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

A comparison of genomically enhanced breeding values predicted by different single-step approaches | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A comparison of genomically enhanced breeding values predicted by different single-step approaches Dawid Słomian, Joanna Szyda This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5260327/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract 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 Figures Figure 1 Figure 2 Figure 3 Introduction 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. Materials and Methods 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 (h 2 = 0.54) and foot angle representing a low heritable trait (h 2 = 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 Males with MACE DRP 141,397 Phenotype data (Foot angle) Females with phenotypes 1,098,766 Males with MACE DRP 117,482 Genotype data Females 70,134 Males (bulls and candidates) 64,826 Pedigree data Females 1,368,487 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) $$\:\mathbf{y}=\mathbf{X}\mathbf{b}+{\mathbf{W}}_{\mathbf{G}}\mathbf{a}+\mathbf{e}$$ 1 , \(\:\mathbf{y}\) 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) $$\:\mathbf{y}=\mathbf{X}\mathbf{b}+{\mathbf{W}}_{\mathbf{S}}\mathbf{a}+\mathbf{e}$$ 2 , 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). Results 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 R 2 corresponding to each model were small. For stature, the highest R 2 of 0.83 was achieved by the GT approach, while R 2 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 R 2 , 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( \(\:{\widehat{\text{b}}}_{1}\) ) R 2 Stature, 1727 validation bulls, h 2 = 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, h 2 = 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. 1 A), however for foot angle a different pattern emerged (Fig. 1 B) 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. 2 A). 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. 2 B). 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. 3 A) was 39, while the highest number of exclusive bulls (3) was observed in the top50 list resulting from APY15000top. Regarding foot angle (Fig. 3 B), 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 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 Discussion 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. Conclusions 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. Declarations 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). References Aguilar I, Misztal I, Johnson DL, Legarra A, Tsuruta S, Lawlor TJ (2010) Hot topic: a unified approach to utilize phenotypic, full pedigree, and genomic information for genetic evaluation of Holstein final score. 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J Dairy Sci. 103: 5314-5326. https://doi.org/10.3168/jds.2019-17754 Mäntysaari EA, Evans RD, Strandén I. (2017) Efficient single-step genomic evaluation for a multibreed beef cattle population having many genotyped animals. J Anim Sci. 95(11):4728-4737. https://doi: 10.2527/jas2017.1912 Masuda M., Misztal I., Tsuruta S., Legarra A., Aguilar I., Lourenco D.A.L., Fragomeni B.O., Lawlor, T.J. (2016) Implementation of genomic recursions in single-step genomic best linear unbiased predictor for US Holsteins with a large number of genotyped animals. J Dairy Sci. 99: 1968-1974 https://doi.org/10.3168/jds.2015-10540 Misztal I. (2016) Inexpensive Computation of the Inverse of the Genomic Relationship Matrix in Populations with Small Effective Population Size. Genetics, Volume 202, Issue 2, Pages 401–409, https://doi.org/10.1534/genetics.115.182089 Misztal I., Legarra A., Aguilar I. (2009) Computing procedures for genetic evaluation including phenotypic, full pedigree, and genomic information. J Dairy Sci. 92: 4648-4655. https://doi.org/10.3168/jds.2009-2064 Misztal I., Legarra A., Aguilar I. (2014) Using recursion to compute the inverse of the genomic relationship matrix. J Dairy Sci., 97: 3943-3952 https://doi.org/10.3168/jds.2013-7752 RStudio Team (2021). RStudio: Integrated Development Environment for R. RStudio, PBC, Boston, MA URL http://www.rstudio.com/ Strandén I., Garrick D.J. (2009) Technical note: Derivation of equivalent computing algorithms for genomic predictions and reliabilities of animal merit. J Dairy Sci. 92: 2971-2975. https://doi.org/10.3168/jds.2008-1929 Strandén I., Matilainen K., Aamand G.P., Mantysaari E.A. (2017) Solving efficiently large single-step genomic best linear unbiased prediction models. JABG, 134: 264-274 https://doi.org/10.1111/jbg.12257 Taylor J.F. (2014) Implementation and accuracy of genomic selection. Aquaculture 420–421: S8-S14 https://doi.org/10.1016/j.aquaculture.2013.02.017 Ten Napel J., Vandenplas J., Lidauer M., Stranden I., Taskinen M., Mäntysaari E.A., Calus M.P., Veerkamp R.F. (2021) MiXBLUP 3.0.1 manual VanRaden P.M. (2008) Efficient methods to compute genomic predictions. J Dairy Sci. 91: 4414-4423 https://doi.org/10.3168/jds.2007-0980 Vandenplas J., Ten Napel J., Darbaghshahi S.N., Evans R., Calus M.P.L., Veerkamp R., Cromie A., Mäntysaari E.A., Strandén I. (2023) Efficient large-scale single-step evaluations and indirect genomic prediction of genotyped selection candidates. Genet. Sel. 55, 37 https://doi.org/10.1186/s12711-023-00808-z Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-5260327","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":366855934,"identity":"fae2b393-f800-495c-8485-2de0615017d1","order_by":0,"name":"Dawid Słomian","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABAElEQVRIiWNgGAWjYDCCA0DEw8Agw8DA2MDAUCEBEjMAYguCWnggWs7AtUjg1cIA0QIEjG0MhLXwHT9jeOANQx0Pv9jhxs+F8yzsGdibt0kw7sCtRfJMjsHBOQyHeSRnJzZLz9wmwczAc6xMgvEMbi0GB3I3HOYBesfgdmKDNO82CTYGiRwzCcY2PFrOvwVpqQNpaf7NO0eCh0H+DQEtN8C2MIO0tEnzNkhIMEjw4NcieeP9h4NzDMB+abPmOSZhwMaTVmyRiMcvfOfTkj+8qaiT45dOf3ybp6bOnp/98MYbH3fY4NQCdR4Smw1EJDYQ0IEJGEnXMgpGwSgYBcMXAABTzUxAnsFifgAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0002-9037-7703","institution":"National Research Institute of Animal Production: Instytut Zootechniki Panstwowy Instytut Badawczy","correspondingAuthor":true,"prefix":"","firstName":"Dawid","middleName":"","lastName":"Słomian","suffix":""},{"id":366855935,"identity":"587aed72-9f82-4f1d-a0df-de3619fc320a","order_by":1,"name":"Joanna Szyda","email":"","orcid":"","institution":"Wrocław University of Environmental and Life Sciences: Uniwersytet Przyrodniczy we Wroclawiu","correspondingAuthor":false,"prefix":"","firstName":"Joanna","middleName":"","lastName":"Szyda","suffix":""}],"badges":[],"createdAt":"2024-10-14 10:28:41","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5260327/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5260327/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":66930958,"identity":"a471da0c-1f92-4b19-b7be-03417fa9073e","added_by":"auto","created_at":"2024-10-18 07:07:48","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":3394494,"visible":true,"origin":"","legend":"\u003cp\u003eMean GEBV difference divided by the genetic standard deviation between SNP-BLUP model and GT, APY models for stature (A) and foot angle (B).