Practical implementation of genetic groups in single-step genomic evaluations with Woodbury matrix identity-based genomic relationship inverse.

The growing amount of genomic information in dairy cattle has increased computational and modeling challenges in the single-step evaluations. The computational challenges are due to the dense inverses of genomic (G) and pedigree (A22) relationship matrices of genotyped animals in the single-step mixed model equations. An equivalent mixed model equation is given by single-step genomic BLUP that are based on the T matrix (ssGTBLUP), where these inverses are avoided by expressing G-1 through a product of 2 rectangular matrices, and (A22)-1 through sparse matrix blocks of the inverse of full relationship matrix A-1. A proper way to account genetic groups through unknown parent groups (UPG) after the Quaas-Pollak transformation (QP) is one key factor in a single-step model. When the UPG effects are incompletely accounted, the iterative solving method may have convergence problems. In this study, we investigated computational and predictive performance of ssGTBLUP with residual polygenic (RPG) effect and UPG. The QP transformation used A-1 and, in the complete form, T and (A22)-1 matrices as well. The models were tested with official Nordic Holstein milk production test-day data and model. The results show that UPG can be easily implemented in ssGTBLUP having RPG. The complete QP transformation was computationally feasible when preconditioned conjugate gradient iteration and iteration on data without explicitly setting up G or A22 matrices were used. Furthermore, for good convergence of the preconditioned conjugate gradient method, a complete QP transformation was necessary.

Symposium review: Single-step genomic evaluations in dairy cattle.

During the last decade, genomic selection has revolutionized dairy cattle breeding. For example, Nordic dairy cows (Denmark, Finland, and Sweden) born in 2018 were >90% sired by young genomically tested bulls. Thus, the average age of sires for Red Dairy Cattle cows born in 2018 was only 3.1 yr, whereas in 2011 it was 5.7 yr. Earlier the key driver of genetic progress was the selection of progeny-tested sires, but now it is the genomic preselection of young sires. This leads to a biased estimation of genetic progress by the traditional genetic evaluations. When these are used as input for multi-step genomic evaluations also they became distorted. The only long-term solution to maintain unbiasedness is to include the genomic information in evaluations. Although means for single-step evaluation models were introduced in 2010, they have not yet been implemented in large-scale national dairy evaluations. At first, single-step evaluations were hindered by computational cost. This has been largely solved, either by sparse presentations of the inverses of the genomic relationship (G) and pedigree relationship (A22) matrices of genotyped animals needed in the single-step evaluation models based on G (ssGBLUP), or by using the single-step marker models. Approaches for G-1 are the APY-G, where the relationships among "young" animals are completely determined by their relationship to the "core" animals, and single-step evaluations where the G-1 is replaced by a computational formula based on the structure of G (ssGTBLUP). The single-step marker models include the marker effects either directly, as effects in the statistical model, or indirectly, to generate genomic relationships among genotyped animals. Concurrently with development of the algorithm, computing resources have evolved in both availability of computer memory and speed. The problems actively studied now are the same for both of the single-step approaches (GBLUP and marker models). Convergence in iterative solving seems to get worse with an increasing number of genotypes. These problems are more pronounced with low-heritability traits and in multi-trait models with high genetic correlations among traits. Problems are also related to the unbalancedness of pedigrees and diverse genetic groups. In many cases, the problem can be solved by properly accounting for contributions of the genotyped animals to genetic groups. The standard solving approach is preconditioned conjugate gradient iteration, in which the convergence has been improved by better preconditioning matrices. Another difficulty to be considered is inflation in genomic evaluations of candidate animals; genomic models seem to overvalue the genomic information. The problem is usually smaller in single-step evaluations than in multi-step evaluations but is more difficult to mitigate by ad hoc adjustments.

Using Monte Carlo method to include polygenic effects in calculation of SNP-BLUP model reliability.

