Algorithm For Proven And Young Research Articles

Short Communication: Reduced GBLUP equations to core animals in the algorithm for proven and young (APY)

The number of animal genotypes is rapidly increasing, and a major challenge for animal models is inverting the genomic relationship matrix (G). Matrix G has a limited dimensionality, and the algorithm for proven and young (APY) makes inverting a large G possible via the inverse of a block diagonal of G with a size equivalent to the dimensionality of G. APY divides genotyped animals into core and non-core groups, and breeding values of non-core animals are conditioned on the breeding values of core animals. Therefore, there is the possibility of opting out equations for non-core animals from the model. A methodology was presented for a reduced APY genomic BLUP (GBLUP) to equations for core animals. Using a small example dataset, the method was validated by the equality of the full and the reduced model analysis results. Absorption of fixed effect equations into random effect equations was successful in reducing the number of equations to solve and producing the same random effect solutions. Extending the method to APY single-step GBLUP (ssGBLUP) was not computationally justifiable. Other reduction techniques exist for ssGBLUP (regardless of APY or non-APY) that work by reducing the number of equations for non-genotyped animals. The number of equations can further be reduced by data pruning.

149 Marker Effect P-Value for Large Genotype Populations with the Algorithm for Proven and Young

Abstract The single-step method (ssGBLUP) allows for simultaneous evaluations of populations composed of genotyped and non-genotyped animals. As a GBLUP-based method, the output of ssGBLUP is breeding values. However, interest might rely on the estimation of SNP effects and on how genome segments are associated with the phenotype of interest. When that is the case, SNP effects can be obtained from a linear transformation of breeding values, and P-values can be used as a measure of estimation certainty. P-values are derived from the prediction error variance of SNP effects, which relies on the prediction error (co)variance matrix of breeding values obtained from the inverse of the left-hand side of the mixed model equations (LHS). However, inverting the LHS becomes unfeasible with large, genotyped populations. Therefore, alternative strategies to overcome this problem and obtain p-values with increasing genotyped populations might be evaluated. In this study, we aimed to estimate marker effect p-values when the algorithm for proven and young (APY) was used to approximate the predictor error (co)variance for animals in a large, genotyped population. We estimate P-values of SNP marker effects for PWG in an Angus cattle population. The dataset was composed of around 844K phenotyped animals, 450K genotypes, and 1.8M pedigree records. Analyses were split into two sets. In the first set, a reduced genotype data of 50K was used, so p-values obtained from a direct-inverse of LHS using the regular G-1 (Exact_G) or APY G-1 (Exact_GAPY) were compared with APY G-1 and an approximation of the prediction error variance that did not require the inversion of the LHS (Approx_GAPY). In the second set of analyses, P-values with Approx_GAPY were estimated with the full genomic set of 450K (Approx_GAPY_450K), and computational requirements were recorded. All analyses were performed in three replicates. In the first set of analyses, genome-wide association with Exact_G uncovered two significant regions in chromosomes 7 and 20. Along all replicates, the same regions were also identified with Exact_GAPY and Approx_GAPY, indicating that similar resolutions are obtained with exact and approximated p-values. Results from Approx_GAPY_450K showed that, with a complete genome set, besides the two regions in chromosomes 7 and 20, two new regions in chromosomes 6 and 14 were identified. On average, the entire procedure for obtaining P-values with 450K genotyped animals had an elapsed wall clock time of 24h with a maximum memory usage of 87.6 GB. Altogether, our results suggest that with APY and the approximation of the prediction error variance, current computational boundaries for obtaining marker effect P-values for large genotyped populations should be lifted.

Optimisation of the core subset for the APY approximation of genomic relationships

BackgroundBy entering the era of mega-scale genomics, we are facing many computational issues with standard genomic evaluation models due to their dense data structure and cubic computational complexity. Several scalable approaches have been proposed to address this challenge, such as the Algorithm for Proven and Young (APY). In APY, genotyped animals are partitioned into core and non-core subsets, which induces a sparser inverse of the genomic relationship matrix. This partitioning is often done at random. While APY is a good approximation of the full model, random partitioning can make results unstable, possibly affecting accuracy or even reranking animals. Here we present a stable optimisation of the core subset by choosing animals with the most informative genotype data.MethodsWe derived a novel algorithm for optimising the core subset based on a conditional genomic relationship matrix or a conditional single nucleotide polymorphism (SNP) genotype matrix. We compared the accuracy of genomic predictions with different core subsets for simulated and real pig data sets. The core subsets were constructed (1) at random, (2) based on the diagonal of the genomic relationship matrix, (3) at random with weights from (2), or (4) based on the novel conditional algorithm. To understand the different core subset constructions, we visualise the population structure of the genotyped animals with linear Principal Component Analysis and non-linear Uniform Manifold Approximation and Projection.ResultsAll core subset constructions performed equally well when the number of core animals captured most of the variation in the genomic relationships, both in simulated and real data sets. When the number of core animals was not sufficiently large, there was substantial variability in the results with the random construction but no variability with the conditional construction. Visualisation of the population structure and chosen core animals showed that the conditional construction spreads core animals across the whole domain of genotyped animals in a repeatable manner.ConclusionsOur results confirm that the size of the core subset in APY is critical. Furthermore, the results show that the core subset can be optimised with the conditional algorithm that achieves an optimal and repeatable spread of core animals across the domain of genotyped animals.

