A comparison of genomically enhanced breeding values predicted by different single-step approaches

doi:10.21203/rs.3.rs-5260327/v1

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A comparison of genomically enhanced breeding values predicted by different single-step approaches

https://doi.org/10.21203/rs.3.rs-5260327/v1

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Many countries are currently adopting the single-step model for national genetic evaluations of dairy cattle. The two most widely applied statistical formulations of the single-step model are Genomic Best Linear Unbiased Prediction (G-BLUP) and Single Nucleotide Polymorphism BLUP (SNP-BLUP), with the main difference being the handling of additive genetic covariance between individuals with genotypes. Using solvers available in the MiXBLUP software, our study aimed to compare both models regarding the quality of Genomically Enhanced Breeding Value (GEBV) prediction, bull rankings, and computational efficiency (memory consumption and computational time). The results demonstrated no marked differences in the quality of GEBV prediction expressed by the metrics underlying the Interbull validation, except for the G-BLUP, APY-based solvers with 3,000 core bulls. However, the ranking of the top 50 bulls differed between models, which has implications for the breeding industry and selection, since the top-ranking bulls are typically the most widely used. 39 and 31 of the top 50 bulls were common to all models for stature and foot angle, respectively. In terms of computational time, SNP-BLUP and G-BLUP with APY solver using 3,000 bulls were the fastest, the GT G-BLUP solver was the slowest. The selection of core individuals for the APY solver was a crucial element that affected the prediction accuracy. Still, the use of the GT G-BLUP or the SNP-BLUP solver can circumvent this issue since no selection of core individuals is required.

APY

G-BLUP

GT-BLUP

GEBV

single-step

SNP-BLUP

Today, many countries are implementing the single-step model for their national genetic evaluations of dairy cattle. The main difference between the currently used (conventional) models for genetic evaluation and the single-step models is the incorporation of genomic information available for the calculation of additive genetic covariances between individuals. Considering a single trait, i.e. univariate formulation, the conventional model utilizes the information contained in pedigree records to build the additive genetic covariance matrix that defines the shrinkage distribution imposed on the random animal additive genetic effect. The single-step model additionally incorporates the identical-by-descent or at least identical-by-state similarity between the genotyped fraction of the evaluated individuals, expressed by the similarity of their genotypes. Genotypic information is typically expressed by single nucleotide polymorphisms (SNPs) identified using the Illumina Bovine 50K oligonucleotide microarray or imputed to this standard. In practical applications for the evaluation of large national populations of dairy cattle, the two equivalent statistical formulations of such a single-step model comprise the genomic best linear unbiased prediction (G-BLUP) and SNP-BLUP (Liu et al. 2016; Strandén and Garrick 2009; Taylor 2014). The only statistical difference between those models lies in the formulation of the additive genetic covariance between individuals, in particular, between individuals with genotypes. The single-step

G-BLUP (Christensen and Lund 2010; Misztal et al. 2009) fits the traditional numerator pedigree relationship matrix between all evaluated individuals with the components related to non-genotyped individuals constructed using the pedigree information and the components corresponding to genotyped animals constructed as a weighted sum of pedigree and SNP genotype similarity (VanRaden 2008). The single-step SNP-BLUP fits the genomic covariance between additive genetic effects of SNPs as a separate component in addition to a pedigree-based relationship (Liu et al. 2014).

However, due to a large number of data records and then the corresponding dimensions of the single-step model underlying genetic evaluations carried out on the national scale, considerable differences among the models arise regarding approaches to the estimation of their effects. In particular, in handling the inverse of the dense genomic relationship matrix. In the context of the G-BLUP model Misztal et al. (2014) proposed an APY (the algorithm for proven and young) procedure that separates the genomic covariance matrix into a smaller, dense component corresponding to a predefined set of genotyped individuals (so-called core individuals) while the remainder, i.e. non-core genotyped individuals are fitted as genomically uncorrelated using a diagonal genomic covariance matrix. So, the system of solving model equations only requires a computationally intensive inverse of the dense genomic submatrix for the core individuals and the submatrices corresponding to the covariance between core and noncore individuals. Another approach to obtaining an inverse of the genomic covariance matrix was proposed, under the acronym GT, by Mäntysaari et al. (2017), who expressed the inverse as a function of the inverse of the numerator relationship matrix corresponding to genotyped individuals, the Cholesky factor, and the SNP genotype design matrix. Regarding the single-step SNP-BLUP, due to an alternative formulation of the covariance structure, as indicated above, solving the system of equations does not require inverting of the part (APY) or full (GT) genomic covariance matrix, but only a computationally simple inverse of the SNP covariance matrix.

