Using recursion to compute the inverse of the genomic relationship matrix

Computing the inverse of the genomic relationship matrix using recursion was investigated. A traditional algorithm to invert the numerator relationship matrix is based on the observation that the conditional expectation for an additive effect of 1 animal given the effects of all other animals depends on the effects of its sire and dam only, each with a coefficient of 0.5. With genomic relationships, such an expectation depends on all other genotyped animals, and the coefficients do not have any set value. For each animal, the coefficients plus the conditional variance can be called a genomic recursion. If such recursions are known, the mixed model equations can be solved without explicitly creating the inverse of the genomic relationship matrix. Several algorithms were developed to create genomic recursions. In an algorithm with sequential updates, genomic recursions are created animal by animal. That algorithm can also be used to update a known inverse of a genomic relationship matrix for additional genotypes. In an algorithm with forward updates, a newly computed recursion is immediately applied to update recursions for remaining animals. The computing costs for both algorithms depend on the sparsity pattern of the genomic recursions, but are lower or equal than for regular inversion. An algorithm for proven and young animals assumes that the genomic recursions for young animals contain coefficients only for proven animals. Such an algorithm generates exact genomic EBV in genomic BLUP and is an approximation in single-step genomic BLUP. That algorithm has a cubic cost for the number of proven animals and a linear cost for the number of young animals. The genomic recursions can provide new insight into genomic evaluation and possibly reduce costs of genetic predictions with extremely large numbers of genotypes.     

Multiple trait genomic evaluation of conception rate in Holsteins

A national data set of artificial inseminations in US Holsteins was used to obtain genetic evaluations for conception rate (CR). The objective of this study was to investigate the feasibility and resulting accuracy from using all available phenotypic, pedigree, and genomic information. Evaluations were performed by regular BLUP or by BLUP with the traditional pedigree and genomic relationships combined in a unified single-step procedure (SSP). Genetic parameters of CR in the first 3 parities were estimated with data from New York State only. Heritability estimates were around 2% and genetic correlations between CR in different parities were >0.73. The R² obtained with the SSP were almost twice as large as those achieved with regular BLUP. Computing the SSP took 2 h, and it was 33% slower than a regular BLUP. A multiple-trait evaluation of CR using the SSP is both possible and advantageous.     

Evidence of biases in genetic evaluations due to genomic preselection in dairy cattle

A genomic preselection step of young sires is now often included in dairy cattle breeding schemes. Young sires are selected based on their genomic breeding values. They have better Mendelian sampling contribution so that the assumption of random Mendelian sampling term in genetic evaluations is clearly violated. When these sires and their progeny are evaluated using BLUP, it is feared that estimated breeding values are biased. The effect of genomic selection on genetic evaluations was studied through simulations keeping the structure of the Holstein population in France. The quality of genetic evaluations was assessed by computing bias and accuracy from the difference and correlation between true and estimated breeding values, respectively, and also the mean square error of prediction. Different levels of heritability, selection intensity, and accuracy of genomic evaluation were tested. After only one generation and whatever the scenario, breeding values of preselected young sires and their daughters were significantly underestimated and their accuracy was decreased. Genomic preselection needs to be accounted for in genetic evaluation models.     

Within- and across-breed genomic predictions and genomic relationships for Western Pyrenees dairy sheep breeds Latxa,Manech, and Basco-Béarnaise

Genotypes, phenotypes and pedigrees of 6 breeds of dairy sheep (including subdivisions of Latxa, Manech, and Basco-Béarnaise) from the Spain and France Western Pyrenees were used to estimate genetic relationships across breeds (together with genotypes from the Lacaune dairy sheep) and to verify by forward cross-validation single-breed or multiple-breed genetic evaluations. The number of rams genotyped fluctuated between 100 and 1,300 but generally represented the 10 last cohorts of progeny-tested rams within each breed. Genetic relationships were assessed by principal components analysis of the genomic relationship matrices and also by the conservation of linkage disequilibrium patterns at given physical distances in the genome. Genomic and pedigree-based evaluations used daughter yield performances of all rams, although some of them were not genotyped. A pseudo-single step method was used in this case for genomic predictions. Results showed a clear structure in blond and black breeds for Manech and Latxa, reflecting historical exchanges, and isolation of Basco-Béarnaise and Lacaune. Relatedness between any 2 breeds was, however, lower than expected. Single-breed genomic predictions had accuracies comparable with other breeds of dairy sheep or small breeds of dairy cattle. They were more accurate than pedigree predictions for 5 out of 6 breeds, with absolute increases in accuracy ranging from 0.05 to 0.30 points. They were significantly better, as assessed by bootstrapping of candidates, for 2 of the breeds. Predictions using multiple populations only marginally increased the accuracy for a couple of breeds. Pooling populations does not increase the accuracy of genomic evaluations in dairy sheep; however, single-breed genomic predictions are more accurate, even for small breeds, and make the consideration of genomic schemes in dairy sheep interesting.     

