Accelerated matrix-vector multiplications for matrices involving genotype covariates with applications in genomic prediction.

In the last decade, a number of methods have been suggested to deal with large amounts of genetic data in genomic predictions. Yet, steadily growing population sizes and the suboptimal use of computational resources are pushing the practical application of these approaches to their limits. As an extension to the C/CUDA library miraculix, we have developed tailored solutions for the computation of genotype matrix multiplications which is a critical bottleneck in the empirical evaluation of many statistical models. We demonstrate the benefits of our solutions at the example of single-step models which make repeated use of this kind of multiplication. Targeting modern Nvidia® GPUs as well as a broad range of CPU architectures, our implementation significantly reduces the time required for the estimation of breeding values in large population sizes. miraculix is released under the Apache 2.0 license and is freely available at https://github.com/alexfreudenberg/miraculix.

Integration of beef cattle international pedigree and genomic estimated breeding values into national evaluations, with an application to the Italian Limousin population.

BACKGROUND: International evaluations combine data from different countries allowing breeders to have access to larger panels of elite bulls and to increase the accuracy of estimated breeding values (EBV). However, international and national evaluations can use different sources of information to compute EBV (EBVINT and EBVNAT, respectively), leading to differences between them. Choosing one of these EBV results in losing the information that is contained only in the discarded EBV. Our objectives were to define and validate a procedure to integrate publishable sires' EBVINT and their associated reliabilities computed from pedigree-based or single-step international beef cattle evaluations into national evaluations to obtain "blended" EBV. The Italian (ITA) pedigree-based national evaluation was used as a case study to validate the integration procedure. METHODS: Publishable sires' international information, i.e. EBVINT and their associated reliabilities, was included in the national evaluation as pseudo-records. Data were available for 444,199 individual age-adjusted weaning weights of Limousin cattle from eight countries and 17,607 genotypes from four countries (ITA excluded). To mimic differences between international and national evaluations, international evaluations included phenotypes (and genotypes) of animals born prior to January 2019, while national evaluations included ITA phenotypes of animals born until April 2019. International evaluations using all available information were considered as reference scenarios. Publishable sires were divided into three groups: sires with ≥ 15, < 15 and no recorded offspring in ITA. RESULTS: Overall, for these three groups, integrating either pedigree-based or single-step international information into national pedigree-based evaluations improved the similarity of the blended EBV with the reference EBV compared to national evaluations without integration. For instance, the correlation with the reference EBV for direct (maternal) EBV went from 0.61 (0.79) for a national evaluation without integration to 0.97 (0.88) when integrating single-step international information, on average across all groups of publishable sires. CONCLUSIONS: Our proposed one-animal-at-a-time integration procedure yields blended EBV that are in close agreement with full international EBV for all groups of animals analysed. The procedure can be directly applied by countries since it does not rely on specific software and is computationally inexpensive, allowing straightforward integration of publishable sires' EBVINT from pedigree-based or single-step based international beef cattle evaluations into national evaluations.

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.

Validation with single-step SNPBLUP shows that evaluations can continue using a single mean of genotyped individuals, even with multiple breeds.

BACKGROUND: In genomic prediction, it is common to centre the genotypes of single nucleotide polymorphisms based on the allele frequencies in the current population, rather than those in the base generation. The mean breeding value of non-genotyped animals is conditional on the mean performance of genotyped relatives, but can be corrected by fitting the mean performance of genotyped individuals as a fixed regression. The associated covariate vector has been referred to as a 'J-factor', which if fitted as a fixed effect can improve the accuracy and dispersion bias of sire genomic estimated breeding values (GEBV). To date, this has only been performed on populations with a single breed. Here, we investigated whether there was any benefit in fitting a separate J-factor for each breed in a three-way crossbred population, and in using pedigree-based expected or genome-based estimated breed fractions to define the J-factors. RESULTS: For body weight at 7 days, dispersion bias decreased when fitting multiple J-factors, but only with a low proportion of genotyped individuals with selective genotyping. On average, the mean regression coefficients of validation records on those of GEBV increased with one J-factor compared to none, and further increased with multiple J-factors. However, for body weight at 35 days this was not observed. The accuracy of GEBV remained unchanged regardless of the J-factor method used. Differences between the J-factor methods were limited with correlations approaching 1 for the estimated covariate vector, the estimated coefficients of the regression on the J-factors, and the GEBV. CONCLUSIONS: Based on our results and in the particular design analysed here, i.e. all the animals with phenotype are of the same type of crossbreds, fitting a single J-factor should be sufficient, to reduce dispersion bias. Fitting multiple J-factors may reduce dispersion bias further but this depends on the trait and genotyping rate. For the crossbred population analysed, fitting multiple J-factors has no adverse consequences and if this is done, it does not matter if the breed fractions used are based on the pedigree-expectation or the genomic estimates. Finally, when GEBV are estimated from crossbred data, any observed bias can potentially be reduced by including a straightforward regression on actual breed proportions.

