A relationship matrix including full pedigree and genomic information

Dense molecular markers are being used in genetic evaluation for parts of the population. This requires a two-step procedure where pseudo-data (for instance, daughter yield deviations) are computed from full records and pedigree data and later used for genomic evaluation. This results in bias and loss of information. One way to incorporate the genomic information into a full genetic evaluation is by modifying the numerator relationship matrix. A naive proposal is to substitute the relationships of genotyped animals with the genomic relationship matrix. However, this results in incoherencies because the genomic relationship matrix includes information on relationships among ancestors and descendants. In other words, using the pedigree-derived covariance between genotyped and ungenotyped individuals, with the pretense that genomic information does not exist, leads to inconsistencies. It is proposed to condition the genetic value of ungenotyped animals on the genetic value of genotyped animals via the selection index (e.g., pedigree information), and then use the genomic relationship matrix for the latter. This results in a joint distribution of genotyped and ungenotyped genetic values, with a pedigree-genomic relationship matrix H. In this matrix, genomic information is transmitted to the covariances among all ungenotyped individuals. The matrix is (semi)positive definite by construction, which is not the case for the naive approach. Numerical examples and alternative expressions are discussed. Matrix H is suitable for iteration on data algorithms that multiply a vector times a matrix, such as preconditioned conjugated gradients.     

Technical note: Automatic scaling in single-step genomic BLUP

Single-step genomic BLUP (ssGBLUP) requires compatibility between genomic and pedigree relationships for unbiased and accurate predictions. Scaling the genomic relationship matrix (G) to have the same averages as the pedigree relationship matrix (i.e., scaling by averages) is one way to ensure compatibility. This requires computing both relationship matrices, calculating averages, and changing G, whereas only the inverses of those matrices are needed in the mixed model equations. Therefore, the compatibility process can add extra computing burden. In the single-step Bayesian regression, the scaling is done by including a mean (μ_g) as a fixed effect in the model. The parameter μ_g can be interpreted as the average of the breeding values of the genotyped animals. In this study, such scaling, called automatic, was implemented in ssGBLUP via Quaas-Pollak transformation of the inverse of the relationship matrix used in ssGBLUP (H), which combines the inverses of the pedigree and genomic relationship matrices. Comparisons involved a simulated data set, and the genomic relationship matrix was computed using different allele frequencies either from the current population (i.e., realized allele frequencies), equal among all the loci, or from the base population. For all of the scenarios, we computed bias [defined as the average difference between true breeding values (TBV) and genomic estimated breeding values (GEBV)], accuracy (defined as the correlation between TBV and GEBV), and dispersion (defined as the regression coefficient of GEBV on TBV). With no scaling, the bias expressed in terms of genetic standard deviations was 0.86, 0.64, and 0.58 with realized, equal, and base population allele frequencies, respectively. With scaling by averages, which is currently used in ssGBLUP, bias was 0.07, 0.08, and 0.03, respectively. With automatic scaling, bias was 0.18 regardless of allele frequencies. Accuracies were similar among scaling methods, but about 0.1 lower in the scenario without scaling. The GEBV were more inflated without any scaling, whereas the automatic scaling performed similarly to the scaling by averages. The average dispersion for those methods was 0.94. When μ_g was treated as random, with the variance equal to differences between pedigree and genomic relationships, the bias was the same as with the scaling by averages. The automatic scaling is biased, especially when μ_g is treated as a fixed effect. The bias may be small in real data with fewer generations, when traits are undergoing weak selection, or when the number of genotyped animals is large.     

Genomic selection of milk fatty acid composition in Sarda dairy sheep: Effect of different phenotypes and relationship matrices on heritability and breeding value accuracy

