Estimating genomic relationships of metafounders across and within breeds using maximum likelihood, pseudo-expectation-maximization maximum likelihood and increase of relationships.

BACKGROUND: The theory of "metafounders" proposes a unified framework for relationships across base populations within breeds (e.g. unknown parent groups), and base populations across breeds (crosses) together with a sensible compatibility with genomic relationships. Considering metafounders might be advantageous in pedigree best linear unbiased prediction (BLUP) or single-step genomic BLUP. Existing methods to estimate relationships across metafounders Γ are not well adapted to highly unbalanced data, genotyped individuals far from base populations, or many unknown parent groups (within breed per year of birth). METHODS: We derive likelihood methods to estimate Γ . For a single metafounder, summary statistics of pedigree and genomic relationships allow deriving a cubic equation with the real root being the maximum likelihood (ML) estimate of Γ . This equation is tested with Lacaune sheep data. For several metafounders, we split the first derivative of the complete likelihood in a term related to Γ , and a second term related to Mendelian sampling variances. Approximating the first derivative by its first term results in a pseudo-EM algorithm that iteratively updates the estimate of Γ by the corresponding block of the H-matrix. The method extends to complex situations with groups defined by year of birth, modelling the increase of Γ using estimates of the rate of increase of inbreeding ( Δ F ), resulting in an expanded Γ and in a pseudo-EM+ Δ F algorithm. We compare these methods with the generalized least squares (GLS) method using simulated data: complex crosses of two breeds in equal or unsymmetrical proportions; and in two breeds, with 10 groups per year of birth within breed. We simulate genotyping in all generations or in the last ones. RESULTS: For a single metafounder, the ML estimates of the Lacaune data corresponded to the maximum. For simulated data, when genotypes were spread across all generations, both GLS and pseudo-EM(+ Δ F ) methods were accurate. With genotypes only available in the most recent generations, the GLS method was biased, whereas the pseudo-EM(+ Δ F ) approach yielded more accurate and unbiased estimates. CONCLUSIONS: We derived ML, pseudo-EM and pseudo-EM+ Δ F methods to estimate Γ in many realistic settings. Estimates are accurate in real and simulated data and have a low computational cost.

Redefining and interpreting genomic relationships of metafounders.

Metafounders are a useful concept to characterize relationships within and across populations, and to help genetic evaluations because they help modelling the means and variances of unknown base population animals. Current definitions of metafounder relationships are sensitive to the choice of reference alleles and have not been compared to their counterparts in population genetics-namely, heterozygosities, FST coefficients, and genetic distances. We redefine the relationships across populations with an arbitrary base of a maximum heterozygosity population in Hardy-Weinberg equilibrium. Then, the relationship between or within populations is a cross-product of the form Γ b , b ' = 2 n 2 p b - 1 2 p b ' - 1 ' with p being vectors of allele frequencies at n markers in populations b and b ' . This is simply the genomic relationship of two pseudo-individuals whose genotypes are equal to twice the allele frequencies. We also show that this coding is invariant to the choice of reference alleles. In addition, standard population genetics metrics (inbreeding coefficients of various forms; FST differentiation coefficients; segregation variance; and Nei's genetic distance) can be obtained from elements of matrix Γ .

Confidence intervals for validation statistics with data truncation in genomic prediction.

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

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.

Recording data for long years is common in animal breeding and genetics. However, the larger the data, the higher the computing cost of the analysis, especially with genomic information. This study aimed to investigate the impact of removing data, namely, genotypes, phenotypes, and pedigree, on the computing performance and prediction ability of genomic breeding values. We tested four scenarios to remove genotyped individuals in pig populations. For each scenario, phenotypes were removed according to birth year, and the pedigree was either kept complete or traced back from two to three generations. Reliabilities for young, genotyped animals did not differ after removing genotypes for older or less important animals. However, using only two generations of data slightly reduces the reliability for young, genotyped animals. The dispersion did not change across the studied scenarios, and its worst value was observed when using only one generation in the pedigree. Using the last ten years of phenotypes, a pedigree depth of three generations, and removing genotyped animals not contributing own or progeny phenotypes reduces computing cost with no change in the ability to predict breeding values.

Impact of interpopulation distance on dominance variance and average heterosis in hybrid populations within species.

