A simple method for weighted bending of genetic (co)variance matrices

Genetic Parameters for Limousine Interbeef Genetic Evaluation of Calving Traits

The aim of this study was to estimate across-country genetic correlations for calving traits (birth weight, calving ease) in the Limousine breed. Correlations were estimated for eight populations (Czech Republic, joint population of Denmark, Finland, and Sweden, France, Great Britain, Ireland, Slovenia, Switzerland, and Estonia). An animal model on raw performance accounting for across-country interactions (AMACI) was used. (Co)variance components were estimated for pairwise combinations of countries. Fixed and random effects were defined by each country according to its national genetic evaluation system. The average across-country genetic correlation for the direct genetic effect was 0.85 for birth weight (0.69-0.96) and 0.75 for calving ease (0.62-0.94). The average correlation for the maternal genetic effect was 0.57 for birth weight and 0.61 for calving ease. After the estimation of genetic parameters, the weighted bending procedure was used to compute the full Interbeef genetic correlation matrix. After bending, direct genetic correlations ranged from 0.62 to 0.84 (with an average of 0.73) for birth weight and from 0.58 to 0.82 (with an average of 0.68) for calving ease.

Genetic Parameters of Performance and Conformation Traits of 3-Year-Old Warmblood Sport Horses in the Czech Republic

The aim of this study was to estimate the genetic parameters of a one-day performance test together with the linear type traits of 3-year-old warmblood horses. The study of genetic parameters was based on 5958 tested horses in the period 1998-2021. A total of 22 traits of linear description, three quantitatively measured traits, and one summary mark from the performance test were tested. The model equation included the fixed effect of gender and combination effects of classifier-year of evaluation-place. A single-trait animal model was used for the estimation of heritability and genetic variance, while the two-trait animal model was applied for the estimation of variance and covariance between all traits. The heritability of the overall score of the performance test was 0.25. The range for heritability was between 0.04 and 0.33 for the linear type traits and between 0.46 and 0.57 for the quantitatively measured traits. Genetic correlations were between -0.47 and 0.92. The estimated genetic parameters suggest that the results from the performance test can be incorporated into genetic evaluation in the Czech Republic.

Multi-environment genomic prediction for soluble solids content in peach (Prunus persica)

Genotype-by-environment interaction (G × E) is a common phenomenon influencing genetic improvement in plants, and a good understanding of this phenomenon is important for breeding and cultivar deployment strategies. However, there is little information on G × E in horticultural tree crops, mostly due to evaluation costs, leading to a focus on the development and deployment of locally adapted germplasm. Using sweetness (measured as soluble solids content, SSC) in peach/nectarine assessed at four trials from three US peach-breeding programs as a case study, we evaluated the hypotheses that (i) complex data from multiple breeding programs can be connected using GBLUP models to improve the knowledge of G × E for breeding and deployment and (ii) accounting for a known large-effect quantitative trait locus (QTL) improves the prediction accuracy. Following a structured strategy using univariate and multivariate models containing additive and dominance genomic effects on SSC, a model that included a previously detected QTL and background genomic effects was a significantly better fit than a genome-wide model with completely anonymous markers. Estimates of an individual's narrow-sense and broad-sense heritability for SSC were high (0.57-0.73 and 0.66-0.80, respectively), with 19-32% of total genomic variance explained by the QTL. Genome-wide dominance effects and QTL effects were stable across environments. Significant G × E was detected for background genome effects, mostly due to the low correlation of these effects across seasons within a particular trial. The expected prediction accuracy, estimated from the linear model, was higher than the realised prediction accuracy estimated by cross-validation, suggesting that these two parameters measure different qualities of the prediction models. While prediction accuracy was improved in some cases by combining data across trials, particularly when phenotypic data for untested individuals were available from other trials, this improvement was not consistent. This study confirms that complex data can be combined into a single analysis using GBLUP methods to improve understanding of G × E and also incorporate known QTL effects. In addition, the study generated baseline information to account for population structure in genomic prediction models in horticultural crop improvement.

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 (ssSNPBLUP_INT) and an international pedigree-based BLUP evaluation (PBLUP_INT). 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. ssSNPBLUP_INT showed the highest accuracies of EBV and, compared to PBLUP_INT, 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 ssSNPBLUP_INT were observed for all countries. Overall, ssSNPBLUP_INT level and dispersion bias remained similar or slightly reduced compared to PBLUP_INT 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.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12711-022-00748-0.

Invited review: A quarter of a century-International genetic evaluation of dairy sires using MACE methodology

For the past few decades, the international exchange of genetic materials has accelerated. This acceleration has been more substantial for dairy cattle compared with other species. The industry faced the need to put international genetic evaluation (IGE) systems in place. The Interbull Centre has been conducting IGE for various dairy cattle breeds and traits. This study reviews the past and the current status of IGE for dairy cattle, emphasizing the most prominent and well-established method of IGE, namely multiple across-country evaluation (MACE), and the challenges that should be addressed in the future of IGE. The first IGE methods were simple conversion equations. Only a limited number of common bulls between pairs of countries were considered. These bulls were a biased sample of highly selected animals, with their daughters under preferential treatment in the importing countries. Genetic relationships among animals were not considered either. The MACE method was the first IGE method based on mixed-model theory that could handle genotype by environment interaction (G × E) between countries. The G × E between countries is handled by treating the same trait in different countries as different traits, with genetic correlations less than unity between the traits. The G × E between countries is not solely due to different genetic expressions in different environments (countries), but is also attributable to different units or ways of measuring the trait, data editing, and statistical approaches and models used in different countries. The MACE method also considers different genetic means, genetic groups for unknown parents, heterogeneous genetic and residual variances among countries, and heterogeneous residual variances (precision weights for observations) within countries. Other IGE methods that came after MACE are rooted in MACE. The genomic revolution of the industry created new needs and opportunities. However, an unwanted aspect of it was genomic preselection bias. Genomic preselection causes directional information loss from pre-culled animals (bias) in statistical models for genetic and genomic evaluations, and preselected progeny of a mating are no longer a random sample of possible progeny from that mating. National genetic evaluations without genotypes are input to MACE, and biases in national evaluations are propagated internationally through MACE. Genomic preselection for the Holstein breed is a source of concern for introducing bias to MACE, especially when genomic preselection is practiced intensively in the population. However, MACE continues to be useful for other breeds, among other species, or for non-IGE purposes. Future methods will need to make optimum use of genomic information and be free of genomic preselection bias.

