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.

Efficient large-scale single-step evaluations and indirect genomic prediction of genotyped selection candidates

BACKGROUND

Single-step genomic best linear unbiased prediction (ssGBLUP) models allow the combination of genomic, pedigree, and phenotypic data into a single model, which is computationally challenging for large genotyped populations. In practice, genotypes of animals without their own phenotype and progeny, so-called genotyped selection candidates, can become available after genomic breeding values have been estimated by ssGBLUP. In some breeding programmes, genomic estimated breeding values (GEBV) for these animals should be known shortly after obtaining genotype information but recomputing GEBV using the full ssGBLUP takes too much time. In this study, first we compare two equivalent formulations of ssGBLUP models, i.e. one that is based on the Woodbury matrix identity applied to the inverse of the genomic relationship matrix, and one that is based on marker equations. Second, we present computationally-fast approaches to indirectly compute GEBV for genotyped selection candidates, without the need to do the full ssGBLUP evaluation.

RESULTS

The indirect approaches use information from the latest ssGBLUP evaluation and rely on the decomposition of GEBV into its components. The two equivalent ssGBLUP models and indirect approaches were tested on a six-trait calving difficulty model using Irish dairy and beef cattle data that include 2.6 million genotyped animals of which about 500,000 were considered as genotyped selection candidates. When using the same computational approaches, the solving phase of the two equivalent ssGBLUP models showed similar requirements for memory and time per iteration. The computational differences between them were due to the preprocessing phase of the genomic information. Regarding the indirect approaches, compared to GEBV obtained from single-step evaluations including all genotypes, indirect GEBV had correlations higher than 0.99 for all traits while showing little dispersion and level bias.

CONCLUSIONS

In conclusion, ssGBLUP predictions for the genotyped selection candidates were accurately approximated using the presented indirect approaches, which are more memory efficient and computationally fast, compared to solving a full ssGBLUP evaluation. Thus, indirect approaches can be used even on a weekly basis to estimate GEBV for newly genotyped animals, while the full single-step evaluation is done only a few times within a year.

Application of single-step single nucleotide polymorphism best linear unbiased predictor model with unknown-parent groups for type traits in Japanese Holsteins

The objectives of this study were to investigate the computational performance and the predictive ability and bias of a single-step SNP BLUP model (ssSNPBLUP) in genotyped young animals with unknown-parent groups (UPG) for type traits, using national genetic evaluation data from the Japanese Holstein population. The phenotype, genotype, and pedigree data were the same as those used in a national genetic evaluation of linear type traits classified between April 1984 and December 2020. In the current study, 2 data sets were prepared: the full data set containing all entries up to December 2020 and a truncated data set ending with December 2016. Genotyped animals were classified into 3 types: sires with classified daughters (S), cows with records (C), and young animals (Y). The computing performance and prediction accuracy of ssSNPBLUP were compared for the following 3 groups of genotyped animals: sires with classified daughters and young animals (SY); cows with records and young animals (CY); and sires with classified daughters, cows with records, and young animals (SCY). In addition, we tested 3 parameters of residual polygenic variance in ssSNPBLUP (0.1, 0.2, or 0.3). Daughter yield deviations (DYD) for the validation bulls and phenotypes adjusted for all fixed effects and random effects other than animal and residual (Y_adj) for the validation cows were obtained using the full data set from the pedigree-based BLUP model. The regression coefficients of DYD for bulls (or Y_adj for cows) on the genomic estimated breeding value (GEBV) using the truncated data set were used to measure the inflation of the predictions of young animals. The coefficient of determination of DYD on GEBV was used to measure the predictive ability of the predictions for the validation bulls. The reliability of the predictions for the validation cows was calculated as the square of the correlation between Y_adj and GEBV divided by heritability. The predictive ability was highest in the SCY group and lowest in the CY group. However, minimal difference was found in predictive abilities with or without UPG models using different parameters of residual polygenic variance. The regression coefficients approached 1.0 as the parameter of residual polygenic variance increased, but regression coefficients were mostly similar regardless of the use of UPG across the groups of genotyped animals. The ssSNPBLUP model, including UPG, was demonstrated as feasible for implementation in the national evaluation of type traits in Japanese Holsteins.

