Application of single-step genomic evaluation using social genetic effect model for growth in pig

Objective Social genetic effects (SGE) are an important genetic component for growth, group productivity, and welfare in pigs. The present study was conducted to evaluate i) the feasibility of the single-step genomic best linear unbiased prediction (ssGBLUP) approach with the inclusion of SGE in the model in pigs, and ii) the changes in the contribution of heritable SGE to the phenotypic variance with different scaling ω constants for genomic relationships. Methods The dataset included performance tested growth rate records (average daily gain) from 13,166 and 21,762 pigs Landrace (LR) and Yorkshire (YS), respectively. A total of 1,041 (LR) and 964 (YS) pigs were genotyped using the Illumina PorcineSNP60 v2 BeadChip panel. With the BLUPF90 software package, genetic parameters were estimated using a modified animal model for competitive traits. Giving a fixed weight to pedigree relationships (τ: 1), several weights (ωxx, 0.1 to 1.0; with a 0.1 interval) were scaled with the genomic relationship for best model fit with Akaike information criterion (AIC). Results The genetic variances and total heritability estimates (T2) were mostly higher with ssGBLUP than in the pedigree-based analysis. The model AIC value increased with any level of ω other than 0.6 and 0.5 in LR and YS, respectively, indicating the worse fit of those models. The theoretical accuracies of direct and social breeding value were increased by decreasing ω in both breeds, indicating the better accuracy of ω0.1 models. Therefore, the optimal values of ω to minimize AIC and to increase theoretical accuracy were 0.6 in LR and 0.5 in YS. Conclusion In conclusion, single-step ssGBLUP model fitting SGE showed significant improvement in accuracy compared with the pedigree-based analysis method; therefore, it could be implemented in a pig population for genomic selection based on SGE, especially in South Korean populations, with appropriate further adjustment of tuning parameters for relationship matrices.


INTRODUCTION
The genetic effect of an individual on the phenotypes of its social partners (i.e., pen mates) is often termed the social genetic effect (SGE) or the indirect genetic effect [1]. The growth rate is a key trait in pig breeding goals because it contributes to economic efficiency. However, negative effects of social interactions, such as tail biting, or excessive aggression can inhibit growth of pen mates, resulting in reduce productivity in pig farming. The report by Bergsma et al [2] on pigs indicated that the heritable social interaction among various group members might play a role in their average daily gain (ADG). In this regard, Bijma et al [1] stated that the total breeding value (TBV), expressed as the combined direct breeding value (DBV) of an individual and social breeding values (SBV) of pen mates, for growth the model. The BLUPF90 software package Misztal et al [14] was used for the estimation of parameters by fitting a classical model with pedigree relationships only (PED classic ) and a social model pedigree relationships only modified for competitive traits (PED social ) [11] as follows: y = Xb+Z DaD +Wl+Vg+e (PED classic ) y = Xb+Z DaD +Z SaS +Wl+Vg+e (PED social ) where y is the vector of observations (ADG), b is the vector of fixed effects, a D is the vector of random direct additive genetic effects, a S is the vector of random SGEs, l is the vector for random birth litter, g is the vector of random group, and e is the vector of residuals. X, Z D , Z S , W, and V are the corresponding incidence matrices. Assumptions for the probability distributions were Bijma et al [1] for traits affected by heritable social effects, the variance of TBV al heritable variation that is exploitable for selection. The TBV of the i th animal is s: e heritable effect of an individual on trait values in the population, which is the sum ic effect ( , ) on its own phenotype and its SGE ( , ) on the phenotypes of its n -1 a et al [1] also stated that the total heritable variance determines the population's nse to selection and can be expressed as: 2 2 , in which N( ) indicates a normal distribution; I is an identity matrix of appropriate dimensions; and its affected by heritable social effects, the variance of TBV hat