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 (r_dm) are usually reported to be negative. In international evaluations, r_dm can differ both within countries (r_{dm_WC}) and between countries (r_{dm_BC}). The r_{dm_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 r_dm 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 r_{dm_BC} (ranging between −0.14 and +0.14) and r_{dm_WC} (between −0.33 and +0.40) with IEBVs obtained either from the current model that assumes r_{dm_BC} to be 0, or from an alternative model that assumes both r_{dm_BC} and r_{dm_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 r_dm on international evaluations, for both the whole set of animals in the evaluation and the domestic ones. Ignoring r_{dm_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 r_{dm_BC} and r_{dm_WC} had less impact on direct IEBVs than on maternal IEBVs. Re-ranking of maternal IEBVs decreased with increasing reliability. Ignoring r_{dm_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 r_{dm_BC} values instead of considering them to be 0 resulted in null to limited increases in population accuracy. Ignoring both r_{dm_BC} and r_{dm_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 r_{dm_BC} in international evaluations for Limousin weaning weight, most likely because the estimated r_{dm_BC} was close to 0. We expect that these conclusions can be extended to other traits that have reported r_dm values in the range of r_{dm_WC} values for weaning weight in Limousin.

mbend: an R package for bending non-positive-definite symmetric matrices to positive-definite

BACKGROUND

R package mbend was developed for bending symmetric non-positive-definite matrices to positive-definite (PD). Bending is a procedure of transforming non-PD matrices to PD. The covariance matrices used in multi-trait best linear unbiased prediction (BLUP) should be PD. Two bending methods are implemented in mbend. The first is an unweighted bending with small positive values in a descending order replacing negative eigenvalues (LRS14), and the second method is a weighted (precision-based) bending with a custom small positive value (ϵ) replacing smaller eigenvalues (HJ03). Weighted bending is beneficial, as it relaxes low precision elements to change and it reduces or prohibits the change in high precision elements. Therefore, a weighted version of LRS14 was developed in mbend. In cases where the precision of matrix elements is unknown, the package provides an unweighted version of HJ03. Another unweighted bending method (DB88) was tested, by which all eigenvalues are changed (eigenvalues less than ϵ replaced with 100 × ϵ), and it is originally designed for correlation matrices.

RESULTS

Different bending procedures were conducted on a 5 × 5 covariance matrix (V), V converted to a correlation matrix (C) and an ill-conditioned 1000 × 1000 genomic relationship matrix (G). Considering weighted distance statistics between matrix elements before and after bending, weighting considerably improved the bending quality. For weighted and unweighted bending of V and C, HJ03-4 (HJ03, ϵ = 10^-4) performed the best. HJ03-2 (HJ03, ϵ = 10^-2) ranked better than LRS14 for V, but not for C. Though the differences were marginal, LRS14 performed the best for G. DB88-4 (DB88, ϵ = 10^-4) was used for unweighted bending and it ranked the last. This method could perform considerably better with a lower ϵ.

CONCLUSIONS

R package mbend provides necessary tools for transforming symmetric non-PD matrices to PD, using different methods and parameters. There were benefits in both weighted bending and small positive values in a descending order replacing negative eigenvalues. Thus, weighted LRS14 was implemented in mbend. Different bending methods might be preferable for different matrices, depending on the matrix type (covariance vs. correlation), number and the magnitude of negative eigenvalues, and the matrix size.

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 (r_g) need to be estimated. However, lack of convergence of the estimated parameters and high standard errors of the r_g are often experienced for beef cattle populations due to limited across-country genetic connections. Furthermore, using all available genetic connections to estimate r_g 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 r_g 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 r_g by using a multi-trait animal model where countries are modelled as different correlated traits. Values of r_g 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 r_g (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 r_g 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 r_g 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 r_g, 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 r_g 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 r_g were consistently smaller (mainly for direct-maternal r_g) and had larger standard errors.

Maternal body composition in seedstock herds. 3. Multivariate analysis using factor analytic models and cluster analysis

Considerable information exists on genetic relationships of body composition and carcass quality of young and finished beef cattle. However, there is a dearth of information on genetic relationships of cow body composition over time and, also, relationships with young-animal body-composition measures. The aim of the present study is to understand genetic relationships among various cow body-composition traits of Angus cows over time, from yearling to weaning of a second calf at ~3.5 years. To determine genetic correlations among various composition traits over time, a multi-trait–multi-time analysis is required. For the Maternal Productivity Project, this necessitates modelling of five traits (namely weight and ultrasound measure for loin eye muscle area (EMA), rib fat, P8 rump fat and intramuscular fat) by five time combinations (recordings at yearling then pre-calving and weaning in first and second parity). The approach was based on including all 25 trait-by-time combinations in an analysis using factor analytic models to approximate the genetic covariance matrix. Various models for the residual covariance structure were investigated. The analyses yielded correlations that could be compared with those of past studies reported in the literature and, also, to a set of bivariate analyses. Clustering of the genetic multi-trait–multi-time correlation structure resulted in a separation of traits (weight and EMA, and the fat traits) and also of time effects into early (heifer = before first lactation) and late (cow = post-first lactation) measurements.

