The accuracy of DGV is critical to determine the utility of DGV in relation to genotyping costs. In simulation studies, the correlation between DGV and true breeding values (TBV) has been used to represent the accuracy of DGV. However, in field data, TBV are not available and the correlation between DGV and the response variable (phenotype records, EBV, DEBV, etc.) typically underestimate the accuracy of DGV due to the contribution of environmental effects and random error to the response variable. Habier et al.  estimated marker effects using daughter yield deviations (DYD) of dairy bulls and divided the correlation between DGV and DYD by the average accuracy of the DYD to estimate the correlation between DGV and TBV. Su et al.  used the average accuracy of EBV to adjust the simple correlation between DGV and EBV (the response variable). VanRaden et al.  divided the GEBV accuracy by the mean accuracy of the DYD and then added the difference between the published and observed accuracy of PA to calculate the realized genomic accuracy. However, using the mean accuracy as an adjustment factor does not consider the heterogeneous error variance, which is associated with the DEBV of different bulls and this may lead to a bias. In this study, accuracy was obtained by standardizing the estimated covariance between DEBV and DGV using the genetic variance.
Reports on the accuracy of DGV for beef cattle are scarce. Rolf et al.  found low accuracies of about 0.3 for average daily feed intake, residual feed intake and average daily gain, when a genomic relationship matrix was used for 2405 genotyped Angus steers and sires. In dairy cattle, Harris et al.  reported accuracies of DGV for young bulls with no daughter information ranging from 0.71 to 0.82 for milk production traits, live weight, fertility, somatic cell count and longevity, compared to an average accuracy of 0.58 for PA in a New Zealand Holstein population. In their study, accuracies of DGV for linear type traits were lower than for production traits and ranged from 0.63 to 0.71, compared to an average of 0.56 for PA for these traits. The average accuracy from combining DGV and PA for 27 traits in the North American Holstein population reported by VanRaden et al.  was 0.71, compared to 0.52 from PA alone. Accuracies for GEBV combining DGV and national EBV for 12 Dutch Holstein traits ranged from 0.52 to 0.82, with an average of 0.71 . Luan et al.  reported accuracies of DGV for milk, fat and protein yields, first lactation mastitis and calving ease ranging from 0.12 to 0.62 using a small sample (500 genotyped bulls) of Norwegian Red cattle. Su et al.  reported simple correlations between DGV and published EBV (as a response variable) ranging from 0.50 to 0.84, with an average of 0.65 and adjusted correlations ranging from 0.70 to 0.85, with an average of 0.74 for 18 traits in a Danish Holstein population. These authors also reported that simple and adjusted accuracies were 0.36 and 0.51 higher than the accuracies of PA. Hayes et al.  reported accuracies of DGV ranging from 0.37 to 0.74 for five simple and index traits in Australian Holstein cattle. In general, however, it is difficult to compare the accuracies from different studies because of differences in trait heritabilities, data types (phenotypes, EPD, DYD or DEBV), training and validation set sizes, validation methods (set definition) and statistical methods to estimate marker effects.
In general, the DGV accuracies obtained here by K-means clustering and 5-fold cross-validation were lower than reported for dairy cattle for traits with similar heritabilities. For example, Su et al.  used 5-fold cross-validation in a genotyped group of 3330 bulls (almost the same size as this study) and reported modified accuracies of 0.71 and 0.72 for birth index and calving index traits. Accuracies obtained for similar traits (birth weight and calving ease direct) in our study were 0.55 and 0.49, respectively. The main reason for the lower accuracies observed in our study is the validation method, where we deliberately tried to minimize the relationship between members of the training and validation sets by K-means clustering. Habier et al.  showed that DGV use realized genetic relationships among individuals to increase the accuracy of DGV (i.e., the accuracy of a DGV on a selection candidate decreases as the average genetic relationship to the training set individuals decreases). Thus, the accuracies of DGV obtained by random clustering or from training in older animals and prediction in younger animals (which can generate larger genetic relationships between members of the training and validation sets) are higher than accuracies of DGV obtained by K-means clustering.