Boundaries for genotype, phenotype, and pedigree truncation in genomic evaluations in pigs

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.

A review of inbreeding depression in dairy cattle: current status, emerging control strategies, and future prospects

Dairy cattle breeding has historically focused on relatively small numbers of elite bulls as sires of sons. In recent years, even if generation intervals were reduced and more diverse sires of sons could have been selected, genomic selection has not fundamentally changed the fact that a large number of individuals are being analyzed. However, a relatively small number of elite bulls are still siring those animals. Therefore inbreeding-derived negative consequences in the gene pool have brought concern. The detrimental effects of non-additive genetic changes such as inbreeding depression and dominance have been widely disseminated while seriously affecting bioeconomically important parameters because of an antagonistic relationship between dairy production and reproductive traits. Therefore, the estimation of benefits and limitations of inbreeding and variance of the selection response deserves to be evaluated and discussed to preserve genetic variability, a significant concern in the selection of individuals for reproduction and production. Short-term strategies for genetic merit improvement through modern breeding programs have severely lowered high-producing dairy cattle fertility potential. Since the current selection programs potentially increase long-term costs, genetic diversity has decreased globally as a consequence. Therefore, a greater understanding of the potential that selection programs have for supporting long-term genetic sustainability and genetic diversity among dairy cattle populations should be prioritized in managing farm profitability. The present review provides a broad approach to current inbreeding-derived problems, identifying critical points to be solved and possible alternative strategies to control selection against homozygous haplotypes while maintaining sustained selection pressure. Moreover, this manuscript explores future perspectives, emphasizing theoretical applications and critical points, and strategies to avoid the adverse effects of inbreeding in dairy cattle. Finally, this review provides an overview of challenges that will soon require multidisciplinary approaches to managing dairy cattle populations, intending to combine increases in productive trait phenotypes with improvements in reproductive, health, welfare, linear conformation, and adaptability traits into the foreseeable future.

Invited review: Unknown-parent groups and metafounders in single-step genomic BLUP

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.

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.

Practical implementation of genetic groups in single-step genomic evaluations with Woodbury matrix identity-based genomic relationship inverse

The growing amount of genomic information in dairy cattle has increased computational and modeling challenges in the single-step evaluations. The computational challenges are due to the dense inverses of genomic (G) and pedigree (A₂₂) relationship matrices of genotyped animals in the single-step mixed model equations. An equivalent mixed model equation is given by single-step genomic BLUP that are based on the T matrix (ssGTBLUP), where these inverses are avoided by expressing G^-1 through a product of 2 rectangular matrices, and (A₂₂)^-1 through sparse matrix blocks of the inverse of full relationship matrix A^-1. A proper way to account genetic groups through unknown parent groups (UPG) after the Quaas-Pollak transformation (QP) is one key factor in a single-step model. When the UPG effects are incompletely accounted, the iterative solving method may have convergence problems. In this study, we investigated computational and predictive performance of ssGTBLUP with residual polygenic (RPG) effect and UPG. The QP transformation used A^-1 and, in the complete form, T and (A₂₂)^-1 matrices as well. The models were tested with official Nordic Holstein milk production test-day data and model. The results show that UPG can be easily implemented in ssGTBLUP having RPG. The complete QP transformation was computationally feasible when preconditioned conjugate gradient iteration and iteration on data without explicitly setting up G or A₂₂ matrices were used. Furthermore, for good convergence of the preconditioned conjugate gradient method, a complete QP transformation was necessary.