Genomic evaluation for a three-way crossbreeding system considering breed-of-origin of alleles

BACKGROUND

Genomic prediction of purebred animals for crossbred performance can be based on a model that estimates effects of single nucleotide polymorphisms (SNPs) in purebreds on crossbred performance. For crossbred performance, SNP effects might be breed-specific due to differences between breeds in allele frequencies and linkage disequilibrium patterns between SNPs and quantitative trait loci. Accurately tracing the breed-of-origin of alleles (BOA) in three-way crosses is possible with a recently developed procedure called BOA. A model that accounts for breed-specific SNP effects (BOA model), has never been tested empirically on a three-way crossbreeding scheme. Therefore, the objectives of this study were to evaluate the estimates of variance components and the predictive accuracy of the BOA model compared to models in which SNP effects for crossbred performance were assumed to be the same across breeds, using either breed-specific allele frequencies ([Formula: see text] model) or allele frequencies averaged across breeds ([Formula: see text] model). In this study, we used data from purebred and three-way crossbred pigs on average daily gain (ADG), back fat thickness (BF), and loin depth (LD).

RESULTS

Estimates of variance components for crossbred performance from the BOA model were mostly similar to estimates from models [Formula: see text] and [Formula: see text]. Heritabilities for crossbred performance ranged from 0.24 to 0.46 between traits. Genetic correlations between purebred and crossbred performance ([Formula: see text]) across breeds ranged from 0.30 to 0.62 for ADG and from 0.53 to 0.74 for BF and LD. For ADG, prediction accuracies of the BOA model were higher than those of the [Formula: see text] and [Formula: see text] models, with significantly higher accuracies only for one maternal breed. For BF and LD, prediction accuracies of models [Formula: see text] and [Formula: see text] were higher than those of the BOA model, with no significant differences. Across all traits, models [Formula: see text] and [Formula: see text] yielded similar predictions.

CONCLUSIONS

The BOA model yielded a higher prediction accuracy for ADG in one maternal breed, which had the lowest [Formula: see text] (0.30). Using the BOA model was especially relevant for traits with a low [Formula: see text]. In all other cases, the use of crossbred information in models [Formula: see text] and [Formula: see text], does not jeopardize predictions and these models are more easily implemented than the BOA model.

Multi-population Genomic Relationships for Estimating Current Genetic Variances Within and Genetic Correlations Between Populations

Different methods are available to calculate multi-population genomic relationship matrices. Since those matrices differ in base population, it is anticipated that the method used to calculate genomic relationships affects the estimate of genetic variances, covariances, and correlations. The aim of this article is to define the multi-population genomic relationship matrix to estimate current genetic variances within and genetic correlations between populations. The genomic relationship matrix containing two populations consists of four blocks, one block for population 1, one block for population 2, and two blocks for relationships between the populations. It is known, based on literature, that by using current allele frequencies to calculate genomic relationships within a population, current genetic variances are estimated. In this article, we theoretically derived the properties of the genomic relationship matrix to estimate genetic correlations between populations and validated it using simulations. When the scaling factor of across-population genomic relationships is equal to the product of the square roots of the scaling factors for within-population genomic relationships, the genetic correlation is estimated unbiasedly even though estimated genetic variances do not necessarily refer to the current population. When this property is not met, the correlation based on estimated variances should be multiplied by a correction factor based on the scaling factors. In this study, we present a genomic relationship matrix which directly estimates current genetic variances as well as genetic correlations between populations.

Covariance Between Genotypic Effects and its Use for Genomic Inference in Half-Sib Families

In livestock, current statistical approaches utilize extensive molecular data, e.g., single nucleotide polymorphisms (SNPs), to improve the genetic evaluation of individuals. The number of model parameters increases with the number of SNPs, so the multicollinearity between covariates can affect the results obtained using whole genome regression methods. In this study, dependencies between SNPs due to linkage and linkage disequilibrium among the chromosome segments were explicitly considered in methods used to estimate the effects of SNPs. The population structure affects the extent of such dependencies, so the covariance among SNP genotypes was derived for half-sib families, which are typical in livestock populations. Conditional on the SNP haplotypes of the common parent (sire), the theoretical covariance was determined using the haplotype frequencies of the population from which the individual parent (dam) was derived. The resulting covariance matrix was included in a statistical model for a trait of interest, and this covariance matrix was then used to specify prior assumptions for SNP effects in a Bayesian framework. The approach was applied to one family in simulated scenarios (few and many quantitative trait loci) and using semireal data obtained from dairy cattle to identify genome segments that affect performance traits, as well as to investigate the impact on predictive ability. Compared with a method that does not explicitly consider any of the relationship among predictor variables, the accuracy of genetic value prediction was improved by 10–22%. The results show that the inclusion of dependence is particularly important for genomic inference based on small sample sizes.

Constraints on geographic variation in fiddler crabs (Ocypodidae: Uca) from the western Atlantic

A key question in evolutionary biology is how intraspecific variation biases the evolution of a population and its divergence from other populations. Such constraints potentially limit the extent to which populations respond to selection, but may endure long enough to have macroevolutionary consequences. Previous studies have focused on the association between covariation patterns and divergence among isolated populations. Few have focused on geographic variation among semi-connected populations, however, even though this may be indicative of early selective pressures that could lead to long-term divergence and speciation. Here, we test whether covariation in the shape of the carapace of fiddler crabs (genus Uca Leach, 1814) is important for structuring geographic variation. We find that morphological divergence among populations is associated with evolvability in the direction of divergence in only a few species. The shape of the ancestral covariation matrix in these species differs from other species in having notably more variation concentrated along fewer directions (i.e. higher eccentricity). For most species, there is some evidence that covariation has constrained the range of directions into which populations have diverged but not the degree of divergence. These results suggest that even though fiddler crab populations have diverged morphologically in directions predicted by covariation, constraints on the extent to which divergence has occurred may only be manifested in species where variation patterns are eccentric enough to limit populations' ability to respond effectively in many directions.