Fatty acid (FA) composition is one of the most important aspects of milk nutritional quality. However, the inclusion of this trait as a breeding goal for dairy species is hampered by the logistics and high costs of phenotype recording. Fourier-transform infrared spectroscopy (FTIR) is a valid and cheap alternative to laboratory gas chromatography (GC) for predicting milk FA composition. Moreover, as for other novel phenotypes, the efficiency of selection for these traits can be enhanced by using genomic data. The objective of this research was to compare traditional versus genomic selection approaches for estimating genetic parameters and breeding values of milk fatty acid composition in dairy sheep using either GC-measured or FTIR-predicted FA as phenotypes. Milk FA profiles were available for a total of 923 Sarda breed ewes. The youngest 100 had their own phenotype masked to mimic selection candidates. Pedigree relationship information and genotypes were available for 923 and 769 ewes, respectively. Three statistical approaches were used: the classical-pedigree-based BLUP, the genomic BLUP that considers the genomic relationship matrix G, and the single-step genomic BLUP (ssGBLUP) where pedigree and genomic relationship matrices are blended into a single H matrix. Heritability estimates using pedigree were lower than ssGBLUP, and very similar between GC and FTIR regarding the statistical approach used. For some FA, mostly associated with animal diet (i.e., C18:2n-6, C18:3n-3), random effect of combination of flock and test date explained a relevant quota of total variance, reducing the heritability estimates accordingly. Genomic approaches (genomic BLUP and ssGBLUP) outperformed the traditional pedigree method both for GC and FTIR FA. Prediction accuracies in the older cohort were larger than the young cohort. Genomic prediction accuracies (obtained using either G or H relationship matrix) in the young cohort of animals, where their own phenotypes were masked, were similar for GC and FTIR. Multiple-trait analysis slightly affected genomic breeding value accuracies. These results suggest that FTIR-predicted milk FA composition could represent a valid option for inclusion in breeding programs.     

Effect of subdivision of the Lacaune dairy sheep breed on the accuracy of genomic prediction

Genomic selection was deployed in Lacaune dairy breed in 2015. Lacaune population split in 1972 into 2 breeding companies with associated flocks, and there have been very few exchanges of animals between the subpopulations, leading to divergence of the 2 subpopulations. In spite of that, there is a joint genomic prediction. The objective of this study is to understand how this structuring affects prediction accuracy. We analyzed all the data available from Lacaune breeding program for milk yield: around 6 million phenotypes, 2 million animals in the pedigree and more than 29,000 genotyped animals, including 3,434 and 2,868 AI rams for each company. To consider missing pedigree, we set up genetic groups using the theory of metafounders. First, we studied the pedigree and genomic structures of the 2 subpopulations calculating F_st, evolution of average pedigree relationships across time and principal components analysis of genomic relationships. In a second part, we compared the reliability between different scenarios: an evaluation with a single reference population (Alone), an evaluation with a joint reference population (Together) and an evaluation of one subpopulation based on the reference population of the other group (Indirect). The low F_st value (0.02) reveals that the 2 subpopulations are still genetically close. Nevertheless, a low and constant average relationship between the animals of the 2 subpopulations confirms the absence of recent connections between them. We can see with principal component analysis results that even if they are close, they diverge over time. Finally, we observe small gains in accuracy of Together versus Alone, in spite of whereas doubling the reference population size in Together. These gains vary across years and subpopulations: less than 0.08 (0.46 to 0.54; ratio of accuracy for the partial and whole evaluations—corresponding to the greatest change in this ratio for breeding company 1, observed for the cohort 2016) for one subpopulation and between 0.03 (0.55 to 0.58) and 0.17 (0.48 to 0.65) for the other. To conclude, the 2 subpopulations remain close enough genetically so that their combined evaluation is advantageous, even if only slightly.     

Hot topic: A unified approach to utilize phenotypic, full pedigree, and genomic information for genetic evaluation of Holstein final score

