Predicting the rate of inbreeding in populations undergoing four-path selection on genomically enhanced breeding values

Objective A formula is needed that is practical for current livestock breeding methods and that predicts the approximate rate of inbreeding (ΔF) in populations where selection is performed according to four-path programs (sires to breed sons, sires to breed daughters, dams to breed sons, and dams to breed daughters). The formula widely used to predict inbreeding neglects selection, we need to develop a new formula that can be applied with or without selection. Methods The core of the prediction is to incorporate the long-tern genetic influence of the selected parents in four-selection paths executed as sires to breed sons, sires to breed daughters, dams to breed sons, and dams to breed daughters. The rate of inbreeding was computed as the magnitude that is proportional to the sum of squared long-term genetic contributions of the parents of four-selection paths to the selected offspring. Results We developed a formula to predict the rate of inbreeding in populations undergoing four-path selection on genomically enhanced breeding values and with discrete generations. The new formula can be applied with or without selection. Neglecting the effects of selection led to underestimation of the rate of inbreeding by 40% to 45%. Conclusion The formula we developed here would be highly useful as a practical method for predicting the approximate rate of inbreeding (ΔF) in populations where selection is performed according to four-path programs.


INTRODUCTION
Deterministic predictions of response to multi-trait genomic selection in a single generation in a population with four-path programs, was developed [1,2]. That is, the selection paths in four-path programs are sires to breed sires (SS), sires to breed dams (SD), dams to breed sires (DS), and dams to breed dams (DD). However, when creating formulas for calculating the asymptotic response to index or single-trait selection in four-path selection programs rather than in a single generation, the initial genetic response in generation 0 overestimated the asymptotic response due to the decrease in equilibrium genetic variance from generation 0 onwards [3]. Consequently, to safeguard the genetic variation of the population over the long term, the rate of inbreeding needs to be restricted to an acceptable level. Therefore, one needs to know the expected rate of inbreeding as well as the equilibrium genetic response before choosing a breeding scheme.
A population with discrete generations under mass selection in a four-path selection program is modeled to predict the rate of inbreeding in the long term. When sires in the SS path are used with constant selection intensity and in equal number throughout the 138 in the linear model is: 6 contribution of individual i in category SS conditional on its select 114 selection is the genomically enhanced breeding value [GEBV]), an 115 the expected lifetime long-term genetic contributions of individua 116 DD, respectively. Furthermore, � �δ �� δ �� δ �� δ �� �, where δ � 117 deviations of the variance of family size from independent Poiss 118 offspring from sires in SS; δ �� , δ �� , and δ �� are corrections for 119 the family size from independent Poisson variances in the selected 120 DS, and DD, respectively. 121 The selective advantage of the i th sire in SS ( �,�� � and in SD (  where α x is the expected contribution of an average parent in x, and β x is the regression coefficient of the contribution of i on its selective advantage where �,�� ��� �� is the breeding value of dam i in DS and DD, respectively; �,�� ��� �� is 139 the breeding value of a sire mated to the i th dam in DS and DD, respectively; the sires mated to 140 the i th dam in DS belong to the SS category; and the sires mated to the i th dam in DD belong to 141 the SD category. 142 Expected contributions ( �,��,��,��,�� �� � are predicted by linear regression on the selective 143 advantage. That is, 144 where α � is the expected contribution of an average parent in , and � is the regression 148 coefficient of the contribution of i on its selective advantage ( �,� �. In addition, α � can be 149 obtained according to Woolliams et al [9]: 150 151 � � , 152 . In addition, α x can be obtained according to Woolliams et al [9]: where �,�� ��� �� is the breeding value of dam i in DS and DD, 139 the breeding value of a sire mated to the i th dam in DS and DD, res 140 the i th dam in DS belong to the SS category; and the sires mated to 141 the SD category. 142 Expected contributions ( �,��,��,��,�� �� � are predicted by linea 143 advantage. That is, 144 where is a 4×4 matrix representing the parental origin of genes of selected offspring in the 154 order of SS, SD, DS, and DD category, i.e., representing rows offspring and columns parental 155 categories. That is, 156 (1) 171 note that the right hand side of (1) is unaffected by the number of parents, so that � is 173 inversely proportional to the number of parents (that is, is a 4×4 matrix of regression coefficients with �� being the 175 regression coefficient of �,� of a selected offspring i of category x (SS, SD, DS, DD) on �,� 176 of its parent j of category y (SS, SD, DS, DD). For example, ��,�� is the regression coefficient 177 of �,�� of a selected offspring i of SD on �,�� of its parent j of SS. Given that SS is the sires 178 to breed sons category, we have non-zero elements, ��,�� and ��,�� , in as elements (1,1) 179 and (2, 1), respectively. In the same way, since SD is the sires to breed daughters category, we 180 have non-zero elements, ��,�� and ��,�� , in as elements (3,2) and (4,2), respectively. 181 Because DS is the dams to breed sons category, we have non-zero elements, ��,�� and 182 is the left eigenvector of G with eigenvalue 1; the left eigenvector is obtained according to Bijma and Woolliams [11] and is equal to (0.25 0.25 0.25 0.25).