Phenotypic Covariation and Morphological Diversification in the Ruminant Skull

Differences among clades in their diversification patterns result from a combination of extrinsic and intrinsic factors. In this study, I examined the role of intrinsic factors in the morphological diversification of ruminants, in general, and in the differences between bovids and cervids, in particular. Using skull morphology, which embodies many of the adaptations that distinguish bovids and cervids, I examined 132 of the 200 extant ruminant species. As a proxy for intrinsic constraints, I quantified different aspects of the phenotypic covariation structure within species and compared them with the among-species divergence patterns, using phylogenetic comparative methods. My results show that for most species, divergence is well aligned with their phenotypic covariance matrix and that those that are better aligned have diverged further away from their ancestor. Bovids have dispersed into a wider range of directions in morphospace than cervids, and their overall disparity is higher. This difference is best explained by the lower eccentricity of bovids' within-species covariance matrices. These results are consistent with the role of intrinsic constraints in determining amount, range, and direction of dispersion and demonstrate that intrinsic constraints can influence macroevolutionary patterns even as the covariance structure evolves.

GenoMatrix: A Software Package for Pedigree-Based and Genomic Prediction Analyses on Complex Traits

Genomic and pedigree-based best linear unbiased prediction methodologies (G-BLUP and P-BLUP) have proven themselves efficient for partitioning the phenotypic variance of complex traits into its components, estimating the individuals' genetic merits, and predicting unobserved (or yet-to-be observed) phenotypes in many species and fields of study. The GenoMatrix software, presented here, is a user-friendly package to facilitate the process of using genome-wide marker data and parentage information for G-BLUP and P-BLUP analyses on complex traits. It provides users with a collection of applications which help them on a set of tasks from performing quality control on data to constructing and manipulating the genomic and pedigree-based relationship matrices and obtaining their inverses. Such matrices will be then used in downstream analyses by other statistical packages. The package also enables users to obtain predicted values for unobserved individuals based on the genetic values of observed related individuals. GenoMatrix is available to the research community as a Windows 64bit executable and can be downloaded free of charge at: http://compbio.ufl.edu/software/genomatrix/.

Using selection index theory to estimate consistency of multi-locus linkage disequilibrium across populations

BACKGROUND

The potential of combining multiple populations in genomic prediction is depending on the consistency of linkage disequilibrium (LD) between SNPs and QTL across populations. We investigated consistency of multi-locus LD across populations using selection index theory and investigated the relationship between consistency of multi-locus LD and accuracy of genomic prediction across different simulated scenarios. In the selection index, QTL genotypes were considered as breeding goal traits and SNP genotypes as index traits, based on LD among SNPs and between SNPs and QTL. The consistency of multi-locus LD across populations was computed as the accuracy of predicting QTL genotypes in selection candidates using a selection index derived in the reference population. Different scenarios of within and across population genomic prediction were evaluated, using all SNPs or only the four neighboring SNPs of a simulated QTL. Phenotypes were simulated using different numbers of QTL underlying the trait. The relationship between the calculated consistency of multi-locus LD and accuracy of genomic prediction using a GBLUP type of model was investigated.

RESULTS

The accuracy of predicting QTL genotypes, i.e. the measure describing consistency of multi-locus LD, was much lower for across population scenarios compared to within population scenarios, and was lower when QTL had a low MAF compared to QTL randomly selected from the SNPs. Consistency of multi-locus LD was highly correlated with the realized accuracy of genomic prediction across different scenarios and the correlation was higher when QTL were weighted according to their effects in the selection index instead of weighting QTL equally. By only considering neighboring SNPs of QTL, accuracy of predicting QTL genotypes within population decreased, but it substantially increased the accuracy across populations.

CONCLUSIONS

Consistency of multi-locus LD across populations is a characteristic of the properties of the QTL in the investigated populations and can provide more insight in underlying reasons for a low empirical accuracy of across population genomic prediction. By focusing in genomic prediction models only on neighboring SNPs of QTL, multi-locus LD is more consistent across populations since only short-range LD is considered, and accuracy of predicting QTL genotypes of individuals from another population is increased.

Genomic selection using beef commercial carcass phenotypes

In this study, an industry terminal breeding goal was used in a deterministic simulation, using selection index methodology, to predict genetic gain in a beef population modelled on the UK pedigree Limousin, when using genomic selection (GS) and incorporating phenotype information from novel commercial carcass traits. The effect of genotype-environment interaction was investigated by including the model variations of the genetic correlation between purebred and commercial cross-bred performance (ρX). Three genomic scenarios were considered: (1) genomic breeding values (GBV)+estimated breeding values (EBV) for existing selection traits; (2) GBV for three novel commercial carcass traits+EBV in existing traits; and (3) GBV for novel and existing traits plus EBV for existing traits. Each of the three scenarios was simulated for a range of training population (TP) sizes and with three values of ρX. Scenarios 2 and 3 predicted substantially higher percentage increases over current selection than Scenario 1. A TP of 2000 sires, each with 20 commercial progeny with carcass phenotypes, and assuming a ρX of 0.7, is predicted to increase gain by 40% over current selection in Scenario 3. The percentage increase in gain over current selection increased with decreasing ρX; however, the effect of varying ρX was reduced at high TP sizes for Scenarios 2 and 3. A further non-genomic scenario (4) was considered simulating a conventional population-wide progeny test using EBV only. With 20 commercial cross-bred progenies per sire, similar gain was predicted to Scenario 3 with TP=5000 and ρX=1.0. The range of increases in genetic gain predicted for terminal traits when using GS are of similar magnitude to those observed after the implementation of BLUP technology in the United Kingdom. It is concluded that implementation of GS in a terminal sire breeding goal, using purebred phenotypes alone, will be sub-optimal compared with the inclusion of novel commercial carcass phenotypes in genomic evaluations.