is exploitable for selection. The TBV of the i th animal is individual on trait values in the population, which is the sum n phenotype and its SGE ( , ) on the phenotypes of its n -1 that the total heritable variance determines the population's be expressed as: itable effect of an individual on trait values in the population, which is the sum ct ( , ) on its own phenotype and its SGE ( , ) on the phenotypes of its n -1 al [1] also stated that the total heritable variance determines the population's selection and can be expressed as: is the variance of direct additive genetic effects. In Model 2, direct and indirect additive genetic effects had the following multivariate normal (MVN) distribution: or of random group, and e is the vector of residuals. X, ZD, ZS, W, and V are the corresponding matrices. Assumptions for the probability distributions were ~N(0, 2 ), 2 ), ~N(0, 2 ), and ~N(0, 2 ), in which N( ) indicates a normal distribution; I is an atrix of appropriate dimensions; and 2 , 2 , 2 , and 2 are the variances of the ding effects. In Model 1, direct additive genetic effects had the following distribution: 2 ), in which A is the numerator relationship matrix and σ a D 2 is the variance of direct enetic effects. In Model 2, direct and indirect additive genetic effects had the following te normal (MVN) distribution: [ ] ~ MVN ( 0, C⊗A ), in which C is defined by the matrix 2 ], 2 is the variance of indirect genetic effects, is the covariance between indirect genetic effects, and C⊗A denotes the Kronecker product of two matrices.
ding to Bijma et al [1] for traits affected by heritable social effects, the variance of TBV the total heritable variation that is exploitable for selection. The TBV of the i th animal is follows: BV is the heritable effect of an individual on trait values in the population, which is the sum ct genetic effect ( , ) on its own phenotype and its SGE ( , ) on the phenotypes of its n -1 tes. Bijma et al [1] also stated that the total heritable variance determines the population's in response to selection and can be expressed as: 2 2 , in which C is defined by the matrix is the vector of random group, and e is the vector of residuals. X, ZD, ZS, W, and V are the corresponding 137 incidence matrices. Assumptions for the probability distributions were ~N(0, 2 ), 138 ~N(0, 2 ), ~N(0, 2 ), and ~N(0, 2 ), in which N( ) indicates a normal distribution; I is an 139 identity matrix of appropriate dimensions; and 2 , 2 , 2 , and 2 are the variances of the 140 corresponding effects. In Model 1, direct additive genetic effects had the following distribution: 141 ~(0, 2 ), in which A is the numerator relationship matrix and σ a D 2 is the variance of direct 142 additive genetic effects. In Model 2, direct and indirect additive genetic effects had the following 143 multivariate normal (MVN) distribution: [ ] ~ MVN ( 0, C⊗A ), in which C is defined by the matrix 144 ], 2 is the variance of indirect genetic effects, is the covariance between 145 direct and indirect genetic effects, and C⊗A denotes the Kronecker product of two matrices. 146 According to Bijma et al [1] for traits affected by heritable social effects, the variance of TBV 147 represents the total heritable variation that is exploitable for selection. The TBV of the i th animal is 148 defined as follows: 149 The TBV is the heritable effect of an individual on trait values in the population, which is the sum 153 of its direct genetic effect ( , ) on its own phenotype and its SGE ( , ) on the phenotypes of its n -1 154 group mates. Bijma et al [1] also stated that the total heritable variance determines the population's 155 potential in response to selection and can be expressed as: 156 is the vector of random group, and e is the vector of residuals. X, ZD, ZS, W, and V are the corresponding 137 incidence matrices. Assumptions for the probability distributions were ~N(0, 2 ), 138 ~N(0, 2 ), ~N(0, 2 ), and ~N(0, 2 ), in which N( ) indicates a normal distribution; I is an 139 identity matrix of appropriate dimensions; and 2 , 2 , 2 , and 2 are the variances of the 140 corresponding effects. In Model 1, direct additive genetic effects had the following distribution: 141 ~(0, 2 ), in which A is the numerator relationship matrix and σ a D 2 is the variance of direct 142 additive genetic effects. In Model 2, direct and indirect additive genetic effects had the following 143 multivariate normal (MVN) distribution: [ ] ~ MVN ( 0, C⊗A ), in which C is defined by the matrix 144 ], 2 is the variance of indirect genetic effects, is the covariance between 145 direct and indirect genetic effects, and C⊗A denotes the Kronecker product of two matrices. 