Another reason for the lower accuracies obtained in our study is that the accuracy of genotyped bulls EBV (used to derive the DEBV response variable) is lower in beef than in dairy cattle because artificial insemination is less used . The average accuracy of EBV for the genotyped bulls across traits was only 0.77 in this study but 0.89 in the study by Su et al. . The accuracy of DGV will increase as the accuracy of EBV increases because the response variable will be closer to the true breeding value. Another reason for the lower accuracies in comparison to those from dairy cattle studies could be the different extents and patterns of LD, which exist among breeds due to differing population histories and effective population sizes (Ne). De Roos et al.  found that, for distances between 100 kb (kilobase) and 1 Mb (Megabase), Dutch Holstein-Friesian (HF) had the highest LD, followed by Dutch Red and White HF, then Australian Angus and New Zealand Jersey, and finally Australian HF and New Zealand HF, demonstrating that the extent of LD differs between subpopulations within a breed such as HF. The subpopulations have different historical backgrounds and effective population sizes. Prasad et al.  showed that there are regions of high and low LD across the chromosomes in both the Angus and Holstein breeds and a clear difference was observed in the pattern of LD between the two breeds. A difference in the extent of LD over different chromosomes has also been reported by McKay et al.  in Angus and other breeds.
Another reason for the lower accuracies of DGV observed in this study could be due to different Ne between breeds. Goddard and Hayes  showed that more animals are needed for training to obtain the same accuracy with increasing effective population size. De Roos et al.  estimated an effective population size of about 100 for Dutch black-and-white Holstein-Friesian bulls, Dutch red-and-white Holstein-Friesian bulls, Australian Holstein-Friesian bulls, Australian Angus animals, New Zealand Friesian cows, and New Zealand Jersey cows. An effective population size less than 100 was estimated for the North American Holstein population by Kim and Kirkpatrick ; Ne = 103 for German Holstein cattle by Qanbari et al. ; and Ne = 49, 53 and 47 for Danish Holstein, Danish Jersey and Danish Red cattle by Sorensen et al. . Marquez et al.  reported a high effective population size (Ne = 445) for American Red Angus beef cattle, whereas a relatively low effective population size (Ne = 85) was estimated for American Hereford beef cattle by Cleveland et al. . We estimated a high effective population size Ne = 654 ± 31 for American Angus beef cattle (data not shown), which is much higher than that found for North American Holstein and American Hereford beef cattle.
DGV were generally less accurate for traits that had fewer animals with DEBV. The importance of training population size on the accuracies of DGV has been shown in several studies [1, 38]. Although training population size and the accuracy of DEBV have a large effect on the accuracy of DGV, the accuracy also depends on other factors such as the genetic architecture of the trait (assumptions about π) and the LD between markers and with genes that affect the trait, which could differ between traits. Hayes et al.  showed that the accuracy of genomic predictions is higher for traits with some loci having large effects than for traits with no loci of large effect. The difference in the accuracy of DGV between low and high heritability traits was relatively small. In most studies using simulated data, the phenotype of genotyped individuals is used to estimate marker effects and in this case heritability has been shown to affect the accuracy of genomic prediction [38, 40]. In this study, we used DEBV to estimate marker effects and DGV. Using DEBV as the response variable is expected to make the DGV accuracy less dependent on heritability and more a function of the EBV accuracy. Here, EBV were predicted from a fairly large dataset, resulting in relatively high accuracies even for traits with a low heritability. Low heritability traits such as fitness traits have been largely ignored in livestock breeding due both to their low heritability and difficulty in recording. However, bulls can have a high accuracy for a low heritability trait if they have sufficient progeny. Thus, these traits could be included in genomic selection programs if suitable training sets could be formed.
Comparing the DGV accuracies obtained from K-means clustering and cross-validation to those for PAadj indicated that the accuracies were similar for most traits. The superiority of DGV accuracies over PAadj accuracies for carcass traits could be due to the lower accuracy of parental EBV for these traits, which are measured in limited numbers of progeny of these parents at slaughter. The PAadj accuracies obtained in this study were higher than those reported in other studies [7, 8] primarily because the available PA information in our dataset does not represent that available on the parents of the genotyped bulls at the time of their birth. The deregression method used here only excluded information for the genotyped bull from the cumulative information available on his parents and did not exclude information from other relatives, including grand-progeny, which are informative for the meioses that produced the bull being deregressed and the majority of the genotyped bulls belonged to large patrilineages. VanRaden et al.  showed that combined predictions (PA and genomic predictions) were more accurate than PA (0.22 to 0.62 greater with nonlinear genomic predictions) in North American Holstein bulls. In this study, the accuracy of GEBV obtained by combining DGV and PAadj information did not increase the accuracy for most traits, suggesting that the PAadj may not be fully independent of the Mendelian sampling effect that produced the bull for which deregression was performed. The gain from combining DGV with PAadj depends on the accuracy of DGV and PAadj and the correlation between them. Less gain in accuracy is expected from combined values if the two information sources are highly correlated. In this study, the accuracies of PA were higher than those available at the time of an animal's birth because the older animals in this population were all ancestors of the younger animals. Thus, in practice, the accuracies of PA on young selection candidates would be lower than found here because the PA would not contain information on grand-progeny and more gain could be expected from combining DGV with PA information. In addition, if the animal's own record is available before the selection decision, we have the advantage of that record in addition to PA. In this situation, less gain could be expected from combining DGV with an animal model EBV that included the individual record. However, in beef cattle, the only observation we typically have on a young bull before it is selected (at castration) is birth weight.
