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The impact of direct-maternal genetic correlations on international beef cattle evaluations for Limousin weaning weight

Abstract In beef cattle maternally influenced traits, estimates of direct-maternal genetic correlations (rdm) are usually reported to be negative. In international evaluations, rdm can differ both within countries (rdm_WC) and between countries (rdm_BC). The rdm_BC are difficult to estimate and are assumed to be zero in the current model for international beef cattle evaluations (Interbeef). Our objective was to investigate re-ranking of international estimated breeding values (IEBVs) in international beef cattle evaluations between models that either used estimated values for rdm or assumed them to be 0. Age-adjusted weaning weights and pedigree data were available for Limousin beef cattle from ten European countries. International EBVs were obtained using a multi-trait animal model with countries modeled as different traits. We compared IEBVs from a model that uses estimated rdm_BC (ranging between −0.14 and +0.14) and rdm_WC (between −0.33 and +0.40) with IEBVs obtained either from the current model that assumes rdm_BC to be 0, or from an alternative model that assumes both rdm_BC and rdm_WC to be 0. Direct and maternal IEBVs were compared across those three scenarios for different groups of animals. The ratio of population accuracies from the linear regression method was used to further investigate the impact of rdm on international evaluations, for both the whole set of animals in the evaluation and the domestic ones. Ignoring rdm_BC, i.e., replacing estimated values with 0, resulted in no (rank correlations > 0.99) or limited (between 0.98 and 0.99) re-ranking for direct and maternal IEBVs, respectively. Both rdm_BC and rdm_WC had less impact on direct IEBVs than on maternal IEBVs. Re-ranking of maternal IEBVs decreased with increasing reliability. Ignoring rdm_BC resulted in no re-ranking for sires with IEBVs that might be exchanged across countries and limited re-ranking for the top 100 sires. Using estimated rdm_BC values instead of considering them to be 0 resulted in null to limited increases in population accuracy. Ignoring both rdm_BC and rdm_WC resulted in considerable re-ranking of animals’ IEBVs in all groups of animals evaluated. This study showed the limited impact of the current practice of ignoring rdm_BC in international evaluations for Limousin weaning weight, most likely because the estimated rdm_BC was close to 0. We expect that these conclusions can be extended to other traits that have reported rdm values in the range of rdm_WC values for weaning weight in Limousin.


