This study applied three GBLUP and two single-step blending methods for genomic prediction in Nordic Holsteins. Predictive abilities of the five methods were compared in terms of reliability and bias. Results indicated that both the original single-step blending and the adjusted single-step blending were more accurate than the three GBLUP methods because the two single-step blending approaches used much more information to predict breeding values. Similar results were reported by Su et al. [18] for the Nordic Red population. In the current study, the size of the training dataset for the single-step blending methods was almost three times as large as that for the three GBLUP methods (Table 1) since DRP of the non-genotyped animals also provided information through a combined relationship matrix. Including pedigree information may also improve genomic predictions because the SNP may not account for all additive genetic variance. As shown in this study, including a residual polygenic effect in the GBLUP methods led to slightly higher reliability of genomic predictions.

A regression coefficient of DRP on genomic predictions less than 1 indicates overestimation of the variance of genomic predictions (inflation), while a coefficient larger than 1 indicates underestimation (deflation). The two single-step blending methods led to less bias than the three GBLUP methods, and the two GBLUP methods with a polygenic effect resulted in less bias than the simple GBLUP method without a polygenic effect. The problem of inflation of genomic predictions is critical in practice [33–35] as it can give an unfair advantage to juvenile over older progeny test bulls [17]. Aguilar et al. [17] showed that this bias was reduced by weighting the **G** and **A** matrices, and Liu et al. [36] found that including a polygenic effect in a GBLUP model (random regressions on SNP genotypes) led to less bias in genomic predictions. The present study showed that the weighting factor had an effect on the bias of genomic predictions for all traits in the single-step blending approaches and the GBLUP methods with a polygenic effect. A weight of 0.40 resulted in the smallest minimum absolute deviation from 1 for the regression of GEBV or DGV_{ω} on DRP, averaged over the 16 traits, but a loss of reliability around 0.8%, compared to a weight of 0.20, which led to highest average reliability and an acceptable average absolute deviation of regression coefficient from 1 (Figure 1,2).

The adjusted single-step blending method resulted in less bias than the original single-step blending for all settings of the weight factor. In a simulation study, Vitezica et al. [19] also found that the single-step method was less biased and more accurate when the genomic relationship matrix was adjusted by a constant. Using chicken data, Chen et al. [20] showed that unbiased evaluations can be obtained by adding a constant to the **G** matrix that is based on current allele frequencies and suggested that the optimal **G** has average of diagonal and off-diagonal elements close to those of **A**
_{11}. Forni et al. [22] also showed that re-scaling the **G** matrix is a reasonable solution to avoid inflation in pig data. However, in the present study, the adjusted **G** matrix did not improve genomic predictions in the GBLUP methods with a polygenic effect. This suggests that, based on the present data, adjustment of **G** has little effect on genomic prediction when only genotyped animals are used, but may be important in other data where there is a large difference in scale between **G** and **A**.

The results from the present study indicate that increasing the weighting factor (0.40) reduces bias and that weighting factors around 0.15 to 0.20 give the highest reliability but the optimal weighting factors differed between traits. Similarly, Liu et al. [36] observed that the optimal residual polygenic variance in a GBLUP model (random regressions on SNP genotypes) with a polygenic effect appears to differ among traits. Therefore, trait-specific weighting factors should be used in the single-step blending methods and the GBLUP methods with a polygenic effect. In the near future, both bulls and heifers may be pre-selected based on genomic EBV. This will lead to biased predictions of breeding values in both conventional and genomic evaluation procedures. In such situations, appropriate methods to correct the bias of predictions are required [37].

Christensen et al. [21] compared the adjusted and original single-step blending methods on pig data. In their study, the improvement of prediction reliabilities by adjustment of **G** matrix is much larger, compared with the results from the current study. This may be because there was more inbreeding in the pig data, which resulted in average values of the diagonal and off-diagonal elements of **A**
_{11} equal to 1.145 and 0.298, and estimates of β and α equal to 0.895 and 0.298, respectively. In the present study, the averages of the diagonal and off-diagonal elements of **A**
_{11}were 1.060 and 0.085, and estimates of β and α were 0.976 and 0.085, i.e. closer to one and zero, respectively. This means that the original **G** matrix was less adjusted in this study compared to the study on pig data by Christensen et al. [21].