Computational strategies for estimation of variance components

Estimates of variances and covariances by restricted maximum likelihood (REML) have desirable properties but can be very expensive to compute. Strategies are presented which may make REML estimates easier to obtain in many models used by animal breeders. A strategy which can greatly reduce costs is to obtain only upper and lower bounds on traces used in computing REML estimates rather than obtaining exact values with inversion. This strategy is effective when the mixed model equations are very large. For smaller sized problems, diagonalization of the system of equations before iteration begins is warranted;An algorithm is developed which guarantees positive definite estimated variance-covariance matrices in multiple-trait problems. By constraining eigenvalues to remain above zero, this algorithm can converge to a point arbitrarily close to the edge of the parameter space, yielding an almost singular matrix, without encountering numerical problems. Similarly, by applying upper constraints to eigenvalues, heritabilities of all traits and all linear combinations of traits can be forced to remain below one. Multiple-trait REML estimates of variances and covariances are produced by this algorithm for about the same cost as would be required to estimate variances only using single-trait REML. A limitation of the algorithm is that all traits must be measured on all animals;A Fortran program was developed which incorporates many of these cost-saving features. The program handles single- or multiple-trait problems, related or unrelated sires, genetic groups or no genetic groups, and computes with either an exact procedure (diagonalization) or approximate procedures (estimates of traces). The program was applied to four data sets of colleagues, the largest one including 49,918 records from 428 sires. Multiple-trait REML estimates of variances and covariances for a model including relationships in this largest data set were obtained with a computing time of 568 CPU seconds and cost of 200. The algorithms presented may make more widespread use of REML estimation possible

International genomic evaluation methods for dairy cattle

<p>Abstract</p> <p>Background</p> <p>Genomic evaluations are rapidly replacing traditional evaluation systems used for dairy cattle selection. Higher reliabilities from larger genotype files promote cooperation across country borders. Genomic information can be exchanged across countries using simple conversion equations, by modifying multi-trait across-country evaluation (MACE) to account for correlated residuals originating from the use of foreign evaluations, or by multi-trait analysis of genotypes for countries that use the same reference animals.</p> <p>Methods</p> <p>Traditional MACE assumes independent residuals because each daughter is measured in only one country. Genomic MACE could account for residual correlations using daughter equivalents from genomic data as a fraction of the total in each country and proportions of bulls shared. MACE methods developed to combine separate within-country genomic evaluations were compared to direct, multi-country analysis of combined genotypes using simulated genomic and phenotypic data for 8,193 bulls in nine countries.</p> <p>Results</p> <p>Reliabilities for young bulls were much higher for across-country than within-country genomic evaluations as measured by squared correlations of estimated with true breeding values. Gains in reliability from genomic MACE were similar to those of multi-trait evaluation of genotypes but required less computation. Sharing of reference genotypes among countries created large residual correlations, especially for young bulls, that are accounted for in genomic MACE.</p> <p>Conclusions</p> <p>International genomic evaluations can be computed either by modifying MACE to account for residual correlations across countries or by multi-trait evaluation of combined genotype files. The gains in reliability justify the increased computation but require more cooperation than in previous breeding programs.</p