Therefore where is the left eigenvector of G with eigenvalue 1; the left eigenvector is obtained 164 according to [11] (1) 171 note that the right hand side of (1) is unaffected by the number of parents, so that � is 173 inversely proportional to the number of parents (that is, is a 4×4 matrix of regression coefficients with �� being the 175 regression coefficient of �,� of a selected offspring i of category x (SS, SD, DS, DD) on �,� 176 of its parent j of category y (SS, SD, DS, DD). For example, ��,�� is the regression coefficient 177 of �,�� of a selected offspring i of SD on �,�� of its parent j of SS. Given that SS is the sires 178 to breed sons category, we have non-zero elements, ��,�� and ��,�� , in as elements (1,1) 179 and (2, 1), respectively. In the same way, since SD is the sires to breed daughters category, we 180 have non-zero elements, ��,�� and ��,�� , in as elements (3,2) and ( (1) 171 172 note that the right hand side of (1) is unaffected by the number of parents, so that � is 173 inversely proportional to the number of parents (that is, is a 4×4 matrix of regression coefficients with �� being the 175 regression coefficient of �,� of a selected offspring i of category x (SS, SD, DS, DD) on �,� 176 of its parent j of category y (SS, SD, DS, DD). For example, ��,�� is the regression coefficient 177 of �,�� of a selected offspring i of SD on �,�� of its parent j of SS. Given that SS is the sires 178 to breed sons category, we have non-zero elements, ��,�� and ��,�� , in as elements (1,1) 179 and (2, 1), respectively. In the same way, since SD is the sires to breed daughters category, we 180 have non-zero elements, ��,�� and ��,�� , in as elements (3,2) and ( (1) 171 172 note that the right hand side of (1) is unaffected by the number of parents, so that � is 173 inversely proportional to the number of parents (that is, is a 4×4 matrix of regression coefficients with �� being the 175 regression coefficient of �,� of a selected offspring i of category x (SS, SD, DS, DD) on �,� 176 of its parent j of category y (SS, SD, DS, DD). For example, ��,�� is the regression coefficient 177 of �,�� of a selected offspring i of SD on �,�� of its parent j of SS. Given that SS is the sires 178 to breed sons category, we have non-zero elements, ��,�� and ��,�� , in as elements (1,1) 179 and (2, 1), respectively. In the same way, since SD is the sires to breed daughters category, we 180 have non-zero elements, ��,�� and ��,�� , in as elements (3,2) and (4,2), respectively. lliams et al [9]: (1) 171 172 note that the right hand side of (1) is unaffected by the number of parents, so that � is 173 inversely proportional to the number of parents (that is, is a 4×4 matrix of regression coefficients with �� being the 175 regression coefficient of �,� of a selected offspring i of category x (SS, SD, DS, DD) on �,� 176 of its parent j of category y (SS, SD, DS, DD). For example, ��,�� is the regression coefficient 177 of �,�� of a selected offspring i of SD on �,�� of its parent j of SS. Given that SS is the sires 178 to breed sons category, we have non-zero elements, ��,�� and ��,�� , in as elements (1,1) 179 and (2, 1), respectively. In the same way, since SD is the sires to breed daughters category, we 180 have non-zero elements, ��,�� and ��,�� , in as elements (3,2) and ( ht hand side of (1) is unaffected by the number of parents, so that � is tional to the number of parents (that is, is a 4×4 matrix of regression coefficients with �� being the icient of �,� of a selected offspring i of category x (SS, SD, DS, DD) on �,� category y (SS, SD, DS, DD). For example, ��,�� is the regression coefficient ected offspring i of SD on �,�� of its parent j of SS. Given that SS is the sires tegory, we have non-zero elements, ��,�� and ��,�� , in as elements (1,1) ctively. In the same way, since SD is the sires to breed daughters category, we lements, ��,�� and ��,�� , in as elements (3,2) and (4,2), respectively.
the dams to breed sons category, we have non-zero elements, ��,�� and elements (1,3) and (2,3), respectively. And given that DD is the dams to breed of its parent j of SS. Given that SS is the sires to breed sons category, we have non-zero elements, with eigenvalue 1; the left eigenvector is obtained 0.25 0.25).

�.