Introduction
In livestock, recorded traits can be influenced in their expression by the mother, i.e., they are influenced by maternal effects (Falconer and Mackay, 1996). Examples of such traits are growth and survival during the early stage of an animal's life (Meyer, 2001;Knol et al., 2002;Hartmann et al., 2003;Eaglen and Bijma, 2009). Maternal effects reflect the mothers' role in providing the environment to survive as well as nourishment for the offspring, starting from uterine development and continuing after birth until weaning (Meyer, 2001;Eaglen and Bijma, 2009), and have both a genetic and an environmental component (Falconer and Mackay, 1996). Therefore, in genetic evaluations of maternally influenced traits, the observed phenotypes are often dissected into a direct genetic effect, a maternal genetic effect, a maternal permanent environment effect, and into environmental effects common to siblings (Bijma, 2006;Mrode, 2014;Schaeffer, 2019). Maternal effects can contribute to phenotypic similarity in multiple offspring of the same dam, e.g., full-sibs and half-sibs, either arising from the same litter or different parities, and variability between families (Falconer and Mackay, 1996).
Models that account for direct and maternal genetic effects allow animal breeders to better estimate breeding values (EBVs) for these components, which are important for selection decisions and genetic progress of maternally influenced traits (Van Vleck et al., 1977;Gerstmayr, 1992;Mrode, 2014). Maternal effects are therefore usually included in the total merit indices for beef cattle (e.g., ICBF, 2020; Institut de l'Élevage, 2020) to reflect the maternal abilities of heifers and cows. Many studies have estimated the co-variance components of direct and maternal genetic effects in chickens, cattle, pigs, sheep, and rabbits (e.g., Hartmann et al., 2003;Koch, 1972;Knol et al., 2002;Abbasi et al., 2012;Krogmeier et al., 1994). The estimation of the magnitude of the genetic covariance and correlation between direct and maternal effects has been the object of study of animal breeders for a long time (Koch, 1972;Baker, 1980;Robinson, 1996a;Robinson, 1996b). In beef cattle, estimates of direct-maternal genetic correlations (r dm ) are usually reported to be negative (Robinson, 1996a;Meyer, 1997). Estimates of r dm could be subject to possible different sources of bias (Meyer, 1997;Clément et al., 2001;Bijma, 2006), and Meyer (1992) showed that large datasets are required for accurate estimation of genetic parameters of maternally affected traits. Later, Schaeffer (2019) suggested that three generations of female data are required to have a proper data structure for accurate estimations of r dm , and when this pedigree depth is not present, to set r dm equal to 0 instead. Given the difficulties associated with the estimation of r dm , David et al. (2015) investigated the impact of ignoring r dm , on direct, maternal, and total EBVs (defined as the sum of direct and maternal EBVs) in sheep, pigs, and rabbits genetic evaluations. The authors showed that r dm had a small influence on the total EBV, recommending, therefore, to set r dm to 0 when their values are uncertain.
International genetic evaluations require estimates of across-country r g (Phocas et al., 2005). However, the estimation process can be challenging (Mark et al., 2005;Venot et al., 2007), especially in beef cattle due to the low number of existing genetic connections between countries (Berry et al., 2016). These connections are established through animals having recorded offspring in more than one country, i.e., mainly international bulls (Jorjani et al., 2005;Bonifazi et al., 2020b). Moreover, this process is even more challenging for maternally influenced traits. Using large national datasets during this process allows to consider all existing genetic connections between countries at the expense of long, or even prohibitive, computational times (Bonifazi et al., 2020b). Therefore, data are usually reduced based on criteria that aim to maximize retained genetic connections across countries (Jorjani et al., 2005;Mark et al., 2005;Bonifazi et al., 2020b). Using a multi-country dataset, Bonifazi et al. (2020b) investigated the impact of data sub-setting on acrosscountry r g in beef cattle international evaluations led by Interbeef (2020) for Limousin weaning weight. While reducing data did not significantly affect across-country direct and maternal r g , within-country direct-maternal genetic correlations (r dm_WC ) and between-country direct-maternal genetic correlations (r dm_BC ) were more negative and were consistently affected by data reduction (−0.12 and −0.11 on average, respectively). When using all data, estimates of r dm_BC were on average 0 and ranged from −0.14 to +0.14. However, they were most often not significantly different from zero, with standard errors being on average 0.14. France, however, had negative r dm_BC (between −0.14 and −0.01) with all other countries except Switzerland which was the only country with a positive r dm_WC . Moreover, France represented 87% of the data and had the strongest connectedness with other countries. Four out of the 8 countries, including France, had an r dm_WC that was significantly different from 0. These r dm_WC being significantly different from 0 suggest that these estimates may be negative. Based on the estimates involving FRA, it seems that r dm_BC do also have a tendency to be negative, albeit with relatively small deviations from zero.
In international evaluations, r dm_BC are assumed to be zero since estimating them can be difficult (Venot et al., 2007;Vesela et al., 2019;Bonifazi et al., 2020b). However, little is known about the impact in international evaluations of assuming r dm_BC and r dm_WC to be zero. Therefore, our objective was, using previously estimated parameters, to investigate the impact on EBVs of using 0 instead of non-zero estimates for r dm in the current international beef cattle evaluations model using Limousin CB  common bulls  EBV  estimated breeding value  IEBV  international estimated breeding  value  LR linear regression REL approximated reliability r g genetic correlations r dm direct-maternal genetic correlations r dm_WC within-country direct-maternal genetic correlations r dm_BC between-country direct-maternal genetic correlations SD standard deviation weaning weight data. We studied the potential impact of this assumption on selection decisions by evaluating the re-ranking of different groups of animals, and the change in population accuracies and dispersion through the linear regression (LR) method (Legarra and Reverter, 2018).

Materials and Methods
Animal Care and Use Committee approval was not requested for this study because commercial data were obtained from existing databases. A more detailed description of the data and pedigree editing criteria applied is provided in Bonifazi et al. (2020b). The number of direct and maternal genetic connections available in the pedigree was quantified in Bonifazi et al. (2020b). In total, 1,436 sires (also called "Common Bulls"-CB) and 3,828 maternal grand-sires (also called "Common MGS"-CMGS) had recorded offspring and grand-offspring in one or more countries, respectively. The number of CB ranged from 1,053 connecting 2 populations to 17 connecting 8 populations. The number of CMGS ranged from 3,040 connecting 2 populations to 19 connecting 8 populations. About 25% of the CMGS were also CB. Hereafter, for simplicity, we will refer to populations as countries, even though the DFS population is composed of more than one country. The AMACI model (Animal Model accounting for Across-Country Interaction) (Phocas et al., 2005) was used for the estimation of an animal's breeding value, which accounts for country-specific fixed and random effects by fitting the national model of each country. The AMACI model is a multi-trait animal model with maternal effects, in which each country is modeled as a different trait:

