Comparison of five methods for genomic breeding value estimation for the common dataset of the 15th QTL-MAS Workshop

Background Genomic breeding value estimation is the key step in genomic selection. Among many approaches, BLUP methods and Bayesian methods are most commonly used for estimating genomic breeding values. Here, we applied two BLUP methods, TABLUP and GBLUP, and three Bayesian methods, BayesA, BayesB and BayesCπ, to the common dataset provided by the 15th QTL-MAS Workshop to evaluate and compare their predictive performances. Results For the 1000 progenies without phenotypic values, the correlations between GEBVs by different methods ranged from 0.812 (GBLUP and BayesCπ) to 0.997 (TABLUP and BayesB). The accuracies of GEBVs (measured as correlations between true breeding values (TBVs) and GEBVs) were from 0.774 (GBLUP) to 0.938 (BayesCπ) and the biases of GEBVs (measure as regressions of TBVs on GEBVs) were from 1.033 (TABLUP) to 1.648 (GBLUP). The three Bayesian methods and TABLUP had similar accuracy and bias. Conclusions BayesA, BayesB, BayesCπ and TABLUP performed similarly and satisfactorily and remarkably outperformed GBLUP for genomic breeding value estimation in this dataset. TABLUP is a promising method for genomic breeding value estimation because of its easy computation of reliabilities of GEBVs and its easy extension to real life conditions such as multiple traits and consideration of individuals without genotypes.

Conclusions: BayesA, BayesB, BayesCπ and TABLUP performed similarly and satisfactorily and remarkably outperformed GBLUP for genomic breeding value estimation in this dataset. TABLUP is a promising method for genomic breeding value estimation because of its easy computation of reliabilities of GEBVs and its easy extension to real life conditions such as multiple traits and consideration of individuals without genotypes.

Background
The goal of genomic selection (GS) [1] is to capture all quantitative trait loci (QTL) influencing a trait by tracing all chromosome segments defined by adjacent markers. With use of highly dense markers, GS is supposed to be able to overcome the problem of traditional maker assisted selection (MAS) that only a limited proportion of the total genetic variance is captured by the markers of QTL. GS has become feasible very recently with the high throughput genotyping technology and the availability of highly dense markers covering whole genome. Genomic breeding value estimation is the key step in GS. A number of approaches have been proposed for estimating genomic breeding values [1][2][3][4][5][6][7][8][9], among which BLUP methods and Bayesian methods are most commonly used. Here, we applied two BLUP methods (GBLUP [3], TABLUP [4]) and three Bayesian methods (BayesA, BayesB [1], BayesCπ [5]) to the common dataset provided by the 15 th QTL-MAS Workshop to evaluate and compare their predictive performances.

Dataset
The common dataset consisted of an outbred population, which had been simulated using the LDSO software [10], with 1000 generations of 1000 individuals, followed by 30 generations of 150 individuals. 9990 SNP markers were distributed on 5 chromosomes. Each chromosome had a size of 1 Morgan and carried 1998 evenly distributed SNPs (1 SNP every 0.05 cM).
The final dataset used for evaluating genomic selection consisted of 3220 individuals, including 20 sires, 200 dams (each sire mated with 10 dams) and 3000 progenies (15 per dam). All individuals were genotyped for the 9990 SNPs without missing or genotyping error. Of the 15 progenies of each dam, 10 were phenotyped for a continuous trait. The 2000 progenies with phenotypic records and the other 1000 individuals (which had simulated true breeding values) without phenotypic records were treated as reference and validation population, respectively.

Estimation of variance components and EBVs
The variance components and the traditional BLUP EBVs were estimated using phenotypes and pedigree and the software DMUv6 [11] based on the following model: where y is the vector of phenotypes of individuals in the reference population, μ is the overall mean, a is the vector of additive genetic effects of the phenotyped individuals and their parents, Z is the incidence matrix of a, and e is the vector of residual errors. The variance-covariance matrices of a and e are Aσ 2 a and Iσ 2 e , respectively, where A is the additive genetic relationship matrix, σ 2 a is the additive genetic variance, and σ 2 e is the residual variance.
The reliabilities of the traditional EBVs were obtained from DMU directly and calculated as the square of the correlation between EBVs and the true unknown breeding values.

Estimation of SNP effects
BayesA, BayesB and BayesCπ were used to estimate SNP effects in the reference population based on the following model: where g is the vector of random SNP effects, X is the matrix of genotype indicators (with values 0, 1, or 2 for genotypes 11, 12, and 22, respectively).
The differences between the three Bayesian methods lay in the assumptions for the prior distribution of SNP effects. BayesA assumes that all SNPs have an effect, but each has a different variance. BayesB and BayesCπ assume that each SNP has either an effect of zero or non-zero with probabilities π and 1-π, respectively, and for those having non-zero effects it is assumed that each SNP has a different variance in BayesB and a common variance in BayesCπ. In addition, in BayesB π is treated as a known parameter, while in BayesCπ it is treated as an unknown parameter with a uniform (0, 1) prior distribution. In this study, we set π = 0.99 for BayesB, and adopted the same prior distributions of g and e for the three Bayesian methods as those in [1,5].
The Markov chain was run for 50,000 cycles of Gibbs sampling (for BayesB, 100 additional cycles of Metropolis-Hastings sampling were performed for the SNP effect variance in each Gibbs sampling cycle), and the first 5000 cycles were discarded as burn-in. All the samples of SNP effects after burn-in were averaged to obtain the SNP effect estimate.
For BayesA, BayesB and BayesCπ, the GEBV of a genotyped individual was calculated as the sum of all marker effects according to its marker genotypes [1].
For GBLUP and TABLUP, the GEBVs were estimated based on the following model: where u is the vector of genomic breeding values of all genotyped individuals with the variance-covariance matrix equal to Gσ 2 u for GBLUP or TAσ 2 u for TABLUP. σ 2 u is the additive genetic variance estimated from the reference population.
The G matrix (realized relationship matrix) was constructed by using genotypes of all markers [3]. The TA matrix (trait-specific marker-derived relationship matrix), was constructed by using genotypes of all markers with each marker being weighted with its estimated effect obtained from BayesB following the rules proposed by Zhang et al. [4].
The accuracies of GEBVs were calculated as the correlation between GEBVs and the simulated true breeding values.