\u003c/p\u003e","description":"","filename":"Figure1.png","url":"https://assets-eu.researchsquare.com/files/rs-5260327/v1/8280a16300b4e950ccbf4718.png"},{"id":66930956,"identity":"33af60b1-24f6-459f-8b52-0a4088b72a62","added_by":"auto","created_at":"2024-10-18 07:07:48","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":2003695,"visible":true,"origin":"","legend":"\u003cp\u003eCorrelation between full and truncated data set for base animals models for stature (A) and foot angle (B).\u003c/p\u003e","description":"","filename":"Figure2.png","url":"https://assets-eu.researchsquare.com/files/rs-5260327/v1/b420a9d8267b84bf149ddc76.png"},{"id":66930957,"identity":"d956a7e1-f8f4-4545-afa7-a7d324a07670","added_by":"auto","created_at":"2024-10-18 07:07:48","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":286923,"visible":true,"origin":"","legend":"\u003cp\u003eUpset plot of top 50 GEBV bull for stature (A) and foot angle (B) across model variants.\u003c/p\u003e","description":"","filename":"Figure3.png","url":"https://assets-eu.researchsquare.com/files/rs-5260327/v1/0a7dd413d1265702e4ab20e4.png"},{"id":70582065,"identity":"a0f464c7-fe34-4065-8609-2ea361407faf","added_by":"auto","created_at":"2024-12-04 15:21:20","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":7162596,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5260327/v1/ea00323b-586f-4281-9d29-9de5481d919f.pdf"}],"financialInterests":"","formattedTitle":"A comparison of genomically enhanced breeding values predicted by different single-step approaches","fulltext":[{"header":"Introduction","content":"\u003cp\u003eToday, 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; Strand\u0026eacute;n 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\u003c/p\u003e \u003cp\u003eG-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).\u003c/p\u003e \u003cp\u003eHowever, 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 \u003cem\u003ecore\u003c/em\u003e 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 M\u0026auml;ntysaari 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.\u003c/p\u003e"},{"header":"Materials and Methods","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003eMaterial\u003c/h2\u003e \u003cp\u003eThe data set analyzed (Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) represented data from the Polish national genetic evaluation of Holstein cattle from December 2021 for stature which represents a highly heritable trait (h\u003csup\u003e2\u003c/sup\u003e\u0026thinsp;=\u0026thinsp;0.54) and foot angle representing a low heritable trait (h\u003csup\u003e2\u003c/sup\u003e\u0026thinsp;=\u0026thinsp;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.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eNumber of animals in the analyzed data sets\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCategory\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSex\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eNumber of animals\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003ePhenotype data (Stature)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFemales with phenotypes\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1,098,611\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMales with MACE DRP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e141,397\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003ePhenotype data (Foot angle)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFemales with phenotypes\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1,098,766\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMales with MACE DRP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e117,482\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eGenotype data\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFemales\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e70,134\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMales (bulls and candidates)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e64,826\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003ePedigree data\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFemales\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1,368,487\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMales\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e187,508\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eModels\u003c/h3\u003e\n\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003eGEBV prediction models\u003c/h2\u003e \u003cp\u003eThe following single-step single-trait models were considered for the prediction of GEBV:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003ethe G-BLUP model (Aguilar et al. 2010; Christensen and Lund 2010)\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003cdiv id=\"Equ1\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:\\mathbf{y}=\\mathbf{X}\\mathbf{b}+{\\mathbf{W}}_{\\mathbf{G}}\\mathbf{a}+\\mathbf{e}$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{y}\\)\u003c/span\u003e \u003c/span\u003e is the vector of dependent variables represented by cows\u0026rsquo; measured phenotypes and bulls' pseudo phenotypes expressed by their MACE DRPs, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{b}\\)\u003c/span\u003e\u003c/span\u003e represents a vector of fixed effects including age at calving, lactation phase, and herd corresponding to cows\u0026rsquo; phenotypes as well as corresponding phantom codes of the fixed effects for bulls\u0026rsquo; DRPs, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{a}\\)\u003c/span\u003e\u003c/span\u003e represents a vector of breeding values, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{e}\\)\u003c/span\u003e\u003c/span\u003e is the residual. The underlying covariance structure of the random effects is given by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{a}\\sim\\mathbf{N}\\left(0,{\\mathbf{H}}_{\\mathbf{G}}{{\\sigma\\:}}_{\\text{a}}^{2}\\right)\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{e}\\sim\\mathbf{N}\\left(0,\\mathbf{R}{{\\sigma\\:}}_{\\text{e}}^{2}\\right)\\)\u003c/span\u003e\u003c/span\u003e. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{H}}_{\\mathbf{G}}\\)\u003c/span\u003e\u003c/span\u003e is given by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left[\\begin{array}{cc}{\\mathbf{A}}_{11}\u0026amp;\\:{\\mathbf{A}}_{12}\\\\\\:{\\mathbf{A}}_{21}\u0026amp;\\:{\\mathbf{A}}_{22}\\end{array}\\right]+\\left[\\begin{array}{cc}0\u0026amp;\\:0\\\\\\:0\u0026amp;\\:\\mathbf{G}-{\\mathbf{A}}_{22}\\end{array}\\right]\\)\u003c/span\u003e\u003c/span\u003e, where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{A}}_{11}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{A}}_{12}/{\\mathbf{A}}_{21}\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{A}}_{22}\\)\u003c/span\u003e\u003c/span\u003e 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 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{G}\\)\u003c/span\u003e\u003c/span\u003e represents the genomic covariance matrix between genotyped animals. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{R}\\)\u003c/span\u003e\u003c/span\u003e is a diagonal matrix containing 1.00 for cows with phenotypes or \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{n}}_{\\text{i}}\\)\u003c/span\u003e\u003c/span\u003e for bulls with MACE DRPs, where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{n}}_{\\text{i}}\\)\u003c/span\u003e\u003c/span\u003e represents a difference in effective daughter contributions of i-th bull between the MACE and the national evaluation. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{X}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{W}}_{\\mathbf{G}}\\)\u003c/span\u003e\u003c/span\u003e denote the corresponding design matrices.\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003eThe SNP-BLUP model (Liu et al. 2014)\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003cdiv id=\"Equ2\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:\\mathbf{y}=\\mathbf{X}\\mathbf{b}+{\\mathbf{W}}_{\\mathbf{S}}\\mathbf{a}+\\mathbf{e}$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{y}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{b}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{a}\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{e}\\)\u003c/span\u003e\u003c/span\u003e are the same as defined above, but \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{a}\\)\u003c/span\u003e\u003c/span\u003e is parameterised as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{Z}\\mathbf{g}+\\:\\mathbf{u}\\)\u003c/span\u003e\u003c/span\u003e with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{g}\\)\u003c/span\u003e\u003c/span\u003e being the vector of random SNP effects and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{u}\\)\u003c/span\u003e\u003c/span\u003e - a vector of random additive residual polygenic effects. The covariance structure imposed on the residual effect (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{e}\\)\u003c/span\u003e\u003c/span\u003e) is the same as defined above, while for the additive genetic effect \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{a}\\)\u003c/span\u003e\u003c/span\u003e the distribution is defined as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{a}\\sim\\mathbf{N}\\left(0,{\\mathbf{H}}_{\\mathbf{S}}{{\\sigma\\:}}_{\\text{a}}^{2}\\right)\\)\u003c/span\u003e\u003c/span\u003e where the structure of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{H}}_{\\mathbf{S}}\\)\u003c/span\u003e\u003c/span\u003e is expressed by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left[\\begin{array}{ccc}\\mathbf{K}\\mathbf{G}{\\mathbf{K}}^{\\mathbf{{\\prime\\:}}}+\\mathbf{D}\u0026amp;\\:\\mathbf{K}\\mathbf{G}\u0026amp;\\:\\mathbf{K}\\mathbf{Z}\\mathbf{B}\\\\\\:\\mathbf{G}{\\mathbf{K}}^{\\mathbf{{\\prime\\:}}}\u0026amp;\\:\\mathbf{G}\u0026amp;\\:\\mathbf{Z}\\mathbf{B}\\\\\\:\\mathbf{B}{\\mathbf{Z}}^{\\mathbf{{\\prime\\:}}}{\\mathbf{K}}^{\\mathbf{{\\prime\\:}}}\u0026amp;\\:\\mathbf{B}\\mathbf{Z}{\\prime\\:}\u0026amp;\\:\\mathbf{B}\\end{array}\\right]\\)\u003c/span\u003e\u003c/span\u003e with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{D}={{(\\mathbf{A}}^{11})}^{-1}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{K}={\\mathbf{A}}_{12}{\\mathbf{A}}_{22}^{-1}\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{B}\\)\u003c/span\u003e\u003c/span\u003e representing a diagonal matrix of the form\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:\\mathbf{I}\\frac{1}{\\sum\\:_{\\text{i}=1}^{\\text{N}}2{\\text{p}}_{\\text{i}}\\left(1-{\\text{p}}_{\\text{i}}\\right)}\\)\u003c/span\u003e\u003c/span\u003e with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{p}}_{\\text{i}}\\)\u003c/span\u003e\u003c/span\u003e denoting the allele frequency of the i-th SNP and N is the number of SNPs. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{X}\\:\\text{a}\\text{n}\\text{d}\\:{\\mathbf{W}}_{\\mathbf{S}}\\:\\)\u003c/span\u003e\u003c/span\u003e 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\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) were used.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eVariance components underlying the analyzed phenotypes\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTrait\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGenetic variance\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eResidual variance\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eResidual polygenic variance\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eStature\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e5.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.63\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e20%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFoot angle\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e20%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eNote that \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\sigma\\:}}_{\\text{a}}^{2}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\sigma\\:}}_{\\text{e}}^{2}\\)\u003c/span\u003e\u003c/span\u003e used in all of the above models, were not estimated, instead values from the national conventional evaluation were used. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\sigma\\:}}_{\\text{a}}^{2}=5.50058\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\sigma\\:}}_{\\text{e}}^{2}=4.63406\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eSolving the GEBV prediction equations\u003c/h3\u003e\n\u003cp\u003eSolutions 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: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\mathbf{D}}^{-1}\\mathbf{M}}^{-1}\\mathbf{C}\\mathbf{x}={{\\mathbf{D}}^{-1}\\mathbf{M}}^{-1}\\mathbf{p},\\)\u003c/span\u003e\u003c/span\u003e where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{C}\\)\u003c/span\u003e\u003c/span\u003e represents the coefficient matrix corresponding to the Mixed Model Equations (MME) for solving (1) or (2), \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{x}\\:\\)\u003c/span\u003e\u003c/span\u003eis the vector of model parameters, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{p}\\)\u003c/span\u003e\u003c/span\u003e is the RHS of MME, while \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{M}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{D}\\)\u003c/span\u003e\u003c/span\u003e 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 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{H}}_{\\mathbf{G}}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{H}}_{\\mathbf{S}}\\)\u003c/span\u003e\u003c/span\u003e matrix, in particular of its component related to genotyped individuals \u0026ndash; the dense sub-matrix \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{G}\\)\u003c/span\u003e\u003c/span\u003e. In our study, the following approaches were considered:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003ethe GT approach (M\u0026auml;ntysaari et al. 2020) in which the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{G}}^{-1}\\)\u003c/span\u003e\u003c/span\u003e matrix is represented by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\frac{1}{\\text{w}}{\\mathbf{A}}_{22}^{-1}-\\frac{1}{\\text{w}}{\\mathbf{T}}^{{\\prime\\:}}\\mathbf{T}\\)\u003c/span\u003e\u003c/span\u003e, with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{T}={\\mathbf{L}}^{-1}{\\mathbf{Z}}^{{\\prime\\:}}{\\mathbf{A}}_{22}^{-1}\\)\u003c/span\u003e\u003c/span\u003e where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:w\\)\u003c/span\u003e\u003c/span\u003e denotes the proportion of a residual polygenic variance and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{L}\\)\u003c/span\u003e\u003c/span\u003e is defined by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{L}{\\mathbf{L}}^{\\mathbf{{\\prime\\:}}}={\\mathbf{Z}}^{{\\prime\\:}}{\\mathbf{A}}_{22}^{-1}\\mathbf{Z}+\\text{w}\\mathbf{I}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003ethe 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 (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{G}}_{\\text{c}}\\)\u003c/span\u003e\u003c/span\u003e) and the covariance between core and non-core individuals, while the part of the matrix corresponding to the non-core genotyped animals (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{G}}_{\\text{n}}\\)\u003c/span\u003e\u003c/span\u003e) is handled as a diagonal matrix: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{G}}^{-1}\\approx\\:\\left[\\begin{array}{cc}{\\mathbf{G}}_{\\text{c}}^{-1}\u0026amp;\\:0\\\\\\:0\u0026amp;\\: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}\u0026amp;\\:\\mathbf{I}\\end{array}\\right]\\)\u003c/span\u003e\u003c/span\u003e where the subscripts \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{n}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{c}\\)\u003c/span\u003e\u003c/span\u003e represent non-core and core individuals respectively, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{G}}_{\\text{n}}\\)\u003c/span\u003e\u003c/span\u003e is a diagonal matrix. In our study, four approaches to the selection of core animals were considered: APY3000top \u0026ndash; where 3,000 genotyped bulls with the highest effective daughter contributions (EDC) were selected as core individuals, APY3000random \u0026ndash; 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.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eThe SNP-BLUP approach (Liu et al. 2014) does not meet the numerical burden of the models based on \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{G}\\)\u003c/span\u003e\u003c/span\u003e, since in terms of the modelling of genomic information only the inverse of the diagonal matrix \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{B}\\)\u003c/span\u003e\u003c/span\u003e is required: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{H}}_{\\mathbf{S}}^{-1}=\\left[\\begin{array}{ccc}{\\mathbf{A}}^{11}\u0026amp;\\:{\\mathbf{A}}^{12}\u0026amp;\\:0\\\\\\:{\\mathbf{A}}^{21}\u0026amp;\\:{\\mathbf{A}}^{22}+\\left(\\frac{1}{\\text{w}}-1\\right){\\mathbf{A}}_{22}^{-1}\u0026amp;\\:-\\frac{1}{\\text{w}}{\\mathbf{A}}_{22}^{-1}\\mathbf{Z}\\\\\\:0\u0026amp;\\:-\\frac{1}{\\text{w}}{\\mathbf{Z}}^{{\\prime\\:}}{\\mathbf{A}}_{22}^{-1}\u0026amp;\\:\\frac{1}{1-\\text{w}}{\\mathbf{B}}^{-1}+\\frac{1}{\\text{w}}{\\mathbf{Z}}^{{\\prime\\:}}{\\mathbf{A}}_{22}^{-1}\\mathbf{Z}\\end{array}\\right]\\)\u003c/span\u003e\u003c/span\u003e where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left[\\begin{array}{cc}{\\mathbf{A}}^{11}\u0026amp;\\:{\\mathbf{A}}^{12}\\\\\\:{\\mathbf{A}}^{21}\u0026amp;\\:{\\mathbf{A}}^{22}\\end{array}\\right]\\)\u003c/span\u003e\u003c/span\u003e represent the blocks corresponding to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\left[\\begin{array}{cc}{\\mathbf{A}}_{11}\u0026amp;\\:{\\mathbf{A}}_{12}\\\\\\:{\\mathbf{A}}_{21}\u0026amp;\\:{\\mathbf{A}}_{22}\\end{array}\\right]}^{-1}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e\n\u003ch3\u003eValidation of GEBV\u003c/h3\u003e\n\u003cp\u003eFor 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 (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\mathbf{y}\\)\u003c/span\u003e\u003c/span\u003e) of equations (\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) and (\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). 