Conventional prediction of dairy cattle merit involves setting up and solving linear equations with the number of unknowns being the number of animals, typically millions, multiplied by the number of traits being simultaneously assessed. The coefficient matrix has been large and sparse and iteration on data has been the method of choice, whereby the coefficient matrix is not stored but recreated as needed. In contrast, genomic prediction involves assessment of the merit of genome fragments characterized by single nucleotide polymorphism genotypes, currently some 50,000, which can then be used to predict the merit of individual animals according to the fragments they have inherited. The prediction equations for chromosome fragments typically have fewer than 100,000 unknowns, but the number of observations used to predict the fragment effects can be one-tenth the number of fragments. The coefficient matrix tends to be dense and the resulting system of equations can be ill behaved. Equivalent computing algorithms for genomic prediction were derived. The number of unknowns in the equivalent system grows with number of genotyped animals, usually bulls, rather than the number of chromosome fragment effects. In circumstances with fewer genotyped animals than single nucleotide polymorphism genotypes, these equivalent computations allow the solving of a smaller system of equations that behaves numerically better. There were 3 solving strategies compared: 1 method that formed and stored the coefficient matrix in memory and 2 methods that iterate on data. Finally, formulas for reliabilities of genomic predictions of merit were developed.

Utility of the preconditioned conjugate gradient algorithm with a diagonal preconditioner for solving mixed-model equations in animal breeding applications was evaluated with 16 test problems. The problems included single- and multiple-trait analyses, with data on beef, dairy, and swine ranging from small examples to national data sets. Multiple-trait models considered low and high genetic correlations. Convergence was based on relative differences between left- and right-hand sides. The ordering of equations was fixed effects followed by random effects, with no special ordering within random effects. The preconditioned conjugate gradient program implemented with double precision converged for all models. However, when implemented in single precision, the preconditioned conjugate gradient algorithm did not converge for seven large models. The preconditioned conjugate gradient and successive overrelaxation algorithms were subsequently compared for 13 of the test problems. The preconditioned conjugate gradient algorithm was easy to implement with the iteration on data for general models. However, successive overrelaxation requires specific programming for each set of models. On average, the preconditioned conjugate gradient algorithm converged in three times fewer rounds of iteration than successive overrelaxation. With straightforward implementations, programs using the preconditioned conjugate gradient algorithm may be two or more times faster than those using successive overrelaxation. However, programs using the preconditioned conjugate gradient algorithm would use more memory than would comparable implementations using successive overrelaxation. Extensive optimization of either algorithm can influence rankings. The preconditioned conjugate gradient implemented with iteration on data, a diagonal preconditioner, and in double precision may be the algorithm of choice for solving mixed-model equations when sufficient memory is available and ease of implementation is essential.

Continuous evaluation of dairy cattle with a random regression test-day model requires a fast solving method and algorithm. A new computing technique feasible in Jacobi and conjugate gradient based iterative methods using iteration on data is presented. In the new computing technique, the calculations in multiplication of a vector by a matrix were recorded to three steps instead of the commonly used two steps. The three-step method was implemented in a general mixed linear model program that used preconditioned conjugate gradient iteration. Performance of this program in comparison to other general solving programs was assessed via estimation of breeding values using univariate, multivariate, and random regression test-day models. Central processing unit time per iteration with the new three-step technique was, at best, one-third that needed with the old technique. Performance was best with the test-day model, which was the largest and most complex model used. The new program did well in comparison to other general software. Programs keeping the mixed model equations in random access memory required at least 20 and 435% more time to solve the univariate and multivariate animal models, respectively. Computations of the second best iteration on data took approximately three and five times longer for the animal and test-day models, respectively, than did the new program. Good performance was due to fast computing time per iteration and quick convergence to the final solutions. Use of preconditioned conjugate gradient based methods in solving large breeding value problems is supported by our findings.

This study investigated the possibility of increasing the reliability of direct genomic values (DGV) by combining reference populations. The data were from 3,735 bulls from Danish, Swedish, and Finnish Red dairy cattle populations. Single nucleotide polymorphism markers were fitted as random variables in a Bayesian model, using published estimated breeding values as response variables. In total, 17 index traits were analyzed. Reliabilities were estimated using a 5-fold cross validation, and calculated as the within-year squared correlation between estimated breeding values and DGV. Marker effects were estimated using reference populations from individual countries, as well as using a combined reference population from all 3 countries. Single-country reference populations gave mean reliabilities across 17 traits of 0.19 to 0.23, whereas the combined reference gave mean reliabilities of 0.26 for all populations. Using marker effects from 1 population to predict the other 2 gave a loss in mean reliability of 0.14 to 0.21 when predicting Swedish or Finnish animals with Danish marker effects, or vice versa. Using Swedish or Finnish marker effects to predict each other only showed a loss in mean reliability of 0.03 to 0.05. A combined Swedish-Finnish reference population led to an average reliability as high as that from the 3-country reference population, but somewhat different for individual traits. The results from this study show that it is possible to increase the reliability of DGV by combining reference populations from related populations.

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.

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%, P0.0625) 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.