146 According to Bijma et al [1] for traits affected by heritable social effects, the variance of TBV 147 represents the total heritable variation that is exploitable for selection. The TBV of the i th animal is 148 defined as follows: 149 The TBV is the heritable effect of an individual on trait values in the population, which is the sum 153 of its direct genetic effect ( , ) on its own phenotype and its SGE ( , ) on the phenotypes of its n -1 154 group mates. Bijma et al [1] also stated that the total heritable variance determines the population's 155 potential in response to selection and can be expressed as: 156 for traits affected by heritable social effects, the variance of TBV tion that is exploitable for selection. The TBV of the i th animal is t of an individual on trait values in the population, which is the sum its own phenotype and its SGE ( , ) on the phenotypes of its n -1 stated that the total heritable variance determines the population's nd can be expressed as: is the covariance between direct and indirect genetic effects, and  According to Bijma et al [1] for traits affected by heritable social effects, the variance of TBV represents the total heritable variation that is exploitable for selection. The TBV of the i th animal is defined as follows: The TBV is the heritable effect of an individual on trait values in the population, which is the sum of its direct genetic effect ( , ) on its own phenotype and its SGE ( , ) on the phenotypes of its n -1 group mates. Bijma et al [1] also stated that the total heritable variance determines the population's potential in response to selection and can be expressed as: denotes the Kronecker product of two matrices.
According to Bijma et al [1] for traits affected by heritable social effects, the variance of TBV represents the total heritable variation that is exploitable for selection. The TBV of the ith animal is defined as follows:  According to Bijma et al [1] for traits affected by heritable social effects, the variance of TBV 147 represents the total heritable variation that is exploitable for selection. The TBV of the i th animal is 148 defined as follows: 149 The TBV is the heritable effect of an individual on trait values in the population, which is the sum 153 of its direct genetic effect ( , ) on its own phenotype and its SGE ( , ) on the phenotypes of its n -1 154 group mates. Bijma et al [1] also stated that the total heritable variance determines the population's 155 potential in response to selection and can be expressed as: 156 According to Bergsma et al [2], the phenotypic variance for such a model can be calculated as 160 The TBV is the heritable effect of an individual on trait values in the population, which is the sum of its direct genetic effect rding to Bijma et al [1] for traits affected by heritable social effects, the variance of TBV s the total heritable variation that is exploitable for selection. The TBV of the i th animal is s follows: BV is the heritable effect of an individual on trait values in the population, which is the sum ct genetic effect ( , ) on its own phenotype and its SGE ( , ) on the phenotypes of its n -1 tes. Bijma et al [1] also stated that the total heritable variance determines the population's in response to selection and can be expressed as: rding to Bergsma et al [2], the phenotypic variance for such a model can be calculated as on its own phenotype and its ording to Bijma et al [1] for traits affected by heritable social effects, the variance of TBV ts the total heritable variation that is exploitable for selection. The TBV of the i th animal is as follows: TBV is the heritable effect of an individual on trait values in the population, which is the sum ect genetic effect ( , ) on its own phenotype and its SGE ( , ) on the phenotypes of its n -1 ates. Bijma et al [1] also stated that the total heritable variance determines the population's l in response to selection and can be expressed as: ording to Bergsma et al [2], the phenotypic variance