g to Woolliams et al [9]: affected by the number of parents, so that � is ents (that is, . to Woolliams et al [9]: affected by the number of parents, so that � is nts (that is, ix of regression coefficients with �� being the offspring i of category x (SS, SD, DS, DD) on �,� D). For example, ��,�� is the regression coefficient �,�� of its parent j of SS. Given that SS is the sires lements, ��,�� and ��,�� , in as elements (1,1) ince SD is the sires to breed daughters category, we �� , in as elements (3,2) and (4,2), respectively. category, we have non-zero elements, ��,�� and espectively. And given that DD is the dams to breed , in Π as elements (1,1) and (2,1), respectively. In the same way, since SD is the sires to breed daughters category, we have non-zero elements, 163 where is the left eigenvector of G with eigenvalue 1; the left eigenvector is obtained 164 according to [11] (1) 171 172 note that the right hand side of (1) is unaffected by the number of parents, so that � is 173 inversely proportional to the number of parents (that is, is a 4×4 matrix of regression coefficients with �� being the 175 regression coefficient of �,� of a selected offspring i of category x (SS, SD, DS, DD) on �,� 176 of its parent j of category y (SS, SD, DS, DD). For example, ��,�� is the regression coefficient 177 of �,�� of a selected offspring i of SD on �,�� of its parent j of SS. Given that SS is the sires 178 to breed sons category, we have non-zero elements, ��,�� and ��,�� , in as elements (1,1) 179 and (2, 1), respectively. In the same way, since SD is the sires to breed daughters category, we 180 have non-zero elements, ��,�� and ��,�� , in as elements (3,2) and (4,2), respectively. 181 Because DS is the dams to breed sons category, we have non-zero elements, �� (1) 171 172 note that the right hand side of (1) is unaffected by the number of parents, so that � is 173 inversely proportional to the number of parents (that is, is a 4×4 matrix of regression coefficients with �� being the 175 regression coefficient of �,� of a selected offspring i of category x (SS, SD, DS, DD) on �,� 176 of its parent j of category y (SS, SD, DS, DD). For example, ��,�� is the regression coefficient 177 of �,�� of a selected offspring i of SD on �,�� of its parent j of SS. Given that SS is the sires 178 to breed sons category, we have non-zero elements, ��,�� and ��,�� , in as elements (1,1) 179 and (2, 1), respectively. In the same way, since SD is the sires to breed daughters category, we 180 have non-zero elements, ��,�� and ��,�� , in as elements (3,2) and (4,2), respectively. 181 Because DS is the dams to breed sons category, we have non-zero elements, ��,�� and 182 ��,�� , in as elements (1,3) and (2,3), respectively. And given that DD is the dams to breed 183 daughters category, we have non-zero elements, ��,�� and ��,�� , in as elements (3,4) and 184 , in Π as elements (3,2) and (4,2), respectively. Because DS is the dams to breed sons category, we have non-zero elements, envector is obtained (1) 171 172 note that the right hand side of (1) is unaffected by the number of parents, so that � is 173 inversely proportional to the number of parents (that is, is a 4×4 matrix of regression coefficients with �� being the 175 regression coefficient of �,� of a selected offspring i of category x (SS, SD, DS, DD) on �,� 176 of its parent j of category y (SS, SD, DS, DD). For example, ��,�� is the regression coefficient 177 of �,�� of a selected offspring i of SD on �,�� of its parent j of SS. Given that SS is the sires 178 to breed sons category, we have non-zero elements, ��,�� and ��,�� , in as elements (1,1) 179 and (2, 1), respectively. In the same way, since SD is the sires to breed daughters category, we 180 have non-zero elements, ��,�� and ��,�� , in as elements (3,2) and (4,2), respectively. 181 Because DS is the dams to breed sons category, we have non-zero elements, ��,�� and 182 ��,�� , in as elements (1,3) and (2,3), respectively. And given that DD is the dams to breed 183 daughters category, we have non-zero elements, ��,�� and ��,�� , in as elements (3,4) and 184 , in Π as elements (1,3) and (2,3), respec-tively. And given that DD is the dams to breed daughters category, we have non-zero elements, 8 have non-zero elements, ��,�� and ��,�� , in as elements (3,2) and (4 181 Because DS is the dams to breed sons category, we have non-zero elem 182 ��,�� , in as elements (1,3) and (2,3), respectively. And given that DD is t 183 daughters category, we have non-zero elements, ��,�� and ��,�� , in as e 184 and 8 have non-zero elements, ��,�� and ��,�� , in as elements (3,2) and (4 181 Because DS is the dams to breed sons category, we have non-zero elem 182 ��,�� , in as elements (1,3) and (2,3), respectively. And given that DD is t 183 daughters category, we have non-zero elements, ��,�� and ��,�� , in as e 184 , in Π as elements (3,4) and (4,4), respectively.
In addition, Λ is a 4×4 matrix of regression coefficients, with λ xy being the regression coefficient of the number of selected offspring of category x on 9 (4,4), respectively. 185 In addition, is a 4×4 matrix of regression coefficients, with �� being the regression 186 coefficient of the number of selected offspring of category x on �,� of its parent j of category y. 187 In the same way as , we have non-zero elements, ��,�� and ��,�� , ��,�� and 188 ��,�� , ��,�� and ��,�� , and ��,�� and ��,�� in as elements (1,1) and (2,1), (3,2) and 189 (4,2), (1,3) and (2,3), and (3,4) and (4,4), respectively. Consequently, 190 194 195 representing rows as offspring and columns as parental categories. 196 In our current study, elements in matrices and were calculated from Woolliams et al 197 [9] and Bijma and Woolliams [11], as outlined in Appendices A and B. 198 The sires in the SS category are included among the sires in SD category. That is, the sires in 199 the SS category are selected not only to breed sons but as sires in the SD category to breed 200 daughters. Similarly the dams in the DS category are included among the dams in the DD 201 category. The dams in the DS category are selected not only to breed sons but as dams in the 202 DD category to breed daughters. Therefore, after applying the procedure of Bijma and 203 Woolliams [6], the number of sires in SD is larger than that of sires in SS, and the number of 204 dams in DD is larger than that of dams in DS. Therefore, E (ΔF) = � � �� � � � ��, where 205 of its parent j of category y. In the same way as Π, we have non-zero elements, 9 (4,4), respectively. 185 In addition, is a 4×4 matrix of regression coefficients, with �� being 186 coefficient of the number of selected offspring of category x on �,� of its paren 187 In the same way as , we have non-zero elements, ��,�� and ��,� 188 ��,�� , ��,�� and ��,�� , and ��,�� and ��,�� in as elements (1,1) and 189 (4,2), (1,3) and (2,3), and (3,4) and (4,4), respectively. Consequently, 190 194 195 representing rows as offspring and columns as parental categories. 196 In our current study, elements in matrices and were calculated from 197 [9] and Bijma and Woolliams [11], as outlined in Appendices A and B. 198 The sires in the SS category are included among the sires in SD category. Tha 199 the SS category are selected not only to breed sons but as sires in the SD ca 200 daughters. Similarly the dams in the DS category are included among the d 201 category. The dams in the DS category are selected not only to breed sons but 202 DD category to breed daughters. Therefore, after applying the procedure 203 Woolliams [6], the number of sires in SD is larger than that of sires in SS, and 204 dams in DD is larger than that of dams in DS. Therefore, E (ΔF) = � � �� � � � �� 205 and 9 (4,4), respectively.