A penalized likelihood approach to pooling estimates of covariance components from analyses by parts

Estimates of covariance matrices for numerous traits are commonly obtained by pooling results from a series of analyses of subsets of traits. A penalized maximum-likelihood approach is proposed to combine estimates from part analyses while constraining the resulting overall matrices to be positive definite. In addition, this provides the scope for 'improving' estimates of individual matrices by applying a penalty to the likelihood aimed at borrowing strength from their phenotypic counterpart. A simulation study is presented showing that the new method performs well, yielding unpenalized estimates closer to results from multivariate analyses considering all traits, than various other techniques used. In particular, combining results for all sources of variation simultaneously minimizes deviations in phenotypic estimates if sampling covariances can be approximated. A mild penalty shrinking estimates of individual covariance matrices towards their sum or estimates of canonical eigenvalues towards their mean proved advantageous in most cases. The method proposed is flexible, computationally undemanding and provides combined estimates with good sampling properties and is thus recommended as alternative to current methods for pooling.

Genome-wide complex trait analysis (GCTA): methods, data analyses, and interpretations

Estimating genetic variance is traditionally performed using pedigree analysis. Using high-throughput DNA marker data measured across the entire genome it is now possible to estimate and partition genetic variation from population samples. In this chapter, we introduce methods and a software tool called Genome-wide Complex Trait Analysis (GCTA) to estimate genomic relationships between pairs of conventionally unrelated individuals using genome-wide single nucleotide polymorphism (SNP) data, to estimate variance explained by all SNPs simultaneously on genomic or chromosomal segments or over the whole genome, and to perform a joint and conditional multiple SNPs association analysis using summary statistics from a meta-analysis of genome-wide association studies and linkage disequilibrium between SNPs estimated from a reference sample.

The effect of linkage disequilibrium and family relationships on the reliability of genomic prediction

Although the concept of genomic selection relies on linkage disequilibrium (LD) between quantitative trait loci and markers, reliability of genomic predictions is strongly influenced by family relationships. In this study, we investigated the effects of LD and family relationships on reliability of genomic predictions and the potential of deterministic formulas to predict reliability using population parameters in populations with complex family structures. Five groups of selection candidates were simulated by taking different information sources from the reference population into account: (1) allele frequencies, (2) LD pattern, (3) haplotypes, (4) haploid chromosomes, and (5) individuals from the reference population, thereby having real family relationships with reference individuals. Reliabilities were predicted using genomic relationships among 529 reference individuals and their relationships with selection candidates and with a deterministic formula where the number of effective chromosome segments (M(e)) was estimated based on genomic and additive relationship matrices for each scenario. At a heritability of 0.6, reliabilities based on genomic relationships were 0.002 ± 0.0001 (allele frequencies), 0.022 ± 0.001 (LD pattern), 0.018 ± 0.001 (haplotypes), 0.100 ± 0.008 (haploid chromosomes), and 0.318 ± 0.077 (family relationships). At a heritability of 0.1, relative differences among groups were similar. For all scenarios, reliabilities were similar to predictions with a deterministic formula using estimated M(e). So, reliabilities can be predicted accurately using empirically estimated M(e) and level of relationship with reference individuals has a much higher effect on the reliability than linkage disequilibrium per se. Furthermore, accumulated length of shared haplotypes is more important in determining the reliability of genomic prediction than the individual shared haplotype length.

Principal component and factor analytic models in international sire evaluation

BACKGROUND

Interbull is a non-profit organization that provides internationally comparable breeding values for globalized dairy cattle breeding programmes. Due to different trait definitions and models for genetic evaluation between countries, each biological trait is treated as a different trait in each of the participating countries. This yields a genetic covariance matrix of dimension equal to the number of countries which typically involves high genetic correlations between countries. This gives rise to several problems such as over-parameterized models and increased sampling variances, if genetic (co)variance matrices are considered to be unstructured.

METHODS

Principal component (PC) and factor analytic (FA) models allow highly parsimonious representations of the (co)variance matrix compared to the standard multi-trait model and have, therefore, attracted considerable interest for their potential to ease the burden of the estimation process for multiple-trait across country evaluation (MACE). This study evaluated the utility of PC and FA models to estimate variance components and to predict breeding values for MACE for protein yield. This was tested using a dataset comprising Holstein bull evaluations obtained in 2007 from 25 countries.

RESULTS

In total, 19 principal components or nine factors were needed to explain the genetic variation in the test dataset. Estimates of the genetic parameters under the optimal fit were almost identical for the two approaches. Furthermore, the results were in a good agreement with those obtained from the full rank model and with those provided by Interbull. The estimation time was shortest for models fitting the optimal number of parameters and prolonged when under- or over-parameterized models were applied. Correlations between estimated breeding values (EBV) from the PC19 and PC25 were unity. With few exceptions, correlations between EBV obtained using FA and PC approaches under the optimal fit were ≥ 0.99. For both approaches, EBV correlations decreased when the optimal model and models fitting too few parameters were compared.

CONCLUSIONS

Genetic parameters from the PC and FA approaches were very similar when the optimal number of principal components or factors was fitted. Over-fitting increased estimation time and standard errors of the estimates but did not affect the estimates of genetic correlations or the predictions of breeding values, whereas fitting too few parameters affected bull rankings in different countries.

