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

Strategies for genomic predictions of an indicine multi-breed population using single-step GBLUP

Brazilian livestock breeding programmes strive to enhance the genetics of beef cattle, with a strong emphasis on the Nellore breed, which has an extensive database and has achieved significant genetic progress in the last years. There are other indicine breeds that are economically important in Brazil; however, these breeds have more modest sets of phenotypes, pedigree and genotypes, slowing down their genetic progress as their predictions are less accurate. Combining several breeds in a multi-breed evaluation could help enhance predictions for those breeds with less information available. This study aimed to evaluate the feasibility of multi-breed, single-step genomic best linear unbiased predictor genomic evaluations for Nellore, Brahman, Guzerat and Tabapua. Multi-breed evaluations were contrasted to the single-breed ones. Data were sourced from the National Association of Breeders and Researchers of Brazil and included pedigree (4,207,516), phenotypic (328,748), and genomic (63,492) information across all breeds. Phenotypes were available for adjusted weight at 210 and 450 days of age, and scrotal circumference at 365 days of age. Various scenarios were evaluated to ensure pedigree and genomic information compatibility when combining different breeds, including metafounders (MF) or building the genomic relationship matrix with breed-specific allele frequencies. Scenarios were compared using the linear regression method for bias, dispersion and accuracy. The results showed that using multi-breed evaluations significantly improved accuracy, especially for smaller breeds like Guzerat and Tabapua. The validation statistics indicated that the MF approach provided accurate predictions, albeit with some bias. While single-breed evaluations tended to have lower accuracy, merging all breeds in multi-breed evaluations increased accuracy and reduced dispersion. This study demonstrates that multi-breed genomic evaluations are proper for indicine beef cattle breeds. The MF approach may be particularly beneficial for less-represented breeds, addressing limitations related to small reference populations and incompatibilities between G and A22. By leveraging genomic information across breeds, breeders and producers can make more informed selection decisions, ultimately improving genetic gain in these cattle populations.

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.

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

BACKGROUND

By entering the era of mega-scale genomics, we are facing many computational issues with standard genomic evaluation models due to their dense data structure and cubic computational complexity. Several scalable approaches have been proposed to address this challenge, such as the Algorithm for Proven and Young (APY). In APY, genotyped animals are partitioned into core and non-core subsets, which induces a sparser inverse of the genomic relationship matrix. This partitioning is often done at random. While APY is a good approximation of the full model, random partitioning can make results unstable, possibly affecting accuracy or even reranking animals. Here we present a stable optimisation of the core subset by choosing animals with the most informative genotype data.

METHODS

We derived a novel algorithm for optimising the core subset based on a conditional genomic relationship matrix or a conditional single nucleotide polymorphism (SNP) genotype matrix. We compared the accuracy of genomic predictions with different core subsets for simulated and real pig data sets. The core subsets were constructed (1) at random, (2) based on the diagonal of the genomic relationship matrix, (3) at random with weights from (2), or (4) based on the novel conditional algorithm. To understand the different core subset constructions, we visualise the population structure of the genotyped animals with linear Principal Component Analysis and non-linear Uniform Manifold Approximation and Projection.

RESULTS

All core subset constructions performed equally well when the number of core animals captured most of the variation in the genomic relationships, both in simulated and real data sets. When the number of core animals was not sufficiently large, there was substantial variability in the results with the random construction but no variability with the conditional construction. Visualisation of the population structure and chosen core animals showed that the conditional construction spreads core animals across the whole domain of genotyped animals in a repeatable manner.

CONCLUSIONS

Our results confirm that the size of the core subset in APY is critical. Furthermore, the results show that the core subset can be optimised with the conditional algorithm that achieves an optimal and repeatable spread of core animals across the domain of genotyped animals.

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

Background

Single-step genomic predictions obtained from a breeding value model require calculating the inverse of 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}$$({\mathbf{G}}^{-1})$$\end{document}(G-1). The Algorithm for Proven and Young (APY) creates a sparse representation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{G}}^{-1}$$\end{document}G-1 with a low computational cost. APY consists of selecting a group of core animals and expressing the breeding values of the remaining animals as a linear combination of those from the core animals plus an error term. The objectives of this study were to: (1) extend APY to marker effects models; (2) derive equations for marker effect estimates when APY is used for breeding value models, and (3) show the implication of selecting a specific group of core animals in terms of a marker effects model.

Results

We derived a family of marker effects models called APY-SNP-BLUP. It differs from the classic marker effects model in that the row space of the genotype matrix is reduced and an error term is fitted for non-core animals. We derived formulas for marker effect estimates that take this error term in account. The prediction error variance (PEV) of the marker effect estimates depends on the PEV for core animals but not directly on the PEV of the non-core animals. We extended the APY-SNP-BLUP to include a residual polygenic effect and accommodate non-genotyped animals. We show that selecting a specific group of core animals is equivalent to select a subspace of the row space of the genotype matrix. As the number of core animals increases, subspaces corresponding to different sets of core animals tend to overlap, showing that random selection of core animals is algebraically justified.

Conclusions

The APY-(ss)GBLUP models can be expressed in terms of marker effect models. When the number of core animals is equal to the rank of the genotype matrix, APY-SNP-BLUP is identical to the classic marker effects model. If the number of core animals is less than the rank of the genotype matrix, genotypes for non-core animals are imputed as a linear combination of the genotypes of the core animals. For estimating SNP effects, only relationships and estimated breeding values for core animals are needed.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12711-022-00741-7.

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

Background

The algorithm for proven and young (APY) has been suggested as a solution for recursively computing a sparse representation for the inverse of a large genomic relationship matrix (G). In APY, a subset of genotyped individuals is used as the core and the remaining genotyped individuals are used as noncore. Size and definition of the core are relevant research subjects for the application of APY, especially given the ever-increasing number of genotyped individuals.

Methods

The aim of this study was to investigate several core definitions, including the most popular animals (MPA) (i.e., animals with high contributions to the genetic pool), the least popular males (LPM), the least popular females (LPF), a random set (Rnd), animals evenly distributed across genealogical paths (Ped), unrelated individuals (Unrel), or based on within-family selection (Fam), or on decomposition of the gene content matrix (QR). Each definition was evaluated for six core sizes based on prediction accuracy of single-step genomic best linear unbiased prediction (ssGBLUP) with APY. Prediction accuracy of ssGBLUP with the full inverse of G was used as the baseline. The dataset consisted of 357k pedigreed Duroc pigs with 111k pigs with genotypes and ~ 220k phenotypic records.

Results

When the core size was equal to the number of largest eigenvalues explaining 50% of the variation of G (n = 160), MPA and Ped core definitions delivered the highest average prediction accuracies (~ 0.41−0.53). As the core size increased to the number of eigenvalues explaining 99% of the variation in G (n = 7320), prediction accuracy was nearly identical for all core types and correlations with genomic estimated breeding values (GEBV) from ssGBLUP with the full inversion of G were greater than 0.99 for all core definitions. Cores that represent all generations, such as Rnd, Ped, Fam, and Unrel, were grouped together in the hierarchical clustering of GEBV.

Conclusions

For small core sizes, the definition of the core matters; however, as the size of the core reaches an optimal value equal to the number of largest eigenvalues explaining 99% of the variation of G, the definition of the core becomes arbitrary.

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

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

Emerging issues in genomic selection

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

Changes in genomic predictions when new information is added

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