Impact of genomic preselection on subsequent ssGBLUP evaluation of preselected animals for scarcely recorded feed intake in pigs.

We have previously shown that single-step genomic best linear unbiased prediction (ssGBLUP) estimates breeding values of genomically preselected animals without preselection bias for widely recorded traits, that is traits recorded for the majority of animals in the breeding population. This study investigated the impact of genomic preselection (GPS) on accuracy and bias in ssGBLUP evaluation of genomically preselected animals for a scarcely recorded trait, that is a trait recorded for only a small proportion of the animals, which generally has a lower prediction accuracy than widely recorded traits, mainly due to having a much smaller number of phenotypes available. We used data from a commercial pig breeding program, considering feed intake as a scarcely recorded target trait, being available for ~30% of the animals with phenotypes for any trait, and average daily gain, backfat thickness and loin depth as widely recorded predictor traits, being available for >95% of the animals with phenotypes for any trait. The data contained the routine GPS implemented by commercial animal breeding programs, and we retrospectively implemented two scenarios with additional layers of GPS by discarding pedigree, genotypes and phenotypes of animals without progeny. The ssGBLUP evaluation following GPS used records only from the target trait, only from the predictor traits, or both. Accuracy for feed intake did not differ statistically across GPS scenarios, although it tended to decrease with more intense GPS. The accuracy had average values of 0.37, 0.44, and 0.45 across all GPS scenarios when, respectively, records from only the target trait, only the predictor traits, or both were used in the ssGBLUP evaluation. Considerable deflation of the genomic breeding values for feed intake was observed in the most stringent GPS scenario, due to the variance components being underestimated as a result of the limited amount of strongly preselected data. As long as (co)variance components were unbiased, no or only marginal bias was observed. These results for accuracy and bias were observed whether records of the scarcely recorded target trait, of the predictor traits, or both were used in the ssGBLUP evaluation. Our results show that for the scarcely recorded feed intake in pigs, ssGBLUP is able to estimate breeding values of preselected animals without preselection bias, similarly as previously observed for widely recorded traits.

Invited review: Reliability computation from the animal model era to the single-step genomic model era.

The calculation of exact reliabilities involving the inversion of mixed model equations poses a heavy computational challenge when the system of equations is large. This has prompted the development of different approximation methods. We give an overview of the various methods and computational approaches in calculating reliability from the era before the animal model to the era of single-step genomic models. The different methods are discussed in terms of modeling, development, and applicability in large dairy cattle populations. The paper also describes the problems faced in reliability computation. Many details dispersed throughout the literature are presented in this paper. It is clear that a universal solution applicable to every model and input data may not be possible, but we point out several efficient and accurate algorithms developed recently for a variety of very large genomic evaluations.

International single-step SNPBLUP beef cattle evaluations for Limousin weaning weight.