The first national single-step, full-information (phenotype, pedigree, and marker genotype) genetic evaluation was developed for final score of US Holsteins. Data included final scores recorded from 1955 to 2009 for 6,232,548 Holsteins cows. BovineSNP50 (Illumina, San Diego, CA) genotypes from the Cooperative Dairy DNA Repository (Beltsville, MD) were available for 6,508 bulls. Three analyses used a repeatability animal model as currently used for the national US evaluation. The first 2 analyses used final scores recorded up to 2004. The first analysis used only a pedigree-based relationship matrix. The second analysis used a relationship matrix based on both pedigree and genomic information (single-step approach). The third analysis used the complete data set and only the pedigree-based relationship matrix. The fourth analysis used predictions from the first analysis (final scores up to 2004 and only a pedigree-based relationship matrix) and prediction using a genomic based matrix to obtain genetic evaluation (multiple-step approach). Different allele frequencies were tested in construction of the genomic relationship matrix. Coefficients of determination between predictions of young bulls from parent average, single-step, and multiple-step approaches and their 2009 daughter deviations were 0.24, 0.37 to 0.41, and 0.40, respectively. The highest coefficient of determination for a single-step approach was observed when using a genomic relationship matrix with assumed allele frequencies of 0.5. Coefficients for regression of 2009 daughter deviations on parent-average, single-step, and multiple-step predictions were 0.76, 0.68 to 0.79, and 0.86, respectively, which indicated some inflation of predictions. The single-step regression coefficient could be increased up to 0.92 by scaling differences between the genomic and pedigree-based relationship matrices with little loss in accuracy of prediction. One complete evaluation took about 2 h of computing time and 2.7 gigabytes of memory. Computing times for single-step analyses were slightly longer (2%) than for pedigree-based analysis. A national single-step genetic evaluation with the pedigree relationship matrix augmented with genomic information provided genomic predictions with accuracy and bias comparable to multiple-step procedures and could account for any population or data structure. Advantages of single-step evaluations should increase in the future when animals are pre-selected on genotypes.     

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.     

A single-step genomic model with direct estimation of marker effects

Compared with the currently widely used multi-step genomic models for genomic evaluation, single-step genomic models can provide more accurate genomic evaluation by jointly analyzing phenotypes and genotypes of all animals and can properly correct for the effect of genomic preselection on genetic evaluations. The objectives of this study were to introduce a single-step genomic model, allowing a direct estimation of single nucleotide polymorphism (SNP) effects, and to develop efficient computing algorithms for solving equations of the single-step SNP model. We proposed an alternative to the current single-step genomic model based on the genomic relationship matrix by including an additional step for estimating the effects of SNP markers. Our single-step SNP model allowed flexible modeling of SNP effects in terms of the number and variance of SNP markers. Moreover, our single-step SNP model included a residual polygenic effect with trait-specific variance for reducing inflation in genomic prediction. A kernel calculation of the SNP model involved repeated multiplications of the inverse of the pedigree relationship matrix of genotyped animals with a vector, for which numerical methods such as preconditioned conjugate gradients can be used. For estimating SNP effects, a special updating algorithm was proposed to separate residual polygenic effects from the SNP effects. We extended our single-step SNP model to general multiple-trait cases. By taking advantage of a block-diagonal (co)variance matrix of SNP effects, we showed how to estimate multivariate SNP effects in an efficient way. A general prediction formula was derived for candidates without phenotypes, which can be used for frequent, interim genomic evaluations without running the whole genomic evaluation process. We discussed various issues related to implementation of the single-step SNP model in Holstein populations with an across-country genomic reference population.     

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.     

Invited review: Unknown-parent groups and metafounders in single-step genomic BLUP

Single-step genomic BLUP (ssGBLUP) is a method for genomic prediction that integrates matrices of pedigree (A) and genomic (G) relationships into a single unified additive relationship matrix whose inverse is incorporated into a set of mixed model equations (MME) to compute genomic predictions. Pedigree information in dairy cattle is often incomplete. Missing pedigree potentially causes biases and inflation in genomic estimated breeding values (GEBV) obtained with ssGBLUP. Three major issues are associated with missing pedigree in ssGBLUP, namely biased predictions by selection, missing inbreeding in pedigree relationships, and incompatibility between G and A in level and scale. These issues can be solved using a proper model for unknown-parent groups (UPG). The theory behind the use of UPG is well established for pedigree BLUP, but not for ssGBLUP. This study reviews the development of the UPG model in pedigree BLUP, the properties of UPG models in ssGBLUP, and the effect of UPG on genetic trends and genomic predictions. Similarities and differences between UPG and metafounder (MF) models, a generalized UPG model, are also reviewed. A UPG model (QP) derived using a transformation of the MME has a good convergence behavior. However, with insufficient data, the QP model may yield biased genetic trends and may underestimate UPG. The QP model can be altered by removing the genomic relationships linking GEBV and UPG effects from MME. This altered QP model exhibits less bias in genetic trends and less inflation in genomic predictions than the QP model, especially with large data sets. Recently, a new model, which encapsulates the UPG equations into the pedigree relationships for genotyped animals, was proposed in simulated purebred populations. The MF model is a comprehensive solution to the missing pedigree issue. This model can be a choice for multibreed or crossbred evaluations if the data set allows the estimation of a reasonable relationship matrix for MF. Missing pedigree influences genetic trends, but its effect on the predictability of genetic merit for genotyped animals should be negligible when many proven bulls are genotyped. The SNP effects can be back-solved using GEBV from older genotyped animals, and these predicted SNP effects can be used to calculate GEBV for young-genotyped animals with missing parents.     