Estimates of variance and covariance components between traits and their respective DGV indicated that heritabilities of the DGV were greater than 0.80 but less than the expected value of 1, when DGV were obtained by K-means clustering and cross-validation (Table 8). The estimated heritabilities for DGV were higher (greater than 0.99) when DGV were obtained by random clustering and cross-validation (data not shown). Heritabilities less than 1 for the DGV obtained by K-means clustering and cross-validation show that the estimated marker effects were not consistent between training sets due to the differences in relatedness between the training and validation groups when five separate models were used to estimate the DGV of animals in each group. However, essentially the same extent of pedigree relatedness is expected when groups are constructed randomly (i.e., groups do not represent subpopulations) which leads to the heritability of DGV being close to 1. The estimated correlations between trait and respective DGV were higher than those reported by MacNeil et al.  for the same traits in Angus cattle, because they used a 384 SNP subset derived from the Illumina BovineSNP50 BeadChip to obtain DGV and validated in a single group (correlations of 0.68, 0.73, and 0.80 in comparison to 0.50, 0.65 and 0.54 for fat thickness, marbling and carcass weight, respectively, Table 8). Estimates of heritability for traits using the bivariate animal model were lower than the corresponding heritabilities reported by AAA or obtained by the weighted univariate animal model using DEBV (results not shown).
This could be due to the dependency between DEBV and DGV when the DGV of animals in one group were predicted from the DEBV of animals in the four other groups. Although five separate models were used to predict DGV, the DGV of individuals in one group are linear combination of the DEBV of individuals in the other groups which makes the covariance matrix between DEBV and DGV close to singular in the bivariate animal model analysis. More studies are needed to overcome this problem.
We used 5-fold cross-validation to evaluate the accuracy of DGV. The advantage of multi-fold cross-validation is that it can retain large training and validation sets. However, in contrast to most previous studies, we used K-means clustering to minimize the genetic relationships between groups. The distribution of amax (maximum additive-genetic relationship) for individuals within each group indicated that amax has a high density around 0.5 (sire-son relationships) and 0.25 (half-sib relationships) but a low density between groups. The distribution of inbreeding coefficients within each group revealed that the Wye population and its descendants (group 5) was distinct from the other groups, with an average inbreeding coefficient of about 0.10 due to the closing of the herd 10 generations ago and this group had low average relationships to the other groups. Accuracies of DGV were generally lower for this group, although it had a larger training set size.
When validation was performed on the younger animals or in groups obtained by random clustering, the accuracies of DGV were much higher than when cross-validation was performed in the K-means defined groups because of the higher genetic relationships between the training and validation set individuals. The lower accuracy of DGV for maternal calving ease in the younger animals is likely the result of low accuracies of EBV (and DEBV) in the younger animals, as these young bulls have few if any daughters of sufficient age to produce calving ease information. The higher accuracy of DGV with random clustering over validation on younger animals is caused by the higher genetic relationships between the training and validation sets within the randomly formed groups. These results demonstrate that validation is sensitive to the choice of the validation sample and to the pedigree relationships between the animals contributing to the validation and training sets, and the accuracies of DGV are dependent on the strength of genetic relationships between the training and validation sets. Thus, on the one hand, a dynamic training population will maintain an approximately constant average genetic relationship between animals in the training set and younger animals available for selection, leading to the largest possible DGV accuracies. On the other hand, future selection candidates, which do not have close relatives in the training set, will have DGV with reduced accuracies. However, we anticipate that there will be greater LD between markers and QTL and thus less dependency of the accuracies of DGV on the genetic relationships between training and validation sets when the recently released Illumina BovineHD and Affymetrix BOS 1 panels are employed for genomic selection.