BACKGROUND: Compared to national evaluations, international collaboration projects further improve accuracies of estimated breeding values (EBV) by building larger reference populations or performing a joint evaluation using data (or proxy of them) from different countries. Genomic selection is increasingly adopted in beef cattle, but, to date, the benefits of including genomic information in international evaluations have not been explored. Our objective was to develop an international beef cattle single-step genomic evaluation and investigate its impact on the accuracy and bias of genomic evaluations compared to current pedigree-based evaluations. METHODS: Weaning weight records were available for 331,593 animals from seven European countries. The pedigree included 519,740 animals. After imputation and quality control, 17,607 genotypes at a density of 57,899 single nucleotide polymorphisms (SNPs) from four countries were available. We implemented two international scenarios where countries were modelled as different correlated traits: an international genomic single-step SNP best linear unbiased prediction (SNPBLUP) evaluation (ssSNPBLUPINT) and an international pedigree-based BLUP evaluation (PBLUPINT). Two national scenarios were implemented for pedigree and genomic evaluations using only nationally submitted phenotypes and genotypes. Accuracies, level and dispersion bias of EBV of animals born from 2014 onwards, and increases in population accuracies were estimated using the linear regression method. RESULTS: On average across countries, 39 and 17% of sires and maternal-grand-sires with recorded (grand-)offspring across two countries were genotyped. ssSNPBLUPINT showed the highest accuracies of EBV and, compared to PBLUPINT, led to increases in population accuracy of 13.7% for direct EBV, and 25.8% for maternal EBV, on average across countries. Increases in population accuracies when moving from national scenarios to ssSNPBLUPINT were observed for all countries. Overall, ssSNPBLUPINT level and dispersion bias remained similar or slightly reduced compared to PBLUPINT and national scenarios. CONCLUSIONS: International single-step SNPBLUP evaluations are feasible and lead to higher population accuracies for both large and small countries compared to current international pedigree-based evaluations and national evaluations. These results are likely related to the larger multi-country reference population and the inclusion of phenotypes from relatives recorded in other countries via single-step international evaluations. The proposed international single-step approach can be applied to other traits and breeds.

Estimation of dam line composition of 3-way crossbred animals using genomic information.

BACKGROUND: In genomic prediction including data of 3- or 4-way crossbred animals, line composition is usually fitted as a regression on expected line proportions, which are 0.5, 0.25 and 0.25, respectively, for 3-way crossbred animals. However, actual line proportions for the dam lines can vary between ~ 0.1 and 0.4, and ignoring this variation may affect the genomic estimated breeding values of purebred selection candidates. Our aim was to validate a proposed gold standard to evaluate different approaches for estimating line proportions using simulated data, and to subsequently use this in actual 3-way crossbred broiler data to evaluate several other methods. RESULTS: Analysis of simulated data confirmed that line proportions computed from assigned breed-origin-of-alleles (BOA) provide a very accurate gold standard, even if the parental lines are closely related. Alternative investigated methods were linear regression of genotypes on line-specific allele frequencies, maximum likelihood estimation using the program ADMIXTURE, and the genomic relationship of crossbred animals with their maternal grandparents. The results from the simulated data showed that the genomic relationship with the maternal grandparent was most accurate, and least affected by closer relationships between the dam lines. Linear regression and ADMIXTURE performed similarly for unrelated lines, but their accuracy dropped considerably when the dam lines were more closely related. In almost all cases, estimates improved after adjusting them to ensure that the sum of dam line contributions within animals was equal to 0.5, and within dam line and across animals the average was equal to 0.25. Results from the broiler data were much more similar between methods. In both cases, stringent linkage disequilibrium pruning of genotype data led to a relatively low accuracy of predicted line proportions, due to the loss of too many single nucleotide polymorphisms. CONCLUSIONS: With relatively unrelated parental lines as typical in crosses in pigs and poultry, linear regression of crossbred genotypes on line-specific allele frequencies and ADMIXTURE are very competitive methods. Thus, linear regression may be the method of choice, as it does not require genotypes of grandparents, is computationally very efficient, and easily implemented and adapted for considering the specific nature of the crossbred animals analysed.

Impact of genomic preselection on subsequent genetic evaluations with ssGBLUP using real data from pigs.