Interpopulation improvement for crosses of close populations in crops and livestock depends on the amount of heterosis and the amount of variance of dominance deviations in the hybrids. It has been intuited that the further the distance between populations, the lower the amount of dominance variation and the higher the heterosis. Although experience in speciation and interspecific crosses shows, however, that this is not the case when populations are so distant-here we confine ourselves to the case of not-too-distant populations typical in crops and livestock. We present equations that relate the distance between 2 populations, expressed as Nei's genetic distance or as correlation of allele frequencies, quadratically to the amount of dominance deviations across all possible crosses and linearly to the expected heterosis averaging all possible crosses. The amount of variation of dominance deviations decreases with genetic distance until the point where allele frequencies are uncorrelated, and then increases for negatively correlated frequencies. Heterosis always increases with Nei's genetic distance. These expressions match well and complete previous theoretical and empirical findings. In practice, and for close enough populations, they mean that unless frequencies are negatively correlated, selection for hybrids will be more efficient when populations are distant.

Reliabilities of estimated breeding values in models with metafounders.

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

Implication of the order of blending and tuning when computing the genomic relationship matrix in single-step GBLUP.

Single-step genomic BLUP (ssGBLUP) relies on the combination of the genomic ( G $$ \mathbf{G} $$ ) and pedigree relationship matrices for all ( A $$ \mathbf{A} $$ ) and genotyped ( A 22 $$ {\mathbf{A}}_{22} $$ ) animals. The procedure ensures G $$ \mathbf{G} $$ and A 22 $$ {\mathbf{A}}_{22} $$ are compatible so that both matrices refer to the same genetic base ('tuning'). Then G $$ \mathbf{G} $$ is combined with a proportion of A 22 $$ {\mathbf{A}}_{22} $$ ('blending') to avoid singularity problems and to account for the polygenic component not accounted for by markers. This computational procedure has been implemented in the reverse order (blending before tuning) following the sequential research developments. However, blending before tuning may result in less optimal tuning because the blended matrix already contains a proportion of A 22 $$ {\mathbf{A}}_{22} $$ . In this study, the impact of 'tuning before blending' was compared with 'blending before tuning' on genomic estimated breeding values (GEBV), single nucleotide polymorphism (SNP) effects and indirect predictions (IP) from ssGBLUP using American Angus Association and Holstein Association USA, Inc. data. Two slightly different tuning methods were used; one that adjusts the mean diagonals and off-diagonals of G $$ \mathbf{G} $$ to be similar to those in A 22 $$ {\mathbf{A}}_{22} $$ and another one that adjusts based on the average difference between all elements of G $$ \mathbf{G} $$ and A 22 $$ {\mathbf{A}}_{22} $$ . Over 6 million Angus growth records and 5.9 million Holstein udder depth records were available. Genomic information was available on 51,478 Angus and 105,116 Holstein animals. Average realized relationship estimates among groups of animals were similar across scenarios. Scatterplots show that GEBV, SNP effects and IP did not noticeably change for all animals in the evaluation regardless of the order of computations and when using blending parameter of 0.05. Formulas were derived to determine the blending parameter that maximizes changes in the genomic relationship matrix and GEBV when changing the order of blending and tuning. Algebraically, the change is maximized when the blending parameter is equal to 0.5. Overall, tuning G $$ \mathbf{G} $$ before blending, regardless of blending parameter used, had a negligible impact on genomic predictions and SNP effects in this study.

Impacts of additive, dominance, and inbreeding depression effects on genomic evaluation by combining two SNP chips in Canadian Yorkshire pigs bred in China.

BACKGROUND: At the beginning of genomic selection, some Chinese companies genotyped pigs with different single nucleotide polymorphism (SNP) arrays. The obtained genomic data are then combined and to do this, several imputation strategies have been developed. Usually, only additive genetic effects are considered in genetic evaluations. However, dominance effects that may be important for some traits can be fitted in a mixed linear model as either 'classical' or 'genotypic' dominance effects. Their influence on genomic evaluation has rarely been studied. Thus, the objectives of this study were to use a dataset from Canadian Yorkshire pigs to (1) compare different strategies to combine data from two SNP arrays (Affymetrix 55K and Illumina 42K) and identify the most appropriate strategy for genomic evaluation and (2) evaluate the impact of dominance effects (classical' and 'genotypic') and inbreeding depression effects on genomic predictive abilities for average daily gain (ADG), backfat thickness (BF), loin muscle depth (LMD), days to 100 kg (AGE100), and the total number of piglets born (TNB) at first parity. RESULTS: The reliabilities obtained with the additive genomic models showed that the strategy used to combine data from two SNP arrays had little impact on genomic evaluations. Models with classical or genotypic dominance effect showed similar predictive abilities for all traits. For ADG, BF, LMD, and AGE100, dominance effects accounted for a small proportion (2 to 11%) of the total genetic variance, whereas for TNB, dominance effects accounted for 11 to 20%. For all traits, the predictive abilities of the models increased significantly when genomic inbreeding depression effects were included in the model. However, the inclusion of dominance effects did not change the predictive ability for any trait except for TNB. CONCLUSIONS: Our study shows that it is feasible to combine data from different SNP arrays for genomic evaluation, and that all combination methods result in similar accuracies. Regardless of how dominance effects are fitted in the genomic model, there is no impact on genetic evaluation. Models including inbreeding depression effects outperform a model with only additive effects, even if the trait is not strongly affected by dominant genes.