Principal component approach in variance component estimation for international sire evaluation

Background

The dairy cattle breeding industry is a highly globalized business, which needs internationally comparable and reliable breeding values of sires. The international Bull Evaluation Service, Interbull, was established in 1983 to respond to this need. Currently, Interbull performs multiple-trait across country evaluations (MACE) for several traits and breeds in dairy cattle and provides international breeding values to its member countries. Estimating parameters for MACE is challenging since the structure of datasets and conventional use of multiple-trait models easily result in over-parameterized genetic covariance matrices. The number of parameters to be estimated can be reduced by taking into account only the leading principal components of the traits considered. For MACE, this is readily implemented in a random regression model.

Methods

This article compares two principal component approaches to estimate variance components for MACE using real datasets. The methods tested were a REML approach that directly estimates the genetic principal components (direct PC) and the so-called bottom-up REML approach (bottom-up PC), in which traits are sequentially added to the analysis and the statistically significant genetic principal components are retained. Furthermore, this article evaluates the utility of the bottom-up PC approach to determine the appropriate rank of the (co)variance matrix.

Results

Our study demonstrates the usefulness of both approaches and shows that they can be applied to large multi-country models considering all concerned countries simultaneously. These strategies can thus replace the current practice of estimating the covariance components required through a series of analyses involving selected subsets of traits. Our results support the importance of using the appropriate rank in the genetic (co)variance matrix. Using too low a rank resulted in biased parameter estimates, whereas too high a rank did not result in bias, but increased standard errors of the estimates and notably the computing time.

Conclusions

In terms of estimation's accuracy, both principal component approaches performed equally well and permitted the use of more parsimonious models through random regression MACE. The advantage of the bottom-up PC approach is that it does not need any previous knowledge on the rank. However, with a predetermined rank, the direct PC approach needs less computing time than the bottom-up PC.

Genetic correlations between measures of Mycobacterium bovis infection and economically important traits in Irish Holstein-Friesian dairy cows

Mycobacterium bovis is the primary agent of tuberculosis (TB) in cattle. The failure of Ireland and some other countries to reach TB-free status indicates a need to investigate complementary control strategies. One such approach would be genetic selection for increased resistance to TB. Previous research has shown that considerable genetic variation exists for susceptibility to the measures of M. bovis infection, confirmed M. bovis infection, and M. bovis-purified protein derivative (PPD) responsiveness. The objective of this study was to estimate the genetic and phenotypic correlations between economically important traits and these measures of M. bovis infection. A total of 20,148 and 17,178 cows with confirmed M. bovis infection and M. bovis-PPD responsiveness records, respectively, were available for inclusion in the analysis. First- to third-parity milk, fat, and protein yields, somatic cell count, calving interval, and survival, as well as first-parity body condition score records, were available on cows that calved between 1985 and 2007. Bivariate linear-linear and threshold-linear sire mixed models were used to estimate (co)variance components. The genetic correlations between economically important traits and the measures of M. bovis infection estimated from the linear-linear and threshold-linear sire models were similar. The genetic correlations between susceptibility to confirmed M. bovis infection and economically important traits investigated in this study were all close to zero. Mycobacterium bovis-PPD responsiveness was positively genetically correlated with fat production (0.39) and body condition score (0.36), and negatively correlated with somatic cell score (-0.34) and survival (-0.62). Hence, selection for increased survival may indirectly reduce susceptibility to M. bovis infection, whereas selection for reduced somatic cell count and increased fat production and body condition score may increase susceptibility to M. bovis infection.

Genetic and environmental contributions to variation in baboon cranial morphology

The development, function, and integration of morphological characteristics are all hypothesized to influence the utility of traits for phylogenetic reconstruction by affecting the way in which morphological characteristics evolve. We use a baboon model to test the hypotheses about phenotypic and quantitative genetic variation of traits in the cranium that bear on a phenotype's propensity to evolve. We test the hypotheses that: 1) individual traits in different functionally and developmentally defined regions of the cranium are differentially environmentally, genetically, and phenotypically variable; 2) genetic covariance with other traits constrains traits in one region of the cranium more than those in others; 3) and regions of the cranium subject to different levels of mechanical strain differ in the magnitude of variation in individual traits. We find that the levels of environmental and genetic variation in individual traits are randomly distributed across regions of the cranium rather than being structured by developmental origin or degree of exposure to strain. Individual traits in the cranial vault tend to be more constrained by covariance with other traits than those in other regions. Traits in regions subject to high degrees of strain during mastication are not any more variable at any level than other traits. If these results are generalizable to other populations, they indicate that there is no reason to suppose that individual traits from any one part of the cranium are intrinsically less useful for reconstructing patterns of evolution than those from any other part.

Marker-based estimation of the coefficient of coancestry in hybrid breeding programmes

Molecular markers allow to estimate the pairwise relatedness between the members of a breeding pool when their selection history is no longer available or has become too complex for a classical pedigree analysis. The field of population genetics has several estimation procedures at its disposal, but when the genotyped individuals are highly selected inbred lines, their application is not warranted as the theoretical assumptions on which these estimators were built, usually linkage equilibrium between marker loci or even Hardy-Weinberg equilibrium, are not met. An alternative approach requires the availability of a genotyped reference set of inbred lines, which allows to correct the observed marker similarities for their inherent upward bias when used as a coancestry measure. However, this approach does not guarantee that the resulting coancestry matrix is at least positive semi-definite (psd), a necessary condition for its use as a covariance matrix. In this paper we present the weighted alikeness in state (WAIS) estimator. This marker-based coancestry estimator is compared to several other commonly applied relatedness estimators under realistic hybrid breeding conditions in a number of simulations. We also fit a linear mixed model to phenotypical data from a commercial maize breeding programme and compare the likelihood of the different variance structures. WAIS is shown to be psd which makes it suitable for modelling the covariance between genetic components in linear mixed models involved in breeding value estimation or association studies. Results indicate that it generally produces a low root mean squared error under different breeding circumstances and provides a fit to the data that is comparable to that of several other marker-based alternatives. Recommendations for each of the examined coancestry measures are provided.