BACKGROUND: Empirically assessing the impact of preselection on genetic evaluation of preselected animals requires comparing scenarios that take different approaches into account, including scenarios without preselection. However, preselection is almost always performed in animal breeding programs, so it is difficult to have a dataset without preselection. Hence, most studies on preselection have used simulated datasets, and have concluded that genomic estimated breeding values (GEBV) from subsequent single-step genomic best linear unbiased prediction (ssGBLUP) evaluations are unbiased. The aim of this study was to investigate the impact of genomic preselection (GPS) on accuracy and bias in subsequent ssGBLUP evaluations, using data from a commercial pig breeding program. METHODS: We used data on average daily gain during performance testing, average daily gain throughout life, backfat thickness, and loin depth from one sire line and one dam line of pigs. As these traits have different weights in the breeding goals of the two lines, we analyzed the lines separately. For each line, we implemented a reference GPS scenario that kept all available data, against which the next two scenarios were compared. We then implemented two other scenarios with additional layers of GPS by removing all animals without progeny either (i) only in the validation generation, or (ii) in all generations. We conducted subsequent ssGBLUP evaluations for each GPS scenario, using all the data remaining after implementing the GPS scenario. Accuracy and bias were computed by comparing GEBV against progeny yield deviations of validation animals. RESULTS: Results for all traits and in both lines showed a marginal loss in accuracy due to the additional layers of GPS. Average accuracies across all GPS scenarios in the two lines were 0.39, 0.47, 0.56, and 0.60, for average daily gain during performance testing and throughout life, backfat thickness, and loin depth, respectively. Biases were largely absent, and when present, did not differ greatly between the GPS scenarios. CONCLUSIONS: We conclude that the impact of preselection on accuracy and bias in subsequent ssGBLUP evaluations of selection candidates in pigs is generally minimal. We expect this conclusion to apply for other animal breeding programs as well, since preselection of any type or intensity generally has the same effect in animal breeding programs.

The impact of direct-maternal genetic correlations on international beef cattle evaluations for Limousin weaning weight.

In beef cattle maternally influenced traits, estimates of direct-maternal genetic correlations (rdm) are usually reported to be negative. In international evaluations, rdm can differ both within countries (rdm_WC) and between countries (rdm_BC). The rdm_BC are difficult to estimate and are assumed to be zero in the current model for international beef cattle evaluations (Interbeef). Our objective was to investigate re-ranking of international estimated breeding values (IEBVs) in international beef cattle evaluations between models that either used estimated values for rdm or assumed them to be 0. Age-adjusted weaning weights and pedigree data were available for Limousin beef cattle from ten European countries. International EBVs were obtained using a multi-trait animal model with countries modeled as different traits. We compared IEBVs from a model that uses estimated rdm_BC (ranging between -0.14 and +0.14) and rdm_WC (between -0.33 and +0.40) with IEBVs obtained either from the current model that assumes rdm_BC to be 0, or from an alternative model that assumes both rdm_BC and rdm_WC to be 0. Direct and maternal IEBVs were compared across those three scenarios for different groups of animals. The ratio of population accuracies from the linear regression method was used to further investigate the impact of rdm on international evaluations, for both the whole set of animals in the evaluation and the domestic ones. Ignoring rdm_BC, i.e., replacing estimated values with 0, resulted in no (rank correlations > 0.99) or limited (between 0.98 and 0.99) re-ranking for direct and maternal IEBVs, respectively. Both rdm_BC and rdm_WC had less impact on direct IEBVs than on maternal IEBVs. Re-ranking of maternal IEBVs decreased with increasing reliability. Ignoring rdm_BC resulted in no re-ranking for sires with IEBVs that might be exchanged across countries and limited re-ranking for the top 100 sires. Using estimated rdm_BC values instead of considering them to be 0 resulted in null to limited increases in population accuracy. Ignoring both rdm_BC and rdm_WC resulted in considerable re-ranking of animals' IEBVs in all groups of animals evaluated. This study showed the limited impact of the current practice of ignoring rdm_BC in international evaluations for Limousin weaning weight, most likely because the estimated rdm_BC was close to 0. We expect that these conclusions can be extended to other traits that have reported rdm values in the range of rdm_WC values for weaning weight in Limousin.

Convergence behavior of single-step GBLUP and SNPBLUP for different termination criteria.