Genomic predictions for New Zealand dairy bulls and integration with national genetic evaluation

A method is described for the prediction of breeding values incorporating genomic information. The first stage involves the prediction of genomic breeding values for genotyped individuals. A novel component of this is the estimation of the genomic relationship matrix in the context of a multi-breed population. Because not all ancestors of genotyped animals are genotyped, a selection index procedure is used to blend genomic predictions with traditional ancestral information that is lost between the process of deregression of the national breeding values and subsequent re-estimation using the genomic relationship matrix. Finally, the genomically enhanced predictions are filtered through to nongenotyped descendants using a regression procedure.     

Use of a single-step approach for integrating foreign information into national genomic evaluation in Holstein cattle

The use of multi-trait across-country evaluation (MACE) and the exchange of genomic information among countries allows national breeding programs to combine foreign and national data to increase the size of the training populations and potentially increase accuracy of genomic prediction of breeding values. By including genotyped and nongenotyped animals simultaneously in the evaluation, the single-step genomic BLUP (GBLUP) approach has the potential to deliver more accurate and less biased genomic evaluations. A single-step genomic BLUP approach, which enables integration of data from MACE evaluations, can be used to obtain genomic predictions while avoiding double-counting of information. The objectives of this study were to apply a single-step approach that simultaneously includes domestic and MACE information for genomic evaluation of workability traits in Canadian Holstein cattle, and compare the results obtained with this methodology with those obtained using a multi-step approach (msGBLUP). By including MACE bulls in the training population, msGBLUP led to an increase in reliability of genomic predictions of 4.8 and 15.4% for milking temperament and milking speed, respectively, compared with a traditional evaluation using only pedigree and phenotypic information. Integration of MACE data through a single-step approach (ssGBLUP_IM) yielded the highest reliabilities compared with other considered methods. Integration of MACE data also helped reduce bias of genomic predictions. When using ssGBLUP_IM, the bias of genomic predictions decreased by half compared with msGBLUP using domestic and MACE information. Therefore, the reliability and bias of genomic predictions for both traits improved substantially when a single-step approach was used for evaluation compared with a multi-step approach. The use of a single-step approach with integration of MACE information provides an alternative to the current method used in Canadian genomic evaluations.     

Genomic evaluation for calf wellness traits in Holstein cattle

Reducing calf morbidity and mortality is important for attaining financial sustainability and improving animal welfare on commercial dairy operations. Zoetis (Kalamazoo, MI) has developed genomic predictions for calf wellness traits in Holsteins that include calf respiratory disease (RESP; recorded between 0 and 365 d of age), calf scours (DIAR; recorded between 2 and 50 d of age), and calf livability (DEAD; recorded between 2 and 365 d of age). Phenotype and pedigree data were from commercial dairies and provided directly by producers upon obtaining their permission. The number of records ranged from 741,484 for DIAR to 1,926,261 for DEAD. The number of genotyped animals was 325,025. All traits were analyzed using a univariate threshold animal model including fixed effect of year of birth × calving season × region, and random effects of herd × year of birth and animal. A total of 45,425 SNP were used in genomic analyses. Animals genotyped with low-density chips were imputed to the required number of SNP. All analyses were conducted using single-step genomic BLUP implementing the “algorithm for proven and young” (APY) animals designed to accommodate very large numbers of genotypes. Estimated heritabilities were 0.042, 0.045, and 0.060 for RESP, DIAR, and DEAD, respectively. The genomic predicted transmitting abilities ranged between ?8.0 and 24.0, ?11.5 and 28.5, and ?6.5 to 22.8 for RESP, DIAR, and DEAD, respectively. Reliabilities of breeding values were obtained by approximation based on partitioning of a function of reliability into contributions from records, pedigree, and genotypes, where the genotype contribution was approximated using the diagonal value of the genomic relationship matrix. The average reliabilities for the genotyped animals were 41.9, 42.6, and 47.3% for RESP, DIAR, and DEAD, respectively. Estimated genomic predicted transmitting abilities and reliabilities were approximately normally distributed for all analyzed traits. Approximated genetic correlations of calf wellness with Zoetis dairy wellness traits and traits included in the US national genetic evaluation were low to moderate. The results indicate that direct evaluation of calf wellness traits under a genomic threshold model is feasible and offers predictions with average reliabilities comparable to other lowly heritable traits. Genetic selection for calf wellness traits presents a compelling opportunity for dairy producers to help manage herd replacement costs and improve overall profitability.     