An SNP-BLUP model is computationally scalable even for large numbers of genotyped animals. When genetic variation cannot be completely captured by SNP markers, a more accurate model is obtained by fitting a residual polygenic effect (RPG) as well. However, inclusion of the RPG effect increases the size of the SNP-BLUP mixed model equations (MME) by the number of genotyped animals. Consequently, the calculation of model reliabilities requiring elements of the inverted MME coefficient matrix becomes more computationally challenging with increasing numbers of genotyped animals. We present a Monte Carlo (MC)-based sampling method to estimate the reliability of the SNP-BLUP model including the RPG effect, where the MME size depends on the number of markers and MC samples. We compared reliabilities calculated using different RPG proportions and different MC sample sizes in analyzing 2 data sets. Data set 1 (data set 2) contained 19,757 (222,619) genotyped animals, with 11,729 (50,240) SNP markers, and 231,186 (13.35 million) pedigree animals. Correlations between the correct and the MC-calculated reliabilities were above 98% even with 5,000 MC samples and an 80% RPG proportion in both data sets. However, more MC samples were needed to achieve a small maximum absolute difference and mean squared error, particularly when the RPG proportion exceeded 20%. The computing time for MC SNP-BLUP was shorter than for GBLUP. In conclusion, the MC-based approach can be an effective strategy for calculating SNP-BLUP model reliability with an RPG effect included.

Metafounder approach for single-step genomic evaluations of Red Dairy cattle.

Single-step genomic BLUP (ssGBLUP) is a powerful approach for breeding value prediction in populations with a limited number of genotyped animals. However, conflicting genomic (G) and pedigree (A22) relationship matrices complicate the implementation of ssGBLUP into practice. The metafounder (MF) approach is a recently proposed solution for this problem and has been successfully used on simulated and real multi-breed pig data. Advantages of the method are easily seen across breed evaluations, where pedigrees are traced to several pure breeds, which are thereafter used as MF. Application of the MF method to ruminants is complicated due to multi-breed pedigree structures and the inability to transmit existing unknown parent groups (UPG) to MF. In this study, we apply the MF approach for ssGBLUP evaluation of Finnish Red Dairy cattle treated as a single breed. Relationships among MF were accounted for by a (co)variance matrix (Γ) computed using estimated base population allele frequencies. The attained Γ was used to calculate a relationship matrix A22Γ for the genotyped animals. We tested the influence of SNP selection on the Γ matrix by applying a minor allele frequency (MAF) threshold (ΓMAF) where accepted markers had an MAF ≥0.05. Elements in the ΓMAF matrix were slightly lower than in the Γ matrix. Correlation between diagonal elements of the genomic and pedigree relationship matrices increased from 0.53 (A22) to 0.76 ( A22Γ and [Formula: see text] ). Average diagonal elements of A22Γ and [Formula: see text] matrices increased to the same level as in the G matrix. The ssGBLUP breeding values (GEBV) were solved using either the original 236 or redefined 8 UPG, or 8 MF computed with or without the MAF threshold. For bulls, the GEBV validation test results for the 8 UPG and 8 MF gave the same validation reliability (R2; 0.31) and over-dispersion (0.73, measured by regression coefficient b1). No significant R2 increase was observed in cows. Thus, the MF greatly influenced the pedigree relationship matrices but not the GEBV. Selection of SNP according to MAF had a notable effect on the Γ matrix and made the A22 and G matrices more similar.

Short communication: Genomic prediction using different single-step methods in the Finnish red dairy cattle population.