In addition, is a 4×4 matrix of regression coefficients, with �� being the regression coefficient of the number of selected offspring of category x on �,� of its parent j of category y.
In our current study, elements in matrices and were calculated from Woolliams et al [9] and Bijma and Woolliams [11], as outlined in Appendices A and B.
The sires in the SS category are included among the sires in SD category. That is, the sires in the SS category are selected not only to breed sons but as sires in the SD category to breed daughters. Similarly the dams in the DS category are included among the dams in the DD category. The dams in the DS category are selected not only to breed sons but as dams in the DD category to breed daughters. Therefore, after applying the procedure of Bijma and Woolliams [6], the number of sires in SD is larger than that of sires in SS, and the number of dams in DD is larger than that of dams in DS. Therefore, (4,4), respectively.
In addition, is a 4×4 matrix of regression coefficients, with �� being the regression coefficient of the number of selected offspring of category x on �,� of its parent j of category y.
In our current study, elements in matrices and were calculated from Woolliams et al [9] and Bijma and Woolliams [11], as outlined in Appendices A and B.
The sires in the SS category are included among the sires in SD category. That is, the sires in the SS category are selected not only to breed sons but as sires in the SD category to breed daughters. Similarly the dams in the DS category are included among the dams in the DD category. The dams in the DS category are selected not only to breed sons but as dams in the DD category to breed daughters. Therefore, after applying the procedure of Bijma and Woolliams [6], the number of sires in SD is larger than that of sires in SS, and the number of dams in DD is larger than that of dams in DS. Therefore, E (ΔF) = In our current study, elements in matric 197 [9] and Bijma and Woolliams [11], as outlin 198 The sires in the SS category are include In our current study, elements in matri 197 [9] and Bijma and Woolliams [11], as outlin 198 The sires in the SS category are include In our current study, elements in matr 197 [9] and Bijma and Woolliams [11], as outl 198 The sires in the SS category are includ In addition, is a 4×4 matrix of regression coefficients, with �� being the regression 186 coefficient of the number of selected offspring of category x on �,� of its parent j of category y. 187 In the same way as , we have non-zero elements, ��,�� and ��,�� , ��,�� and 188 ��,�� , ��,�� and ��,�� , and ��,�� and ��,�� in as elements (1,1) and (2,1), (3,2) and 189 (4,2), (1,3) and (2,3), and (3,4) and (4,4), respectively. Consequently, 190 194 195 representing rows as offspring and columns as parental categories. 196 In our current study, elements in matrices and were calculated from Woolliams et al 197 [9] and Bijma and Woolliams [11], as outlined in Appendices A and B. 198 The sires in the SS category are included among the sires in SD category. That is, the sires in 199 the SS category are selected not only to breed sons but as sires in the SD category to breed 200 daughters. Similarly the dams in the DS category are included among the dams in the DD 201 category. The dams in the DS category are selected not only to breed sons but as dams in the 202 DD category to breed daughters. Therefore, after applying the procedure of Bijma and 203 Woolliams [6], the number of sires in SD is larger than that of sires in SS, and the number of 204 dams in DD is larger than that of dams in DS. Therefore, E (ΔF) = In addition, is a 4×4 matrix of regression coefficients, with �� being the regression 186 coefficient of the number of selected offspring of category x on �,� of its parent j of category y. 187 In the same way as , we have non-zero elements, ��,�� and ��,�� , ��,�� and 188 ��,�� , ��,�� and ��,�� , and ��,�� and ��,�� in as elements (1,1) and (2,1), (3,2) and 189 (4,2), (1,3) and (2,3), and (3,4) and (4,4), respectively. Consequently, 190 194 195 representing rows as offspring and columns as parental categories. 196 In our current study, elements in matrices and were calculated from Woolliams et al 197 [9] and Bijma and Woolliams [11], as outlined in Appendices A and B. 198 The sires in the SS category are included among the sires in SD category. That is, the sires in 199 the SS category are selected not only to breed sons but as sires in the SD category to breed 200 daughters. Similarly the dams in the DS category are included among the dams in the DD 201 category. The dams in the DS category are selected not only to breed sons but as dams in the 202 DD category to breed daughters. Therefore, after applying the procedure of Bijma and 203 Woolliams [6], the number of sires in SD is larger than that of sires in SS, and the number of 204 dams in DD is larger than that of dams in DS. Therefore, E (ΔF) = � � �� � � � ��, where 205 in Λ as elements (1,1) and (2,1), (3,2) and (4,2), (1,3) and (2,3), and (3,4) and (4,4), respectively. Consequently, 9 (4,4), respectively. 185 In addition, is a 4×4 matrix of regression coefficients, with 186 coefficient of the number of selected offspring of category x on �,� o 187 In the same way as , we have non-zero elements, ��,�� 188 ��,�� , ��,�� and ��,�� , and ��,�� and ��,�� in as elements 189 (4,2), (1,3) and (2,3), and (3,4) and (4,4), respectively. Consequently, 190 194 195 representing rows as offspring and columns as parental categories. 196 In our current study, elements in matrices and were calcula 197 [9] and Bijma and Woolliams [11], as outlined in Appendices A and B. In our current study, elements in matrices and were calcula 197 [9] and Bijma and Woolliams [11], as outlined in Appendices A and B. , representing rows as offspring and columns as parental categories.