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: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\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}\\)\u003c/span\u003e\u003c/span\u003e, with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{G}\\mathbf{E}\\mathbf{B}\\mathbf{V}}_{\\text{f}}\\)\u003c/span\u003e\u003c/span\u003e representing the vector of GEBVs predicted based on the full data set with all available individuals while \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\mathbf{G}\\mathbf{E}\\mathbf{B}\\mathbf{V}}_{\\text{t}}\\)\u003c/span\u003e\u003c/span\u003e contains the GEBVs predicted based on the truncated data set. For i-th bull weights were defined as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:\\frac{{\\text{E}\\text{D}\\text{C}}_{\\text{i}}}{{\\text{E}\\text{D}\\text{C}}_{\\text{i}}+\\text{k}}\\)\u003c/span\u003e\u003c/span\u003e with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{k}=\\:\\frac{4-{\\text{h}}^{2}}{{\\text{h}}^{2}}\\)\u003c/span\u003e\u003c/span\u003e. The above linear regression equation was fitted using the \u003cem\u003elm\u003c/em\u003e function in R software (Rstudio Team 2021).\u003c/p\u003e"},{"header":"Results","content":"\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003eModel validation\u003c/h2\u003e \u003cp\u003e1,727 and 1,725 bulls with EDC\u0026thinsp;\u0026gt;\u0026thinsp;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 R\u003csup\u003e2\u003c/sup\u003e corresponding to each model were small. For stature, the highest R\u003csup\u003e2\u003c/sup\u003e of 0.83 was achieved by the GT approach, while R\u003csup\u003e2\u003c/sup\u003e 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 R\u003csup\u003e2\u003c/sup\u003e, 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\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e). Pearson\u0026rsquo;s linear correlations (Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e) 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.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eValidation of GEBV prediction\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eModel variants\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\widehat{\\text{b}}}_{0}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eSE(\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\widehat{\\text{b}}}_{0}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\widehat{\\text{b}}}_{1}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003eSE(\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\widehat{\\text{b}}}_{1}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eR\u003csup\u003e2\u003c/sup\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"5\" nameend=\"c5\" namest=\"c1\"\u003e \u003cp\u003e\u003cb\u003eStature, 1727 validation bulls, h\u003c/b\u003e\u003csup\u003e\u003cb\u003e2\u003c/b\u003e\u003c/sup\u003e\u0026thinsp;\u003cb\u003e=\u0026thinsp;0.54\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c7\" namest=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSNP-BLUP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-3.90\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.32\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e0.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.77\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGT\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-4.40\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.27\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e0.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.83\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY3000top\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-1.81\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.44\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.94\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e0.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.57\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY3000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-2.20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.44\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.99\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e0.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.60\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY10000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-3.74\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.36\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e0.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.72\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY15000top\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-2.59\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.32\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.96\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e0.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.75\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY15000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-2.32\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.33\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.96\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e0.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.73\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"5\" nameend=\"c5\" namest=\"c1\"\u003e \u003cp\u003e\u003cb\u003eFoot angle, 1725 validation bulls, h\u003c/b\u003e\u003csup\u003e\u003cb\u003e2\u003c/b\u003e\u003c/sup\u003e\u0026thinsp;\u003cb\u003e=\u0026thinsp;0.09\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c7\" namest=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSNP-BLUP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-2.03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e0.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.76\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGT\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-1.96\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.04\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e0.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.76\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY3000top\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-0.68\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.97\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e0.