基于XML的种猪遗传评估系统设计与实现

针对关系模式在养猪管理方面存在内容整齐划一、数据分散、访问数据效率低等不足之处,提出采用XML文档存储种猪数据的方案。根据XML的自我描述性特点,采用一个XML文档保存一头猪的所有数据,同时在XML文档中添加种猪谱系信息,设计出基于XML的种猪遗传评估系统。通过采用XML文档集中地存储种猪数据,管理人员能够方便地查询、分析及传输种猪数据,从而实现种猪精细化管理以及提高遗传育种计算效率。     

Autoregressive single-step test-day model for genomic evaluations of Portuguese Holstein cattle

The multiple-lactation autoregressive test-day (AR) model is the adopted model for the national genetic evaluation of dairy cattle in Portugal. Under this model, animals' permanent environment effects are assumed to follow a first-order autoregressive process over the long (auto-correlations between parities) and short (auto-correlations between test-days within lactation) terms. Given the relevance of genomic prediction in dairy cattle, it is essential to include marker information in national genetic evaluations. In this context, we aimed to evaluate the feasibility of applying the single-step genomic (G)BLUP to analyze milk yield using the AR model in Portuguese Holstein cattle. In total, 11,434,294 test-day records from the first 3 lactations collected between 1994 and 2017 and 1,071 genotyped bulls were used in this study. Rank correlations and differences in reliability among bulls were used to compare the performance of the traditional (A-AR) and single-step (H-AR) models. These 2 modeling approaches were also applied to reduced data sets with records truncated after 2012 (deleting daughters of tested bulls) to evaluate the predictive ability of the H-AR. Validation scenarios were proposed, taking into account young and proven bulls. Average EBV reliabilities, empirical reliabilities, and genetic trends predicted from the complete and reduced data sets were used to validate the genomic evaluation. Average EBV reliabilities for H-AR (A-AR) using the complete data set were 0.52 (0.16) and 0.72 (0.62) for genotyped bulls with no daughters and bulls with 1 to 9 daughters, respectively. These results showed an increase in EBV reliabilities of 0.10 to 0.36 when genomic information was included, corresponding to a reduction of up to 43% in prediction error variance. Considering the 3 validation scenarios, the inclusion of genomic information improved the average EBV reliability in the reduced data set, which ranged, on average, from 0.16 to 0.26, indicating an increase in the predictive ability. Similarly, empirical reliability increased by up to 0.08 between validation tests. The H-AR outperformed A-AR in terms of genetic trends when unproven genotyped bulls were included. The results suggest that the single-step GBLUP AR model is feasible and may be applied to national Portuguese genetic evaluations for milk yield.     

National single-step genomic method that integrates multi-national genomic information

The aim of this paper was to develop a national single-step genomic BLUP that integrates multi-national genomic estimated breeding values (EBV) and associated reliabilities without double counting dependent data contributions from the different evaluations. Simultaneous use of all data, including phenotypes, pedigree, and genotypes, is a condition to obtain unbiased EBV. However, this condition is not always fully met, mainly due to unavailability of foreign raw data for imported animals. In dairy cattle genetic evaluations, this issue is traditionally tackled through the multiple across-country evaluation (MACE) of sires, performed by Interbull Centre (Uppsala, Sweden). Multiple across-country evaluation regresses all the available national information onto a joint pedigree to obtain country-specific rankings of all sires without sharing the raw data. In the context of genomic selection, the issue is handled by exchanging sire genotypes and by using MACE information (i.e., MACE EBV and reliabilities), as a valuable source of “phenotypic” data. Although all the available data are considered, these “multi-national” genomic evaluations use multi-step methods assuming independence of various sources of information, which is not met in all situations. We developed a method that handles this by single-step genomic evaluation that jointly (1) uses national phenotypic, genomic, and pedigree data; (2) uses multi-national genomic information; and (3) avoids double counting dependent data contributions from an animal’s own records and relatives’ records. The method was demonstrated by integrating multi-national genomic EBV and reliabilities of Brown Swiss sires, included in the InterGenomics consortium at Interbull Centre, into the national evaluation in Slovenia. The results showed that the method could (1) increase reliability of a national (genomic) evaluation; (2) provide consistent ranking of all animals: bulls, cows, and young animals; and (3) increase the size of a genomic training population. These features provide more efficient and transparent selection throughout a breeding program.     