Multiple-trait multiple-country genetic evaluations for udder health traits

International genetic bull evaluations of somatic cell counts (SCC) from 8 different Holstein populations and clinical mastitis from 3 of these populations were inferred simultaneously using a multiple-trait-multiple-country evaluation (MT-MACE) model. This model considered effective independent weighting factors and multivariately deregressed national genetic evaluations for countries with multiple-trait national models. Predictions of genetic merit from MT-MACE and their reliabilities were compared with the corresponding results from 2 separate single-trait-multiple-country evaluations (ST-MACE) for different groups of bulls. The assumed heritabilities for clinical mastitis (h(2) = 0.02 to 0.05) were substantially lower than the heritabilities for SCC (h(2) = 0.08 to 0.27). The predictive ability of MT-MACE was essentially equal to or better than the predictive ability of ST-MACE for all country-trait combinations, but both methods yielded effectively unbiased and consistent consecutive predictions (correlation > 0.93). Both sets of predictions also agreed well with future national genetic evaluations for bulls receiving additional daughter information (correlation > 0.96), except for evaluations for which within-country correlations were utilized internationally, but not nationally (correlation = 0.86 to 0.97). The reliabilities for MT-MACE were essentially equal to or higher than reliabilities for ST-MACE, depending on the trait and group of bulls in question. Reliabilities increased most for young bulls, and for clinical mastitis in countries that did not use the within-country correlations with SCC in the national evaluation (up to a 23% increase in average reliability).

National cattle evaluation system for combined analysis of carcass characteristics and indicator traits recorded by using ultrasound in Angus cattle

The objectives were to 1) evaluate genetic relationships of sex-specific indicators of carcass merit obtained by using ultrasound with carcass traits of steers; 2) estimate genetic parameters needed to implement combined analyses of carcass and indicator traits to produce unified national cattle evaluations for LM area, subcutaneous fat depth (SQF), and marbling (MRB), with the ultimate goal of publishing only EPD for the carcass traits; and 3) compare resulting evaluations with previous ones. Four data sets were extracted from the records of the American Angus Association from 33,857 bulls, 33,737 heifers, and 1,805 steers that had measures of intramuscular fat content (IMF), LM area (uLMA), and SQF derived from interpretation of ultrasonic imagery, and BW recorded at the time of scanning. Also used were 38,296 records from steers with MRB, fat depth at the 12th to 13th rib interface (FD), carcass weight, and carcass LM area (cLMA) recorded on slaughter. (Co)variance components were estimated with ASREML by using the same models as used for national cattle evaluations by the American Angus Association. Heritability estimates for carcass measures were 0.45 +/- 0.03, 0.34 +/- 0.02, 0.40 +/- 0.02, and 0.33 +/- 0.02 for MRB, FD, carcass weight, and cLMA, respectively. Genetic correlations of carcass measures from steers with ultrasonic measures from bulls and heifers indicated sex-specific relationships for IMF (0.66 +/- 0.05 vs. 0.52 +/- 0.06) and uLMA (0.63 +/- 0.06 vs. 0.78 +/- 0.05), but not for BW at scanning (0.46 +/- 0.07 vs. 0.40 +/- 0.07) or SQF (0.53 +/- 0.06 vs. 0.55 +/- 0.06). For each trait, estimates of genetic correlations between bulls and heifers measured by using ultrasound were greater than 0.8. Prototype national cattle evaluations were conducted by using the estimated genetic parameters, resulting in some reranking of sires relative to previous analyses. Rank correlations of high-impact sires were 0.91 and 0.84 for the joint analysis of MRB and IMF with previous separate analyses of MRB and IMF, respectively. Corresponding results for FD and SQF were 0.90 and 0.90, and for cLMA and uLMA were 0.79 and 0.89. The unified national cattle evaluation for carcass traits using measurements from slaughtered animals and ultrasonic imagery of seed stock in a combined analysis appropriately weights information from these sources and provides breeders estimates of genetic merit consistent with traits in their breeding objectives on which to base selection decisions.

Genetic parameters for a maternal breeding goal in beef production

New maternal breeding values have been developed for use in UK beef evaluations. To undertake multitrait BLUP evaluations, it is necessary to have a full covariance matrix. This study outlines the approach taken to construct the full covariance matrices for the four beef breeds that most widely contribute to suckler beef cows in the United Kingdom. The maternal traits investigated were age at first calving, calving interval, lifespan, mature cow weight, 200-d weight, and calving difficulty. Three terminal sire traits (weight at 400 d, ultrasonic fat depth, and muscle score) were included to estimate covariances between the new and existing traits. A sire-maternal-grandsire model was used for the estimation procedure in a series of bivariate and multivariate models. A weighted bending procedure was employed to construct positive definite covariance matrices. Parameter estimates broadly agreed with literature values, although for some traits, literature information was very scarce. Some differences between parameters for different breeds were evident.

Prediction of Genetic Correlations and International Breeding Values for Missing Traits

Prediction of genetic merit for missing traits is possible by combining available indicator traits. Indicator traits were combined using genetic correlations obtained from multiple regression equations of estimated genetic correlations among available indicator traits on variables explaining production circumstances and trait definitions. This prediction of missing traits was closer to actual breeding values than breeding values for any of the indicator traits. This was verified by evaluating clinical mastitis in each of the Nordic countries as a missing trait. The derived methodology was used to predict breeding values for clinical mastitis in the United States for local and international bulls with an average reliability of 43%.