Material

The data set analyzed (Table 1) represented data from the Polish national genetic evaluation of Holstein cattle from December 2021 for stature which represents a highly heritable trait (h² = 0.54) and foot angle representing a low heritable trait (h² = 0.09). It comprised 1,098,611 cows with phenotypes for stature and 1,098,766 cows with phenotypes for foot angle born from 1992 to 2019 as well as 141,397 bulls (stature), and 117,482 bulls (foot angle) born from 1986 to 2017 with pseudo-phenotypes expressed by their de-regressed proofs (DRP) from the multiple across country evaluation (MACE) carried out by the Interbull (interbull.org). Genomic data in the form of genotypes of 46,118 SNPs were available for 134,960 individuals, including 70,134 cows with phenotypes born from 2009 to 2021 as well as 64,826 bulls born from 1985 to 2021, of which 26,471 represented young individuals without pseudo-phenotypes and 38,355 were bulls with MACE-DRP. In our study, the pedigree of animals with phenotype data was truncated after the 5th generation resulting in 1,555,995 individuals and 33 genetic groups.

Table 1

Number of animals in the analyzed data sets
Category	Sex	Number of animals
Phenotype data (Stature)	Females with phenotypes	1,098,611
Phenotype data (Stature)	Males with MACE DRP	141,397
Phenotype data (Foot angle)	Females with phenotypes	1,098,766
Phenotype data (Foot angle)	Males with MACE DRP	117,482
Genotype data	Females	70,134
Genotype data	Males (bulls and candidates)	64,826
Pedigree data	Females	1,368,487
Pedigree data	Males	187,508

Models

GEBV prediction models

The following single-step single-trait models were considered for the prediction of GEBV:

the G-BLUP model (Aguilar et al. 2010; Christensen and Lund 2010)

$$\:\mathbf{y}=\mathbf{X}\mathbf{b}+{\mathbf{W}}_{\mathbf{G}}\mathbf{a}+\mathbf{e}$$

$\:\mathbf{y}$ is the vector of dependent variables represented by cows’ measured phenotypes and bulls' pseudo phenotypes expressed by their MACE DRPs, $\:\mathbf{b}$ represents a vector of fixed effects including age at calving, lactation phase, and herd corresponding to cows’ phenotypes as well as corresponding phantom codes of the fixed effects for bulls’ DRPs, $\:\mathbf{a}$ represents a vector of breeding values, and $\:\mathbf{e}$ is the residual. The underlying covariance structure of the random effects is given by $\:\mathbf{a}\sim\mathbf{N}\left(0,{\mathbf{H}}_{\mathbf{G}}{{\sigma\:}}_{\text{a}}^{2}\right)$ and $\:\mathbf{e}\sim\mathbf{N}\left(0,\mathbf{R}{{\sigma\:}}_{\text{e}}^{2}\right)$. $\:{\mathbf{H}}_{\mathbf{G}}$ is given by $\:\left[\begin{array}{cc}{\mathbf{A}}_{11}&\:{\mathbf{A}}_{12}\\\:{\mathbf{A}}_{21}&\:{\mathbf{A}}_{22}\end{array}\right]+\left[\begin{array}{cc}0&\:0\\\:0&\:\mathbf{G}-{\mathbf{A}}_{22}\end{array}\right]$, where $\:{\mathbf{A}}_{11}$, $\:{\mathbf{A}}_{12}/{\mathbf{A}}_{21}$, and $\:{\mathbf{A}}_{22}$ being the components of the numerator relationship matrix constructed based on the pedigree information corresponding to non-genotyped animals, the covariance between non-genotyped and genotyped animals, as well as to genotyped animals, while $\:\mathbf{G}$ represents the genomic covariance matrix between genotyped animals. $\:\mathbf{R}$ is a diagonal matrix containing 1.00 for cows with phenotypes or $\:{\text{n}}_{\text{i}}$ for bulls with MACE DRPs, where $\:{\text{n}}_{\text{i}}$ represents a difference in effective daughter contributions of i-th bull between the MACE and the national evaluation. $\:\mathbf{X}$ and $\:{\mathbf{W}}_{\mathbf{G}}$ denote the corresponding design matrices.

