A computationally efficient method for approximating reliabilities in large-scale single-step genomic prediction.

BACKGROUND: In this study, computationally efficient methods to approximate the reliabilities of genomic estimated breeding values (GEBV) in a single-step genomic prediction model including a residual polygenic (RPG) effect are described. In order to calculate the reliabilities of the genotyped animals, a single nucleotide polymorphism best linear unbiased prediction (SNPBLUP) or a genomic BLUP (GBLUP), was used, where two alternatives to account for the RPG effect were tested. In the direct approach, the genomic model included the RPG effect, while in the blended method, it did not but an index was used to weight the genomic and pedigree-based BLUP (PBLUP) reliabilities. In order to calculate the single-step GBLUP reliabilities for the breeding values for the non-genotyped animals, a simplified weighted-PBLUP model that included a general mean and additive genetic effects with weights accounting for the non-genomic and genomic information was used. We compared five schemes for the weights. Two datasets, i.e., a small (Data 1) one and a large (Data 2) one were used. RESULTS: For the genotyped animals in Data 1, correlations between approximate reliabilities using the blended method and exact reliabilities ranged from 0.993 to 0.996 across three lactations. The slopes observed by regressing the reliabilities of GEBV from the exact method on those from the blended method were 1.0 for all three lactations. For Data 2, the correlations and slopes ranged, respectively, from 0.980 to 0.986 and from 0.91 to 0.96, and for the non-genotyped animals in Data 1, they ranged, respectively, from 0.987 to 0.994 and from 0.987 to 1, which indicate that the approximations were in line with the exact results. The best approach achieved correlations of 0.992 to 0.994 across lactations. CONCLUSIONS: Our results demonstrate that the approximated reliabilities calculated using our proposed approach are in good agreement with the exact reliabilities. The blended method for the genotyped animals is computationally more feasible than the direct method when RPG effects are included, particularly for large-scale datasets. The approach can serve as an effective strategy to estimate the reliabilities of GEBV in large-scale single-step genomic predictions.

Breeding value reliabilities for multiple-trait single-step genomic best linear unbiased predictor.

Approximate multistep methods to calculate reliabilities for estimated breeding values in large genetic evaluations were developed for single-trait (ST-R2A) and multitrait (MT-R2A) single-step genomic BLUP (ssGBLUP) models. First, a traditional animal model was used to estimate the amount of nongenomic information for the genotyped animals. Second, this information was used with genomic data in a genomic BLUP model (genomic BLUP/SNP-BLUP) to approximate the total amount of information and ssGBLUP reliabilities for the genotyped animals. Finally, reliabilities for the nongenotyped animals were calculated using a traditional animal model where the increased information due to genomic data for the genotyped animals is accounted for by including pseudo-record counts for the genotyped animals. The approaches were tested using a multiple-trait ssGBLUP model on 2 data sets. The first data set (data 1) was small enough such that exact ssGBLUP model reliabilities could be computed by inversion and compared with the approximation method reliabilities. Data 1 had 46,535 first-, 35,290 second-, and 23,780 third-lactation 305-d milk yield records from 47,124 Finnish Red dairy cows. The pedigree comprised 64,808 animals, of which 19,757 were genotyped. We examined the efficiency of the MT-R2A approximation on a large data set (data 2) derived from the joint Nordic (Danish, Finnish, and Swedish) Holstein dairy cattle data. Data 2 had 17.8 million 305-d milk records from 8.3 million cows and first 3 lactations. The pedigree had 11 million animals of which 274,145 were genotyped on 46,342 SNP markers. For data 1, correlations between the exact ssGBLUP model and the ST-R2A for the genotyped (nongenotyped) animals were 0.995 (0.987), 0.965 (0.984), and 0.950 (0.983) for first, second, and third lactation, respectively. Correspondingly, correlations between exact ssGBLUP reliabilities and MT-R2A for the genotyped (nongenotyped) animals were 0.995 (0.993), 0.992 (0.991), and 0.990 (0.990) for first, second, and third lactation, respectively. The regression coefficients (b1) of ssGBLUP reliability on ST-R2A for the genotyped (nongenotyped) animals ranged from 0.87 (0.94) for first lactation to 0.68 (0.93) for third lactation, whereas for MT-R2A they were between 0.91 (0.99) for first lactation to 0.89 (0.99) for third lactation. Correspondingly, the intercepts varied from 0.11 (0.05) to 0.3 (0.06) for ST-R2A and from 0.06 (0.01) to 0.07 (0.02) for MT-R2A. The computing time for the approximation method was approximately 12% of that required by the direct exact approach. In conclusion, the developed approximate approach allows calculating estimated breeding value reliabilities in the ssGBLUP model even for large data sets.

Single-step SNP-BLUP with on-the-fly imputed genotypes and residual polygenic effects.

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.