for such a model can be calculated as on the phenotypes of its n -1 group mates. Bijma et al [1] also stated that the total heritable variance determines the population's potential in response to selection and can be expressed as: According to Bijma et al [1] for traits affected by heritable social effects, the variance of TBV 147 represents the total heritable variation that is exploitable for selection. The TBV of the i th animal is 148 defined as follows: 149 The TBV is the heritable effect of an individual on trait values in the population, which is the sum 153 of its direct genetic effect ( , ) on its own phenotype and its SGE ( , ) on the phenotypes of its n -1 154 group mates. Bijma et al [1] also stated that the total heritable variance determines the population's 155 potential in response to selection and can be expressed as: 156 According to Bergsma et al [2], the phenotypic variance for such a model can be calculated as 160 follows: 161 162 According to Bergsma et al [2], the phenotypic variance for such a model can be calculated as follows: where n indicates the average size of social groups. The total heritable var 165 to phenotypic variance [2] as follows: 166 Estimation using single-step method: The relationship matrix H, in 170 the relationship among genotyped and nongenotyped animals. The inv 171 simple in structure [15,16] and can be given as: 172 where A22 is the matrix for genotyped animals only (a submatrix der 176 relationship matrix, A and G is the relationship matrix among individuals 177 The G matrix was constructed according to VanRaden [4]. Both A22 and 178 combined. Thus, overall, the two matrices represented similar diagonals 179 write it in full form.) for a single step mainly differed from the pedigree-b 180 τ (0.95 G+0.05 A22) -1 -ω A22 -1 , given to the genotyped animals [17,18]. 181 proportion of polygenic variances that were unexplained by markers. The 182 size of the genomic and pedigree relationships, respectively. The weig 183 between 0.1 and 1.0, whereas τ was fixed at 1. The models including gen 184 as ω1.0, ω0.9, ω0.8, ω0.7, ω0.6, ω0.5, ω0.4, ω0.3, ω0.2, and ω0.1, in accordance w 185 ω, whereas a model with pedigree information only is denoted as PED in 186 Validation process: Accuracy of breeding value was calculated in tw 187 accuracy [3] and cross validation [6]). The last 2 years were masked 188 where n indicates the average size of social groups. The total heritable variance can be expressed relative to phenotypic variance [2] as follows:

164
where n indicates the average size of social groups. The total heritable v 165 to phenotypic variance [2] as follows: 166 Estimation using single-step method: The relationship matrix H, i 170 the relationship among genotyped and nongenotyped animals. The in 171 simple in structure [15,16] and can be given as: 172 where A22 is the matrix for genotyped animals only (a submatrix d 176 relationship matrix, A and G is the relationship matrix among individua 177 The G matrix was constructed according to VanRaden [4]. Both A22 an 178 combined. Thus, overall, the two matrices represented similar diagon 179 write it in full form.) for a single step mainly differed from the pedigree 180 τ (0.95 G+0.05 A22) -1 -ω A22 -1 , given to the genotyped animals [17,18 181 proportion of polygenic variances that were unexplained by markers. T 182 size of the genomic and pedigree relationships, respectively. The we 183 between 0.1 and 1.0, whereas τ was fixed at 1. The models including g 184 as ω1.0, ω0.9, ω0.8, ω0.7, ω0.6, ω0.5, ω0.4, ω0.3, ω0.2, and ω0.1, in accordance 185 ω, whereas a model with pedigree information only is denoted as PED 186 Validation process: Accuracy of breeding value was calculated in 187 accuracy [3] and cross validation [6]). The last 2 years were maske 188 Estimation using single-step method: The relationship matrix H, in single-step evaluation, defines the relationship among genotyped and nongenotyped animals. The inverse of the H matrix is rather simple in structure [15,16] and can be given as: where n indicates the average size of social groups. The total heritable variance 165 to phenotypic variance [2] as follows: 166 Estimation using single-step method: The relationship matrix H, in single 170 the relationship among genotyped and nongenotyped animals. The inverse o 171 simple in