Data and model
where i is the country; y i is the vector of observations for country i; b i and r i are the vectors of fixed and random environmental effects, respectively, for country i (as detailed below); u i and m i are the vectors of direct and maternal random additive genetic effects, respectively, for country i (i.e., corresponding to the vectors of IEBVs for each individual on each of the 8 country scales); pe i is the vector of random maternal permanent environmental effects (provided by the dam) for country i; and e i is the vector of random residual effects for country i. X i and C i are incidence matrices linking records to fixed and random environmental effects, respectively. Z i , W i , and P i are incidence matrices linking records to the animal, maternal genetic, and maternal permanent environmental effects, respectively. Fixed and random effects for each country are reported in Supplementary Table S1. In particular, random environmental effects were modeled for only three countries: CZE (herd-year-season), DEU (herd-year), and CHE (herd-year and sire-herd). The maternal permanent environmental effect was not fitted for the DEU population. It is assumed that where u is the vector of random direct additive genetic effects for all countries; m is the vector of random maternal additive genetic effects for all countries; G is the across-country genetic (co)variance matrix of order 16 by 16 in which G d,d is the across-country direct additive genetic (co)variance matrix, G m,m is the across-country maternal additive genetic (co)variance matrix, and G d,m (G m,d ) contains additive genetic covariances between direct and maternal effects within-country (diagonal elements) and additive genetic covariances between direct and maternal effects between-country (off-diagonal elements); A is the numerator relationship matrix; and ⊗ indicates the Kronecker product. Random environmental effects, random maternal permanent environmental effects, and residuals were fitted using block-diagonal variance matrices.
To closely represent current Interbeef evaluations, the genetic variance-covariance matrix with additive direct and maternal genetic effects (G) was built as where S is the diagonal matrix with national genetic standard deviations for direct and maternal genetic effects and Φ is the across-country estimated genetic correlation matrix (of order 16 × 16 with diagonal values of 1). The genetic correlation matrix Φ was previously estimated in Bonifazi et al. (2020b) (Scenario ALL) and was used for all scenarios implemented in this study. Both Φ and the obtained G (co)variance matrix are reported in Table 2. Both genetic and environmental variances were the same as those used in the national genetic evaluations  Table S2). Interbeef uses this procedure to compute the genetic variance-covariance matrix under the assumption that the national estimates of genetic variances are more accurate (e.g., when not all national data are submitted for international evaluations; Michenet, Interbull Centre, personal communication). The presence of genotype-by-country interactions due to differences in production systems and management conditions, as well as differences in trait and model definitions between countries, is accounted for in the AMACI model by modeling AWW of different countries as different correlated traits. These factors are reflected in estimated genetic correlations between countries lower than unity (Mark, 2004;Bonifazi et al., 2020b).

Scenarios
Breeding values were estimated for the following scenarios, where r dm within and between countries were either used or replaced by zero, while between-country direct and maternal r g were used in all scenarios. This led to the following three scenarios, hereafter referred to as "international scenarios" (Table 3 summarizes the implemented (co)variance structure in each scenario): • Scenario REF: both r dm_WC and r dm_BC are used in the evaluation, i.e., the covariance structure across countries is as shown in Table 2. In this scenario, the information between direct and maternal breeding values is exchanged at the national level, through r dm_WC , and at an international level through r dm_BC . We considered this scenario as the reference scenario. • Scenario CUR: r dm_WC are used in the evaluation, but r dm_BC are set to zero. This scenario represents the current-inuse methodology for Interbeef evaluations (Bonifazi et al., 2020b), where r dm information is used at its minimum since information between direct and maternal breeding values is exchanged only at a national level through r dm_WC . • Scenario NONE: both r dm_WC and r dm_BC are set to 0. In this scenario, there is no usage of r dm and it was used as an extreme case to understand the effects of completely ignoring r dm .
Between-country direct-maternal r g • 1 Not used r g were replaced by 0. 2 Scenario: NAT = national single-trait evaluations, NONE = both r dm_WC and r dm_BC set to 0, CUR = r dm_WC used in the evaluation, and r dm_BC set to 0, REF = both r dm_WC and r dm_BC used in the evaluation. With r dm_WC = within-country direct-maternal genetic correlations, and r dm_BC = between-country direct-maternal genetic correlations.  Next to the international scenarios, breeding values were estimated with the following scenario aiming to represent pseudo-national single-trait evaluations: • Scenario NAT: the AMACI model was run with all betweencountry r g set to 0 (i.e., for r d , r m , and r dm_BC ), while r dm_WC were used. With this setting, the AMACI model is equivalent to run eight separate single-trait national evaluations. Once the model was run, for each country, EBVs were retained for all animals with phenotypes on their own, or any of their ancestors, or both. Hereafter, we will refer to those animals as the domestic set of animals. The number of retained domestic animals per country is reported in Table 1.
It should be noted that the resulting genetic correlation matrix in scenario CUR was non-positive definite, and therefore, it was bended using the unweighted bending approach of Jorjani et al. (2003) (using the R package "mbend" [Nilforooshan, 2020] with a threshold of 10 −3 ). Initial analyses showed that the effects of bending on all the estimated r g were minimal with an unweighted bending approach and allowed to keep the r dm_BC close to 0. Moreover, bending the genetic correlation matrix allows to keep the same national genetic variances between international scenarios, which is often desired in international evaluations (Jorjani et al., 2003;Bonifazi et al., 2020b). Due to the bending process, some r dm_BC showed small deviations from the fixed value of zero: ranging from −0.03 to 0.02. Changes due to bending in direct r g , maternal r g and r dm_WC ranged between −0.03 and 0.02, −0.03 and 0.01, and −0.04 and 0.09, respectively.