Controlling bias in genomic breeding values for young genotyped bulls

The objectives of this study were to investigate bias in genomic predictions for dairy cattle and to find a practical approach to reduce the bias. The simulated data included phenotypes, pedigrees, and genotypes, mimicking a dairy cattle population (i.e., cows with phenotypes and bulls with no phenotypes) and assuming selection by breeding values or no selection. With the simulated data, genomic estimated breeding values (GEBV) were calculated with a single-step genomic BLUP and compared with true breeding values. Phenotypes and genotypes were simulated in 10 generations and in the last 4 generations, respectively. Phenotypes in the last generation were removed to predict breeding values for those individuals using only genomic and pedigree information. Complete pedigrees and incomplete pedigrees with 50% missing dams were created to construct the pedigree-based relationship matrix with and without inbreeding. With missing dams, unknown parent groups (UPG) were assigned in relationship matrices. Regression coefficients (b₁) and coefficients of determination (R²) of true breeding values on (G)EBV were calculated to investigate inflation and accuracy in GEBV for genotyped animals, respectively. In addition to the simulation study, 18 linear type traits of US Holsteins were examined. For the 18 type traits, b₁ and R² of GEBV with full data sets on GEBV with partial data sets for young genotyped bulls were calculated. The results from the simulation study indicated inflation in GEBV for genotyped males that were evaluated with only pedigree and genomic information under BLUP selection. However, when UPG for only pedigree-based relationships were included, the inflation was reduced, accuracy was highest, and genetic trends had no bias. For the linear type traits, when UPG for only pedigree-based relationships were included, the results were generally in agreement with those from the simulation study, implying less bias in genetic trends. However, when including no UPG, UPG in pedigree-based relationships, or UPG in genomic relationships, inflation and accuracy in GEBV were similar. The results from the simulation and type traits suggest that UPG must be defined accurately to be estimable and inbreeding should be included in pedigree-based relationships. In dairy cattle, known pedigree information with inbreeding and estimable UPG plays an important role in improving compatibility between pedigree-based and genomic relationship matrices, resulting in more reliable genomic predictions.     

Accuracy of breeding values in small genotyped populations using different sources of external information—A simulation study

Genetically linked small and large dairy cattle populations were simulated to test the effect of different sources of information from foreign populations on the accuracy of predicting breeding values for young animals in a small population. A large dairy cattle population (P_L) with >20 generations was simulated, and a small subpopulation (P_S) with 3 generations was formed as a related population, including phenotypes and genomic information. Predicted breeding values for young animals in the small population were calculated using BLUP and single-step genomic BLUP (ssGBLUP) in 4 different scenarios: (S1) 3,166 phenotypes, 22,855 pedigree animals, and 1,000 to 6,000 genotypes for P_S; (S2) S1 plus genomic estimated breeding value (GEBV) for 4,475 sires from P_L as external information; (S3) S1 plus 221,580 phenotypes, 402,829 pedigree animals, and 53,558 genotypes for P_L; and (S4) single nucleotide polymorphism (SNP) effects calculated based on P_L data. The ability to predict true breeding value was assessed in the youngest third of the genotyped animals in the small population. When data only from the small population were used and 1,000 animals were genotyped, the accuracy of GEBV was only 1 point greater than the estimated breeding value accuracy (0.32 vs. 0.31). Adding external GEBV for sires from P_L did not considerably increase accuracy (0.33 vs. 0.32 in S1). Combining phenotypes, pedigree, and genotypes for P_S and P_L was beneficial for predicting accuracy of GEBV in the small population, and the prediction accuracy of GEBV in this scenario was 0.38 compared with 0.31 from estimated breeding values. When SNP effects from P_L were used to predict GEBV for young genotyped animals from P_S, accuracy was greatest (0.56). With 6,000 genotyped animal in P_S, accuracy was greatest (0.61) with the combined populations. In a small population with few genotypes, the highest accuracy of evaluation may be obtained by using SNP effects derived from a related large population.     