BACKGROUND: The preconditioned conjugate gradient (PCG) method is the current method of choice for iterative solving of genetic evaluations. The relative difference between two successive iterates and the relative residual of the system of equations are usually chosen as a termination criterion for the PCG method in animal breeding. However, our initial analyses showed that these two commonly used termination criteria may report that a PCG method applied to a single-step single nucleotide polymorphism best linear unbiased prediction (ssSNPBLUP) is not converged yet, whereas the solutions are accurate enough for practical use. Therefore, the aim of this study was to propose two termination criteria that have been (partly) developed in other fields, but are new in animal breeding, and to compare their behavior to that of the two termination criteria widely used in animal breeding for the PCG method applied to ssSNPBLUP. The convergence patterns of ssSNPBLUP were also compared to the convergence patterns of single-step genomic BLUP (ssGBLUP). RESULTS: Building upon previous work, we propose two termination criteria that take the properties of the system of equations into account. These two termination criteria are directly related to the relative error of the iterates with respect to the true solutions. Based on pig and dairy cattle datasets, we show that the preconditioned coefficient matrices of ssSNPBLUP and ssGBLUP have similar properties when using a second-level preconditioner for ssSNPBLUP. Therefore, the PCG method applied to ssSNPBLUP and ssGBLUP converged similarly based on the relative error of the iterates with respect to the true solutions. This similar convergence behavior between ssSNPBLUP and ssGBLUP was observed for both proposed termination criteria. This was, however, not the case for the termination criterion defined as the relative residual when applied to the dairy cattle evaluations. CONCLUSION: Our results showed that the PCG method can converge similarly when applied to ssSNPBLUP and to ssGBLUP. The two proposed termination criteria always depicted these similar convergence behaviors, and we recommend them for comparing convergence properties of different models and for routine evaluations.

Technical note: Genetic groups in single-step single nucleotide polymorphism best linear unbiased predictor.

Genetic groups, also called unknown or phantom parents groups, are often used in dairy cattle genetic evaluations to account for selection that cannot be accounted for by known genetic relationships. With the advent of genomic evaluations, the theory of genetic groups was extended to the so-called single-step genomic BLUP (ssGBLUP). In short, genetic groups can be fitted in ssGBLUP through regression effects, or by including them in the pedigree and computing the adequate combined pedigree and genomic relationship matrix. In this study, we applied the so-called Quaas and Pollak transformation to a system of equations for single-step SNP BLUP (ssSNPBLUP), such that genetic groups can thereafter be included in the pedigree. The example in this study showed that including the genetic groups in the pedigree for ssSNPBLUP allowed reduced memory burden and computational costs in comparison to genetic groups fitted as covariates.

Avoiding preselection bias in subsequent single-step genomic BLUP evaluations of genomically preselected animals.

In animal breeding, parents of the next generation are usually selected in multiple stages, and the initial stages of this selection are called preselection. Preselection reduces the information available for subsequent evaluation of preselected animals and this sometimes leads to bias. The objective of this study was to establish the minimum information required to subsequently evaluate genomically preselected animals without bias arising from preselection, with single-step genomic best linear unbiased prediction (ssGBLUP). We simulated a nucleus of a breeding program in which a recent population of 15 generations was produced. In each generation, parents of the next generation were selected in a single-stage selection based on pedigree BLUP. However, in generation 15, 10% of male and 15% of female offspring were preselected on their genomic estimated breeding values (GEBV). These GEBV were estimated using ssGBLUP, including the pedigree of all animals in generations 0-15, genotypes of all animals in generations 13-15 and phenotypes of all animals in generations 11-14. In subsequent ssGBLUP evaluation of these preselected animals, genotypes and phenotypes from various groups of animals were excluded one after another. We found that GEBV of the preselected animals were only estimated without preselection bias when genotypes and phenotypes of all animals in generations 13 and 14 and of the preselected animals were included in the subsequent evaluation. We also found that genotypes of the animals discarded at preselection only helped in reducing preselection bias in GEBV of their preselected sibs when genotypes of their parents were absent or excluded from the subsequent evaluation. We concluded that to prevent preselection bias in subsequent ssGBLUP evaluation of genomically preselected animals, information representative of the reference data used in the evaluation at preselection and genotypes and phenotypes of the preselected animals are needed in the subsequent evaluation.

Single-step genome-wide association studies (GWAS) and post-GWAS analyses to identify genomic regions and candidate genes for milk yield in Brazilian Girolando cattle.

Milk production is economically important to the Brazilian agribusiness, and the majority of the country's milk production derives from Girolando (Gir × Holstein) cows. This study aimed to identify quantitative trait loci (QTL) and candidate genes associated with 305-d milk yield (305MY) in Girolando cattle. In addition, we investigated the SNP-specific variances for Holstein and Gir breeds of origin within the sequence of candidate genes. A single-step genomic BLUP procedure was used to identify QTL associated with 305MY, and the most likely candidate genes were identified through follow-up analyses. Genomic breeding values specific for Holstein and Gir were estimated in the Girolando animals using a model that uses breed-specific partial relationship matrices, which were converted to breed of origin SNP effects. Differences between breed of origin were evaluated by comparing estimated SNP variances between breeds. From 10 genome regions explaining most additive genetic variance for 305MY in Girolando cattle, 7 candidate genes were identified on chromosomes 1, 4, 6, and 26. Within the sequence of these 7 candidate genes, Gir breed of origin SNP alleles showed the highest genetic variance. These results indicated QTL regions that could be further explored in genomic selection panels and which may also help in understanding the gene mechanisms involved in milk production in the Girolando breed.

