The first national single-step, full-information (phenotype, pedigree, and marker genotype) genetic evaluation was developed for final score of US Holsteins. Data included final scores recorded from 1955 to 2009 for 6,232,548 Holsteins cows. BovineSNP50 (Illumina, San Diego, CA) genotypes from the Cooperative Dairy DNA Repository (Beltsville, MD) were available for 6,508 bulls. Three analyses used a repeatability animal model as currently used for the national US evaluation. The first 2 analyses used final scores recorded up to 2004. The first analysis used only a pedigree-based relationship matrix. The second analysis used a relationship matrix based on both pedigree and genomic information (single-step approach). The third analysis used the complete data set and only the pedigree-based relationship matrix. The fourth analysis used predictions from the first analysis (final scores up to 2004 and only a pedigree-based relationship matrix) and prediction using a genomic based matrix to obtain genetic evaluation (multiple-step approach). Different allele frequencies were tested in construction of the genomic relationship matrix. Coefficients of determination between predictions of young bulls from parent average, single-step, and multiple-step approaches and their 2009 daughter deviations were 0.24, 0.37 to 0.41, and 0.40, respectively. The highest coefficient of determination for a single-step approach was observed when using a genomic relationship matrix with assumed allele frequencies of 0.5. Coefficients for regression of 2009 daughter deviations on parent-average, single-step, and multiple-step predictions were 0.76, 0.68 to 0.79, and 0.86, respectively, which indicated some inflation of predictions. The single-step regression coefficient could be increased up to 0.92 by scaling differences between the genomic and pedigree-based relationship matrices with little loss in accuracy of prediction. One complete evaluation took about 2h of computing time and 2.7 gigabytes of memory. Computing times for single-step analyses were slightly longer (2%) than for pedigree-based analysis. A national single-step genetic evaluation with the pedigree relationship matrix augmented with genomic information provided genomic predictions with accuracy and bias comparable to multiple-step procedures and could account for any population or data structure. Advantages of single-step evaluations should increase in the future when animals are pre-selected on genotypes.

An algorithm is described to estimate variance components for a univariate animal model using REML. Sparse matrix techniques are employed to calculate those elements of the inverse of the coefficient matrix required for the first derivatives of the likelihood. Residuals and fitted values for random effects can be used to derive additional right-hand sides for which the mixed model equations can be repeatedly solved in turn to yield an average of the observed and expected second derivatives of the likelihood function.This Newton method, using average information, generally converges in 4 0 iterations. Although the time required per iteration is two to three times greater than that required per likelihood evaluation for derivative-free methods, the total time to convergence is generally much less. An example of a complex model, involving correlated direct and maternal genetic effects, and an additional uncorrelated random effect, indicates that REML, using average information, is about five times faster than a derivativefree algorithm, using the simplex method, which is about three times faster than an expectation-maximization algorithm.

A method is described for the prediction of breeding values incorporating genomic information. The first stage involves the prediction of genomic breeding values for genotyped individuals. A novel component of this is the estimation of the genomic relationship matrix in the context of a multi-breed population. Because not all ancestors of genotyped animals are genotyped, a selection index procedure is used to blend genomic predictions with traditional ancestral information that is lost between the process of deregression of the national breeding values and subsequent re-estimation using the genomic relationship matrix. Finally, the genomically enhanced predictions are filtered through to nongenotyped descendants using a regression procedure.

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