In our current study, elements in matrices Π and Λ were calculated from Woolliams et al [9] and Bijma and Woolliams [11], as outlined in Appendices A and B.
The sires in the SS category are included among the sires in SD category. That is, the sires in the SS category are selected not only to breed sons but as sires in the SD category to breed daughters. Similarly the dams in the DS category are included among the dams in the DD category. The dams in the DS category are selected not only to breed sons but as dams in the DD category to breed daughters. Therefore, after applying the procedure of Bijma and Woolliams [6], the number of sires in SD is larger than that of sires in SS, and the number of dams in DD is larger than that of dams in DS. Therefore, 9 (4,4), respectively. 185 In addition, is a 4×4 matrix of regression coefficients, with �� being the regression 186 coefficient of the number of selected offspring of category x on �,� of its parent j of category y. 187 In the same way as , we have non-zero elements, ��,�� and ��,�� , ��,�� and 188 ��,�� , ��,�� and ��,�� , and ��,�� and ��,�� in as elements (1,1) and (2,1), (3,2) and 189 In our current study, elements in matrices and were calculated from Woolliams et al 197 [9] and Bijma and Woolliams [11], as outlined in Appendices A and B. 198 The sires in the SS category are included among the sires in SD category. That is, the sires in expectation with respect to the selective advantage,   General predictions of expected genetic contributions was developed using equilibrium genetic variances instead of second generation genetic variances [9]. Therefore, variances thereafter are the equilibrium reliability of GEBV in the male and female populations, respectively; and are variance reduction coefficients for offspring selection in SS, SD, DS, and DD, respectively. Note that covariances of mates between SS and SD and be-tween DS and DD are zero, because of random mating. General predictions of expected genetic contributions was developed using equilibrium genetic variances instead of second generation genetic variances [9]. Therefore, variances thereafter refer to those in equilibrium.
The accounting percentage derived from SS, SD, DS, and DD for the rate of inbreeding (ΔF) is obtained,

Correction of E (ΔF) from Poisson variances 261
The correction for deviations of the variance of the family size 262 variances in the selected offspring from SS, SD, DS, and DD parents, 263 can be approximated by Woolliams and Bijma [10]. 264 According to Woolliams and Bijma [10],

Correction of E (ΔF) from Poisson variances 261
The correction for deviations of the variance of the family size 262 variances in the selected offspring from SS, SD, DS, and DD parents, 263 can be approximated by Woolliams and Bijma [10]. 264 According to Woolliams and Bijma [10], When the effect of selection on inbreeding is ignored, i.e., � � 0 , E (ΔF) 256 This result is in agreement with the formula from Gowe et al [8], which likewise neglects 258 the effects of selection on ΔF. 259 260

Example applications of the formula
To demonstrate the application of our formula, we assumed only two quantitative traits: trait 1 was assumed to be moderately heritable, with h 2 = 0.3, whereas trait 2 was assumed to have low heritability, with h 2 = 0.1. These traits are selected as single traits expressed as GEBV. Furthermore, we assumed an aggregate genotype as a linear combination of genetic values, each weighted by the relative economic weights, which was expressed as To demonstrate the application of our formula, we assumed only two quantitative 282 was assumed to be moderately heritable, with h 2 = 0.3, whereas trait 2 was assumed 283 heritability, with ℎ � � 0.1. These traits are selected as single traits expressed 284 Furthermore, we assumed an aggregate genotype as a linear combination of genetic 285 weighted by the relative economic weights, which was expressed as � � � � � , 286 the true genetic value for trait i, � is the relative economic weight for trait i, and 287 correlation between traits 1 and 2 was assumed as 0.4. Index selection was perform 288 � � � � � , that is, breeding goal (H), under the assumption that the relative econ 289 between traits 1 and 2 is 1: 1. Breeding value (A) was defined as described e 290 Methods; for example, the breeding value of sire i in SS was defined as �,�� . S 291 breeding goal value (H) of sire i in SS can be expressed as �,�� ; note that the for 292 developed in Methods can be applied not only to breeding value ( ) but also to b 293 + 13 deviated from Poisson variance by applying binomial distribution to the family size from the parents of SS,SD,DS, and DD, respectively, and � is the selective advantage of parents in category �SS, SD, DS, DD�. Elements of ∆ , , ,�� are shown in Appendix C.