Groups of animals evaluated
Changes in across-country r g may result in re-ranking of animals, e.g., top bulls. We computed Spearman rank correlations of direct and maternal IEBVs for different groups of animals between the tested international scenarios that all used the full international pedigree: 1.
All (3,431,742) animals included in the international evaluation. 2.
Reliability (REL) class. Three groups of animals were formed based on their direct and maternal IEBV REL: REL ≤ 0.3, 0.3 < REL ≤ 0.6, and REL > 0.6. Approximated REL were computed using Tier and Meyer (2004) methodology under scenario REF.

3.
Common Bulls (CB). In the analyzed dataset, 1,436 CB, i.e., sires with recorded offspring in more than one country, were present. Re-ranking of CB is an indicator of the r dm_BC effect on the IEBV of animals having recorded offspring in two or more countries.
Sire's direct IEBV should be associated with the following: 1) a REL ≥ 0.5 on at least one country scale and 2) ≥ 25 recorded progeny across all countries. Sire's maternal IEBV should fulfil the following conditions: 1) have a publishable direct IEBV; 2) an associated REL ≥ 0.3 on at least one country scale; and 3) ≥ 15 daughters with recorded progeny and ≥ 25 recorded grand-progeny from daughters across all countries. Sires with IEBVs that fulfil the above requirements under the scenario REF were selected. 5.
Young sires. Sires that fulfil the following conditions under the scenario REF were selected: 1) a publishable direct IEBV (following the same rules as described in group 4) and 2) a REL associated with a maternal IEBV < 0.3 on all country scales.
For each of the above groups of animals, we considered that no re-ranking was present between scenarios REF and CUR, and scenarios REF and NONE when the rank correlation was equal to or greater than 0.990. When the rank correlation was between 0.980 and 0.990, we considered that small re-ranking was present. Furthermore, for each scenario, we selected the top 100 sires on each country scale from those with a publishable IEBV (following the same rules as described in group 4) and calculated the number of commonly selected top 100 sires between scenarios.

Increases in population accuracies and dispersion
To investigate the impact of modeling across-country r dm on the population accuracies (acc), we used the LR method (Legarra and Reverter, 2018). Population accuracy is defined as the correlation between the true breeding values (TBVs) and the EBVs across individuals in a population (Legarra and Reverter, 2018). Following Legarra and Reverter (2018), the ratio of population accuracies of two evaluations, evaluation p with partial information, and evaluation w with all (i.e., whole) information is defined as ρ p,w = accp accw and is computed as the Pearson correlation between EBVs from evaluations p and w, where û p is the vector of animals' EBVs from the partial evaluation and û w is the vector of animals' EBVs from the whole evaluation. Thus, ρ p,w is a direct estimator of the increase in population accuracy of EBVs from an evaluation with partial data to an evaluation with whole data (Legarra and Reverter, 2018).
In the context of international evaluations, national evaluations can be seen as evaluations with partial information, where recorded phenotypes are available only at the country level. International evaluations provide a new source of information through related animals being recorded in other countries and, therefore, represent evaluation w. Thus, the relative increases in population accuracy when moving from the partial (i.e., national; NAT) to the whole (i.e., international; INT) evaluations were calculated for each country and for international scenarios CUR and REF as the reciprocal of ρ NAT d ,INT d (Macedo et al., 2020), , by using only domestic animals' EBVs (denoted by d) of each country from scenario NAT, i.e., national animals. For example, when ρ NAT d ,INT d is 0.8, the additional information from the international evaluation increased the accuracy by 25% relative to the national evaluation (1/ρ NAT d ,INT d = 1.25).
The change in the r dm structure implemented in the international scenarios can also be viewed as increasingly adding data to a "partial" international evaluation when moving from scenario NONE to CUR or REF, since non-zero values for the r dm effectively change the amount of information contributing to the domestic direct and maternal animal's IEBVs in a specific country. Thus, using all 3,431,742 IEBVs expressed on each country scale, we computed 1/ρ NONE,REF and 1/ρ NONE,CUR. For instance, the latter represents the increase in population accuracy when fitting r dm_WC .
Using the LR method, we also evaluated any changes in dispersion of the EBVs between partial and whole evaluations.
The estimator for dispersion (bw,p) is defined as the slope of the regression of û w on û p, computed as b w,p = cov (ûw,ûp) var(ûp) (Legarra and Reverter, 2018). The expectation of b w,p is 1 if û w and û p have the same dispersion, while b w,p > 1 indicates less dispersion in û p compared to û w, and b w,p < 1 indicates more dispersion in û p compared to û w. We computed b w,p between CUR or REF (whole evaluations) and NAT (partial evaluation) using only domestic animals' EBV, and between REF (whole evaluation) and NONE or CUR (partial evaluations) using all IEBVs.