Genomic Analysis, Progress and Future Perspectives in Dairy Cattle Selection: A Review

Genomics comprises a set of current and valuable technologies implemented as selection tools in dairy cattle commercial breeding programs. The intensive progeny testing for production and reproductive traits based on genomic breeding values (GEBVs) has been crucial to increasing dairy cattle productivity. The knowledge of key genes and haplotypes, including their regulation mechanisms, as markers for productivity traits, may improve the strategies on the present and future for dairy cattle selection. Genome-wide association studies (GWAS) such as quantitative trait loci (QTL), single nucleotide polymorphisms (SNPs), or single-step genomic best linear unbiased prediction (ssGBLUP) methods have already been included in global dairy programs for the estimation of marker-assisted selection-derived effects. The increase in genetic progress based on genomic predicting accuracy has also contributed to the understanding of genetic effects in dairy cattle offspring. However, the crossing within inbred-lines critically increased homozygosis with accumulated negative effects of inbreeding like a decline in reproductive performance. Thus, inaccurate-biased estimations based on empirical-conventional models of dairy production systems face an increased risk of providing suboptimal results derived from errors in the selection of candidates of high genetic merit-based just on low-heritability phenotypic traits. This extends the generation intervals and increases costs due to the significant reduction of genetic gains. The remarkable progress of genomic prediction increases the accurate selection of superior candidates. The scope of the present review is to summarize and discuss the advances and challenges of genomic tools for dairy cattle selection for optimizing breeding programs and controlling negative inbreeding depression effects on productivity and consequently, achieving economic-effective advances in food production efficiency. Particular attention is given to the potential genomic selection-derived results to facilitate precision management on modern dairy farms, including an overview of novel genome editing methodologies as perspectives toward the future.

Comparison of models for missing pedigree in single-step genomic prediction

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.

Impact of missing pedigrees in single-step genomic evaluation

Context A common problem in mixed model-based genetic evaluation schemes for livestock is that cohorts of animals differ systematically in mean genetic merit, for example, due to missing pedigree. This can be modelled by fitting genetic groups. Single-step genomic evaluation (ssGBLUP) combining information from genotyped and non-genotyped individuals has become routine, but little is known of the effects of unknown parents in this context. Aims To investigate the effects of missing pedigrees on accuracy and bias of predicted breeding values for ssGBLUP analyses. Methods A simulation study was used to examine alternative ways to account for genetic groups in ssGBLUP, for multi-generation data with strong selection and rapidly increasing numbers of genotyped animals in the most recent generations. Key results Results demonstrated that missing pedigrees can markedly impair predicted breeding values. With selection, alignment of genomic and pedigree relationship matrices is essential when fitting unknown parent groups (UPG). Genomic relationships are complete; that is, they ‘automatically’ reference the genomic base, which typically differs from the genetic base for pedigreed animals. This can lead to biased comparisons between genotyped and non-genotyped animals with unknown parents when the two categories of animals are assigned to the same UPG. Allocating genotyped individuals to a separate UPG across all generations for each strain or breed was shown to be a simple and effective way to reduce misalignment bias. In contrast, fitting metafounders modified pedigree-based relationships to account for ancestral genomic relationships and inbreeding rather than the genomic relationship matrix. Thus, no bias due to different types of animals assigned to the same metafounders was apparent. Overall, fitting metafounders yielded slightly higher correlations between true and predicted breeding values than did UPG models, which assume genetic groups to be unrelated. Conclusions Missing pedigrees are more problematic with ssGBLUP than for analyses considering pedigree-based relationships only. UPG models with separation of genotyped and non-genotyped individuals and analyses fitting metafounders yielded comparable predictions of breeding values in terms of accuracy and bias. Implications A previously unidentified incompatibility between alignment of founder populations and assignment of genotyped and non-genotyped animals to the same UPG has been reported. Implementation of the proposed strategy to reduce ‘double counting’ is straightforward and can improve results of ssGBLUP analyses.