The SNP-BLUP model (Liu et al. 2014)

$$\:\mathbf{y}=\mathbf{X}\mathbf{b}+{\mathbf{W}}_{\mathbf{S}}\mathbf{a}+\mathbf{e}$$

where $\:\mathbf{y}$, $\:\mathbf{b}$, $\:\mathbf{a}$, and $\:\mathbf{e}$ are the same as defined above, but $\:\mathbf{a}$ is parameterised as $\:\mathbf{Z}\mathbf{g}+\:\mathbf{u}$ with $\:\mathbf{g}$ being the vector of random SNP effects and $\:\mathbf{u}$ - a vector of random additive residual polygenic effects. The covariance structure imposed on the residual effect ($\:\mathbf{e}$) is the same as defined above, while for the additive genetic effect $\:\mathbf{a}$ the distribution is defined as $\:\mathbf{a}\sim\mathbf{N}\left(0,{\mathbf{H}}_{\mathbf{S}}{{\sigma\:}}_{\text{a}}^{2}\right)$ where the structure of $\:{\mathbf{H}}_{\mathbf{S}}$ is expressed by $\:\left[\begin{array}{ccc}\mathbf{K}\mathbf{G}{\mathbf{K}}^{\mathbf{{\prime\:}}}+\mathbf{D}&\:\mathbf{K}\mathbf{G}&\:\mathbf{K}\mathbf{Z}\mathbf{B}\\\:\mathbf{G}{\mathbf{K}}^{\mathbf{{\prime\:}}}&\:\mathbf{G}&\:\mathbf{Z}\mathbf{B}\\\:\mathbf{B}{\mathbf{Z}}^{\mathbf{{\prime\:}}}{\mathbf{K}}^{\mathbf{{\prime\:}}}&\:\mathbf{B}\mathbf{Z}{\prime\:}&\:\mathbf{B}\end{array}\right]$ with $\:\mathbf{D}={{(\mathbf{A}}^{11})}^{-1}$, $\:\mathbf{K}={\mathbf{A}}_{12}{\mathbf{A}}_{22}^{-1}$, and $\:\mathbf{B}$ representing a diagonal matrix of the form$\:\:\mathbf{I}\frac{1}{\sum\:_{\text{i}=1}^{\text{N}}2{\text{p}}_{\text{i}}\left(1-{\text{p}}_{\text{i}}\right)}$ with $\:{\text{p}}_{\text{i}}$ denoting the allele frequency of the i-th SNP and N is the number of SNPs. $\:\mathbf{X}\:\text{a}\text{n}\text{d}\:{\mathbf{W}}_{\mathbf{S}}\:$ denote the corresponding design matrices (Liu et al. 2016). Note that for none of the models, the variance components and the proportion of residual additive polygenic variance were estimated, instead their values corresponding to the parameters used in the Polish national genetic and genomic evaluation for a given trait (Table 2) were used.

Table 2

Variance components underlying the analyzed phenotypes
Trait	Genetic variance	Residual variance	Residual polygenic variance
Stature	5.01	4.63	20%
Foot angle	0.11	1.06	20%

Note that $\:{{\sigma\:}}_{\text{a}}^{2}$ and $\:{{\sigma\:}}_{\text{e}}^{2}$ used in all of the above models, were not estimated, instead values from the national conventional evaluation were used. $\:{{\sigma\:}}_{\text{a}}^{2}=5.50058$, $\:{{\sigma\:}}_{\text{e}}^{2}=4.63406$.