Single-step genomic prediction models utilizing both genotyped and nongenotyped animals are likely to become the prevailing tool in genetic evaluations of livestock. Various single-step prediction models have been proposed, based either on estimation of individual marker effects or on direct prediction via a genomic relationship matrix. In this study, a classical pedigree-based animal model, a regular single-step genomic BLUP (ssGBLUP) model, algorithm for proven and young (APY) with 2 strategies for choosing core animals, and a single-step Bayesian regression (ssBR) model were compared for 305-d production traits (milk, fat, protein) in the Finnish red dairy cattle population. A residual polygenic effect with 10% of total genetic variance was included in the single-step models to reduce inflation of genomic predictions. Validation reliability was calculated as the squared Pearson correlation coefficient between genomically enhanced breeding value (GEBV) and yield deviation for masked records for 2,056 validation cows from the last year in the data set investigated. The results showed that gains of 0.02 to 0.04 on validation reliability were achieved by using single-step methods compared with the classical animal model. The regular ssGBLUP model and ssBR model with an extra polygenic effect yielded the same results. The APY methods yielded similar reliabilities as the regular ssGBLUP and ssBR. Exact prediction error variance of GEBV could be obtained by ssBR to avoid any approximation methods used for ssGBLUP when inversion left-hand side of mixed model equations is computationally infeasible for large data sets.

Reducing bias in the dairy cattle single-step genomic evaluation by ignoring bulls without progeny.

The number of genotyped animals has increased rapidly creating computational challenges for genomic evaluation. In animal model BLUP, candidate animals without progeny and phenotype do not contribute information to the evaluation and can be discarded. In theory, genotyped candidate animal without progeny can bring information into single-step BLUP (ssGBLUP) and affect the estimation of other breeding values. We studied the effect of including or excluding genomic information of culled bull calves on genomic breeding values (GEBV) from ssGBLUP. In particular, GEBVs of genotyped bulls with daughters and GEBVs of young bulls selected into AI to be progeny tested (test bulls) were studied. The ssGBLUP evaluation was computed using Nordic test day (TD) model and TD data for the Nordic Red Dairy Cattle. The results indicate that genomic information of culled bull calves does not affect the GEBVs of progeny tested reference animals, but if genotypes of the culled bulls are used in the TD ssGBLUP, the genetic trend in the test bulls is considerably higher compared to the situation when genomic information of the culled bull calves is excluded. It seems that by discarding genomic information of culled bull calves without progeny, upward bias of GEBVs of test bulls is reduced.

Solving efficiently large single-step genomic best linear unbiased prediction models.

Single-step genomic BLUP (ssGBLUP) requires a dense matrix of the size equal to the number of genotyped animals in the coefficient matrix of mixed model equations (MME). When the number of genotyped animals is high, solving time of MME will be dominated by this matrix. The matrix is the difference of two inverse relationship matrices: genomic (G) and pedigree (A22 ). Different approaches were used to ease computations, reduce computing time and improve numerical stability. Inverse of A22 can be computed as A22-1=A22-A21A11-1A12 where Aij , i, j = 1,2, are sparse sub-matrices of A-1 , and numbers 1 and 2 refer to non-genotyped and genotyped animals, respectively. Inversion of A11 was avoided by three alternative approaches: iteration on pedigree (IOP), matrix iteration in memory (IM), and Cholesky decomposition by CHOLMOD library (CM). For the inverse of G, the APY (algorithm for proven and young) approach using Cholesky decomposition was formulated. Different approaches to choose the APY core were compared. These approaches were tested on a joint genetic evaluation of the Nordic Holstein cattle for fertility traits and had 81,031 genotyped animals. Computing time per iteration was 1.19 min by regular ssGBLUP, 1.49 min by IOP, 1.32 min by IM, and 1.21 min by CM. In comparison with the regular ssGBLUP, the total computing time decreased due to omitting the inversion of the relationship matrix A22 . When APY used 10,000 (20,000) animals in the core, the computing time per iteration was at most 0.44 (0.63) min by all the APY alternatives. A core of 10,000 animals in APY gave GEBVs sufficiently close to those by regular ssGBLUP but needed only 25% of the total computing time. The developed approaches to invert the two relationship matrices are expected to allow much higher number of genotyped animals than was used in this study.

Breeding for better eye health in Finnish blue fox (Vulpes lagopus).