Genomic prediction ability for feed efficiency traits using different models and pseudo-phenotypes under several validation strategies in Nelore cattle

There is a growing interest to improve feed efficiency (FE) traits in cattle. The genomic selection was proposed to improve these traits since they are difficult and expensive to measure. Up to date, there are scarce studies about the implementation of genomic selection for FE traits in indicine cattle under different scenarios of pseudo-phenotypes, models, and validation strategies on a commercial large scale. Thus, the aim was to evaluate the feasibility of genomic selection implementation for FE traits in Nelore cattle applying different models and pseudo-phenotypes under validation strategies. Phenotypic and genotypic information from 4 329 and 3 467 animals were used, respectively, which were tested for residual feed intake, DM intake, feed efficiency, feed conversion ratio, residual BW gain, and residual intake and BW gain. Six prediction methods were used: single-step genomic best linear unbiased prediction, Bayes A, Bayes B, Bayes Cπ, Bayesian least absolute shrinkage and selection operator (BLASSO), and Bayes R. Phenotypes adjusted for fixed effects (Y*), estimated breeding value (EBV), and EBV deregressed (DEBV) were used as pseudo-phenotypes. The validation approaches used were: (1) random: the data was randomly divided into ten subsets and the validation was done in each subset at a time; (2) age: the partition into training and testing sets was based on year of birth and testing animals were born after 2016; and (3) EBV accuracy: the data was split into two groups, being animals with accuracy above 0.45 the training set; and below 0.45 the validation set. In the analyses that used the Y* as pseudo-phenotype, prediction ability (PA) was obtained by dividing the correlation between pseudo-phenotype and genomic EBV (GEBV) by the square root of the heritability of the trait. When EBV and DEBV were used as the pseudo-phenotype, the simple correlation of this quantity with the GEBV was considered as PA. The prediction methods show similar results for PA and bias. The random cross-validation presented higher PA (0.17) than EBV accuracy (0.14) and age (0.13). The PA was higher for Y* than for EBV and DEBV (30.0 and 34.3%, respectively). Random validation presented the highest PA, being indicated for use in populations composed mainly of young animals and traits with few generations of data recording. For high heritability traits, the validation can be done by age, enabling the prediction of the next-generation genetic merit. These results would support breeders to identify genomic approaches that are more viable for genomic prediction for FE-related traits.

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.

Application of multivariate single-step SNP best linear unbiased predictor model and revised SNP list for genomic evaluation of dairy cattle in Australia

The objectives of this study were (1) to evaluate the computational feasibility of the multitrait test-day single-step SNP-BLUP (ssSNP-BLUP) model using phenotypic records of genotyped and nongenotyped animals, and (2) to compare accuracies (coefficient of determination; R²) and bias of genomic estimated breeding values (GEBV) and de-regressed proofs as response variables in 3 Australian dairy cattle breeds (i.e., Holstein, Jersey, and Red breeds). Additive genomic random regression coefficients for milk, fat, protein yield and somatic cell score were predicted in the first, second, and third lactation. The predicted coefficients were used to derive 305-d GEBV and were compared with the traditional parent averages obtained from a BLUP model without genomic information. Cow fertility traits were evaluated from the 5-trait repeatability model (i.e., calving interval, days from calving to first service, pregnancy diagnosis, first service nonreturn rate, and lactation length). The de-regressed proofs were only for calving interval. Our results showed that ssSNP-BLUP using multitrait test-day model increased reliability and reduced bias of breeding values of young animals when compared with parent average from traditional BLUP in Australian Holsten, Jersey, and Red breeds. The use of a custom selection of approximately 46,000 SNP (custom XT SNP list) increased the reliability of GEBV compared with the results obtained using the commercial Illumina 50K chip (Illumina, San Diego, CA). The use of the second preconditioner substantially improved the convergence rate of the preconditioned conjugate gradient method, but further work is needed to improve the efficiency of the computation of the Kronecker matrix product by vector. Application of ssSNP-BLUP to multitrait random regression models is computationally feasible.

Computational strategies for the preconditioned conjugate gradient method applied to ssSNPBLUP, with an application to a multivariate maternal model

Background

The single-step single nucleotide polymorphism best linear unbiased prediction (ssSNPBLUP) is one of the single-step evaluations that enable a simultaneous analysis of phenotypic and pedigree information of genotyped and non-genotyped animals with a large number of genotypes. The aim of this study was to develop and illustrate several computational strategies to efficiently solve different ssSNPBLUP systems with a large number of genotypes on current computers.