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.62\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY3000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-0.52\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.26\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e0.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.60\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY10000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-2.04\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e0.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.69\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY15000top\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-2.15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e0.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.75\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY15000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-2.10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.19\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e0.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.73\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003ePearson correlation coefficients of GEBV predicted by different model variants\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eModel variants\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eStature\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eFoot angle\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSNP-BLUP \u0026ndash; GT\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.996\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.999\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSNP-BLUP \u0026ndash; APY3000top\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.834\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.871\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSNP-BLUP \u0026ndash; APY3000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.858\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.872\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSNP-BLUP \u0026ndash; APY10000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.958\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.965\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSNP-BLUP \u0026ndash; APY15000top\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.972\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.989\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSNP-BLUP \u0026ndash; APY15000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.961\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.983\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGT \u0026ndash; APY3000top\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.824\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.870\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGT \u0026ndash; APY3000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.850\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.870\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGT \u0026ndash; APY10000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.950\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.965\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGT \u0026ndash; APY15000top\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.970\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.989\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGT \u0026ndash; APY15000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.958\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.983\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY3000top \u0026ndash; APY3000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.810\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.870\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY3000top \u0026ndash; APY10000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.851\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.889\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY3000top \u0026ndash; APY15000top\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.810\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.873\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY3000top \u0026ndash; APY15000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.825\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.880\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY3000random \u0026ndash; APY10000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.880\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.908\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY3000random \u0026ndash; APY15000top\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.837\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.874\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY3000random \u0026ndash; APY15000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.842\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.883\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY15000top \u0026ndash; APY10000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.942\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.965\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY15000top \u0026ndash; APY15000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.947\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.984\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY15000random \u0026ndash; APY10000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.946\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.970\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eDifferences in GEBV prediction\u003c/h3\u003e\n\u003cp\u003eFigure \u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e 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.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eA), however for foot angle a different pattern emerged (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eB) 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.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eA). 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.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003eB). Finally, we compared the overlap of bulls with the top 50 GEBV predictions resulting from different models (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e). The number of bulls common across all models for stature (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003eA) was 39, while the highest number of exclusive bulls (3) was observed in the top50 list resulting from APY15000top. Regarding foot angle (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003eB), the number of common bulls was even lower \u0026ndash; 31 and 8 bulls were exclusive for the APY3000random.