Solving the GEBV prediction equations

Solutions for the effects fitted in the above models were obtained using the MiXBLUP 3.0 software (Ten Napel et al. 2020) that optimise the following equation: $\:{{\mathbf{D}}^{-1}\mathbf{M}}^{-1}\mathbf{C}\mathbf{x}={{\mathbf{D}}^{-1}\mathbf{M}}^{-1}\mathbf{p},$ where $\:\mathbf{C}$ represents the coefficient matrix corresponding to the Mixed Model Equations (MME) for solving (1) or (2), $\:\mathbf{x}\:$is the vector of model parameters, and $\:\mathbf{p}$ is the RHS of MME, while $\:\mathbf{M}$ and $\:\mathbf{D}$ respectively represent the first level and the second level preconditioning matrices. The computations were performed on a dedicated computing server running the Linux Red operating system with 260GB of RAM, 16 Intel Xeon CPUs with 2.20GHz, and 600GB hard disk space. For a large national dairy population, as the one used in our study, the numerically severe challenge is to obtain the inverse of the $\:{\mathbf{H}}_{\mathbf{G}}$ and $\:{\mathbf{H}}_{\mathbf{S}}$ matrix, in particular of its component related to genotyped individuals – the dense sub-matrix $\:\mathbf{G}$. In our study, the following approaches were considered:

the GT approach (Mäntysaari et al. 2020) in which the $\:{\mathbf{G}}^{-1}$ matrix is represented by $\:\frac{1}{\text{w}}{\mathbf{A}}_{22}^{-1}-\frac{1}{\text{w}}{\mathbf{T}}^{{\prime\:}}\mathbf{T}$, with $\:\mathbf{T}={\mathbf{L}}^{-1}{\mathbf{Z}}^{{\prime\:}}{\mathbf{A}}_{22}^{-1}$ where $\:w$ denotes the proportion of a residual polygenic variance and $\:\mathbf{L}$ is defined by $\:\mathbf{L}{\mathbf{L}}^{\mathbf{{\prime\:}}}={\mathbf{Z}}^{{\prime\:}}{\mathbf{A}}_{22}^{-1}\mathbf{Z}+\text{w}\mathbf{I}$.
the APY approach (Misztal et al. 2014) that divides the genotyped individuals into a core and non-core sub-groups for which the inverse is handled differently in a way that the exact inverse is computed only for the core animal sub-group ($\:{\mathbf{G}}_{\text{c}}$) and the covariance between core and non-core individuals, while the part of the matrix corresponding to the non-core genotyped animals ($\:{\mathbf{G}}_{\text{n}}$) is handled as a diagonal matrix: $\:{\mathbf{G}}^{-1}\approx\:\left[\begin{array}{cc}{\mathbf{G}}_{\text{c}}^{-1}&\:0\\\:0&\:0\end{array}\right]+\left[\begin{array}{c}-{\mathbf{G}}_{\text{c}}^{-1}{\mathbf{G}}_{\text{c}\text{n}}\\\:\mathbf{I}\end{array}\right]{\mathbf{M}}_{\text{n}}^{-1}\left[\begin{array}{cc}-{\mathbf{G}}_{\text{n}\text{c}}{\mathbf{G}}_{\text{c}}^{-1}&\:\mathbf{I}\end{array}\right]$ where the subscripts $\:\text{n}$ and $\:\text{c}$ represent non-core and core individuals respectively, and $\:{\mathbf{G}}_{\text{n}}$ is a diagonal matrix. In our study, four approaches to the selection of core animals were considered: APY3000top – where 3,000 genotyped bulls with the highest effective daughter contributions (EDC) were selected as core individuals, APY3000random – where the 3,000 core individuals were selected randomly from the genotyped population and the corresponding versions termed APY15000top, APY10000random, and APY15000random implementing 10,000 and 15,000 core individuals respectively.
The SNP-BLUP approach (Liu et al. 2014) does not meet the numerical burden of the models based on $\:\mathbf{G}$, since in terms of the modelling of genomic information only the inverse of the diagonal matrix $\:\mathbf{B}$ is required: $\:{\mathbf{H}}_{\mathbf{S}}^{-1}=\left[\begin{array}{ccc}{\mathbf{A}}^{11}&\:{\mathbf{A}}^{12}&\:0\\\:{\mathbf{A}}^{21}&\:{\mathbf{A}}^{22}+\left(\frac{1}{\text{w}}-1\right){\mathbf{A}}_{22}^{-1}&\:-\frac{1}{\text{w}}{\mathbf{A}}_{22}^{-1}\mathbf{Z}\\\:0&\:-\frac{1}{\text{w}}{\mathbf{Z}}^{{\prime\:}}{\mathbf{A}}_{22}^{-1}&\:\frac{1}{1-\text{w}}{\mathbf{B}}^{-1}+\frac{1}{\text{w}}{\mathbf{Z}}^{{\prime\:}}{\mathbf{A}}_{22}^{-1}\mathbf{Z}\end{array}\right]$ where $\:\left[\begin{array}{cc}{\mathbf{A}}^{11}&\:{\mathbf{A}}^{12}\\\:{\mathbf{A}}^{21}&\:{\mathbf{A}}^{22}\end{array}\right]$ represent the blocks corresponding to $\:{\left[\begin{array}{cc}{\mathbf{A}}_{11}&\:{\mathbf{A}}_{12}\\\:{\mathbf{A}}_{21}&\:{\mathbf{A}}_{22}\end{array}\right]}^{-1}$.