Leveraging low-density crossbred genotypes to offset crossbred phenotypes and their impact on purebred predictions.

The objectives of this study were to 1) investigate the predictability and bias of genomic breeding values (GEBV) of purebred (PB) sires for CB performance when CB genotypes imputed from a low-density panel are available, 2) assess if the availability of those CB genotypes can be used to partially offset CB phenotypic recording, and 3) investigate the impact of including imputed CB genotypes in genomic analyses when using the algorithm for proven and young (APY). Two pig populations with up to 207,375 PB and 32,893 CB phenotypic records per trait and 138,026 PB and 32,893 CB genotypes were evaluated. PB sires were genotyped for a 50K panel, whereas CB animals were genotyped for a low-density panel of 600 SNP and imputed to 50K. The predictability and bias of GEBV of PB sires for backfat thickness (BFX) and average daily gain recorded (ADGX) recorded on CB animals were assessed when CB genotypes were available or not in the analyses. In the first set of analyses, direct inverses of the genomic relationship matrix (G) were used with phenotypic datasets truncated at different time points. In the next step, we evaluated the APY algorithm with core compositions differing in the CB genotype contributions. After that, the performance of core compositions was compared with an analysis using a random PB core from a purely PB genomic set. The number of rounds to convergence was recorded for all APY analyses. With the direct inverse of G in the first set of analyses, adding CB genotypes imputed from a low-density panel (600 SNP) did not improve predictability or reduce the bias of PB sires' GEBV for CB performance, even for sires with fewer CB progeny phenotypes in the analysis. That indicates that the inclusion of CB genotypes primarily used for inferring pedigree in commercial farms is of no benefit to offset CB phenotyping. When CB genotypes were incorporated into APY, a random core composition or a core with no CB genotypes reduced bias and the number of rounds to convergence but did not affect predictability. Still, a PB random core composition from a genomic set with only PB genotypes resulted in the highest predictability and the smallest number of rounds to convergence, although bias increased. Genotyping CB individuals for low-density panels is a valuable identification tool for linking CB phenotypes to pedigree; however, the inclusion of those CB genotypes imputed from a low-density panel (600 SNP) might not benefit genomic predictions for PB individuals or offset CB phenotyping for the evaluated CB performance traits. Further studies will help understand the usefulness of those imputed CB genotypes for traits with lower PB-CB genetic correlations and traits not recorded in the PB environment, such as mortality and disease traits.

Theoretical accuracy for indirect predictions based on SNP effects from single-step GBLUP

BackgroundAlthough 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.ResultsCorrelations 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.ConclusionsTheoretical 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.

35 Young Scholar Award Talk: Computing Strategies for National Beef Cattle Evaluations

Abstract The constantly growing size of the national beef cattle evaluations datasets creates computational challenges to estimate breeding values and their accuracies under the single-step methodology timely and in a robust way. This study aims to present strategies for making weekly national single-step beef cattle evaluations computationally feasible. The process for estimating breeding values in single-step can be divided into two parts. In the first part, hereafter called genomic setup, the genotype file is read, and the inverse of the genomic relationship matrix is calculated using the Algorithm for Proven and Young (APY). The genotype file is read in a stream-unformatted format, and the markers are stored bit-wise. A residual polygenic effect can be fitted with an algorithm whose computing cost is constant with the number of genotyped animals. The calculation of is done using parallel computing and optimized matrix algebra procedures. The second part estimates breeding values by iteration on data with a block diagonal preconditioner. Finally, the accuracy of the estimated breeding values is calculated by combining the pedigree and genomic information in terms of effective daughter contributions. For an Angus dataset with 10 million animals in the pedigree and 500,000 genotypes, the genomic setup took 4 hours. With the same dataset, solutions for a three-trait model with maternal effects including birth weight, weaning weight, and post-weaning gain were obtained after 12 hours. The approximation of the accuracies of breeding values took less than one hour. Our results show that weekly national beef cattle evaluations are computationally feasible.