Example applications of the formula
To demonstrate the application of our formula, we assumed only two quantitative traits: trait 1 was assumed to be moderately heritable, with h 2 = 0.3, whereas trait 2 was assumed to have low heritability, with ℎ � � 0.1. These traits are selected as single traits expressed as GEBV.
Furthermore, we assumed an aggregate genotype as a linear combination of genetic values, each weighted by the relative economic weights, which was expressed as � � � � � , where � is the true genetic value for trait i, � is the relative economic weight for trait i, and the genetic correlation between traits 1 and 2 was assumed as 0.4. Index selection was performed to select � � � � � , that is, breeding goal (H), under the assumption that the relative economic weight between traits 1 and 2 is 1: 1. Breeding value (A) was defined as described earlier in the Methods; for example, the breeding value of sire i in SS was defined as �,�� . Similarly, the breeding goal value (H) of sire i in SS can be expressed as �,�� ; note that the formula that we developed in Methods can be applied not only to breeding value ( ) but also to breeding goal , that is, breeding goal (H), under the assumption that the relative economic weight between traits 1 and 2 is 1:1. Breeding value (A) was defined as described earlier in the Methods; for example, the breeding value of sire i in SS was defined as l distribution to the family size from the is the selective advantage of parents in ; note that the formula that we developed in Methods can be applied not only to breeding value (A) but also to breeding goal value (H).
In our example, we assumed the reliabilities of the GEBVs for traits 1 and 2 to be 0.5721 and 0.4836, respectively [3]. Index selection (I) was performed as I = assumed the reliabilities of the GEBVs for traits 1 and 2 to be 0.5721 y [3]. Index selection (I) was performed as I = � � � � � , ssumed to be derived from multiple-trait BLUP (MT BLUP) genetic the current study (as done for single-step genomic BLUP, [13]). We genetic variances and reliabilities based on Togashi et al [3]. The initial librium genetic variances and reliabilities for single-trait selection (h 2 = ex selection are shown in Table 1 be adjusted more easily in male selection paths (SS and SD) than in + ssumed the reliabilities of the GEBVs for traits 1 and 2 to be 0.5721 [3]. Index selection (I) was performed as I = � � � � � , umed to be derived from multiple-trait BLUP (MT BLUP) genetic e current study (as done for single-step genomic BLUP, [13]). We netic variances and reliabilities based on Togashi et al [3]. The initial rium genetic variances and reliabilities for single-trait selection (h 2 = x selection are shown in Table 1 , because GEBVs are assumed to be derived from multipletrait BLUP (MT BLUP) genetic evaluation methods in the current study (as done for single-step genomic BLUP [13]). We calculated equilibrium genetic variances and reliabilities based on Togashi et al [3]. The initial (generation 0) and equilibrium genetic variances and reliabilities for single-trait selection (h 2 = 0.3 or h 2 = 0.1) and index selection are shown in Table 1. Rates of inbreeding were calculated based on equilibrium genetic variances and reliabilities, because regression coefficients of the number or breeding value of selected offspring on the breeding value of the parent are equal for the parental and offspring generations under equilibrium genetic variances and reliabilities.
We considered two scenarios for the selection percentages for SS, SD, DS, and DD-5%-12.5%-1%-70% and 1%-5%-1%-70%-and three scenarios for the numbers of selected parents of SS, SD, DS, and DD-namely 20-50-100-7,000, 40-100-200-14,000, and 60-150-300-21,000. Therefore, we considered six scenarios (two scenarios of selection percentage and three scenarios of the number of parents in SS, SD, DS, DD) in total. Note that the two scenarios for selection percentage for SS, SD, DS, and DD differ only in the selection percentage along the SS and SD selection paths, because under actual breeding conditions, selection intensity can be adjusted more easily in male selection paths (SS and SD) than in female selection paths (DS and DD). The numbers of male and female offspring from a dam of DS, i.e., fmds and ffds, were set at 4. The number of female offspring from a dam of DD, i.e., ffdd, was set at 1.4. These numbers are derived from the years of usage of a dam and the reproduction method (ovum collection, in vitro fertilization, or embryo transfer). When DS and DD parents are used with constant selection intensity and in equal numbers over several years, they belong to a single or exclusive category. The numbers are used to compute the deviation of the variance of the family size from Poisson variance.