structure [15,16] and can be given as: 172 where A22 is the matrix for genotyped animals only (a submatrix derived 176 relationship matrix, A and G is the relationship matrix among individuals based 177 The G matrix was constructed according to VanRaden [4]. Both A22 and G ma 178 combined. Thus, overall, the two matrices represented similar diagonals. Ho 179 write it in full form.) for a single step mainly differed from the pedigree-based 180 τ (0.95 G+0.05 A22) -1 -ω A22 -1 , given to the genotyped animals [17,18]. The 181 proportion of polygenic variances that were unexplained by markers. The para 182 size of the genomic and pedigree relationships, respectively. The weights f 183 between 0.1 and 1.0, whereas τ was fixed at 1. The models including genomic 184 as ω1.0, ω0.9, ω0.8, ω0.7, ω0.6, ω0.5, ω0.4, ω0.3, ω0.2, and ω0.1, in accordance with th 185 ω, whereas a model with pedigree information only is denoted as PED in later 186 Validation process: Accuracy of breeding value was calculated in two di 187 accuracy [3] and cross validation [6]). The last 2 years were masked as th 188 where A 22 is the matrix for genotyped animals only (a submatrix derived from the pedigree-based relationship matrix, A and G is the relationship matrix among individuals based on genomic information. The G matrix was constructed according to VanRaden [4]. Both A 22 and G matrices were subsequently combined. Thus, overall, the two matrices represented similar diagonals. However, the mixed model equations for a single step mainly differed from the pedigree-based model by a matrix block, τ (0.95 G+0.05 A 22 ) -1 -ω A 22 -1 , given to the genotyped animals [17,18]. The constant ω represents the proportion of polygenic variances that were unexplained by markers. The parameters τ and ω scaled the size of the genomic and pedigree relationships, respectively. The weights for the ω parameter were between 0.1 and 1.0, whereas τ was fixed at 1. The models including genomic information are denoted as ω 1.0 , ω 0.9 , ω 0.8 , ω 0.7 , ω 0.6 , ω 0.5 , ω 0.4 , ω 0.3 , ω 0.2 , and ω 0.1 , in accordance with their values of the constant ω, whereas a model with pedigree information only is denoted as PED in later sections.
Validation process: Accuracy of breeding value was calculated in two different ways (theoretical accuracy [3] and cross validation [6]). The last 2 years were masked as the validation data set and predictions were made using the first 9 years as the training data set. The validation data set for LR and YS contained 10% and 8% of the observations, respectively. The theoretical accuracy of the estimated breeding value for the ith individual with the mth model was calculated as follows: where PEV is the prediction error variance of its breeding value, F is the inbreeding coefficient of an individual as computed from the pedigree, and σ 2 is the additive genetic variance of the model. We also calculated correlation between cor-rected phenotype and the combined breeding value (CBV) for the validation pigs. Accuracy was defined as where PEV is the prediction error variance of its breeding value, F is the inbreeding coefficient of an 195 individual as computed from the pedigree, and 2 is the additive genetic variance of the model. We also 196 calculated correlation between corrected phenotype and the combined breeding value (CBV) for the 197

Model fitness 207
The variances, covariances, and various model parameters obtained from the studied models for LR and 208 YSs are presented in Tables 1 and 2, respectively. The Akaike information criterion (AIC) parameter of 209 the pedigree-classical model was higher than the pedigree-social model in both breeds. This result 210 showed that model including SGE fitted the data significantly better than a classical animal model. In 211 addition, AIC parameter of the pedigree-social model was the highest in both breeds compared with 212 those of all ssGBLUP methods. The AIC as an indicator of the goodness fit of the models indicates that 213 the ssGBLUP models performed better in general, which was as expected due to the addition of genomic 214 information alongside the pedigree relationship. This is a feasible approach with a single-step method 215 where CBV is the sum of pig's own direct breeding value and SBVs of pen mates, y c is corrected ADG for fixed effects.