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

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

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

Genomic evaluation and genome-wide association studies for total number of teats in a combined American and Danish Yorkshire pig populations selected in China.

Joint genomic evaluation by combining data recordings and genomic information from different pig herds and populations is of interest for pig breeding companies because the efficiency of genomic selection (GS) could be further improved. In this work, an efficient strategy of joint genomic evaluation combining data from multiple pig populations is investigated. Total teat number (TTN), a trait that is equally recorded on 13,060 American Yorkshire (AY) populations (~14.68 teats) and 10,060 Danish Yorkshire (DY) pigs (~14.29 teats), was used to explore the feasibility and accuracy of GS combining datasets from different populations. We first estimated the genetic correlation (rg) of TTN between AY and DY pig populations (rg = 0.79, se = 0.23). Then we employed the genome-wide association study to identify quantitative trait locus (QTL) regions that are significantly associated with TTN and investigate the genetic architecture of TTN in different populations. Our results suggested that the genomic regions controlling TTN are slightly different in the two Yorkshire populations, where the candidate QTL regions were on SSC 7 and SSC 8 for the AY population and on SSC 7 for the DY population. Finally, we explored an optimal way of genomic prediction for TTN via three different genomic best linear unbiased prediction models and we concluded that when TTN across populations are regarded as different, but correlated, traits in a multitrait model, predictive abilities for both Yorkshire populations improve. As a conclusion, joint genomic evaluation for target traits in multiple pig populations is feasible in practice and more accurate, provided a proper model is used.

This study aimed at investigating joint genomic evaluation by combining data from multiple pig populations. Genomic evaluation is commonly applied in the pig industry to select the best animals to be the parents for the next generation. A bottleneck of genomic evaluation is that the selection accuracy is not high enough. To increase the selection accuracy, in theory, larger datasets are needed. In this article, multiple pig populations were considered together and we explored the feasibility and accuracy of genomic evaluation combining datasets from different populations. To realize the objective, total teat number (TTN), a trait that is equally recorded across different populations, was chosen. We first estimated the genetic correlation of TTN between American and Danish Yorkshire pig populations. Then to interpret why such genetic correlation was obtained, we employed the genome-wide association study to identify quantitative trait locus regions that are significantly associated with TTN and investigated the genetic architecture of TTN in different populations. Finally, we explored an optimal way of genomic prediction for TTN via three different genomic models and we concluded that when TTN across populations are regarded as different, but correlated, traits in a multitrait model, predictive abilities for both Yorkshire populations improve.

Genomic Prediction Methods Accounting for Nonadditive Genetic Effects.

The use of genomic information for prediction of future phenotypes or breeding values for the candidates to selection has become a standard over the last decade. However, most procedures for genomic prediction only consider the additive (or substitution) effects associated with polymorphic markers. Nevertheless, the implementation of models that consider nonadditive genetic variation may be interesting because they (1) may increase the ability of prediction, (2) can be used to define mate allocation procedures in plant and animal breeding schemes, and (3) can be used to benefit from nonadditive genetic variation in crossbreeding or purebred breeding schemes. This study reviews the available methods for incorporating nonadditive effects into genomic prediction procedures and their potential applications in predicting future phenotypic performance, mate allocation, and crossbred and purebred selection. Finally, a brief outline of some future research lines is also proposed.

Unfavorable genetic correlations between fecal egg count and milk production traits in the French blond-faced Manech dairy sheep breed.