Investigating the impact of preselection on subsequent single-step genomic BLUP evaluation of preselected animals.

BACKGROUND: Preselection of candidates, hereafter referred to as preselection, is a common practice in breeding programs. Preselection can cause bias and accuracy loss in subsequent pedigree-based best linear unbiased prediction (PBLUP). However, the impact of preselection on subsequent single-step genomic BLUP (ssGBLUP) is not completely clear yet. Therefore, in this study, we investigated, across different heritabilities, the impact of intensity and type of preselection on subsequent ssGBLUP evaluation of preselected animals. METHODS: We simulated a nucleus of a breeding programme, in which a recent population of 15 generations was produced with PBLUP-based selection. In generation 15 of this recent population, the parents of the next generation were preselected using several preselection scenarios. These scenarios were combinations of three intensities of preselection (no, high or very high preselection) and three types of preselection (genomic, parental average or random), across three heritabilities (0.5, 0.3 or 0.1). Following each preselection scenario, a subsequent evaluation was performed using ssGBLUP by excluding all the information from the preculled animals, and these genetic evaluations were compared in terms of accuracy and bias for the preselected animals, and in terms of realized genetic gain. RESULTS: Type of preselection affected selection accuracy at both preselection and subsequent evaluation stages. While preselection accuracy decreased, accuracy in the subsequent ssGBLUP evaluation increased, from genomic to parent average to random preselection scenarios. Bias was always negligible. Genetic gain decreased from genomic to parent average to random preselection scenarios. Genetic gain also decreased with increasing intensity of preselection, but only by a maximum of 0.1 additive genetic standard deviation from no to very high genomic preselection scenarios. CONCLUSIONS: Using ssGBLUP in subsequent evaluations prevents preselection bias, irrespective of intensity and type of preselection, and heritability. With GPS, in addition to reducing the phenotyping effort considerably, the use of ssGBLUP in subsequent evaluations realizes only a slightly lower genetic gain than that realized without preselection. This is especially the case for traits that are expensive to measure (e.g. feed intake of individual broiler chickens), and traits for which phenotypes can only be measured at advanced stages of life (e.g. litter size in pigs).

Impact of sub-setting the data of the main Limousin beef cattle population on the estimates of across-country genetic correlations.

BACKGROUND: Cattle international genetic evaluations allow the comparison of estimated breeding values (EBV) across different environments, i.e. countries. For international evaluations, across-country genetic correlations (rg) need to be estimated. However, lack of convergence of the estimated parameters and high standard errors of the rg are often experienced for beef cattle populations due to limited across-country genetic connections. Furthermore, using all available genetic connections to estimate rg is prohibitive due to computational constraints, thus sub-setting the data is necessary. Our objective was to investigate and compare the impact of strategies of data sub-setting on estimated across-country rg and their computational requirements. METHODS: Phenotype and pedigree information for age-adjusted weaning weight was available for ten European countries and 3,128,338 Limousin beef cattle males and females. Using a Monte Carlo based expectation-maximization restricted maximum likelihood (MC EM REML) methodology, we estimated across-country rg by using a multi-trait animal model where countries are modelled as different correlated traits. Values of rg were estimated using the full data and four different sub-setting strategies that aimed at selecting the most connected herds from the largest population. RESULTS: Using all available data, direct and maternal rg (standard errors in parentheses) were on average equal to 0.79 (0.14) and 0.71 (0.19), respectively. Direct-maternal within-country and between-country rg were on average equal to - 0.12 (0.09) and 0.00 (0.14), respectively. Data sub-setting scenarios gave similar results: on average, estimated rg were smaller compared to using all data for direct (0.02) and maternal (0.05) genetic effects. The largest differences were obtained for the direct-maternal within-country and between-country rg, which were, on average 0.13 and 0.12 smaller compared to values obtained by using all data. Standard errors always increased when reducing the data, by 0.02 to 0.06, on average. The proposed sub-setting strategies reduced the required computing time up to 22% compared to using all data. CONCLUSIONS: Estimating all 120 across-country rg that are required for beef cattle international evaluations, using a multi-trait MC EM REML approach, is feasible but involves long computing time. We propose four strategies to reduce computational requirements while keeping a multi-trait estimation approach. In all scenarios with data sub-setting, the estimated rg were consistently smaller (mainly for direct-maternal rg) and had larger standard errors.