Software and settings
In all scenarios, estimated breeding values and approximated reliabilities were obtained using MiX99 software (MiX99 Development Team, 2017). The convergence criterion of the preconditioned conjugate gradient (PCG) algorithm for the mixedmodel equation solutions was defined as the square root of the relative difference between solutions of two consecutive PCG iterations and was set to 10 −7 . Convergence was also monitored for two other criteria, i.e., the relative difference between two consecutive solutions for the additive genetic animal effects and the relative residual of the mixed model equations.

Results
First, we present the distribution of the EBV for the implemented scenarios, followed by the results on rank correlations for the different groups of animals evaluated and for the LR method.

Distribution of EBV
We computed the standard deviation (SD) across the EBV for the group of domestic animals for each country and scenario ( Figure 1). The SD of domestic animals' direct and maternal EBVs remained equal or increased when moving from national (NAT) to international evaluations (i.e., NONE, CUR, and REF) for all countries (Figure 1). The increase in EBV SD when comparing national with international evaluations was larger for smaller countries in terms of phenotypes (CZE, IRL, and CHE). CZE had the largest increase of EBV SD when moving from scenario NAT to REF, being 102% for direct EBV and 84% for maternal EBV. The SD of FRA domestic animals' direct EBV remained almost the same under NAT and international scenarios, while slightly increased (5%) for maternal EBV under scenario NONE.
In general, there were no large differences in terms of EBV SD between international scenarios.

Impact on re-ranking of animals' IEBV
Regarding the direct IEBV, no re-ranking was observed when considering all animals for all international evaluations in any country (rank correlations > 0.990   Young sires: 1,561 sires obtained under scenario REF. for maternal IEBVs, with an average rank correlation of 0.988 across countries, and a minimum rank correlation of 0.980 for DFS (Table 4). For common bulls, re-ranking was more severe for maternal IEBVs than for direct IEBVs (Table 4). The comparison between scenarios REF and NONE showed re-ranking: rank correlations were on average 0.979 for direct IEBVs and 0.965 for maternal IEBVs, across countries. The comparison between scenarios REF and CUR showed no re-ranking for CB's direct IEBV on any country scale (rank correlations > 0.990), and small re-ranking for maternal IEBV (rank correlations between 0.980 and 0.990) for CZE, DFS, GBR, IRL, and DEU (Table 4).
When grouped by their individual approximated reliabilities, the majority of the animals (> 78%) had a direct IEBV REL between 0.3 and 0.6, with the exception for CHE, where 64% of the animals had direct IEBV REL ≤ 0.3 (Supplementary Table S3). When grouped by maternal IEBV REL, almost all animals had a REL ≤ 0.6 and the majority of them had REL ≤ 0.3. FRA was the only country that had about 1% of the animals with maternal IEBV REL > 0.6 (Supplementary Table S3). Rank correlations between scenarios REF and NONE for direct IEBVs were 0.990, 0.990, and 0.985, for the three REL classes, respectively (Table 5). There was more variation in rank correlations between countries in the class of REL > 0.6 compared to the other two classes, with the smallest direct IEBV rank correlations for CHE (0.973) and DEU (0.975) (  (Table 5).
The number of sires with publishable IEBVs was 32,208 and 13,016 for direct and maternal IEBVs, respectively. These sires were mainly recorded in FRA (89% and 90% of the total, for both direct and maternal IEBVs, respectively), followed by GBR, DFS, and DEU (each accounting for 2% to 4% of the total sires) and less than 1% in other countries (results not shown). The mean REL of sires with publishable IEBVs was 0.64 and 0.50 on average across countries for direct and maternal IEBVs, respectively (Supplementary Table S4). Re-ranking was present for sires with publishable IEBVs between scenarios REF and NONE for direct IEBVs on all country scales except for FRA and IRL: average rank correlation across countries of 0.985 and minimum rank correlation of 0.971 for CHE (Table 4). Similarly, re-ranking was present between scenarios REF and NONE for sires' maternal IEBVs on all country scales: average rank correlation across countries of 0.980 and minimum rank correlation of 0.969 for ESP (Table 4). No re-ranking was observed, on any country scale, for sires with publishable IEBVs between scenarios REF and CUR, for both direct and maternal IEBVs (rank correlations > 0.995 and > 0.990, respectively; Table 4).
There were in total 1,561 young sires, the majority of which were recorded in FRA (78%), followed by GBR (9%), DFS (6%), DEU and CHE (2%), and other countries (1% or less) (results not shown). Young sires showed no re-ranking between scenarios REF and NONE for direct IEBVs, with the only rank correlations below 0.99 for CHE (0.986) ( Table 4). Young sires showed re-ranking between scenarios REF and NONE for maternal IEBVs, with average rank correlations of 0.947 and minimum rank correlation of 0.831 for FRA. No re-ranking was observed for young sires between scenarios REF and CUR for direct IEBVs in any country (rank Table 5. Spearman rank correlations of animals direct (Dir) and maternal (Mat) international estimated breeding values (IEBVs) between scenarios 1 per class of reliability (REL) 3 , per country, and averaged across countries correlations > 0.997), while re-ranking was present for maternal IEBVs (average rank correlation of 0.979 and minimum rank correlation of 0.968 for DFS), with the exception of CHE (rank correlation of 0.992). The number of commonly selected top 100 publishable sires between scenarios REF and NONE was 81 and 79 on average across countries for direct and maternal IEBVs, respectively ( Table 4). The minimum number of commonly selected top 100 sires was 73 for CHE for direct IEBVs and 71 for ESP for maternal IEBVs. We further quantified the re-ranking between these scenarios using the absolute mean of the change in position of the list of top 100 sires selected under scenario REF when ranked based on their IEBVs under scenario NONE. Across countries, the top 100 sires moved on average by 16 and 23 positions, for direct and maternal IEBVs, respectively (Supplementary Table  S5). The number of commonly selected top 100 publishable sires between scenarios REF and CUR was 94 and 90 on average across countries for direct and maternal IEBVs, respectively ( Table 4). The minimum number of commonly selected top 100 sires was 91 for CHE for direct IEBV and 84 for DFS for maternal IEBV. Across countries, the top 100 sires, comparing scenario REF to CUR, moved on average by 2 and 5 positions, for direct and maternal IEBVs, respectively (Supplementary Table S5).