Model fitness
The variances, covariances, and various model parameters obtained from the studied models for LR and YSs are pre-sented in Tables 1 and 2, respectively. The Akaike information criterion (AIC) parameter of the pedigree-classical model was higher than the pedigree-social model in both breeds. This result showed that model including SGE fitted the data significantly better than a classical animal model. In addition, AIC parameter of the pedigree-social model was the highest in both breeds compared with those of all ssGBLUP methods. The AIC as an indicator of the goodness fit of the models indicates that the ssGBLUP models performed better in general, which was as expected due to the addition of genomic information alongside the pedigree relationship. This is a feasible approach with a single-step method as it provides more accurate predictions for both genotyped and nongenotyped , phenotypic variance; T 2 , total heritability for model including social genetic effects; ΔAIC, change in Akaike's information criteria from the best (minimum) model; PED classic , the classic model with pedigree relationships only; PED social , the social model with pedigree relationships only; ω xx , the model with weighted  , covariance between direct and social genetic effects; , phenotypic variance; T 2 , total heritability for model including social genetic effects; ΔAIC, change in Akaike's information criteria from the best (minimum) model; PED classic , the classic model with pedigree relationships only; PED social , the social model with pedigree relationships only; ω xx , the model with weighted animals [6,11,18,19]. Therefore, a ssGBLUP analysis including SGE in the model would be a better choice for the prediction of traits in pigs. However, differences were observed among the various model fits with different scaling factors in the single-step methods. Among the ssGBLUP models, the model with ω of 1.0 showed the worst fit, regardless of the breed. The best fitting models in this study were those with ω 0.6 and ω 0.5 in LR (Table 1) and YR (Table 2), respectively, as indicated by them having the lowest AIC estimates. The model AIC value increased with any level of ω other than 0.6 and 0.5 in LR and YS, respectively, indicating the worse fit of those models. Our results obtained through testing different levels of ω (0.1 to 1.0) indicate that a ssGBLUP method essentially relies on tuning the scales of matrices related to pedigree and genotype relationships, which will lead to less biased model estimates [6,15,17,[20][21][22]. This study strongly coincides with many previous reports in that the choices of appropriate levels of constants (τ and ω) are rather arbitrary, and are to be determined through fine tuning. For instance, Misztal et al [17] reported the best combination of τ = 1.5 and ω = 0.6 in their study on dairy cattle. Another study in dairy cattle by Harris et al [23] also used both parameters at levels as low as 0.5. Likewise, Koivula et al [20] reported using various combinations of A and G matrices to find the best option in their study. In pig, Christensen et al [6] suggested a single-step method that is adjusted for the genomic relationship matrix. In another study by Misztal et al [24], a model with slower convergence at ω values greater than 1 was reported, as their H matrix was found to be nonpositive at higher values of this constant. In this context, it is crucial to find appropriate scaling parameters that will ensure better accuracy, lower bias, and easier convergence. It is also important to consider appropriate weights for relationship matrices through scaling factors as any smaller constant for ω is likely to decrease the emphasis on the genomic relationships and increase the importance of the pedigree relationships [20]. This might explain our estimates obtained with levels of ω lower than those in best fit models, where model estimates might have been associated with some biases due to the lower weight in genotyped animals through their genomic relationships.