The frequency of eye infections in the Finnish blue fox population has increased during the past decade. Eye infection may incur economic losses to producers due to reduced selection intensity, but ethical aspects need to be considered as well because eye infection can be quite painful and reduce animal well-being. The purpose of this study was to determine the potential for genetic selection against susceptibility to eye infection. The data were collected from 2076 blue foxes at the MTT fur animal research station. Genetic parameters were estimated using single- and multiple-trait animal models. The heritability estimate for eye infection was analysed as a binary trait (EYE) and was moderate (0.24 ± 0.07). EYE had a moderate antagonistic genetic correlation (-0.49 ± 0.20) with grading density (thick underfur). The genetic correlation of EYE with grading size or body condition score was estimated without precision, but all size traits had a low antagonistic phenotypic correlation with EYE. Our results suggest that there is genetic variance in susceptibility to EYE, indicating that eye health can be improved through selection. The current recommendation is that the sick animals should be culled immediately. If more efficient selection is needed, the selection index and multiple-trait animal models can be applied in breeding for better eye health.

Risk aversion affects economic values of blue fox breeding scheme.

The profit and production of an average Finnish blue fox farm was simulated using a deterministic bio-economic farm model. Risk was included using Arrow-Prat absolute risk aversion coefficient and profit variance. Risk-rated economic values were calculated for pregnancy rate, litter loss, litter size, pelt size, pelt quality, pelt colour clarity, feed efficiency and eye infection. With high absolute risk aversion, economic values were lower than with low absolute risk aversion. Economic values were highest for litter loss (18.16 and 26.42 EUR), litter size (13.27 and 19.40 EUR), pregnancy (11.99 and 18.39 EUR) and eye infection (12.39 and 13.81 EUR). Sensitivity analysis showed that selection pressure for improved eye health depended strongly on proportion of culled animals among infected animals and much less on the proportion of infected animals. The economic value of feed efficiency was lower than expected (6.06 and 8.03 EUR). However, it was almost the same magnitude as pelt quality (7.30 and 7.30 EUR) and higher than the economic value of pelt size (3.37 and 5.26 EUR). Risk factors should be considered in blue fox breeding scheme because they change the relative importance of traits.

Single-step genomic evaluation using multitrait random regression model and test-day data.

The objectives of this study were to evaluate the feasibility of use of the test-day (TD) single-step genomic BLUP (ssGBLUP) using phenotypic records of Nordic Red Dairy cows. The critical point in ssGBLUP is how genomically derived relationships (G) are integrated with population-based pedigree relationships (A) into a combined relationship matrix (H). Therefore, we also tested how different weights for genomic and pedigree relationships affect ssGBLUP, validation reliability, and validation regression coefficients. Deregressed proofs for 305-d milk, protein, and fat yields were used for a posteriori validation. The results showed that the use of phenotypic TD records in ssGBLUP is feasible. Moreover, the TD ssGBLUP model gave considerably higher validation reliabilities and validation regression coefficients than the TD model without genomic information. No significant differences were found in validation reliability between the different TD ssGBLUP models according to bootstrap confidence intervals. However, the degree of inflation in genomic enhanced breeding values is affected by the method used in construction of the H matrix. The results showed that ssGBLUP provides a good alternative to the currently used multi-step approach but there is a great need to find the best option to combine pedigree and genomic information in the genomic matrix.

Using the unified relationship matrix adjusted by breed-wise allele frequencies in genomic evaluation of a multibreed population.

The observed low accuracy of genomic selection in multibreed and admixed populations results from insufficient linkage disequilibrium between markers and trait loci. Failure to remove variation due to the population structure may also hamper the prediction accuracy. We verified if accounting for breed origin of alleles in the calculation of genomic relationships would improve the prediction accuracy in an admixed population. Individual breed proportions derived from the pedigree were used to estimate breed-wise allele frequencies (AF). Breed-wise and across-breed AF were estimated from the currently genotyped population and also in the base population. Genomic relationship matrices (G) were subsequently calculated using across-breed (GAB) and breed-wise (GBW) AF estimated in the currently genotyped and also in the base population. Unified relationship matrices were derived by combining different G with pedigree relationships in the evaluation of genomic estimated breeding values (GEBV) for genotyped and ungenotyped animals. The validation reliabilities and inflation of GEBV were assessed by a linear regression of deregressed breeding value (deregressed proofs) on GEBV, weighted by the reliability of deregressed proofs. The regression coefficients (b1) from GAB ranged from 0.76 for milk to 0.90 for protein. Corresponding b1 terms from GBW ranged from 0.72 to 0.88. The validation reliabilities across 4 evaluations with different G were generally 36, 40, and 46% for milk, protein, and fat, respectively. Unexpectedly, validation reliabilities were generally similar across different evaluations, irrespective of AF used to compute G. Thus, although accounting for the population structure in GBW tends to simplify the blending of genomic- and pedigree-based relationships, it appeared to have little effect on the validation reliabilities.