Rates of inbreeding
The rates of inbreeding without correction for deviation from Poisson variances (that is, the rates of inbreeding with Poisson family size) are shown in Table 2. Because the rates from Gowe et al [8] do not account for selection, ΔF is the same between two selection percentages in SS, SD, DS, and DD, i.e., 1%-5%-1%-70% and 5%-12.5%-1%-70%. In contrast, ΔF derived from the method developed in the current study increased with the increase in selection intensity. When we applied our formula, ΔF was lower when selection was ignored than when it was included, suggesting that ΔF was underestimated when selection was ignored. The ratio of ΔF when selection was ignored to that when it was included was about 0.61 under the selection percentages of 5%-12.5%-1%-70% for the SS, SD, DS, and DD selection paths, whereas the ΔF ratio was 0.53 to 0.56 under the selection percentage condition of 1%-5%-1%-70%. That is, calculation according to Gowe et al [8] underestimated ΔF by approximately 40% and 45% under selection percentages of 5%-12.5%-1%-70% and 1%-5%-1%-70% for the SS, SD, DS, and DD selection paths, respectively. In contrast, the rates of inbreeding under selection estimated by using our formula were 63% to 87% greater than those calculated according to the current working formula, which does not consider selection [8]. The ratio of ΔF for 5%-12.5%-1%-70% to that for 1%-5%-1%-70% was 0.88 to 0.89, resulting in an approximately 12% decrease in ΔF due to increasing the selection percentage or decreasing the selection intensity for SS and SD for all three scenarios compared in the numbers of parents in SS, SD, DS, and DD (20-50-100-7,000, 40-100-200-14,000, and 60-150-300-21,000). In contrast, the decrease in ∆F due to the increase in the number of parents was proportional to the numbers.
The ∆F under the number of parents in SS, SD, DS, and DD (40-100-200-14,000 and 60-150-300-21,000) was approximately half and one third of the ∆F under the number of parents (20-50-100-7,000), respectively, for all two scenarios compared in the selection percentage of parents in SS, SD, DS, and DD (5%-12.5%-1%-70% and 1%-5%-1%-70%). Consequently, the decrease in the rate of inbreeding likely would be greater with an increase in the number of parents than with a decrease in selection intensity; however, we need to perform more trials at different selection intensities to confirm this association. In general, both genetic gain and ΔF increase with an increase in selection intensity. However, because the number of parents has a greater effect on inbreeding than does selection intensity, increasing the number of parents is one option for offsetting the increase in ΔF due to an increase in selection intensity.
The rate of inbreeding was slightly lower in single-trait selection with a low heritable trait (h 2 = 0.1) than the other selection methods (i.e., single-trait selection with a trait (h 2 = 0.3) and index selection [ Table 2]). However, the difference was not so remarkable. Consequently, we consider the major factors in the rate of inbreeding to be the number of parents and the selection intensity in each of the four selection paths.
Values for effective population size expressed as 16 parents in SS, SD, DS, and DD (5%-12.5%-1%-70% and 1%-5%-1%-70% 353 decrease in the rate of inbreeding likely would be greater with an increa 354 parents than with a decrease in selection intensity; however, we need to p 355 different selection intensities to confirm this association. 356 In general, both genetic gain and ΔF increase with an increase in 357 However, because the number of parents has a greater effect on inbreeding 358 intensity, increasing the number of parents is one option for offsetting the i 359 an increase in selection intensity. 360 The rate of inbreeding was slightly lower in single-trait selection with 361 (h 2 = 0.1) than the other selection methods (i.e., single-trait selection with 362 index selection [ Table 2]). However, the difference was not so remarkabl 363 consider the major factors in the rate of inbreeding to be the number of pare 364 intensity in each of the four selection paths. 365 Values for effective population size expressed as � � � �∆� are shown 366 method that ignores selection [8] overestimated the effective populati 367 compared with that computed by using our formula, which accounts 368 overestimation was greater when the selection percentage in SS, SD 369 1%-5%-1%-70% than when it was 5%-12.5%-1%-70%. The rat 370 5%-12.5%-1%-70% condition to that for 1%-5%-1%-70% became greate 371 are shown in Table 3. Using a method that ignores selection [8] overestimated the effective population size due to ∆F compared with that computed by using our formula, which accounts for selection. The overestimation was greater when the selection percentage in SS, SD, DS, and DD was 1%-5%-1%-70% than when it was 5%-12.5%-1%-70%. The ratio of NE for the 5%-12.5%-1%-70% condition to that for 1%-5%-1%-70% became greater as the numbers of parents in SS, SD, DS, and DD increased from 20-50-100-7,000 to 40-100- ΔF under selection of 1%-5%-1%-70%

Ratio of ΔF ignoring selection to that including selection
Ratio of ΔF at 5%-12.5%-1%-70% to that at 1%-5%-1%-70% 5%-12.5%-1%-70% 1%-5%-1%-70%   Table 4. The expectation of the square of long-term contribution of an individual was the greatest in SS of all the four selection paths (SS, SD, DS, and DD), since selection intensity is the highest and the number of parents is the smallest of all the four selection paths. On the contrary, the square of long-term contribution of an individual was the smallest in DD of all the four selection paths, since selection intensity in DD is the lowest and the number of parents is the largest of all the four selection paths. The square of long-term contribution of an individual in SD was greater than that in DS, mainly because the number of parents in SD is smaller than those SS, sires to breed sons; SD, sires to breed daughters; DS, dams to breed sons; DD, dams to breed daughters. 1) Selection percentages in SS, SD, DS, and DD, respectively.
2) Numbers of parents in SS, SD, DS, and DD, respectively.