Genetic parameters
The genetic variances and total heritability estimates (T 2 ) were mostly higher with ssGBLUP than in the pedigree-based analysis (Tables 1, 2). Among the ssGBLUP models, the genetic variances and T 2 were increased by decreasing ω in both breeds. Therefore, the T 2 of ω 0.1 model was the highest in both breeds (LR, 0.64; YS, 0.88). The best single-step models (ΔAIC = 0) showed larger estimates of direct and social variances than pedigree-based methods, and thus also larger covariance estimates, resulting in higher total heritability estimates with those models. The T 2 estimates with the best fitting models were 0.54 and 0.80 in LR and YS, respectively. They were also greater than those of the pedigree-based analysis method by 0.13 and 0.22 in these two breeds, respectively. However, our T 2 estimates for LR with the ω 10 model coincided strongly with those of Bergsma et al [25] and Duijvesteijn [3]. Comparing the breeds, both direct and social genetic contributions were higher in YS than in LR, so their T 2 estimates also exhibited the same trend. Note that even when the social variance is markedly smaller than direct genetic variance, its contribution to increased by decreasing ω in both breeds. Therefore, the T of ω0.1 model was the 245 (LR, 0.64; YS, 0.88). The best single-step models (ΔAIC = 0) showed larger e 246 social variances than pedigree-based methods, and thus also larger covariance e 247 higher total heritability estimates with those models. The T 2 estimates with the be 248 0.54 and 0.80 in LR and YS, respectively. They were also greater than those of the 249 and 0.22 in these two breeds, respectively. However, our T 2 estimates for LR 250 coincided strongly with those of Bergsma et al [25] and Duijvesteijn [3]. Comp 251 direct and social genetic contributions were higher in YS than in LR, so their T 2 es 252 the same trend. Note that even when the social variance is markedly smaller than d 253 its contribution to 2 TBV σ would be substantial due to the factor (n-1) 2 , especially 254 large, as was the case with YS. The lower T 2 estimates in LR could also be due to 255 litter effects and negative covariances between direct and social effects. Accordi 256 the positive covariance between direct and social genetic variances is likely to incr 257 variation, which coincides well with the present study. The correlation coefficien 258 SBV were somewhat weaker in LR (−0.05 to 0.09) than in YS (0.28 to 31). Some e 259 also stated somewhat similar correlations, mostly positive but not significant. In t 260 correlation in YS could indicate that their pen mates might also have stimulated 261 that SBV is passed on to pen mates, the positive genetic correlation between the 262 effects indicates that pigs with a high DBV will also have a high SBV. In other w 263 our study may show more positive responses to selection for social interactio 264 Nonetheless, breed differences for social interactions are not unlikely. Bergsma e 265 the absence of conflict between an individual's own growth and mate growth mi 266 of neutral or slightly cooperative social interactions. For the negative or neutral 267 LR pigs in this study, it is possible that these pigs were in less competition for fo 268 amount of space that each of them had on average (3 to 8 pigs/9 m 2 pen) was low 269 to 10 pigs/9 m 2 pen). 270 would be substantial due to the factor (n-1) 2 , especially when group sizes are large, as was the case with YS. The lower T 2 estimates in LR could also be due to the larger nongenetic litter effects and negative covariances between direct and social effects. According to Bijma et al [1], the positive covariance between direct and social genetic variances is likely to increase the total heritable variation, which coincides well with the present study. The correlation coefficients between DBV and SBV were somewhat weaker in LR (−0.05 to 0.09) than in YS (0.28 to 31). Some earlier reports [2,3,26] also stated somewhat similar correlations, mostly positive but not significant. In this study, the positive correlation in YS could indicate that their pen mates might also have stimulated a greater ADG. Given that SBV is passed on to pen mates, the positive genetic correlation between the direct and associative effects indicates that pigs with a high DBV will also have a high SBV. In other words, the YR pigs in our study may show more positive responses to selection for social interactions than the LR pigs. Nonetheless, breed differences for social interactions are not unlikely. Bergsma et al [2] suggested that the absence of conflict between an individual's own growth and mate growth might be a consequence of neutral or slightly cooperative social interactions. For the negative or neutral associative effects in LR pigs in this study, it is possible that these pigs were in less competition for food and growth as the amount of space that each of them had on average (3 to 8 pigs/9 m 2 pen) was lower than that of YR (4 to 10 pigs/9 m 2 pen). Table 3 illustrates the accuracy for breeding values obtained with different models. The levels of theoretical accuracy obtained for DBV with PED classic and PED social models in each breed were same and also