Comparison of multiplicative heterogeneous variance adjustment models for genetic evaluations.

Two heterogeneous variance adjustment methods and two variance models were compared in a simulation study. The method used for heterogeneous variance adjustment in the Nordic test-day model, which is a multiplicative method based on Meuwissen (J. Dairy Sci., 79, 1996, 310), was compared with a restricted multiplicative method where the fixed effects were not scaled. Both methods were tested with two different variance models, one with a herd-year and the other with a herd-year-month random effect. The simulation study was built on two field data sets from Swedish Red dairy cattle herds. For both data sets, 200 herds with test-day observations over a 12-year period were sampled. For one data set, herds were sampled randomly, while for the other, each herd was required to have at least 10 first-calving cows per year. The simulations supported the applicability of both methods and models, but the multiplicative mixed model was more sensitive in the case of small strata sizes. Estimation of variance components for the variance models resulted in different parameter estimates, depending on the applied heterogeneous variance adjustment method and variance model combination. Our analyses showed that the assumption of a first-order autoregressive correlation structure between random-effect levels is reasonable when within-herd heterogeneity is modelled by year classes, but less appropriate for within-herd heterogeneity by month classes. Of the studied alternatives, the multiplicative method and a variance model with a random herd-year effect were found most suitable for the Nordic test-day model for dairy cattle evaluation.

Genetic associations of test-day fat:protein ratio with milk yield, fertility, and udder health traits in Nordic Red cattle.

Interest is growing in finding indicator traits for the evaluation of nutritional or tissue energy status of animals directly at the individual animal level. The development and subsequent use of such traits in practice demands a clear understanding of the genetic and phenotypic associations with the various production and functional traits. In this study, the relationships during lactation between milk fat:protein ratio (FPR) and production and functional traits were estimated for Nordic Red cattle, in which published information is scarce. The objectives of this study were to estimate genetic associations of FPR with milk yield (MY), fertility, and udder health traits during different stages of lactation. Traits included in the analyses were MY, 4 fertility traits-days from calving to insemination (DFI), days open (DO), number of inseminations (NI), and nonreturn rate to 56 d (NRR)-and 2 udder health traits-test-day somatic cell score (SCS) and clinical mastitis (CM). Data were from a total of 22,422 first-lactation cows. Random regression models were used to estimate genetic parameters and associations between traits. The mean FPR in first-lactation cows was 1.28 and ranged from 1.25 to 1.45. During first lactation, the heritability of FPR ranged from 0.14 to 0.25. Genetic correlations between FPR and MY in early lactation (until 50 d in milk) were positive and ranged from 0.05 to 0.22; later in lactation, they were close to zero or negative, indicating that cows may have come out of the negative state of energy balance. The strength of genetic associations between FPR and fertility traits varied during lactation. In early lactation, correlations between FPR and the interval fertility traits DFI and DO were positive and ranged from 0.14 to 0.28. Genetic correlations between FPR and the udder health traits SCS and CM in early lactation ranged from 0.09 to 0.20. Milk fat:protein ratio is a heritable trait and easily available from routine milk-recording schemes. It can be used as a low-cost monitoring tool of poor health and fertility in the most critical phases of lactation and as an important indicator trait to improve robustness in dairy cows through selection.