Results

The different developed strategies were based on simplified computations of some terms of the preconditioner, and on splitting the coefficient matrix of the different ssSNPBLUP systems into multiple parts to perform its multiplication by a vector more efficiently. Some matrices were computed explicitly and stored in memory (e.g. the inverse of the pedigree relationship matrix), or were stored using a compressed form (e.g. the Plink 1 binary form for the genotype matrix), to permit the use of efficient parallel procedures while limiting the required amount of memory. The developed strategies were tested on a bivariate genetic evaluation for livability of calves for the Netherlands and the Flemish region in Belgium. There were 29,885,286 animals in the pedigree, 25,184,654 calf records, and 131,189 genotyped animals. The ssSNPBLUP system required around 18 GB Random Access Memory and 12 h to be solved with the most performing implementation.

Conclusions

Based on our proposed approaches and results, we showed that ssSNPBLUP provides a feasible approach in terms of memory and time requirements to estimate genomic breeding values using current computers.

Using Monte Carlo method to include polygenic effects in calculation of SNP-BLUP model reliability

An SNP-BLUP model is computationally scalable even for large numbers of genotyped animals. When genetic variation cannot be completely captured by SNP markers, a more accurate model is obtained by fitting a residual polygenic effect (RPG) as well. However, inclusion of the RPG effect increases the size of the SNP-BLUP mixed model equations (MME) by the number of genotyped animals. Consequently, the calculation of model reliabilities requiring elements of the inverted MME coefficient matrix becomes more computationally challenging with increasing numbers of genotyped animals. We present a Monte Carlo (MC)-based sampling method to estimate the reliability of the SNP-BLUP model including the RPG effect, where the MME size depends on the number of markers and MC samples. We compared reliabilities calculated using different RPG proportions and different MC sample sizes in analyzing 2 data sets. Data set 1 (data set 2) contained 19,757 (222,619) genotyped animals, with 11,729 (50,240) SNP markers, and 231,186 (13.35 million) pedigree animals. Correlations between the correct and the MC-calculated reliabilities were above 98% even with 5,000 MC samples and an 80% RPG proportion in both data sets. However, more MC samples were needed to achieve a small maximum absolute difference and mean squared error, particularly when the RPG proportion exceeded 20%. The computing time for MC SNP-BLUP was shorter than for GBLUP. In conclusion, the MC-based approach can be an effective strategy for calculating SNP-BLUP model reliability with an RPG effect included.

Current status of genomic evaluation

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.

Effects of ignoring inbreeding in model-based accuracy for BLUP and SSGBLUP

Model-based accuracy, defined as the theoretical correlation between true and estimated breeding value, can be obtained for each individual as a function of its prediction error variance (PEV) and inbreeding coefficient F, in BLUP, GBLUP and SSGBLUP genetic evaluations. However, for computational convenience, inbreeding is often ignored in two places. First, in the computation of reliability = 1-PEV/(1 + F). Second, in the set-up, using Henderson's rules, of the inverse of the pedigree-based relationship matrix A. Both approximations have an effect in the computation of model-based accuracy and result in wrong values. In this work, first we present a reminder of the theory and extend it to SSGBLUP. Second, we quantify the error of ignoring inbreeding with real data in three scenarios: BLUP evaluation and SSGBLUP in Uruguayan dairy cattle, and BLUP evaluations in a line of rabbit closed for >40 generations with steady increase of inbreeding up to an average of 0.30. We show that ignoring inbreeding in the set-up of the A-inverse is equivalent to assume that non-inbred animals are actually inbred. This results in an increase of apparent PEV that is negligible for dairy cattle but considerable for rabbit. Ignoring inbreeding in reliability = 1-PEV/(1 + F) leads to underestimation of reliability for BLUP evaluations, and this underestimation is very large for rabbit. For SSGBLUP in dairy cattle, it leads to both underestimation and overestimation of reliability, both for genotyped and non-genotyped animals. We strongly recommend to include inbreeding both in the set-up of A-inverse and in the computation of reliability from PEVs.