Single- and multiple-breed genomic evaluations for conformation traits in Canadian Alpine and Saanen dairy goats

Conformation traits are functional traits known to affect longevity, production efficiency, and profitability of dairy goats. However, genetic progress for these traits is expected to be slower than for milk production traits due to the limited number of herds participating in type classification programs, and often lower heritability estimates. Genomic selection substantially accelerates the rate of genetic progress in many species and industries, especially for lowly heritable, difficult, or expensive to measure traits. Therefore, the main objectives of this study were (1) to evaluate the potential benefits of the implementation of single-step genomic evaluations for conformation traits in Canadian Alpine and Saanen dairy goats, and (2) to investigate the effect of the use of single- and multiple-breed training populations. The phenotypes used in this study were linear conformation scores, on a 1-to-9 scale, for 8 traits (i.e., body capacity, dairy character, fore udder, feet and legs, general appearance, rear udder, medial suspensory ligament, and teats) of 5,158 Alpine and 2,342 Saanen does. Genotypes were available for 833 Alpine and 874 Saanen animals. Averaged across all traits, the use of multiple-breed analyses increased validation accuracy for Saanen, and reduced bias of genomically enhanced breeding values (GEBV) for both Alpine and Saanen compared with single-breed analyses. Little benefit was observed from the use of GEBV relative to pedigree-based EBV in terms of validation accuracy and bias, possibly due to limitations in the validation design, but substantial gains of 0.14 to 0.21 (32–50%) were observed in the theoretical accuracy of validation animals when averaged across traits for single- and multiple-breed analyses. Across the whole genotyped population, average gains in theoretical accuracy for GEBV compared with EBV across all traits ranged from 0.15 to 0.17 (32–37%) for Alpine and 0.17 to 0.19 (40–41%) for Saanen, depending on the model used. The largest gains were observed for does without classification records (0.19–0.22 or 50–55%) and bucks without daughter classification records (0.20–0.27 or 57–82%), which have the least information contributing to their traditional EBV. The use of multiple-breed rather than single-breed models was most beneficial for the Saanen breed, which had fewer phenotypic records available for the analyses. These results suggest that the implementation of genomic selection could increase the accuracy of breeding values for conformation traits in Canadian dairy goats.     

Validation of single-step genomic BLUP random regression test-day models and SNP effects analysis on milk yield in French Saanen goats

The shape of the lactation curve is linked to an animal's health, feed requirements, and milk production throughout the year. Random regression models (RRM) are widely used for genetic evaluation of total milk production throughout the lactation and for milk yield persistency. Genomic information used with the single-step genomic BLUP method (ssGBLUP) substantially improves the accuracy of genomic prediction of breeding values in the main dairy cattle breeds. The aim of this study was to implement an RRM using ssGBLUP for milk yield in Saanen dairy goats in France. The data set consisted of 7,904,246 test-day records from 1,308,307 lactations of Saanen goats collected in France between 2000 and 2017. The performance of this type of evaluation was assessed by applying a validation step with data targeting candidate bucks. The model was compared with a nongenomic evaluation and a traditional evaluation that use cumulated performance throughout the lactation model (LM). The incorporation of genomic information increased correlations between daughter yield deviations (DYD) and estimated breeding values (EBV) obtained with a partial data set for candidate bucks. The LM and the RRM had similar correlation between DYD and EBV. However, the RRM reduced overestimation of EBV and improved the slope of the regression of DYD on EBV obtained at birth. This study shows that a genomic evaluation from a ssGBLUP RRM is possible in dairy goats in France and that RRM performance is comparable to a LM but with the additional benefit of a genomic evaluation of persistency. Variance of adjacent SNPs was studied with LM and RRM following the ssGBLUP. Both approaches converged on approximately the same regions explaining more than 1% of total variance. Regions associated with persistency were also found.     

Short communication: Calculating analytical reliabilities for single-step predictions

It has been shown that single-step genomic BLUP (ssGBLUP) can be reformulated, resulting in an equivalent SNP model that includes the explicit imputation of gene contents of all ungenotyped animals in the pedigree. This reformulation reveals the underlying mechanism enabling ungenotyped animals to contribute information to genotyped animals via estimates of marker effects and consequently to the reliability of genomic predictions, a key feature generally associated with the single-step approach. Irrespective of which BLUP formulation is used for genomic prediction, with increasing numbers of genotyped animals, the marker-oriented model is recommended when calculating the reliabilities of genomic predictions. This approach has the advantage of a manageable and stable size of the model matrix that needs to be inverted to calculate analytical prediction error variances of marker effects, an advantage that also holds for prediction with the single-step model. However, when including imputed genotypes in the design matrix of marker effects, an additional imputation residual term has to be considered to account for the prediction error of imputation. We summarize some of the theoretical aspects associated with the calculation of analytical reliabilities of single-step predictions. Derivations are based on the equivalent reformulation of ssGBLUP as a marker-oriented model and the calculation of prediction error variances of marker effects. We propose 2 approximations that allow for a substantial reduction of the complexity of the matrix operations involved, while retaining most of the relevant information required for reliability calculations. We additionally provide a general framework for an implementation of single-step reliability approximation using standard animal model reliabilities as a starting point. Finally, we demonstrate the effectiveness of the proposed approach using a small example extracted from data of the routine evaluation on dual-purpose Fleckvieh (Simmental) cattle.