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

Detection and parameter estimation for quantitative trait loci using regression models and multiple markers

A strategy of multi-step minimal conditional regression analysis has been developed to determine the existence of statistical testing and parameter estimation for a quantitative trait locus (QTL) that are unaffected by linked QTLs. The estimation of marker-QTL recombination frequency needs to consider only three cases: 1) the chromosome has only one QTL, 2) one side of the target QTL has one or more QTLs, and 3) either side of the target QTL has one or more QTLs. Analytical formula was derived to estimate marker-QTL recombination frequency for each of the three cases. The formula involves two flanking markers for case 1), two flanking markers plus a conditional marker for case 2), and two flanking markers plus two conditional markers for case 3). Each QTL variance and effect, and the total QTL variance were also estimated using analytical formulae. Simulation data show that the formulae for estimating marker-QTL recombination frequency could be a useful statistical tool for fine QTL mapping. With 1 000 observations, a QTL could be mapped to a narrow chromosome region of 1.5 cM if no linked QTL is present, and to a 2.8 cM chromosome region if either side of the target QTL has at least one linked QTL

Design of a Bovine Low-Density SNP Array Optimized for Imputation

The Illumina BovineLD BeadChip was designed to support imputation to higher density genotypes in dairy and beef breeds by including single-nucleotide polymorphisms (SNPs) that had a high minor allele frequency as well as uniform spacing across the genome except at the ends of the chromosome where densities were increased. The chip also includes SNPs on the Y chromosome and mitochondrial DNA loci that are useful for determining subspecies classification and certain paternal and maternal breed lineages. The total number of SNPs was 6,909. Accuracy of imputation to Illumina BovineSNP50 genotypes using the BovineLD chip was over 97% for most dairy and beef populations. The BovineLD imputations were about 3 percentage points more accurate than those from the Illumina GoldenGate Bovine3K BeadChip across multiple populations. The improvement was greatest when neither parent was genotyped. The minor allele frequencies were similar across taurine beef and dairy breeds as was the proportion of SNPs that were polymorphic. The new BovineLD chip should facilitate low-cost genomic selection in taurine beef and dairy cattle

Genome wide CNV analysis reveals additional variants associated with milk production traits in Holsteins

Milk production is an economically important sector of global agriculture. Much attention has been paid to the identification of quantitative trait loci (QTL) associated with milk, fat, and protein yield and the genetic and molecular mechanisms underlying them. Copy number variation (CNV) is an emerging class of variants which may be associated with complex traits. In this study, we performed a genome-wide association between CNVs and milk production traits in 26,362 Holstein bulls and cows. A total of 99 candidate CNVs were identified using Illumina BovineSNP50 array data, and association tests for each production trait were performed using a linear regression analysis with PCA correlation. A total of 34 CNVs on 22 chromosomes were significantly associated with at least one milk production trait after false discovery rate (FDR) correction. Some of those CNVs were located within or near known QTL for milk production traits. We further investigated the relationship between associated CNVs with neighboring SNPs. For all 82 combinations of traits and CNVs (less than 400 kb in length), we found 17 cases where CNVs directly overlapped with tag SNPs and 40 cases where CNVs were adjacent to tag SNPs. In 5 cases, CNVs located were in strong linkage disequilibrium with tag SNPs, either within or adjacent to the same haplotype block. There were an additional 20 cases where CNVs did not have a significant association with SNPs, suggesting that the effects of those CNVs were probably not captured by tag SNPs. We conclude that combining CNV with SNP analyses reveals more genetic variations underlying milk production traits than those revealed by SNPs alone.https://doi.org/10.1186/1471-2164-15-68