BACKGROUND: Genetic selection has proven to be a successful strategy for the sustainable control of gastrointestinal parasitism in sheep. However, little is known on the relationship between resistance to parasites and production traits in dairy breeds. In this study, we estimated the heritabilities and genetic correlations for resistance to parasites and milk production traits in the blond-faced Manech breed. The resistance to parasites of 951 rams from the selection scheme was measured through fecal egg counts (FEC) at 30 days post-infection under experimental conditions. Six milk production traits [milk yield (MY), fat yield (FY), protein yield (PY), fat content (FC), protein content (PC) and somatic cell score (LSCS)], were used in this study and were collected on 140,127 dairy ewes in first lactation, as part of the official milk recording. These ewes were related to the 951 rams (65% of the ewes were daughters of the rams). RESULTS: Fecal egg counts at the end of the first and second infections were moderately heritable (0.19 and 0.37, respectively) and highly correlated (0.93). Heritabilities were moderate for milk yields (ranging from 0.24 to 0.29 for MY, FY and PY) and high for FC (0.35) and PC (0.48). MY was negatively correlated with FC and PC (- 0.39 and - 0.45, respectively). FEC at the end of the second infection were positively correlated with MY, FY and PY (0.28, 0.29 and 0.24, respectively with standard errors of ~ 0.10). These slightly unfavorable correlations indicate that the animals with a high production potential are genetically more susceptible to gastrointestinal parasite infections. A low negative correlation (- 0.17) was also found between FEC after the second infection and LSCS, which suggests that there is a small genetic antagonism between resistance to gastrointestinal parasites and resistance to mastitis, which is another important health trait in dairy sheep. CONCLUSIONS: Our results indicate an unfavorable but low genetic relationship between resistance to gastrointestinal parasites and milk production traits in the blond-faced Manech breed. These results will help the breeders' association make decisions about how to include resistance to parasites in the selection objective.

Computing strategies for multi-population genomic evaluation.

BACKGROUND: Multiple breed evaluation using genomic prediction includes the use of data from multiple populations, or from parental breeds and crosses, and is expected to lead to better genomic predictions. Increased complexity comes from the need to fit non-additive effects such as dominance and/or genotype-by-environment interactions. In these models, marker effects (and breeding values) are modelled as correlated between breeds, which leads to multiple trait formulations that are based either on markers [single nucleotide polymorphism best linear unbiased prediction (SNP-BLUP)] or on individuals [genomic(G)BLUP)]. As an alternative, we propose the use of generalized least squares (GLS) followed by backsolving of marker effects using selection index (SI) theory. RESULTS: All investigated options have advantages and inconveniences. The SNP-BLUP yields marker effects directly, which are useful for indirect prediction and for planned matings, but is very large in number of equations and is structured in dense and sparse blocks that do not allow for simple solving. GBLUP uses a multiple trait formulation and is very general, but results in many equations that are not used, which increase memory needs, and is also structured in dense and sparse blocks. An alternative formulation of GBLUP is more compact but requires tailored programming. The alternative of solving by GLS + SI is the least consuming, both in number of operations and in memory, and it uses only single dense blocks. However, it requires dedicated programming. Computational complexity problems are exacerbated when more than additive effects are fitted, e.g. dominance effects or genotype x environment interactions. CONCLUSIONS: As multi-breed predictions become more frequent and non-additive effects are more often included, standard equations for genomic prediction based on Henderson's mixed model equations become less practical and may need to be replaced by more efficient (although less general) approaches such as the GLS + SI approach proposed here.

Genetic evaluation including intermediate omics features.

In animal and plant breeding and genetics, there has been an increasing interest in intermediate omics traits, such as metabolomics and transcriptomics, which mediate the effect of genetics on the phenotype of interest. For inclusion of such intermediate traits into a genetic evaluation system, there is a need for a statistical model that integrates phenotypes, genotypes, pedigree, and omics traits, and a need for associated computational methods that provide estimated breeding values. In this paper, a joint model for phenotypes and omics data is presented, and a formula for the breeding values on individuals is derived. For complete omics data, three equivalent methods for best linear unbiased prediction of breeding values are presented. In all three cases, this requires solving two mixed model equation systems. Estimation of parameters using restricted maximum likelihood is also presented. For incomplete omics data, extensions of two of these methods are presented, where in both cases, the extension consists of extending an omics-related similarity matrix to incorporate individuals without omics data. The methods are illustrated using a simulated data set.

Dissecting genetic trends to understand breeding practices in livestock: a maternal pig line example.

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

The correlation of substitution effects across populations and generations in the presence of nonadditive functional gene action.

Allele substitution effects at quantitative trait loci (QTL) are part of the basis of quantitative genetics theory and applications such as association analysis and genomic prediction. In the presence of nonadditive functional gene action, substitution effects are not constant across populations. We develop an original approach to model the difference in substitution effects across populations as a first order Taylor series expansion from a "focal" population. This expansion involves the difference in allele frequencies and second-order statistical effects (additive by additive and dominance). The change in allele frequencies is a function of relationships (or genetic distances) across populations. As a result, it is possible to estimate the correlation of substitution effects across two populations using three elements: magnitudes of additive, dominance, and additive by additive variances; relationships (Nei's minimum distances or Fst indexes); and assumed heterozygosities. Similarly, the theory applies as well to distinct generations in a population, in which case the distance across generations is a function of increase of inbreeding. Simulation results confirmed our derivations. Slight biases were observed, depending on the nonadditive mechanism and the reference allele. Our derivations are useful to understand and forecast the possibility of prediction across populations and the similarity of GWAS effects.