Increases in population accuracies and dispersion
The increases in population accuracy when moving from scenario NONE to CUR or to REF, measured using 1/ρ NONE,CUR and 1/ρ NONE,REF , respectively, are reported in Table 6. When comparing scenarios NONE and CUR, for direct IEBVs, 1/ρ NONE,CUR ranged from 0% (for CZE, DFS, GBR, IRL, and FRA) to 1% (for ESP, DEU, and CHE), while for maternal IEBVs, it was on average 2%, ranging from 1% (for DFS, IRL, and DEU) to 4% (for FRA) ( Table 6). When comparing scenarios NONE and REF, 1/ρ NONE,REF was on average 1% for direct IEBVs across all countries, ranging from 0% (for FRA and IRL) to 2% (for CHE and DEU), while for maternal IEBVs, it was on average 3% across all counties, ranging from 2% (for GBR, IRL, and DFS) to 8% (for FRA) ( Table 6). Comparison of 1/ρ NONE,REF with 1/ρ NONE,CUR shows similar gains of population accuracy when moving from scenario NONE to REF instead of moving to CUR, i.e., on average 0% for direct IEBVs and 1% for maternal IEBVs across countries. Only FRA benefits from using all correlations in the scenario REF in comparison to CUR (increase of 4%).
The increases in population accuracy for domestic animals obtained when moving from scenario NAT to CUR or REF are reported in Table 6. When comparing scenarios NAT and CUR, the increases in population accuracy for direct EBVs were on average 19% across countries, ranging from 0% of FRA to 59% of CZE. For maternal EBVs, the increases in population accuracy were on average 27% across countries, ranging from 1% of FRA to 101% of CZE. When comparing scenarios NAT and REF, the increases in population accuracy for direct EBVs were on average 21% across countries, ranging from 0% of FRA to 63% of CZE. For maternal EBVs, the increases in population accuracy were on average 29% across all countries, ranging from 0% of FRA to 106% of CZE. Comparison of 1/ρ NAT d ,REF d with 1/ρ NAT d ,CUR d shows the increase in population accuracy for domestic animals when moving from scenario NAT to REF instead of moving to CUR. For direct EBVs, differences of 1/ρ NAT d ,REF d with 1/ρ NAT d ,CUR d were on average 2%, ranging from 0% of FRA to 5% of CZE, while for maternal EBVs, differences were on average 2%, ranging from −1% for FRA to 5% of IRL.
Regression coefficients of IEBVs of whole on partial evaluations for all animals in the international evaluation are reported in Table 7. The regression coefficients b w,p for direct IEBVs of REF-NONE were close to 1 in all countries, ranging from 0.99 of FRA to 1.06 of CHE. Direct IEBVs b w,p of REF-CUR were also close to 1 in all countries, ranging from 1.00 of FRA to 1.04 of DFS and DEU. For maternal IEBVs, the b w,p of REF-NONE were close to 1 in most of the countries, except for FRA and CHE for which b w,p were 0.88 and 1.15, respectively. Maternal IEBVs b w,p of REF-CUR were close to 1 in all countries, ranging from 0.96 of DFS to 1.06 of CHE.
Regression coefficients of EBVs of whole on partial evaluations for domestic animals are also reported in Table 7. The regression coefficients b w,p for direct and maternal EBVs of NAT-CUR and NAT-REF were similar across countries. In NAT-CUR, direct EBVs b w,p were close to 1 in almost all countries, ranging between 1.01 of FRA Table 6. Increase in population accuracy 1 of moving from partial evaluation (p) to whole evaluation (w) 2 for all animals included in the international evaluations (All) and domestic animals 3