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.

Variance estimates are similar using pedigree or genomic relationships with or without the use of metafounders or the algorithm for proven and young animals1.

With an increase in the number of animals genotyped there has been a shift from using pedigree relationship matrices (A) to genomic ones. As the use of genomic relationship matrices (G) has increased, new methods to build or approximate G have developed. We investigated whether the way variance components are estimated should reflect these changes. We estimated variance components for maternal sow traits by solving with restricted maximum likelihood, with four methods of calculating the inverse of the relationship matrix. These methods included using just the inverse of A (A-1), combining A-1 and the direct inverse of G (HDIRECT-1), including metafounders (HMETA-1), or combining A-1 with an approximated inverse of G using the algorithm for proven and young animals (HAPY-1). There was a tendency for higher additive genetic variances and lower permanent environmental variances estimated with A-1 compared with the three H-1 methods, which supports that G-1 is better than A-1 at separating genetic and permanent environmental components, due to a better definition of the actual relationships between animals. There were limited or no differences in variance estimates between HDIRECT-1, HMETA-1, and HAPY-1. Importantly, there was limited differences in variance components, repeatability or heritability estimates between methods. Heritabilities ranged between <0.01 to 0.04 for stayability after second cycle, and farrowing rate, between 0.08 and 0.15 for litter weight variation, maximum cycle number, total number born, total number still born, and prolonged interval between weaning and first insemination, and between 0.39 and 0.44 for litter birth weight and gestation length. The limited differences in heritabilities suggest that there would be very limited changes to estimated breeding values or ranking of animals across models using the different sets of variance components. It is suggested that variance estimates continue to be made using A-1, however including G-1 is possibly more appropriate if refining the model, for traits that fit a permanent environmental effect.

Assessment of sire contribution and breed-of-origin of alleles in a three-way crossbred broiler dataset.

Broiler breeding programs rely on crossbreeding. With genomic selection, widespread use of crossbred performance in breeding programs comes within reach. Commercial crossbreds, however, may have unknown pedigrees and their genomes may include DNA from 2 to 4 different breeds. Our aim was, for a broiler dataset with a limited number of sires having both purebred and crossbred offspring generated using natural mating, to rapidly derive parentage, assess the distribution of the sire contribution to the offspring generation, and to assess breed-of-origin of alleles in crossbreds. The dataset contained genotypes for 56,075 SNPs for 5,882 purebred and 10,943 3-way crossbred offspring generated by natural mating of 164 purebred sires to 1,016 purebred and 1,386 F1 crossbred hens. Using our algorithm FindParents, joint parentage derivation for the offspring and parent generations required only 1 m 29 s to retrieve parentage for 20,253 animals considering 4,504 possible parents. FindParents was similarly accurate as a maximum likelihood based method, apart from situations where settings of FindParents did not match the genotyping error rate in the data. Numbers of offspring per sire had a very skewed distribution, ranging from 1 to 270 crossbreds and 1 to 154 purebreds. Derivation of breed-of-origin of alleles relied on phasing all genotypes, including 8,205, 372, and 720 animals from the 3 pure lines involved, and allocating haplotypes in the crossbreds to purebred lines based on observed frequencies in the purebred lines. Breed-of-origin could be derived for 96.94% of the alleles of the 1,386 F1 crossbred hens and for 91.88% of the alleles of the 10,943 3-way crossbred offspring, of which 49.49% to the sire line. The achieved percentage of assignment to the sire line was sufficient to proceed with subsequent analyses requiring only the breed-of-origin of the paternal alleles to be known. Although required number of animals may be population dependent, to increase the total percentage of assigned alleles, it seems advisable to use at least approx. 1,000 genotyped purebred animals for each of the lines involved.