On the equivalence between marker effect models and breeding value models and direct genomic values with the Algorithm for Proven and Young

BackgroundSingle-step genomic predictions obtained from a breeding value model require calculating the inverse of the genomic relationship matrix ({mathbf{G}}^{-1}). The Algorithm for Proven and Young (APY) creates a sparse representation of {mathbf{G}}^{-1} 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.ResultsWe 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.ConclusionsThe 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.

A comprehensive study on size and definition of the core group in the proven and young algorithm for single-step GBLUP

BackgroundThe 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.MethodsThe 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.ResultsWhen 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.ConclusionsFor 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.

Is single-step genomic REML with the algorithm for proven and young more computationally efficient when less generations of data are present?

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 () 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 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.

Multibreed genomic evaluation for production traits of dairy cattle in the United States using single-step genomic best linear unbiased predictor

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.

Efficient approximation of reliabilities for single-step genomic best linear unbiased predictor models with the Algorithm for Proven and Young.

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.

A novel genomic prediction method combining randomized Haseman-Elston regression with a modified algorithm for Proven and Young for large genomic data

Computational efficiency has become a key issue in genomic prediction (GP) owing to the massive historical datasets accumulated. We developed hereby a new super-fast GP approach (SHEAPY) combining randomized Haseman-Elston regression (RHE-reg) with a modified Algorithm for Proven and Young (APY) in an additive-effect model, using the former to estimate heritability and then the latter to invert a large genomic relationship matrix for best linear prediction. In simulation results with varied sizes of training population, GBLUP, HEAPY|A and SHEAPY showed similar predictive performance when the size of a core population was half that of a large training population and the heritability was a fixed value, and the computational speed of SHEAPY was faster than that of GBLUP and HEAPY|A. In simulation results with varied heritability, SHEAPY showed better predictive ability than GBLUP in all cases and than HEAPY|A in most cases when the size of a core population was 4/5 that of a small training population and the training population size was a fixed value. As a proof of concept, SHEAPY was applied to the analysis of two real datasets. In an Arabidopsis thaliana F2 population, the predictive performance of SHEAPY was similar to or better than that of GBLUP and HEAPY|A in most cases when the size of a core population (200) was 2/3 of that of a small training population (300). In a sorghum multiparental population, SHEAPY showed higher predictive accuracy than HEAPY|A for all of three traits, and than GBLUP for two traits. SHEAPY may become the GP method of choice for large-scale genomic data.

PSXII-9 Modeling genetic differences of combined broiler chicken populations in single-step GBLUP

Abstract The objectives of this study were to model the inclusion of a group of external birds into a local broiler chicken population for the purpose of genomic evaluations and evaluating the behavior of two accuracy estimators under different model specifications. The pedigree was composed by 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 UPG or metafounders. Genomic estimated breeding values (GEBV) were obtained with single-step GBLUP (ssGBLUP) using the Algorithm for Proven and Young (APY). Bias, dispersion, and accuracy of GEBV for the validation birds, i.e., from the most recent generation, were computed. The bias and dispersion were estimated with the LR-method, whereas accuracy was estimated by the LR-method and predictive ability. Models with fixed UPG and estimated inbreeding or random UPG resulted in similar GEBV. The inclusion of an extra fixed effect in the model made the GEBV unbiased and reduced the inflation, while models without such an effect were significantly biased. Genomic predictions with metafounders 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 and smallest bias 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 models, whereas predictive ability greatly depends on the model specification, that is, on the fixed effects included in the model. When changing model specification, the largest variation for the LR-method was 20%, while for predictive ability was 110%.

Reducing computational cost of large-scale genomic evaluation by using indirect genomic prediction