Validation of GEBV

For the validation of GEBV prediction, bulls born after 2013 (7,296 bulls for stature; 6,200 bulls for foot angle) and cows born after 2016 (96,772 cows for stature; 96,771 cows for foot angle) were removed from the phenotype vector ($\:\mathbf{y}$) of equations (1) and (2). Based on such truncated data sets, the GEBVs of bulls with EDC greater than 20 were then predicted by models (1) and (2). Prediction accuracy was assessed by a weighted linear regression: $\:{\mathbf{G}\mathbf{E}\mathbf{B}\mathbf{V}}_{\text{f}}={\text{b}}_{0}+{\text{b}}_{1}{\mathbf{G}\mathbf{E}\mathbf{B}\mathbf{V}}_{\text{t}}+\mathbf{e}$, with $\:{\mathbf{G}\mathbf{E}\mathbf{B}\mathbf{V}}_{\text{f}}$ representing the vector of GEBVs predicted based on the full data set with all available individuals while $\:{\mathbf{G}\mathbf{E}\mathbf{B}\mathbf{V}}_{\text{t}}$ contains the GEBVs predicted based on the truncated data set. For i-th bull weights were defined as $\:\:\frac{{\text{E}\text{D}\text{C}}_{\text{i}}}{{\text{E}\text{D}\text{C}}_{\text{i}}+\text{k}}$ with $\:\text{k}=\:\frac{4-{\text{h}}^{2}}{{\text{h}}^{2}}$. The above linear regression equation was fitted using the lm function in R software (Rstudio Team 2021).

Model validation

1,727 and 1,725 bulls with EDC > 20 were included in the validation data set for stature and foot angle, respectively. For stature, generally, no marked differences in the estimated slope of the linear regression were observed between the models, varying between 0.94 (APY3000top) and 1.02 (APY10000random). Furthermore, except for APY3000, the differences between the R² corresponding to each model were small. For stature, the highest R² of 0.83 was achieved by the GT approach, while R² for 3,000 core animal scenarios where low amounting to 0.57 (APY3000top) and 0.60 (APY3000random). Similar results were observed for foot angle, although with lower R², with the highest value of 0.76 for GT and SNP-BLUP, and the lowest of 0.60 for APY3000random and 0.62 for APY3000top. Details of the validation results were summarised in (Table 3). Pearson’s linear correlations (Table 4) in GEBVs predicted for the validation bulls from the truncated dataset calculated for all pairs of models, for stature, demonstrated a good agreement only between SNP-BLUP and GT, as expressed by correlations of 0.996. Interestingly, the lowest correlations of 0.810 were observed between both models implementing APY3000 (i.e. APY3000random and APY3000top), as well as between APY3000top and APY15000top. For foot angle correlations are generally higher and the pattern is very similar with the correlation between SNP-BLUP and GT reaching 0.999 and the lowest correlation of 0.870 observed between both APY3000 scenarios, between GT and APY3000top, as well as between GP and APY3000random.