The estimation of genomic relationships using breedwise allele frequencies among animals in multibreed populations.

Different approaches of calculating genomic measures of relationship were explored and compared with pedigree relationships (A) within and across base breeds in a crossbreed population, using genotypes for 38,194 loci of 4,106 Nordic Red dairy cattle. Four genomic relationship matrices (G) were calculated using either observed allele frequencies (AF) across breeds or within-breed AF. The G matrices were compared separately when the AF were estimated in the observed and in the base population. Breedwise AF in the current and base population were estimated using linear regression models of individual genotypes on breed composition. Different G matrices were further used to predict direct estimated genomic values using a genomic BLUP model. Higher variability existed in the diagonal elements of G across breeds (standard deviation=0.06, on average) compared with A (0.01). The use of simple observed AF across base breeds to compute G increased coefficients for individuals in distantly related populations. Estimated breedwise AF reduced differences in coefficients similarly within and across populations. The variability of the current adjusted G matrix decreased from 0.055 to 0.035 when breedwise AF were estimated from the base breed population. The direct estimated genomic values and their validation reliabilities were, however, unaffected by AF used to compute G when estimated with a genomic BLUP model, due to inclusion of breed means in the model. In multibreed populations, G adjusted with breedwise AF from the founder population may provide more consistency among relationship coefficients between genotyped and ungenotyped individuals in an across-breed single-step evaluation.

Use of random regression model as an alternative for multibreed relationship matrix.

A random regression model is presented as an approximation for multibreed variance model. The approximation is derived using the splitted multibreed model where the single breeding value is split to the breed specific and their segregation terms. The random regression model allows extending the multibreed information easily to genomic data models. We present the approach by a simple example.

Genetic parameters of pelt character, feed efficiency and size traits in Finnish blue fox (Vulpes lagopus).

Pelt character traits (size, quality, colour clarity, darkness) are important economic traits in blue fox breeding. Better feed efficiency (FE) is another economically important and new breeding goal for fur animals. The purpose of this study was to determine the correlations between pelt character traits, FE and size traits and to estimate genetic parameters for pelt character traits. Pelt size (pSIcm ) had a high positive genetic correlation with animal grading size (gSI), final body weight (BWFin), body length and daily gain (DG), and a moderate correlation with body condition score (BCS). Animal body length and BCS (describing fatness) were considered as genetically different traits. Genetic correlations between pelt quality and size traits were estimated without precision and did not differ from zero, but colour clarity (pCL) had a low antagonistic genetic correlation with FE. Pelt size and DG had a favourable genetic correlation with FE but a fairly high unfavourable genetic correlation with dry matter feed intake. The current emphasis on selection for larger animal and pelt size improves FE indirectly, but selection for larger pelt size favours fast-growing and fat individuals and simultaneously increases feed intake. The detected genetic connections between FE, size, feed intake and pCL should be taken into account in the Finnish blue fox breeding programme.

Across breed multi-trait random regression genomic predictions in the Nordic Red dairy cattle.

The current study evaluates reliability of genomic predictions in selection candidates using multi-trait random regression model, which accounts for interactions between marker effects and breed of origin in the Nordic Red dairy cattle (RDC). The population structure of the RDC is admixed. Data consisted of individual animal breed proportions calculated from the full pedigree, deregressed proofs (DRP) of published estimated breeding values (EBV) for yield traits and genotypic data for 37,595 single nucleotide polymorphic markers. The analysed data included 3330 bulls in the reference population and 812 bulls that were used for validation. Direct genomic breeding values (DGV) were estimated using the model under study, which accounts for breed effects and also with GBLUP, which assume uniform population. Validation reliability was calculated as a coefficient of determination from weighted regression of DRP on DGV (rDRP,DGV 2), scaled by the mean reliability of DRP. Using the breed-specific model increased the reliability of DGV by 2 and 3% for milk and protein, respectively, when compared to homogeneous population GBLUP. The exception was for fat, where there was no gain in reliability. Estimated validation reliabilities were low for milk (0.32) and protein (0.32) and slightly higher (0.42) for fat.