Discussion
In this study, we investigated the impact of three different definitions of across countries r dm on the ranking of international beef cattle IEBVs. We further explored the impact of r dm on population accuracies and dispersion in international evaluations using the LR method. To the best of our knowledge, this is the first study that investigates the impact of the r dm structure in the context of beef cattle international evaluations. Hereafter, we first discuss the impact of r dm_BC and r dm_WC , followed by the possible implications of this study for beef cattle international evaluations.

Impact of r dm_BC on international evaluations
Ignoring r dm_BC had a small impact on international evaluations. Ignoring r dm_BC did not result in direct IEBV re-ranking for any country when considering all animals, while it did result in limited maternal IEBV re-ranking. This re-ranking was not associated with particular countries, but it was mainly related to animals with low maternal IEBV REL (REL ≤ 0.3). Ignoring r dm_BC had also a small impact on all country scales for CB, with rank correlations between 0.980 and 0.990, while publishable sires showed no re-ranking on any country scale. In the latter group, Interbeef publication rules may have mitigated the impact of r dm_BC by requiring sires to have records across two or three generations. As expected, young sires with low maternal IEBV REL (< 0.3) showed re-ranking for maternal IEBV on all country scales when r dm_BC were ignored. Similarly, the impact of r dm_BC on the re-ranking of the top 100 sires, assessed as the absolute mean change in ranking of the sires, was mostly limited to the maternal IEBV. Ignoring r dm_BC had limited impact on domestic animals EBV SD: on average, the increase in EBV SD for scenario REF compared to CUR was 1% and 0% for direct and maternal EBVs, respectively, and mostly related to countries with smaller national populations (CZE, CHE, and IRL). These results are supported by the increases in population accuracy, when considering the whole set of IEBVs on each country scale, which hardly increased from using r dm_BC . Similar results were obtained for domestic animals: increases in population accuracy from modeling r dm_BC were on average 2% for both direct and maternal EBVs, and at maximum 5% (associated with smaller countries such as CZE and IRL