More animals than markers: a study into the application of the single step T-BLUP model in large-scale multi-trait Australian Angus beef cattle genetic evaluation

Multi-trait single step genetic evaluation is increasingly facing the situation of having more individuals with genotypes than markers within each genotype. This creates a situation where the genomic relationship matrix (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{G }$$\end{document}G) is not of full rank and its inversion is algebraically impossible. Recently, the SS-T-BLUP method was proposed as a modified version of the single step equations, providing an elegant way to circumvent the inversion of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{G }$$\end{document}G and therefore accommodate the situation described. SS-T-BLUP uses the Woodbury matrix identity, thus it requires an add-on matrix, which is usually the covariance matrix of the residual polygenic effet. In this paper, we examine the application of SS-T-BLUP to a large-scale multi-trait Australian Angus beef cattle dataset using the full BREEDPLAN single step genetic evaluation model and compare the results to the application of two different methods of using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{G }$$\end{document}G in a single step model. Results clearly show that SS-T-BLUP outperforms other single step formulations in terms of computational speed and avoids approximation of the inverse of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{G }$$\end{document}G.

Genomic prediction for crossbred performance using metafounders

Future genomic evaluation models to be used routinely in breeding programs for pigs and poultry need to be able to optimally use information of crossbred (CB) animals to predict breeding values for CB performance of purebred (PB) selection candidates. Important challenges in the commonly used single-step genomic best linear unbiased prediction (ssGBLUP) model are the definition of relationships between the different line compositions and the definition of the base generation per line. The use of metafounders (MFs) in ssGBLUP has been proposed to overcome these issues. When relationships between lines are known to be different from 0, the use of MFs generalizes the concept of genetic groups relying on the genotype data. Our objective was to investigate the effect of using MFs in genomic prediction for CB performance on estimated variance components, and accuracy and bias of GEBV. This was studied using stochastic simulation to generate data representing a three-way crossbreeding scheme in pigs, with the parental lines being either closely related or unrelated. Results show that using MFs, the variance components should be scaled appropriately, especially when basing them on estimates obtained with, for example a pedigree-based model. The accuracies of GEBV that were obtained using MFs were similar to accuracies without using MFs, regardless whether the lines involved in the CB were closely related or unrelated. The use of MFs resulted in a model that had similar or somewhat better convergence properties compared to other models. We recommend the use of MFs in ssGBLUP for genomic evaluations in crossbreeding schemes.

Efficient and accurate computation of base generation allele frequencies

Allele frequencies are used for several aspects of genomic prediction, with the assumption that these are equal to the allele frequency in the base generation of the pedigree. The current standard method, however, calculates allele frequencies from the current genotyped population. We compared the current standard method with BLUP and general least squares (GLS) methods explicitly targeting the base population to determine whether there is a more accurate and still efficient method of calculating allele frequencies that better represents the base generation. A data set based on a typical dairy population was simulated for 325,266 animals; the last 100,078 animals in generations 9 to 12 of the population were genotyped, with 1,670 SNP markers. For the BLUP method, several SNP genotypes were analyzed with a multitrait model by assuming a heritability of 0.99 and no genetic correlation among them. This method was limited by the time required for each BLUP to converge (approximately 6 min per BLUP run of 15 SNP). The GLS method had 2 implementations. The first implementation, using imputation on the fly and multiplication of sparse matrices, was very efficient and required just 49 s and 1.3 GB of random access memory. The second implementation, using a dense full A₂₂^-1 matrix, was very inefficient and required more than 1 d of wall clock time and more than 118.2 GB of random access memory. When no selection was considered in the simulations, all methods predicted equally well. When selection was introduced, higher correlations between the estimated allele frequency and known base generation allele frequency were observed for BLUP (0.96 ± 0.01) and GLS (0.97 ± 0.01) compared with the current standard method (0.87 ± 0.01). The GLS method decreased in accuracy when introducing incomplete pedigree, with 25% of sires in the first 5 generations randomly replaced as unknown to erroneously identify founder animals (0.93 ± 0.01) and a further decrease for 8 generations (0.91 ± 0.01). There was no change in accuracy when introducing 5% genotyping errors (0.97 ± 0.01), 5% missing genotypes (0.97 ± 0.01), or both 5% genotyping errors and missing genotypes (0.97 ± 0.01). The GLS method provided the most accurate estimates of base generation allele frequency and was only slightly slower compared with the current method. The efficient implementation of the GLS method, therefore, is very well suited for practical application and is recommended for implementation.