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003eDifferences in computational resources\u003c/h2\u003e \u003cp\u003eThe 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\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e), 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.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eComputational resources utilized by the model variants\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eModel variants\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c3\" namest=\"c2\"\u003e \u003cp\u003eWall clock time (min)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c5\" namest=\"c4\"\u003e \u003cp\u003ePeak RAM consumption (GB)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c7\" namest=\"c6\"\u003e \u003cp\u003eNumber of iterations\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eStature\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eFoot angle\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eStature\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eFoot angle\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eStature\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eFoot angle\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSNP-BLUP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e23\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e32\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e5.81\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e5.81\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e673\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e1027\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGT\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e138\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e143\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e63.89\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e63.88\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e477\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e629\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY3000top\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e23\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e29\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e49.48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e49.47\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e335\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e811\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY3000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e23\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e32\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e49.48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e49.47\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e390\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e499\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY10000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e32\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e34\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e56.53\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e56.53\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e469\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e652\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY15000top\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e68\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e70\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e61.56\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e61.56\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e425\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e625\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAPY15000random\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e54\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e61.56\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e61.56\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e477\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e551\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"Discussion","content":"\u003cp\u003eThere 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\u0026rsquo; 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; Strand\u0026eacute;n 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 \u0026ndash; 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.\u003c/p\u003e"},{"header":"Conclusions","content":"\u003cp\u003eRegarding 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.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eEthics approval\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot applicable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData and model availability statement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe data set corresponds to the genetic evaluation from December 2021 from the National Research Institute of Animal Production.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eDeclaration of generative AI and AI-assisted technologies in the writing process\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare that they did not use AI technology.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor ORCIDs\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eDawid Słomian: https://orcid.org/0000-0002-9037-7703\u003c/p\u003e\n\u003cp\u003eJoanna Szyda: https://orcid.org/0000-0001-9688-0193\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eD. 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.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eDeclaration of interest\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare that they have no competing interests.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgments\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot applicable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFinancial support statement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis study was supported by a grant from the Ministry of Agriculture and Rural Development (DŻW.pp.862.1.2023).