Different methods to calculate genomic predictions--comparisons of BLUP at the single nucleotide polymorphism level (SNP-BLUP), BLUP at the individual level (G-BLUP), and the one-step approach (H-BLUP).

Several strategies to use genomic data in predictions have been proposed. The aim of this study was to compare different genomic prediction methods. The response variables used in the genomic predictions were deregressed proofs, which were derived from 2 estimated breeding value (EBV) data sets. The full EBV data set from March 2010 included the EBV for production and mastitis traits for all Nordic red bulls. The reduced data set included the same animals as the full data set, but the EBV were predicted from a data set that excluded the last 5 yr of observations. Genomic predictions were obtained using different BLUP models: BLUP at the single nucleotide polymorphism level (SNP-BLUP), BLUP at the individual level (G-BLUP), and the one-step approach (H-BLUP). For the selection candidate bulls, the SNP-BLUP and G-BLUP models gave the same direct genomic breeding values (e.g., correlation of direct genomic breeding values between SNP-BLUP and G-BLUP for protein was 0.99), but slightly different from genomic EBV obtained from H-BLUP (correlations of SNP-BLUP or G-BLUP with H-BLUP were about 0.96). For all traits, SNP-BLUP and G-BLUP gave the same validation reliability, whereas H-BLUP led to slightly higher reliability. Therefore, the results support a slight advantage of using H-BLUP for genomic evaluation.

Employing a Monte Carlo algorithm in expectation maximization restricted maximum likelihood estimation of the linear mixed model.

Multiple-trait and random regression models have multiplied the number of equations needed for the estimation of variance components. To avoid inversion or decomposition of a large coefficient matrix, we propose estimation of variance components by Monte Carlo expectation maximization restricted maximum likelihood (MC EM REML) for multiple-trait linear mixed models. Implementation is based on full-model sampling for calculating the prediction error variances required for EM REML. Performance of the analytical and the MC EM REML algorithm was compared using a simulated and a field data set. For field data, results from both algorithms corresponded well even with one MC sample within an MC EM REML round. The magnitude of the standard errors of estimated prediction error variances depended on the formula used to calculate them and on the MC sample size within an MC EM REML round. Sampling variation in MC EM REML did not impair the convergence behaviour of the solutions compared with analytical EM REML analysis. A convergence criterion that takes into account the sampling variation was developed to monitor convergence for the MC EM REML algorithm. For the field data set, MC EM REML proved far superior to analytical EM REML both in computing time and in memory need.

Influence of model specifications on the reliabilities of genomic prediction in a Swedish-Finnish red breed cattle population.

Using a combined multi-breed reference population, this study explored the influence of model specification and the effect of including a polygenic effect on the reliability of genomic breeding values (DGV and GEBV). The combined reference population consisted of 2986 Swedish Red Breed (SRB) and Finnish Ayrshire (FAY) dairy cattle. Bayesian methodology (common prior and mixture models with different prior distribution settings for the marker effects) as well as a best linear unbiased prediction with a genomic relationship matrix [genomic best linear unbiased predictor (GBLUP)] was used in the prediction of DGV. Mixture models including a polygenic effect were used to predict GEBV. In total, five traits with low, high and medium heritability were analysed. For the models using a mixture prior distribution, reliabilities of DGV tended to decrease with an increasing proportion of markers with small effects. The influence of the inclusion of a polygenic effect on the reliability of DGV varied across traits and model specifications. Average correlation between DGV with the Mendelian sampling term, across traits, was highest (R(2) = 0.25) for the GBLUP model and decreased with increasing proportion of markers with large effects. Reliabilities increased when DGV and parent average information were combined in an index. The GBLUP model with the largest gain across traits in the reliability of the index achieved the highest DGV mean reliability. However, the polygenic models showed to be less biased and more consistent in the estimation of DGV regardless of the model specifications compared with the mixture models without the polygenic effect.