Estimates of genetic trend for single-step genomic evaluations

Background

A common measure employed to evaluate the efficacy of livestock improvement schemes is the genetic trend, which is calculated as the means of predicted breeding values for animals born in successive time periods. This implies that different cohorts refer to the same base population. For genetic evaluation schemes integrating genomic information with records for all animals, genotyped or not, this is often not the case: expected means for pedigree founders are zero whereas values for genotyped animals are expected to sum to zero at the (mean) time corresponding to the frequencies that are used to center marker allele counts when calculating genomic relationships.

Methods

The paper examines estimates of genetic trends from single-step genomic evaluations. After a review of methods which propose to align pedigree-based and genomic relationship matrices, simulation is used to illustrate the effects of alignments and choice of assumed gene frequencies on trajectories of genetic trends.

Results

The results show that methods available to alleviate differences between the founder populations implied by the two types of relationship matrices perform well; in particular, the meta-founder approach is advantageous. An application to data from routine genetic evaluation of Australian sheep is shown, confirming their effectiveness for practical data.

Conclusions

Aligning pedigree and genomic relationship matrices for single step genetic evaluation for populations under selection is essential. Fitting meta-founders is an effective and simple method to avoid distortion of estimates of genetic trends.

Sparse single-step genomic BLUP in crossbreeding schemes

The algorithm for proven and young animals (APY) efficiently computes an approximated inverse of the genomic relationship matrix, by dividing genotyped animals in the so-called core and noncore animals. The APY leads to computationally feasible single-step genomic Best Linear Unbiased Prediction (ssGBLUP) with a large number of genotyped animals and was successfully applied to real single-breed or line datasets. This study aimed to assess the quality of genomic estimated breeding values (GEBV) when using the APY (GEBVAPY), in comparison to GEBV when using the directly inverted genomic relationship matrix (GEBVDIRECT), for situations based on crossbreeding schemes, including F1 and F2 crosses, such as the ones for pigs and chickens. Based on simulations of a 3-way crossbreeding program, we compared different approximated inverses of a genomic relationship matrix, by varying the size and the composition of the core group. We showed that GEBVAPY were accurate approximations of GEBVDIRECT for multivariate ssGBLUP involving different breeds and their crosses. GEBVAPY as accurate as GEBVDIRECT were obtained when the core groups included animals from different breed compositions and when the core groups had a size between the numbers of the largest eigenvalues explaining 98% and 99% of the variation in the raw genomic relationship matrix.

A fast indirect method to compute functions of genomic relationships concerning genotyped and ungenotyped individuals, for diversity management

Background

Pedigree-based management of genetic diversity in populations, e.g., using optimal contributions, involves computation of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{Ax}}$$\end{document}Ax type yielding elements (relationships) or functions (usually averages) of relationship matrices. For pedigree-based relationships \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{A}}$$\end{document}A, a very efficient method exists. When all the individuals of interest are genotyped, genomic management can be addressed using the genomic relationship matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{G}}$$\end{document}G; however, to date, the computational problem of efficiently computing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{Gx}}$$\end{document}Gx has not been well studied. When some individuals of interest are not genotyped, genomic management should consider the relationship matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{H}}$$\end{document}H that combines genotyped and ungenotyped individuals; however, direct computation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{Hx}}$$\end{document}Hx is computationally very demanding, because construction of a possibly huge matrix is required. Our work presents efficient ways of computing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{Gx}}$$\end{document}Gx and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{Hx}}$$\end{document}Hx, with applications on real data from dairy sheep and dairy goat breeding schemes.