the lowest among the different models (LR, 0.52; YS, 0.55). The ω 1.0 models also performed poorly in DBV prediction (LR, 0.55; YS, 0.58). Among the ssGBLUP models, the theoretical accuracy of DBV was increased by decreasing ¬ω in both breeds (LR, 0.55 to 0.66; YS, 0.58 to 0.64). The best fit models based on AIC exhibited an increase of accuracy by 5% to 8% compared with the ω 1.0 models in both breeds. The ranges of SBV accuracies with the PED social in LR and in YS were 0.16 and 0.31, respectively. Simi-lar to DBV, both PED social and ω 1.0 models performed poorly in SBV prediction. However, unlike the DBVs from the single-step methods, the best fitting models exhibited notable increases in SBV accuracies by 39% (LR) and 19% (YS) with ω 0.6 and ω 0.5 , respectively, compared with each of the breed's worst fit (ω 1.0 ) model. In cross validation, the correlations between CBV and corrected phenotype were also mostly higher with ssGBLUP than in the pedigree-based analysis. However, there were little differences among the ssGBLUP models. The ranges of correlations between CBV and corrected phenotype in LR and YS were 0.31 to 0.33 and 0.21 to 0.22, respectively. The correlative prediction methods showed more variability in terms of ranking of models across traits and replicates so care should be taken interpreting these results with small sample sizes [27]. Putz et al [27] also suggested that for withinbreed selection, theoretical accuracy using the prediction error variance was consistent and accurate in ssGBLUP. However, selection programmes should be careful which validation method they choose and should inspect multiple methods if possible [27]. Therefore, to minimize AIC and to increase theoretical accuracy in this study, the optimal values of ω in LR and YS were 0.6 and 0.5, respectively. Martini [28] reported that increasing τ or decreasing ω may mainly decrease inflation by decreasing the variance of the estimated breeding values, which indicate the possibility of further adjustment of τ in the H matrix.

Prospect of social genetic effects
The phenotypic variability of some traits that are expressed in the social environment could be significantly influenced by SGEs. Earlier reports on such traits, for instance, social dominance or aggressiveness, also suggested that SGEs can substantially influence total phenotypic variability [29][30][31][32].
The importance of SGEs can also be recognized from many previous reports [33][34][35], which showed that the higher SBV and some desirable characteristics in pigs i.e., fearlessness, stress-tolerance are associated to each other. These characteristics in commercial pig production are particularly beneficial for ease of farm management. For this reason, appropriate attention to such socially influenced traits alongside the pig population structure is vital when genomic selection is considered [36]. Certain strategies could also be applied during selection to achieve a high SBV for a desirable trait. One such approach is to select animals with higher TBVs to improve group performance, especially for growth traits [1][2][3]. Direct selection of pigs for SBV could be another strategy to alter their social behavior. Earlier evidence suggested that high SBV, due to apathy of the animal, could reduce negative social effects on the growth of others [37][38][39]. Moreover, the inclusion of SNP effects with SGEs in the model could provide better predictions [40]. For successful realization of TBV, it is also important to consider social environments, such as the mixing method of suckling piglets [41].

CONCLUSION
For SGEs, our study showed greater improvement in parameter estimates through ssGBLUP over the traditional pedigree-based method. Both breeds differed to some extent for their estimated parameters. The value of ω used for adjusting A 22 matrix also differed between the best fitting models for the LR and YS breeds. But it was clear that the models with ω of 1.0 in the H matrix were the worst fitting. Our study also indicated the possibility of further adjustment of other model parameters (α, β, τ) in the H matrix to reduce inflation of the estimated breeding values. Our results DBV acc , the theoretical accuracy of direct breeding value; SBV acc , the theoretical accuracy of social breeding value; Cor, the correlation between corrected phenotype and the combined breeding value (CBV); PED classic , the classic model with pedigree relationships only; PED social , the social model with pedigree relationships only; ω xx , the model with weighted Estimates of variances, covariances, genetic parameters, and accuracies for different models in pigs , covariance between direct and social genetic effects; 2 , social genetic 2 , random group variance; 2 , random litter variance; 2 , random residual variance; r, correlation direct and social genetic effects; 2 , phenotypic variance; 2 , total heritability for model including etic effects; ΔAIC, change in Akaike's information criteria from the best (minimum) model; PEDclassic, c model with pedigree relationships only; PEDsocial, the social model with pedigree relationships only; odel with weighted 22 −1 matrix by different ω constants. matrix by different ω constants.