Technical note: Methods for interim prediction of single-step breeding values for young animals

Single-step genomic evaluations have the advantage of simultaneously combining all pedigree, phenotypic, and genotypic information available. However, systems with a large number of genotyped animals have some computational challenges. In many genomic breeding programs, genomic predictions of young animals should become available for selection decisions in the shortest time possible, which requires either a very effective estimation or an approximation with negligible loss in accuracy. We investigated different procedures for predicting breeding values of young genotyped animals without setting up the full single-step system augmented for the additional genotypes. Methods were based on transmitting the information from single-step breeding values of genotyped animals that took part in the previous full run to young animals, either through genomic relationships or through a marker-based model. The different procedures were tested on real data from the April 2017 run of the German-Austrian official genomic evaluation for Fleckvieh. The data set included 62,559 genotyped animals and was used to run single-step evaluations for 23 conformation traits. A further data set comprising 1,768 young animals was used for interim prediction and we called it the validation set. The reference values for validation were the predicted breeding values of the young animals from a full single-step run containing the genotypes of all 64,327 animals. Correlations between the approximated predictions and those from the full single-step run also containing genotypes from young animals averaged 0.9932 for the best method (from 0.990 to 0.995 across traits). In conclusion, prediction of single-step breeding values for young animals can be well approximated using systems of size equal to the number of markers.     

Genomic breeding value estimation using genetic markers, inferred ancestral haplotypes, and the genomic relationship matrix

With the introduction of new single nucleotide polymorphism (SNP) chips of various densities, more and more genotype data sets will include animals genotyped for only a subset of the SNP. Imputation techniques based on unobserved ancestral haplotypes may be used to infer missing genotypes. These ancestral haplotypes may also be used in the genomic prediction model, instead of using the SNP. This may increase the reliability of predictions because the ancestral haplotype may capture more linkage disequilibrium with quantitative trait loci than SNP. The aim of this paper was to study whether using unobserved ancestral haplotypes in a genomic prediction model would provide more reliable genomic predictions than using SNP, and to determine how many loci in the genomic prediction model would be redundant. Genotypes of 8,960 bulls and cows for 39,557 SNP were analyzed with a hidden Markov model to associate each individual at each locus to 2 ancestral haplotypes. The number of ancestral haplotypes per locus was fixed at 10, 15, or 20. Subsequently, a validation study was performed in which the phenotypes of 3,251 progeny-tested bulls for 16 traits were used in a genomic prediction model to predict the estimated breeding values of at least 753 validation bulls. The squared correlation between genomic prediction and deregressed daughter performance estimated breeding value, when averaged across traits, was slightly higher when 15 or 20 ancestral haplotypes per locus were used in the prediction model instead of the SNP genotypes, whereas the prediction model using a genomic relationship matrix gave the lowest squared correlations. The number of redundant loci [i.e., loci that had less than 18 jumps (0.1%) from one ancestral haplotype to another ancestral haplotype at the next locus], was 18,793 (48%), which means that only 20,764 loci would need to be included in the genomic prediction model. This provides opportunities for greatly decreasing computer requirements of genomic evaluations with very large numbers of markers.     

Modeling missing pedigree in single-step genomic BLUP

The objective was to compare methods of modeling missing pedigree in single-step genomic BLUP (ssGBLUP). Options for modeling missing pedigree included ignoring the missing pedigree, unknown parent groups (UPG) based on A (the numerator relationship matrix) or H (the unified pedigree and genomic relationship matrix), and metafounders. The assumptions for the distribution of estimated breeding values changed with the different models. We simulated data with heritabilities of 0.3 and 0.1 for dairy cattle populations that had more missing pedigrees for animals of lesser genetic merit. Predictions for the youngest generation and UPG solutions were compared with the true values for validation. For both traits, ssGBLUP with metafounders provided accurate and unbiased predictions for young animals while also appropriately accounting for genetic trend. Accuracy was least and bias was greatest for ssGBLUP with UPG for H for the trait with heritability of 0.3 and with UPG for A for the trait with heritability of 0.1. For the trait with heritability of 0.1 and UPG for H, the UPG accuracy (SD) was ?0.49 (0.12), suggesting poor estimates of genetic trend despite having little bias for validations on young, genotyped animals. Problems with UPG estimates were likely caused by the lesser amount of information available for the lower heritability trait. Hence, UPG need to be defined differently based on the trait and amount of information. More research is needed to investigate accounting for UPG in A₂₂ to better account for missing pedigrees for genotyped animals.     