Results

For genomic relationships, an efficient indirect computation with quadratic instead of cubic cost is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{x}} = {\mathbf{Z}}\left( {{\mathbf{Z^{\prime}x}}} \right)/k$$\end{document}x=ZZ′x/k, where Z is a matrix relating animals to genotypes. For the relationship matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{H}}$$\end{document}H, we propose an indirect method based on the difference between vectors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{Hx}} - {\mathbf{Ax}}$$\end{document}Hx-Ax, which involves computation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{Ax}}$$\end{document}Ax and of products such as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{Gw}}$$\end{document}Gw and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{A}}_{22}^{ - 1} {\mathbf{w}}$$\end{document}A22-1w, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{w}}$$\end{document}w is a working vector derived from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{x}}$$\end{document}x. The latter computation is the most demanding but can be done using sparse Cholesky decompositions of matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{A}}^{ - 1}$$\end{document}A-1, which allows handling very large genomic and pedigree data files. Studies based on simulations reported in the literature show that the trends of average relationships in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{H}}$$\end{document}H and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{A}}$$\end{document}A differ as genomic selection proceeds. When selection is based on genomic relationships but management is based on pedigree data, the true genetic diversity is overestimated. However, our tests on real data from sheep and goat obtained before genomic selection started do not show this.

Conclusions

We present efficient methods to compute elements and statistics of the genomic relationships \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{G}}$$\end{document}G and of matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{H}}$$\end{document}H that combines ungenotyped and genotyped individuals. These methods should be useful to monitor and handle genomic diversity.

Efficient single-step genomic evaluation for a multibreed beef cattle population having many genotyped animals

An equivalent computational approach called ssGTBLUP was formulated for the original single-step GBLUP (ssGBLUP). In ssGTBLUP, the genomic relationship matrix has the form = ' + , where the (centered and scaled) marker matrix has size x (numbers of genotypes and markers), and the matrix can be easily inverted. The inverse can be written as = - ' where is an by matrix. When the preconditioned conjugate gradient (PCG) method is used to solve the mixed model equations, a matrix vector product needs to be computed. In ssGBLUP, this requires multiplications, but in ssGTBLUP, the product ' has 2 multiplications and has multiplications with the constant independent of or . In an approximate approach called ssGTBLUP(p), the eigendecomposition of ' is used to reduce the number of rows in the matrix. Here, p is the percentage of total variance explained by the accepted eigenvalues. The objective of this study was to compare the performance of ssGBLUP, ssGTBLUP, ssGTBLUP(p), and the APY (algorithm for proven and young) method. In APY, the core had 50,000 (APY50K), 30,000 (APY30K), or 10,000 (APY10K) animals. The approaches were tested on the Irish beef carcass conformation genetic evaluation which has a heterogeneous multibreed population. The pedigree had 13.3 million animals. There were = 54,620 markers available from = 163,277 genotyped animals. For genotyped animals, the correlations of breeding values between ssGBLUP and ssGTBLUP(p) for the 11 traits in the model ranged from 0.999-1.000 for p = 99, 0.998-1.000 for p = 98, and 0.992-0.998 for p = 95 but were 0.994-1.000 for APY50K, 0.969-0.997 for APY30K, and 0.899-0.967 for APY10K. Computing times per iteration were 4.43, 3.30, 2.69, 2.29, 1.55, 1.76, 1.27, and 0.55 min for ssGBLUP, ssGTBLUP, ssGTBLUP(99), ssGTBLUP(98), ssGTBLUP(95), APY50K, APY30K, and APY10K, respectively. The ssGTBLUP(p) approach allowed a well-defined approximation to ssGBLUP and fast computations.