Table 3

Validation of GEBV prediction
Model variants	$\:{\widehat{\text{b}}}_{0}$	SE($\:{\widehat{\text{b}}}_{0}$)	$\:{\widehat{\text{b}}}_{1}$	SE($\:{\widehat{\text{b}}}_{1}$)	R²
Stature, 1727 validation bulls, h² = 0.54
SNP-BLUP	-3.90	0.32	1.01	0.01	0.77
GT	-4.40	0.27	1.01	0.01	0.83
APY3000top	-1.81	0.44	0.94	0.02	0.57
APY3000random	-2.20	0.44	0.99	0.02	0.60
APY10000random	-3.74	0.36	1.02	0.02	0.72
APY15000top	-2.59	0.32	0.96	0.01	0.75
APY15000random	-2.32	0.33	0.96	0.01	0.73
Foot angle, 1725 validation bulls, h² = 0.09
SNP-BLUP	-2.03	0.18	1.03	0.01	0.76
GT	-1.96	0.18	1.04	0.01	0.76
APY3000top	-0.68	0.21	0.97	0.02	0.62
APY3000random	-0.52	0.26	1.03	0.02	0.60
APY10000random	-2.04	0.21	1.02	0.02	0.69
APY15000top	-2.15	0.18	1.02	0.01	0.75
APY15000random	-2.10	0.19	1.03	0.02	0.73

Table 4

Pearson correlation coefficients of GEBV predicted by different model variants
Model variants	Stature	Foot angle
SNP-BLUP – GT	0.996	0.999
SNP-BLUP – APY3000top	0.834	0.871
SNP-BLUP – APY3000random	0.858	0.872
SNP-BLUP – APY10000random	0.958	0.965
SNP-BLUP – APY15000top	0.972	0.989
SNP-BLUP – APY15000random	0.961	0.983
GT – APY3000top	0.824	0.870
GT – APY3000random	0.850	0.870
GT – APY10000random	0.950	0.965
GT – APY15000top	0.970	0.989
GT – APY15000random	0.958	0.983
APY3000top – APY3000random	0.810	0.870
APY3000top – APY10000random	0.851	0.889
APY3000top – APY15000top	0.810	0.873
APY3000top – APY15000random	0.825	0.880
APY3000random – APY10000random	0.880	0.908
APY3000random – APY15000top	0.837	0.874
APY3000random – APY15000random	0.842	0.883
APY15000top – APY10000random	0.942	0.965
APY15000top – APY15000random	0.947	0.984
APY15000random – APY10000random	0.946	0.970

Differences in GEBV prediction

Figure 1 depicts differences between GEBVs predicted by SNP-BLUP in comparison to GEBVs predicted by the six G-BLUP implementations. For the GEBV difference between SNP-BLUP and GT all genotyped animals were considered, while for all comparisons involving APY only the core individuals were used. For each model, original solutions were rescaled by subtracting the mean of a cow base population to provide GEBVs comparable across all modes. For stature, the differences were generally small across bull birth years (Fig. 1A), however for foot angle a different pattern emerged (Fig. 1B) as for all comparisons, except GT and the most informative APY15000top, the differences were unstable across bull birth years. Furthermore, we compared correlations of GEBVs predicted by the full and the truncated models for individuals representing base cows (i.e. cows with phenotype records) and bulls (i.e. bulls, which have a minimum of one daughter). The dashed line divides the plot into animals defined as old and young in the validation process. For stature, regardless of the model, correlations for old animals were very close to unity, for young candidates we observed a declining trend with the lowest correlations calculated for APY3000random (Fig. 2A). For foot angle, similar results were obtained, except for APY3000random for females, where we observed a decreasing correlation trend already from 2004. So from 2008 correlations dropped below 0.9 (Fig. 2B). Finally, we compared the overlap of bulls with the top 50 GEBV predictions resulting from different models (Fig. 3). The number of bulls common across all models for stature (Fig. 3A) was 39, while the highest number of exclusive bulls (3) was observed in the top50 list resulting from APY15000top. Regarding foot angle (Fig. 3B), the number of common bulls was even lower – 31 and 8 bulls were exclusive for the APY3000random.

Differences in computational resources

The wall clock time corresponding to setting up and solving models (1) or (2) using the MiXBLUP software, as well as the peak memory consumption varied considerably between solvers. The exact values are specified in (Table 5), but generalizing, for both traits, the SNP-BLUP and APY3000 were the fastest, closely followed by the APY10000random. The wall clock time of GT was the longest with twice the time of APY15000top. The peak memory consumption by SNP-BLUP was in the order of ten lower than for the remaining solvers. In the case of iterations, SNP-BLUP required the most iterations (673 for stature and 1027 for foot angle) and an average of 2.3 seconds per iteration. The lowest number of 335 iterations and an average of 0.18 seconds per iteration was used for stature by APY3000top and 499 and 0.18 seconds for foot angle by APY3000random.

