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Background: Single-step genomic best linear unbiased prediction (BLUP) evaluation combines relationship information from pedigree and genomic marker data. The inclusion of the genomic information into mixed model equations requires the inverse of the combined relationship matrix [Formula: see text], which has a dense matrix block for genotyped animals., Methods: To avoid inversion of dense matrices, single-step genomic BLUP can be transformed to single-step single nucleotide polymorphism BLUP (SNP-BLUP) which have observed and imputed marker coefficients. Simple block LDL type decompositions of the single-step relationship matrix [Formula: see text] were derived to obtain different types of linearly equivalent single-step genomic mixed model equations with different sets of reparametrized random effects. For non-genotyped animals, the imputed marker coefficient terms in the single-step SNP-BLUP were calculated on-the-fly during the iterative solution using sparse matrix decompositions without storing the imputed genotypes. Residual polygenic effects were added to genotyped animals and transmitted to non-genotyped animals using relationship coefficients that are similar to imputed genotypes. The relationships were further orthogonalized to improve convergence of iterative methods., Results: All presented single-step SNP-BLUP models can be solved efficiently using iterative methods that rely on iteration on data and sparse matrix approaches. The efficiency, accuracy and iteration convergence of the derived mixed model equations were tested with a small dataset that included 73,579 animals of which 2885 were genotyped with 37,526 SNPs., Conclusions: Inversion of the large and dense genomic relationship matrix was avoided in single-step evaluation by using fully orthogonalized single-step SNP-BLUP formulations. The number of iterations until convergence was smaller in single-step SNP-BLUP formulations than in the original single-step GBLUP when heritability was low, but increased above that of the original single-step when heritability was high.

Three random regression models were developed for routine genetic evaluation of Danish, Finnish, and Swedish dairy cattle. Data included over 169 million test-day records with milk, protein, and fat yield observations from over 8.7 million dairy cows of all breeds. Variance component analyses showed significant differences in estimates between Holstein, Nordic Red Cattle, and Jersey, but only small to moderate differences within a breed across countries. The obtained variance component estimates were used to build, for each breed, their own set of covariance functions. The covariance functions describe the animal effects on milk, protein, and fat yields of the first 3 lactations as 9 different traits, assuming the same heritabilities and a genetic correlation of unity across countries. Only 15, 27, and 7 eigenfunctions with the largest eigenvalues were used to describe additive genetic animal effects and nonhereditary animal effects across lactations and within later lactations, respectively. These reduced-rank covariance functions explained 99.0 to 99.9% of the original variances but reduced the number of animal equations to be solved by 44%. Moderate rank reduction for nonhereditary animal effects and use of one-third-smaller measurement error correlations than obtained from variance component estimation made the models more robust against extreme observations. Estimation of the genetic levels of the countries' subpopulations within a breed was found sensitive to the way the breed effects were modeled, especially for the genetically heterogeneous Nordic Red Cattle. Means to ensure that only additive genetic effects entered the estimated breeding values were to describe the crossbreeding effects by fixed and random cofactors and the calving age effect by an age × breed proportion interaction, and to model phantom parent groups as random effects. To ensure that genetic variances were the same across the 3 countries in breeding value estimation, as suggested by the variance component estimates, the applied multiplicative heterogeneous variance adjustment method had to be tailored using country-specific reference measurement error variances. Results showed the feasibility of across-country genetic evaluation of cows and sires based on original test-day phenotypes. Nevertheless, applying a thorough model validation procedure is essential throughout the model building process to obtain reliable breeding values., (Copyright © 2015 American Dairy Science Association. Published by Elsevier Inc. All rights reserved.)

Background: Although the X chromosome is the second largest bovine chromosome, markers on the X chromosome are not used for genomic prediction in some countries and populations. In this study, we presented a method for computing genomic relationships using X chromosome markers, investigated the accuracy of imputation from a low density (7K) to the 54K SNP (single nucleotide polymorphism) panel, and compared the accuracy of genomic prediction with and without using X chromosome markers., Methods: The impact of considering X chromosome markers on prediction accuracy was assessed using data from Nordic Holstein bulls and different sets of SNPs: (a) the 54K SNPs for reference and test animals, (b) SNPs imputed from the 7K to the 54K SNP panel for test animals, (c) SNPs imputed from the 7K to the 54K panel for half of the reference animals, and (d) the 7K SNP panel for all animals. Beagle and Findhap were used for imputation. GBLUP (genomic best linear unbiased prediction) models with or without X chromosome markers and with or without a residual polygenic effect were used to predict genomic breeding values for 15 traits., Results: Averaged over the two imputation datasets, correlation coefficients between imputed and true genotypes for autosomal markers, pseudo-autosomal markers, and X-specific markers were 0.971, 0.831 and 0.935 when using Findhap, and 0.983, 0.856 and 0.937 when using Beagle. Estimated reliabilities of genomic predictions based on the imputed datasets using Findhap or Beagle were very close to those using the real 54K data. Genomic prediction using all markers gave slightly higher reliabilities than predictions without X chromosome markers. Based on our data which included only bulls, using a G matrix that accounted for sex-linked relationships did not improve prediction, compared with a G matrix that did not account for sex-linked relationships. A model that included a polygenic effect did not recover the loss of prediction accuracy from exclusion of X chromosome markers., Conclusions: The results from this study suggest that markers on the X chromosome contribute to accuracy of genomic predictions and should be used for routine genomic evaluation.