3) 0, 1, 2, and 3 refer to rates of inbreeding from Gowe et al [8], which does not account for selection; single-trait selection (h 2 = 0.3); single-trait selection (h 2 = 0.1); and index selection based on two traits (h 2 = 0.3 and h 2 = 0.1) with equal economic weights, respectively.  in DS. With the increase in selection intensity or decrease in selection percentage in the four selection paths (SS-SD-DS-DD), i.e., from 5%-12.5%-1%-70% to 1%-5%-1%-70%, the increase in the square of long-term contribution of individuals in SS and DS was greater than that in SD and DD, because the selective advantage of an individual in DS was the sum of its breeding value and the breeding value of its mate in SS category with the greatest long-term contribution of all the four selection paths. The increase in the number of parents decreased the square of long-term contribution of an individual in SS, SD, DS, and DD, because the expected contribution of an average parent (α) in each of the four selection paths decreased with the increase in the number of parents. The square of long-term contribution of an individual was slightly lower in single-trait selection with a low heritable trait (h 2 = 0.1) than the other selection methods (i.e., single-trait selection with a trait (h 2 = 0.3) and index selection). However, the difference was not so remarkable in all selection methods (single-trait selection with a trait (h 2 = 0.1 or 0.3) and index selection), which was consistent with the trend that the rate of inbreeding was almost the same in all selection methods ( Table 2). The accounting percentage derived from SS, SD, DS, and DD for the rate of inbreeding (ΔF) when the numbers of parents in SS, SD, DS, and DD are 40-100-200-14,000 is shown in Table 5. The accounting percentage in SS was the greatest of all the four selection paths for all two scenarios compared in the selection percentage in SS, SD, DS, and DD (that is, 5%-12.5%-1%-70% and 1%-5%-1%-70%), because the expectation of the square of lifetime long-term contribution of an individual was the greatest in SS of all the four selection paths ( Table 4). The sum of accounting percentage in SS and SD was approximately 90% for ΔF, because the number of male parents in SS and SD was smaller than that of female parents in DS and DD and selection intensity in male parents is generally higher than that in female parents. In addition, the accounting percentage in each of the four selection paths when the numbers of parents in SS, SD, DS, and DD were 40-100-200-14,000 (Table 5) was approximately the same as the other scenario when the numbers of parents in SS, SD, DS, and DD were 20-50-100-7,000 or 60-150-300-21,000, although the accounting percentage in SS, SD, DS, and DD in the other scenarios was not shown. This is mainly because the expected contribution of an average parent (α) and the regression coefficient of the contribution of an individual on its selective advantage (β) are inversely proportional to the number of parents as explained previously in equation (1). Consequently, the accounting percentage derived from SS, SD, DS, and DD for the rate of inbreeding (ΔF), (that is, the relative magnitude of ΔF in SS, SD, DS, and DD), resulted in almost the same for all three scenarios compared in the numbers of parents in SS, SD, DS, and DD, even if the absolute magnitude of ΔF derived from each of the four selection paths differed in the number of parents in each of the four selection paths.

Correction derived from deviation from Poisson variance
Corrections for deviations in the variance of the family size from independent Poisson variances (×10 -4 ) approximated by binomial distribution are shown in Table 6. The magnitude approximated by binomial distribution under the assumed selection percentages in the SS, SD, DS, and DD selection paths of 5%-12.5%-1%-70% and 1%-5%-1%-70% varied from -0.29×10 -4 to -0.88×10 -4 , and -0.04×10 -4 to -0.12×10 -4 ,   respectively. In comparison, the rates of inbreeding with Poisson family size without correction shown in Table 2 varied from 0.2×10 -2 to 0.7×10 -2 . Therefore, because the magnitude of correction was much smaller than that of the rates of inbreeding with Poisson family size without correction, correction is unnecessary; thus the rates of inbreeding without correction (Table 2) are reasonable rates of inbreeding. However, the method in terms of the factorial moments [10] should be examined to confirm that the magnitude of correction is much smaller than those of ΔF with Poisson family size without correction. Selection intensities and variance reduction coefficients should be adjusted by using the procedure from Wray and Thompson [4] in situations of few families with numerous candidates per family, for example, when the number of selected parents is only 5 or 10 [6]. Because we set the number of parents in SS at 20, 40, and 60, we did not adjust the selection intensity in the SS path. In addition, selection intensity in DD generally is much smaller than those in SS, SD, and DD selection paths. Consequently, when selection in DD is not performed, the selection intensity and reduction factor of the variance need to be set at zero in the DD selection path in the formula developed in the current study.

CONCLUSION
We here developed a formula for calculating the rates of inbreeding in populations under selection based on GEBV. The population is selected along the four selection paths of SS (sires to breed sons), SD (sires to breed daughters), DS (dams to breed sons), and DD (dams to breed daughters). Assuming that the number and selection intensity of parents remained the same over the period of usage (several years) enabled us to regard generations as discrete generations. The effect on decreasing the rate of inbreeding was greater when the number of parents was increased than when the selection intensity was decreased, and both number of parents and the selection intensity in four-path selection emerged as major factors affecting the rate of inbreeding. In general, both genetic gain and ΔF tended to increase in line with any increase in selection intensity. Therefore, increasing the number of parents is one option for offsetting the increase in ΔF due to an increase in selection intensity. Especially, increasing the number of male parents would be effective, since the accounting percentage for the increase in ΔF from male parents is greater than that from female parents. When applied without correction for deviation of family size from Poisson distributions, the formula we developed here would be highly useful as a practical method for predicting the approximate rate of inbreeding (∆F) in populations where selection is performed according to four-path programs.