Impact of r dm_WC on international evaluations
Our results suggest that ignoring r dm_WC affects international evaluations. Ignoring r dm_WC resulted in re-ranking for both direct and maternal IEBVs. As expected, the largest impact on Scenario: NAT = national single-trait evaluations, NONE = both r dm_WC and r dm_BC set to 0, CUR = r dm_WC used in the evaluation, and r dm_BC set to 0, REF = both r dm_WC and r dm_BC used in the evaluation. With r dm_WC = within-country direct-maternal genetic correlations, and r dm_BC = betweencountry direct-maternal genetic correlations.
IEBVs re-ranking was observed for those countries with strong negative estimated r dm_WC , i.e., ESP and FRA. For example, FRA showed the lowest rank correlation for maternal IEBVs (0.917) when considering all animals, but grouping animals by their REL revealed that this re-ranking of FRA was mainly related to animals with REL ≤ 0.3. In general, when animals were grouped by REL, re-ranking within each group of REL was more severe in those countries with absolute values of r dm_WC greater than 0.2 (i.e., FRA, ESP, DEU, and CHE). These results are in agreement with those of  which also observed an increased re-ranking for maternal IEBVs when ignoring r dm_WC in international evaluations. Ignoring r dm_WC , in addition to r dm_BC , also gave more severe re-ranking for publishable sires and CB than ignoring only r dm_BC . Re-ranking in the latter group may be due to the majority of the CB originating from FRA (82.5%) (Bonifazi et al., 2020b). These FRA CB may have mainly recorded domestic offspring and, ignoring r dm_WC for FRA, would lead to a different ranking for this group of animals on all country scales. Similarly, ignoring r dm_WC led to high re-ranking of maternal IEBVs on all country scales for young sires, with the lowest rank correlation being for FRA. Re-ranking in this group may be related to the majority of young sires originating from FRA (78%). Ignoring r dm_WC also affected the top 100 sires with publishable IEBVs, both for direct and maternal IEBVs. This was also confirmed by the results for the absolute mean change in position of top 100 sires. FRA contributes with the largest amount of data in the Limousin Interbeef evaluation (87% of all phenotypes) and has the strongest pedigree connections with other countries. Thus, ignoring r dm_WC in FRA may have contributed to generating re-ranking for the categories of publishable sires, young sires, and CB not only on the FRA scale, but also on other countries scales since the majority of the animals in these categories originates from FRA. Ignoring r dm_WC affected the increases of domestic animals EBV SD, for both small countries like CHE and CZE (increases in EBV SD of CUR compared to NONE higher than 5%), and large countries like FRA (maternal EBV SD 5% smaller in scenario NONE compared to CUR). The importance of using r dm_WC in international evaluations was also reflected in the increases in population accuracy for the whole set of IEBVs.  (1977) showed that when r dm are negative, these correlations need to be considered for selection decisions. Thus, the results of our study support the suggestion of  that when estimates of r dm_WC are considerably different from 0, as was the case for most countries, they should be used in international evaluations and not ignored.
level may be biased due to data structure or modeling issues as reported in the literature (Meyer, 1997;Clément et al., 2001;Bijma, 2006). This could potentially also affect Interbeef evaluations as the current Interbeef AMACI model adopts each participating countries' national model. The results of this study show that r dm_WC have an impact on the re-ranking of IEBVs, underlining the importance of validation procedures of participating countries' national models. Domestic animals' EBV are expected to re-rank when data from relatives recorded in other countries are accounted for through international evaluations . In popular breeds such as Limousin, including information from countries as France where most connections and founders can be traced (Bouquet et al., 2011;Bonifazi et al., 2020b) is beneficial for two reasons as shown in previous studies Bonifazi et al., 2020a). First, accounting for information from relatives recorded in other countries leads to higher EBV's reliabilities of domestic animals, especially for smaller countries. Second, breeders get access to elite foreign bulls IEBV expressed on their own country scale and that rank similar or higher than domestic ones, giving breeders the opportunity to achieve higher genetic gains and better meet their selection objectives (Bonifazi et al., 2020a).
The LR method has been applied in evaluations with multiple traits, maternally affected traits, and traits where phenotypes are not available on all individuals for all environments (Chu et al., 2019;Macedo et al., 2020;Picard Druet et al., 2020). As such, the LR method appears to be a useful choice to evaluate the increase in population accuracy when ignoring r dm in the context of beef cattle international evaluations. The LR statistics ρ p,w relies on two assumptions (Legarra and Reverter, 2018). The first assumption, which does not affect ρ p,w, is that the regression of û p and of û w on the TBV is equal to one. The second assumption is that the regression of û w on û p is also equal to one, i.e., cov û w ,û p = var û p . When the LR method was applied, this latter assumption did hold for most of the comparisons between scenarios as shown by the regression coefficients b w,p being mostly between 0.9 and 1.1. We further noticed that the assumption of the LR method was met very closely, especially for domestic animals with reliable EBVs under the evaluation p (i.e., with country recorded phenotypes and with direct or maternal REL for evaluation p > 0.6; results not shown). Recently, Macedo et al. (2020) reported that some of the LR method statistics may be sensitive to incorrectly estimated genetic parameters; nonetheless, the same authors reported that the ratio of population accuracies ρ p,w performed well in their study. Results of the reported ρ p,w statistic should be treated with caution since estimates of r g between countries are prone to sampling error during the estimation process and often reported with high SE (Bonifazi et al., 2020b). Nevertheless, we expect that the observed trends in increases of population accuracy across the scenarios are not affected by departures of the underlying assumptions of the LR method or inaccuracy of estimated variance components.

Conclusions
Results of this study, based on Limousin weaning weight data, provide support that the current practice of ignoring r dm_BC in international beef cattle evaluations results in limited decreases in population accuracies, negligible impact on dispersion of EBVs, and no or limited re-ranking of animals' direct and maternal IEBVs, respectively. Re-ranking for maternal IEBVs was mainly related to animals with REL ≤ 0.3. No re-ranking was present for sires with publishable IEBVs. Moreover, results show that fixing r dm_WC to 0 would result in considerable re-ranking of animals.

Supplementary Data
Supplementary data are available at Journal of Animal Science online.