the marker, gene content has an additive heritability of 1 by construction. The method uses restricted maximum likelihood (REML) to estimate heritability of gene content at each SNP and also builds a likelihood-ratio test statistic to test for zero error variance in genotyping. As a by-product, estimates of the allele frequencies of markers at the base population are obtained. Using simulated data with 10% permutation error (4% actual error) in genotyping, the method had a specificity of 0.96 (4% of correct markers are rejected) and a sensitivity of 0.99 (1% of wrong markers are accepted) if markers with heritability lower than 0.975 are discarded. Checking of Mendelian errors resulted in a lower sensitivity (0.84) for the same simulation. The proposed method is further illustrated with a real data set with genotypes from 3534 animals genotyped for 50,433 markers from the Illumina PorcineSNP60 chip and a pedigree of 6473 individuals; those markers underwent very little quality control. A total of 4099 markers with P-values lower than 0.01 were discarded based on our method, with associated estimates of heritability as low as 0.12. Contrary to other techniques, our method uses all information in the population simultaneously, can be used in any population with markers and pedigree recordings, and is simple to implement using standard software for REML estimation. Scripts for its use are provided., (Copyright © 2015 by the Genetics Society of America.)