Genomic and pedigree-based genetic parameters for scarcely recorded traits when some animals are genotyped

Genetic parameters were estimated using relationships between animals that were based either on pedigree, 43,011 single nucleotide polymorphisms, or a combination of these, considering genotyped and non-genotyped animals. The standard error of the estimates and a parametric bootstrapping procedure was used to investigate sampling properties of the estimated variance components. The data set contained milk yield, dry matter intake and body weight for 517 first-lactation heifers with genotypes and phenotypes, and another 112 heifers with phenotypes only. Multivariate models were fitted using the different relationships in ASReml software. Estimates of genetic variance were lower based on genomic relationships than using pedigree relationships. Genetic variances from genomic and pedigree relationships were, however, not directly comparable because they apply to different base populations. Standard errors indicated that using the genomic relationships gave more accurate estimates of heritability but equally accurate estimates of genetic correlation. However, the estimates of standard errors were affected by the differences in scale between the 2 relationship matrices, causing differences in values of the genetic parameters. The bootstrapping results (with genetic parameters at the same level), confirmed that both heritability and genetic correlations were estimated more accurately with genomic relationships in comparison with using the pedigree relationships. Animals without genotype were included in the analysis by merging genomic and pedigree relationships. This allowed all phenotypes to be used, including those from non-genotyped animals. This combination of genomic and pedigree relationships gave the most accurate estimates of genetic variance. When a small data set is available it might be more advantageous for the estimation of genetic parameters to genotype existing animals, rather than collecting more phenotypes.     

A mating advice system in dairy cattle incorporating genomic information

This study investigated the effects of alternative mating programs that incorporate genomic information on expected progeny herd performance and inbreeding, as well as methods to include un-genotyped animals in such mating programs. A total of 54,535 Holstein-Friesian cattle with imputed high-density genotypes (547,650 SNP after edits) were available. First, to quantify the accuracy of imputing un-genotyped animals (often an issue in populations), a sub-population of 729 genotyped animals had their genotypes masked, and their allele dosages were imputed, using linear regression exploiting information on genotyped relatives. The reference population for imputation included all genotyped animals, excluding the 729 selected animals and their sires, dams, and grandsires, and had either (1) their sires' genotypes, (2) their dams' genotypes (3) both their sires' and their dams' genotypes, or (4) both their sires' and maternal grandsires' genotypes introduced into the reference population. The correlations between true genotypes and the imputed allele dosages ranged from 0.58 (sire only) to 0.68 (both sire and dam). A herd of 100 cows was then simulated (1,000 replicates) from the sub-population of 729 imputed animals. The top 10 bulls from the genotyped population, based on their total genetic merit index (TMI) were selected to be used as sires. Three mating allotment methods were investigated: (1) random mating, (2) sequential mating based on maximizing only the expected TMI of the progeny, and (3) linear programming to maximize a generated index constructed to maximize genetic merit and minimize expected progeny inbreeding as well as intra- and inter-progeny variability in genetic merit. Relationships among candidate parents were calculated using either the pedigree relationship matrix or the genomic relationship matrix; the latter was constructed using either the true genotypes of both parents or the true genotypes of the sire plus the imputed allele dosages of the dam. Using the genomic co-ancestry estimates resulted in lower average herd expected genomic inbreeding levels compared with using the pedigree-based co-ancestry estimates. Additionally, if the dams were not genotyped, using their imputed allele dosages also resulted in lower average herd expected inbreeding levels compared with using the pedigree co-ancestry estimates. The inter-progeny coefficient of variation for selected traits, milk and fertility, estimated breeding values were reduced by 12 to 65% using the linear programing method compared with sequential mating.     

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.