Table 5

Computational resources utilized by the model variants
Model variants	Wall clock time (min)		Peak RAM consumption (GB)		Number of iterations
Model variants	Stature	Foot angle	Stature	Foot angle	Stature	Foot angle
SNP-BLUP	23	32	5.81	5.81	673	1027
GT	138	143	63.89	63.88	477	629
APY3000top	23	29	49.48	49.47	335	811
APY3000random	23	32	49.48	49.47	390	499
APY10000random	32	34	56.53	56.53	469	652
APY15000top	68	70	61.56	61.56	425	625
APY15000random	54	57	61.56	61.56	477	551

There have been many studies related to the comparison of single-step genomic prediction models with conventional, two-step approaches. However, the literature related to the comparisons within the frame of single-step modeling is scarce. Koivula et al. (2012) considered genomic prediction models similar to (1) and (2), however, applied only to the genotyped part of the population. Similarly to the results of our study, those authors observed very high correlations between models for predicted bulls’ GEBVs or DGVs (Direct Genetic Values) as well as similar validation results. The validation performance of single-step models was also considered by Gao et al. (2018) and although no marked differences were observed the authors indicated that for the APY-based solvers not only the number but also the selection of core individuals was a crucial step influencing the prediction accuracy (Fragomeni et al. 2015; Gao et al. 2018; Masuda et al. 2016; Strandén et al. 2017). The differences due to the core animal set composition were not demonstrated in our study while comparing the prediction performance of APYtop and APYranom models. However, our results were well in line by demonstrating differences arising from the number of the core individual sets, with a clear indication that a large number of core individuals is recommended, provided the availability of computational resources – especially RAM. Similarly to the results of Misztal et al. (2016) we also observed no marked gain in GEBV prediction quality when using more than 10,000 core individuals. We observed very large differences in RAM consumption between SNP-BLUP and other models. GT needed over 10 times more RAM and fewer iterations than SNP-BLUP, the same as demonstrated by (Vandenplass et al. 2023). Two possible ways to circumvent the problem of the optimal choice of core individuals are either the use of GT solver-based prediction, the application however, comes with the price of high memory requirements and long computing times, or using of SNP-BLUP solver-based prediction, which does not consume much memory and computing time.

Regarding the prediction of GEBV on a whole active population scope, no marked differences between solvers (except the APY with only 3,000 core individuals) were observed. Hence the major factor influencing the choice of the solver is its computational efficiency expressed by computing time and memory resources required, which was also indicated by Koivula et al. 2012). Still, it should be kept in mind that the ranking of top bulls is not identical between solvers, which has implications for the breeding industry in terms of semen pricing, as well as for selection since typically the top-ranking bulls are the most widely used.

Ethics approval

Not applicable.

Data and model availability statement

The data set corresponds to the genetic evaluation from December 2021 from the National Research Institute of Animal Production.

Declaration of generative AI and AI-assisted technologies in the writing process

The authors declare that they did not use AI technology.

Author ORCIDs

Dawid Słomian: https://orcid.org/0000-0002-9037-7703

Joanna Szyda: https://orcid.org/0000-0001-9688-0193

Author contributions

D. Słomian performed all data analyses, and was involved in creating the study concept, J. Szyda conceptualized the study, interpreted the data, and drafted the manuscript. All authors read and approved the final manuscript.

Declaration of interest

The authors declare that they have no competing interests.

Acknowledgments

Not applicable.

Financial support statement

This study was supported by a grant from the Ministry of Agriculture and Rural Development (DŻW.pp.862.1.2023).

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A comparison of genomically enhanced breeding values predicted by different single-step approaches

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Abstract

Figures

Introduction

Materials and Methods

Material

Models

GEBV prediction models

Solving the GEBV prediction equations

Validation of GEBV

Results

Model validation

Differences in GEBV prediction

Differences in computational resources

Discussion

Conclusions

Declarations

References

Status:

Version 1