Estimation of variance components by Monte Carlo (MC) expectation maximization (EM) restricted maximum likelihood (REML) is computationally efficient for large data sets and complex linear mixed effects models. However, efficiency may be lost due to the need for a large number of iterations of the EM algorithm. To decrease the computing time we explored the use of faster converging Newton-type algorithms within MC REML implementations. The implemented algorithms were: MC Newton-Raphson (NR), where the information matrix was generated via sampling; MC average information(AI), where the information was computed as an average of observed and expected information; and MC Broyden's method, where the zero of the gradient was searched using a quasi-Newton-type algorithm. Performance of these algorithms was evaluated using simulated data. The final estimates were in good agreement with corresponding analytical ones. MC NR REML and MC AI REML enhanced convergence compared to MC EM REML and gave standard errors for the estimates as a by-product. MC NR REML required a larger number of MC samples, while each MC AI REML iteration demanded extra solving of mixed model equations by the number of parameters to be estimated. MC Broyden's method required the largest number of MC samples with our small data and did not give standard errors for the parameters directly. We studied the performance of three different convergence criteria for the MC AI REML algorithm. Our results indicate the importance of defining a suitable convergence criterion and critical value in order to obtain an efficient Newton-type method utilizing a MC algorithm. Overall, use of a MC algorithm with Newton-type methods proved feasible and the results encourage testing of these methods with different kinds of large-scale problem settings.

Background: Genomic data are used in animal breeding to assist genetic evaluation. Several models to estimate genomic breeding values have been studied. In general, two approaches have been used. One approach estimates the marker effects first and then, genomic breeding values are obtained by summing marker effects. In the second approach, genomic breeding values are estimated directly using an equivalent model with a genomic relationship matrix. Allele coding is the method chosen to assign values to the regression coefficients in the statistical model. A common allele coding is zero for the homozygous genotype of the first allele, one for the heterozygote, and two for the homozygous genotype for the other allele. Another common allele coding changes these regression coefficients by subtracting a value from each marker such that the mean of regression coefficients is zero within each marker. We call this centered allele coding. This study considered effects of different allele coding methods on inference. Both marker-based and equivalent models were considered, and restricted maximum likelihood and Bayesian methods were used in inference., Results: Theoretical derivations showed that parameter estimates and estimated marker effects in marker-based models are the same irrespective of the allele coding, provided that the model has a fixed general mean. For the equivalent models, the same results hold, even though different allele coding methods lead to different genomic relationship matrices. Calculated genomic breeding values are independent of allele coding when the estimate of the general mean is included into the values. Reliabilities of estimated genomic breeding values calculated using elements of the inverse of the coefficient matrix depend on the allele coding because different allele coding methods imply different models. Finally, allele coding affects the mixing of Markov chain Monte Carlo algorithms, with the centered coding being the best., Conclusions: Different allele coding methods lead to the same inference in the marker-based and equivalent models when a fixed general mean is included in the model. However, reliabilities of genomic breeding values are affected by the allele coding method used. The centered coding has some numerical advantages when Markov chain Monte Carlo methods are used.

Genome-wide SNP data provide a powerful tool to estimate pairwise relatedness among individuals and individual inbreeding coefficient. The aim of this study was to compare methods for estimating the two parameters in a Finnsheep population based on genome-wide SNPs and genealogies, separately. This study included ninety-nine Finnsheep in Finland that differed in coat colours (white, black, brown, grey, and black/white spotted) and were from a large pedigree comprising 319 119 animals. All the individuals were genotyped with the Illumina Ovine SNP50K BeadChip by the International Sheep Genomics Consortium. We identified three genetic subpopulations that corresponded approximately with the coat colours (grey, white, and black and brown) of the sheep. We detected a significant subdivision among the colour types (F(ST) = 5.4%, P<0.05). We applied robust algorithms for the genomic estimation of individual inbreeding (F(SNP)) and pairwise relatedness (Φ(SNP)) as implemented in the programs KING and PLINK, respectively. Estimates of the two parameters from pedigrees (F(PED) and Φ(PED)) were computed using the RelaX2 program. Values of the two parameters estimated from genomic and genealogical data were mostly consistent, in particular for the highly inbred animals (e.g. inbreeding coefficient F>0.0625) and pairs of closely related animals (e.g. the full- or half-sibs). Nevertheless, we also detected differences in the two parameters between the approaches, particularly with respect to the grey Finnsheep. This could be due to the smaller sample size and relative incompleteness of the pedigree for them.We conclude that the genome-wide genomic data will provide useful information on a per sample or pairwise-samples basis in cases of complex genealogies or in the absence of genealogical data.

A method based on Taylor series expansion for estimation of location parameters and variance components of non-linear mixed effects models was considered. An attractive property of the method is the opportunity for an easily implemented algorithm. Estimation of non-linear mixed effects models can be done by common methods for linear mixed effects models, and thus existing programs can be used after small modifications. The applicability of this algorithm in animal breeding was studied with simulation using a Gompertz function growth model in pigs. Two growth data sets were analyzed: a full set containing observations from the entire growing period, and a truncated time trajectory set containing animals slaughtered prematurely, which is common in pig breeding. The results from the 50 simulation replicates with full data set indicate that the linearization approach was capable of estimating the original parameters satisfactorily. However, estimation of the parameters related to adult weight becomes unstable in the case of a truncated data set.