Estimation of Heritability and Genetic Correlations for the Major Fatty Acids in Bovine Milk

The current cattle selection program for dairy cattle in the Walloon region of Belgium does not consider the relative content of the different fatty acids (FA) in milk. However, interest by the local dairy industry in differentiated milk products is increasing. Therefore, farmers may be interested in selecting their animals based on the fat composition. The aim of this study was to evaluate the feasibility of genetic selection to improve the nutritional quality of bovine milk fat. The heritabilities and correlations among milk yield, fat, protein, and major FA contents in milk were estimated. Heritabilities for FA in milk and fat ranged from 5 to 38%. The genetic correlations estimated among FA reflected the common origin of several groups of FA. Given these results, an index including FA contents with the similar metabolic process of production in the mammary gland could be used, for example, to increase the monounsaturated and conjugated fatty acids in milk. Moreover, the genetic correlations between the percentage of fat and the content of C14:0, C12:0, C16:0, and C18:0 in fat were -0.06, 0.55, 0.60, and 0.84, respectively. This result demonstrates that an increase in fat content is not directly correlated with undesirable changes in FA profile in milk for human health. Based on the obtained genetic parameters, a future selection program to improve the FA composition of milk fat could be initiated.

Prior (Co)Variances Can Improve Multiple-Trait Across-Country Evaluations of Weakly Linked Bull Populations

National genetic evaluation results for fore udder attachment from 9 Ayrshire populations were used to assess the impact of different uses of prior genetic correlations in multiple-trait across-country evaluations (MACE) on predicted international genetic merit. These Ayrshire populations were poorly connected; that is, 2% of the bulls had evaluations in 2 or more countries. Genetic correlations from the Holstein populations in the same countries were used as prior information to improve inferences of location parameters and international genetic merits. Fully Bayesian analyses using Gibbs sampling and computationally less demanding traditional MACE assuming a weighted average of prior and estimated Ayrshire genetic correlations were compared for 3 different prior degrees of belief and for different groups of bulls. Posterior means of genetic correlations estimated by Gibbs sampling were on average higher (+0.2) than those estimated by REML. Posterior heritabilities differed up to 0.2 units from those assumed in national genetic evaluations. Predicted genetic merit and international sire rankings of bulls with daughter information in the country of interest were not affected substantially by method of analysis and even less by varying prior degree of belief. Method of analysis had a larger impact on predicted genetic merit for bulls without daughter information in the country of interest. Here the average correlation between predicted genetic merit in different analyses ranged from 0.62 to 0.99. The predictive ability for young and randomly chosen bulls favored Bayesian MACE. The prior degree of belief did not have much impact on sire rankings and predictive ability, but intermediate prior degree of belief tended to perform best. All MACE analyses yielded nearly unbiased predictions. Traditional MACE assuming a simple weighted average of prior and estimated Ayrshire genetic correlations has been implemented by Interbull for routine international genetic evaluations.

Principal components and factorial approaches for estimating genetic correlations in international sire evaluation

The increasing number of participating countries and the lack of genetic links among some of them lead to statistical and computational difficulties in estimating the genetic (co)variance matrix needed for international sire evaluation of milk yield. Reparameterization using principal components or factorial approaches is proposed to exploit patterns in the genetic correlation matrix in order to reduce the number of parameters to be estimated without much loss of information. A 2-step approach was used. First, the genetic matrix between 8 or 9 "base" countries was used to determine a reduced number of principal components or factors. Then, the contributions of the remaining countries to these principal components or factors were computed. The resulting genetic correlations for the 18 countries were compared with the "reference" genetic correlations obtained with a classical model. The impact of using reparameterized genetic correlation matrices on breeding value prediction was investigated for both approaches. A better agreement between predicted breeding values and stability of their rankings was found when an approximate factor analysis was used, whatever the number of factors considered. The estimation of genetic correlations among 18 countries using an approximate factorial approach with 5 factors taken into account led to a reduction of the number of parameters to estimate from 171 to 80. The average absolute deviation of the correlations estimated with an approximate factorial approach from the "reference" genetic correlations was 0.014, which is considered very satisfactory in light of the computational ease.

Data Subsetting Strategies for Estimation of Across-Country Genetic Correlations

International genetic evaluation of dairy cattle requires estimation of genetic correlations among populations to account for genotype-environment interaction. Simultaneous estimation of across-country genetic correlations among all populations of a widespread breed, such as the Holstein breed is, however, hampered by connectedness problems and computational challenges. The purpose of this study was to examine the effects of using bulls with across-country, balanced distribution of daughters on estimates of genetic correlations. For this purpose, dairy cattle populations undergoing selection in 6 countries were simulated. Two population-size settings were used. In the small population-size setting (S-populations), the 6 simulated countries had 2000 cows and 20 young progeny testing bulls per generation. In the larger population-size setting (L-populations), the 6 simulated countries had between 2000 and 64,000 cows and 20 to 640 young progeny testing bulls per generation. The simulated (true) across-country genetic correlations, depending on the country combination, varied between 0.5 and 0.9. Simulations comprised a base population and 10 generations and were replicated 16 times. Results for the S-populations were not conclusive. For the L-populations, results indicated that by use of data from a relatively small subset of bulls with distribution of daughters balanced across countries, genetic correlations could be estimated with very small bias (overall average of absolute value of bias across replicates was 0.03 for the L-populations). The suggested bull subsetting strategy would allow simultaneous estimation of across-country genetic correlations to be computed for a larger number of countries and in a shorter window of time than was possible previously.