\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eAguilar I, Misztal I, Johnson DL, Legarra A, Tsuruta S, Lawlor TJ (2010) Hot topic: a unified approach to utilize phenotypic, full pedigree, and genomic information for genetic evaluation of Holstein final score. J Dairy Sci. 93(2):743-52. https://doi.org/10.3168/jds.2009-2730\u003c/li\u003e\n\u003cli\u003eChristensen O.F., Lund M.S., (2010) Genomic prediction when some animals are not genotyped. Genet. Sel. 42\u003cstrong\u003e:\u003c/strong\u003e 2. https://doi.org/10.1186/1297-9686-42-2\u003c/li\u003e\n\u003cli\u003eFragomeni B.O., Lourenco D.A.L., Tsuruta S., Masuda Y., Aguilar I., Legarra A., Lawlor T.J., Misztal I. (2015) Hot topic: Use of genomic recursions in the single-step genomic best linear unbiased predictor (BLUP) with a large number of genotypes. J Dairy Sci. 98: 4090-4094 https://doi.org/10.3168/jds.2014-9125\u003c/li\u003e\n\u003cli\u003eGao H., Koivula M., Jensen J., Strand\u0026eacute;n I., Madsen P., Pitk\u0026auml;nen T., Aamand G.P., M\u0026auml;ntysaari E.A. (2018) Short communication: Genomic prediction using different single-step methods in the Finnish red dairy cattle population, J Dairy Sci. 101\u003cstrong\u003e:\u003c/strong\u003e 10082-10088. https://doi.org/10.3168/jds.2018-14913\u003c/li\u003e\n\u003cli\u003eHenderson C.R. (1975) Rapid method for computing the inverse of a relationship matrix. Biometrics 58: 1727\u0026ndash;1730. https://doi.org/10.3168/jds.S0022-0302(75)84776-X\u003c/li\u003e\n\u003cli\u003eKoivula M., Strand\u0026eacute;n I., Su G., M\u0026auml;ntysaari E.A. (2012) Different methods to calculate genomic predictions - Comparisons of BLUP at the single nucleotide polymorphism level (SNP-BLUP), BLUP at the individual level (G-BLUP), and the one-step approach (H-BLUP). J Dairy Sci.95: 4065-4073. https://doi.org/10.3168/jds.2011-4874\u003c/li\u003e\n\u003cli\u003eLiu Z., Goddard M.E., Reinhardt F., Reents R. (2014) A single-step genomic model with direct estimation of marker effects. J Dairy Sci.97: 5833-5850. https://doi.org/10.3168/jds.2014-7924\u003c/li\u003e\n\u003cli\u003eLiu Z., Goddard M.E., Hayes B.J., Reinhardt F., Reents R. (2016) Technical note: Equivalent genomic models with a residual polygenic effect. J Dairy Sci. 99: 2016-2025. https://doi.org/10.3168/jds.2015-10394\u003c/li\u003e\n\u003cli\u003eM\u0026auml;ntysaari E.A., Koivula M., Strand\u0026eacute;n I. (2020) Symposium review: Single-step genomic evaluations in dairy cattle. J Dairy Sci. 103: 5314-5326. https://doi.org/10.3168/jds.2019-17754\u003c/li\u003e\n\u003cli\u003eM\u0026auml;ntysaari EA, Evans RD, Strand\u0026eacute;n I. (2017) Efficient single-step genomic evaluation for a multibreed beef cattle population having many genotyped animals. J Anim Sci. 95(11):4728-4737. https://doi: 10.2527/jas2017.1912\u003c/li\u003e\n\u003cli\u003eMasuda M., Misztal I., Tsuruta S., Legarra A., Aguilar I., Lourenco D.A.L., Fragomeni B.O., Lawlor, T.J. (2016) Implementation of genomic recursions in single-step genomic best linear unbiased predictor for US Holsteins with a large number of genotyped animals. J Dairy Sci. 99: 1968-1974 https://doi.org/10.3168/jds.2015-10540\u003c/li\u003e\n\u003cli\u003eMisztal I. (2016) Inexpensive Computation of the Inverse of the Genomic Relationship Matrix in Populations with Small Effective Population Size. Genetics, Volume 202, Issue 2, Pages 401\u0026ndash;409, https://doi.org/10.1534/genetics.115.182089\u003c/li\u003e\n\u003cli\u003eMisztal I., Legarra A., Aguilar I. (2009) Computing procedures for genetic evaluation including phenotypic, full pedigree, and genomic information. J Dairy Sci. 92: 4648-4655. https://doi.org/10.3168/jds.2009-2064\u003c/li\u003e\n\u003cli\u003eMisztal I., Legarra A., Aguilar I. (2014) Using recursion to compute the inverse of the genomic relationship matrix. J Dairy Sci., 97: 3943-3952 https://doi.org/10.3168/jds.2013-7752\u003c/li\u003e\n\u003cli\u003eRStudio Team (2021). RStudio: Integrated Development Environment for R. RStudio, PBC, Boston, MA URL http://www.rstudio.com/\u003c/li\u003e\n\u003cli\u003eStrand\u0026eacute;n I., Garrick D.J. (2009) Technical note: Derivation of equivalent computing algorithms for genomic predictions and reliabilities of animal merit. J Dairy Sci. 92: 2971-2975. https://doi.org/10.3168/jds.2008-1929\u003c/li\u003e\n\u003cli\u003eStrand\u0026eacute;n I., Matilainen K., Aamand G.P., Mantysaari E.A. (2017) Solving efficiently large single-step genomic best linear unbiased prediction models. JABG, 134: 264-274 https://doi.org/10.1111/jbg.12257\u003c/li\u003e\n\u003cli\u003eTaylor J.F. (2014) Implementation and accuracy of genomic selection. Aquaculture 420\u0026ndash;421: S8-S14 https://doi.org/10.1016/j.aquaculture.2013.02.017\u003c/li\u003e\n\u003cli\u003eTen Napel J., Vandenplas J., Lidauer M., Stranden I., Taskinen M., M\u0026auml;ntysaari E.A., Calus M.P., Veerkamp R.F. (2021) MiXBLUP 3.0.1 manual\u003c/li\u003e\n\u003cli\u003eVanRaden P.M. (2008) Efficient methods to compute genomic predictions. J Dairy Sci. 91: 4414-4423 https://doi.org/10.3168/jds.2007-0980\u003c/li\u003e\n\u003cli\u003eVandenplas J., Ten Napel J., Darbaghshahi S.N., Evans R., Calus M.P.L., Veerkamp R., Cromie A., M\u0026auml;ntysaari E.A., Strand\u0026eacute;n I. (2023) Efficient large-scale single-step evaluations and indirect genomic prediction of genotyped selection candidates. Genet. Sel. 55, 37 https://doi.org/10.1186/s12711-023-00808-z\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"APY, G-BLUP, GT-BLUP, GEBV, single-step, SNP-BLUP","lastPublishedDoi":"10.21203/rs.3.rs-5260327/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5260327/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eMany 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.\u003c/p\u003e","manuscriptTitle":"A comparison of genomically enhanced breeding values predicted by different single-step approaches","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-10-18 07:07:43","doi":"10.21203/rs.3.rs-5260327/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"1b26e241-8802-4cfa-b1e1-51a99a5faadd","owner":[],"postedDate":"October 18th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-12-04T15:13:07+00:00","versionOfRecord":[],"versionCreatedAt":"2024-10-18 07:07:43","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-5260327","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-5260327","identity":"rs-5260327","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}