Genomic evaluations with many more genotypes

<p>Abstract</p> <p>Background</p> <p>Genomic evaluations in Holstein dairy cattle have quickly become more reliable over the last two years in many countries as more animals have been genotyped for 50,000 markers. Evaluations can also include animals genotyped with more or fewer markers using new tools such as the 777,000 or 2,900 marker chips recently introduced for cattle. Gains from more markers can be predicted using simulation, whereas strategies to use fewer markers have been compared using subsets of actual genotypes. The overall cost of selection is reduced by genotyping most animals at less than the highest density and imputing their missing genotypes using haplotypes. Algorithms to combine different densities need to be efficient because numbers of genotyped animals and markers may continue to grow quickly.</p> <p>Methods</p> <p>Genotypes for 500,000 markers were simulated for the 33,414 Holsteins that had 50,000 marker genotypes in the North American database. Another 86,465 non-genotyped ancestors were included in the pedigree file, and linkage disequilibrium was generated directly in the base population. Mixed density datasets were created by keeping 50,000 (every tenth) of the markers for most animals. Missing genotypes were imputed using a combination of population haplotyping and pedigree haplotyping. Reliabilities of genomic evaluations using linear and nonlinear methods were compared.</p> <p>Results</p> <p>Differing marker sets for a large population were combined with just a few hours of computation. About 95% of paternal alleles were determined correctly, and > 95% of missing genotypes were called correctly. Reliability of breeding values was already high (84.4%) with 50,000 simulated markers. The gain in reliability from increasing the number of markers to 500,000 was only 1.6%, but more than half of that gain resulted from genotyping just 1,406 young bulls at higher density. Linear genomic evaluations had reliabilities 1.5% lower than the nonlinear evaluations with 50,000 markers and 1.6% lower with 500,000 markers.</p> <p>Conclusions</p> <p>Methods to impute genotypes and compute genomic evaluations were affordable with many more markers. Reliabilities for individual animals can be modified to reflect success of imputation. Breeders can improve reliability at lower cost by combining marker densities to increase both the numbers of markers and animals included in genomic evaluation. Larger gains are expected from increasing the number of animals than the number of markers.</p

Estimating genomic breeding values and detecting QTL using univariate and bivariate models

Background Genomic selection is particularly beneficial for difficult or expensive to measure traits. Since multi-trait selection is an important tool to deal with such cases, an important question is what the added value is of multi-trait genomic selection. Methods The simulated dataset, including a quantitative and binary trait, was analyzed with four univariate and bivariate linear models to predict breeding values for juvenile animals. Two models estimated variance components with REML using a numerator (A), or SNP based relationship matrix (G). Two SNP based Bayesian models included one (BayesA) or two distributions (BayesC) for estimated SNP effects. The bivariate BayesC model sampled QTL probabilities for each SNP conditional on both traits. Genotypes were permuted 2,000 times against phenotypes and pedigree, to obtain significance thresholds for posterior QTL probabilities. Genotypes were permuted rather than phenotypes, to retain relationships between pedigree and phenotypes, such that polygenic effects could still be estimated. Results Correlations between estimated breeding values (EBV) of different SNP based models, for juvenile animals, were greater than 0.93 (0.87) for the quantitative (binary) trait. Estimated genetic correlation was 0.71 (0.66) for model G (A). Accuracies of breeding values of SNP based models were for both traits highest for BayesC and lowest for G. Accuracies of breeding values of bivariate models were up to 0.08 higher than for univariate models. The bivariate BayesC model detected 14 out of 32 QTL for the quantitative trait, and 8 out of 22 for the binary trait. Conclusions Accuracy of EBV clearly improved for both traits using bivariate compared to univariate models. BayesC achieved highest accuracies of EBV and was also one of the methods that found most QTL. Permuting genotypes against phenotypes and pedigree in BayesC provided an effective way to derive significance thresholds for posterior QTL probabilitie

Fitting and validating the genomic evaluation model to Polish Holstein-Friesian cattle

The aim of the study was to fit the genomic evaluation model to Polish Holstein-Friesian dairy cattle. A training data set for the estimation of additive effects of single nucleotide polymorphisms (SNPs) consisted of 1227 Polish Holstein-Friesian bulls. Genotypes were obtained by the use of Illumina BovineSNP50 Genotyping BeadChip. Altogether 29 traits were considered: milk-, fat- and protein- yields, somatic cell score, four female fertility traits, and 21 traits describing conformation. The prediction of direct genomic values was based on a mixed model containing deregressed national proofs as a dependent variable and random SNP effects as independent variables. The correlations between direct genomic values and conventional estimated breeding values estimated for the whole data set were overall very high and varied between 0.98 for production traits and 0.78 for non return rates for cows. For the validation data set of 232 bulls the corresponding correlations were 0.38 for milk-, 0.37 for protein-, and 0.32 for fat yields, while the correlations between genomic enhanced breeding values and conventional estimated breeding values for the four traits were: 0.43, 0.44, 0.31, and 0.35. This model was able to pass the interbull validation criteria for genomic selection, which indicates that it is realistic to implement genomic selection in Polish Holstein-Friesian cattle