Variance components
The estimated additive genetic variance and residual variance were 24.82 and 58.65, respectively. Therefore, the estimated heritability was 0.30. These estimates were used for the subsequent estimation of SNP effects and GEBVs. Figure 1 includes the profiles of SNP effects estimated by BayesA ( Figure 1A), BayesB ( Figure 1B) and BayesCπ ( Figure 1C). These estimated effects, which are obviously not evenly distributed, reflect the underlying architecture of the trait. The estimated value of π in BayesCπ is 0.9986. In general, the SNP effect profiles from the three Bayesian methods are quite similar. In particular, all of the three methods show a big peak on chromosome 1, two peaks on chromosome 2, and a peak on chromosome 3. In addition, BayesCπ shows another peak on chromosome 3 and a peak on chromosome 4. No peaks appear on chromosome 5 for all of the three methods. The peak positions and the corresponding SNP effect estimates are given in Table 1. For chromosomes 1, 2 and 3, where one, two and two additive QTL were simulated, respectively, these peak positions match all the simulated QTL positions quite well, except that BayesA and BayesB missed one QTL on chromosome 3. For chromosomes 4 and 5, where an imprinted QTL and two epistatic QTL were simulated, respectively, either no peak was detected or the detected peak is far away from the simulated position. From these results, it seems that these methods could also serve as tools for QTL mapping and BayesCπ performed better in this respect. The drawback of BayesA and BayesB regarding the impact of prior hyperparameters and treating the prior probability π as known has been addressed by Gianola et al. [12] and Habier et al. [5]. Our results partially confirmed their arguments.

Correlations between GEBVs by different methods and between EBVs and GEBVs for the 20 sires
For the 20 sires, the reliability of traditional EBVs was 0.95. Table 2 shows the correlations between GEBVs by different methods and between EBVs and GEBVs of the  Correlations between GEBVs by different methods for the 1000 progenies without phenotypic values Table 3 shows the correlations between GEBVs by different methods for the 1000 progenies without phenotypic values. The correlations ranged from 0.812 to 0.997, and the highest correlation was between TABLUP and BayesB, and the lowest between GBLUP and BayesCπ. The correlations among the three Bayesian methods and TABLUP are all very high (over 0.97), indicating high similarity in GEBVs from these methods, while the correlations between them and GBLUP are all less than 0.9, indicating some differences in GEBVs exist herein.

Accuracies and biases of GEBVs
The availability of true breeding values (TBVs) of the 1000 progenies without phenotypic values allowed a more efficient assessment for methods. Table 4 shows the correlations of TBVs and GEBVs, which measure the accuracies of GEBVs, and regressions of TBVs on GEBVs, which measure the biases of GEBVs, by different methods. In terms of both accuracy and bias, the three Bayesian methods and TABLUP performed similarly with correlations over 0.92 and slightly downward bias. BayesB and BayesCπ were slightly more accurate than BayesA and TABLUP, while TABLUP yielded smallest bias.
GBLUP gave the lowest accuracy and the highest downward bias. TABLUP is an improvement of GBLUP in the way that the G matrix is replaced with TA matrix. In construction of the TA matrix, not only the marker genotypes, but also the marker effects are taken into account. The advantage of the TA matrix over the G matrix is that it not only accounts for the Mendelian sampling term, but also puts greater weight on loci explaining more of genetic variance for the trait of interest. This makes TABLUP more accurate than GBLUP. On the other hand, although TABLUP and the Bayesian methods gave similar accuracies, TABLUP has two important features that Bayesian methods lack. The first is that the reliability of an individual's GEBV can be calculated by TABLUP through the method outlined for GBLUP by VanRaden [3] and Strandén et al. [13]. The second is that TABLUP can be extended to estimate GEBVs for individuals without genotypes by constructing a joint pedigree-genomic relationship matrix according to the rule proposed by Legarra et al. [14].

Conclusions
BayesA, BayesB, BayesCπ and TABLUP performed similarly and satisfactorily and remarkably outperformed GBLUP for genomic breeding value estimation in this dataset. TABLUP is a promising method for genomic breeding value estimation because of its easy computation of reliabilities of GEBVs and its easy extension to real life conditions such as multiple traits and consideration of individuals without genotypes.    Authors' contributions CLW, PPM and ZZ contributed the data analyses and the manuscript. XDD and JFL contributed the modification of manuscript. WXF and ZQW carried out the data analyses. QZ coordinated the analyses and revised the manuscript. All authors have read and contributed to the final text of the manuscript.