Over half a million Holsteins are being genotyped annually in the United States. The computational cost of including all genotypes in single-step genomic (ssG)BLUP is high, although it is feasible to conduct large-scale genomic prediction using an efficient algorithm such as APY (algorithm for proven and young). An effective method to further reduce the computing cost could be the use of indirect genomic predictions (IGP) for genotyped animals when they have neither progeny nor phenotypes. These young genotyped animals have no effect on the other genotyped animals and could have their genomic prediction done indirectly. The main objective of this study was to calculate IGP for various groups of genotyped animals and investigate the reduction in computing time as well as bias and accuracy of the IGP. We compared IGP with genomic (G)EBV for 18 linear type traits in US Holsteins, including 2.3 million (M) genotyped animals. The full data set consisted of 10.9M records for 18 linear type traits up to 2018 calving, 13.6M animals in the pedigree, and 2.3M animals genotyped for 79K SNP. For IGP, ssGBLUP included all genotyped animals except those with neither progeny nor phenotypes by year from 2014 to 2018 (i.e., the target animals). The SNP marker effects were computed based on GEBV for genotyped animals that had progeny, or phenotypes, or both. Further, IGP were calculated for target genotyped animals in each year group. For all genotyped animal groups from 2014 to 2018, the coefficients of determination (R2) of a linear regression of GEBV on IGP were 0.960 for males and 0.954 for females for 18 traits on average. To reduce computing costs, the SNP marker effects were calculated based on GEBV from randomly selected genotyped animals from 15K to 60K. By randomly selecting a small number of genotyped animals, the computing time was dramatically reduced. As more genotyped animals were randomly selected to calculate SNP effects, R2 was higher (more accurate) and the regression coefficient was lower (more inflated IGP). In a practical genomic evaluation in US Holsteins, to get sufficient contributions from GEBV, 25K to 35K is a rational number of genotyped animals that can be randomly selected to compute SNP effects and obtain accurate and unbiased IGP. Considering the computing time and both unbiasedness and accuracy of IGP, genomic evaluation can be conducted separately in GEBV for genotyped animals with phenotypes or progeny and in IGP for young genotyped animals. This can be a practical solution when conducting a large-scale genomic evaluation and would enable more frequent evaluation at lower cost, especially when many genotyped animals have neither phenotypes nor progeny.

Changes in genomic predictions when new information is added.

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.

22 Accuracy of indirect predictions for large datasets based on prediction error covariance of SNP effects from single-step GBLUP

Abstract With an ever-increasing number of genotyped animals, there is a question of whether to include all genotypes into single-step GBLUP (ssGBLUP) evaluations or to include only genotyped animals with phenotypes and use indirect predictions (IP) for the remaining young genotyped animals. Under ssGBLUP, SNP effects can be backsolved from GEBV, and IP can be calculated as the sum of SNP effects weighted by the gene content. To publish IP, a measure of accuracy that reflects the standard error of prediction, and that is comparable to GEBV accuracy, is needed. Our first objective was to test formulas to compute accuracy of IP by backsolving prediction error covariance (PEC) of GEBV into PEC of SNP effects. The second objective was to investigate the number of genotyped animals needed to obtain robust IP accuracy. Data were provided by the American Angus Association, with 38,000 post-weaning gain phenotypes and 60,000 genotyped animals. Correlations between GEBV and IP were ≥0.99. When all genotyped animals were used for PEC computations, accuracy correlations were also ≥0.99. Additionally, GEBV and IP accuracies were compatible, with both direct inversion of the genomic relationship matrix (G) or using the algorithm for proven and young (APY) to obtain G inverse. As the number of genotyped animals in PEC computations decreased to 15,000, accuracy correlations were still high (≥0.96), but IP accuracies were biased downwards. Indirect prediction accuracy can be successfully obtained from ssGBLUP without running an extra SNP-BLUP evaluation to compute SNP PEC. It is possible to reduce the number of genotyped animals in PEC computations, but accuracies may be slightly underestimated. When the amount of genomic and phenotypic data is large, the polygenic part of GEBV becomes small and IP can be very accurate. Further research is needed to approximate SNP PEC with a large number of genotyped animals.

Core-dependent changes in genomic predictions using the Algorithm for Proven and Young in single-step genomic best linear unbiased prediction.

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.

Indirect predictions with a large number of genotyped animals using the algorithm for proven and young.

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.

Correction to: A rapid genomic selection method combining Haseman-Elston (HE) model and algorithm for proven and young (APY)

Correction to: A rapid genomic selection method combining Haseman-Elston (HE) model and algorithm for proven and young (APY)

A rapid genomic selection method combining Haseman-Elston (HE) model and algorithm for proven and young (APY)

To increase the predictive accuracy of genomic selection, large-scale phenotypic and genomic data is needed, which results in enormous consumption of computational resources, hampering the feasibility of genomic selection in return. In order to address computational limitation, a rapid genomic selection method HEAPY|A is developed in this study, combining Haseman-Elston (HE) model and algorithm for proven and young (APY). The proposed approach utilizes (i) HE regression to estimate the heritability and then (ii) APY to solve the inverse of the large matrix in best linear prediction (BLP); both HE and APY can reduce the computational cost compared with conventional techniques. When the size of a core population is half of that of a large training population, GBLUP|A, HEAPY|A, and HEBLP|A have similar performance in simulation studies (the core population can be further reduced if the training population size is further increased). When the size of a core population is half of that of a small training population, the predictive accuracy of HEAPY|A is a little lower than that of GBLUP|A and HEBLP|A in simulation study and empirical data—an Arabidopsis thaliana F2 population. HEAPY|A helps in predicting large genomic selection dataset with comparable accuracy without significant expense of time seen in traditional genomic selection algorithm.