### INTRODUCTION

### MATERIALS AND METHODS

### Prediction of expected long-term genetic contributions

*r*

*) in generation t*

_{i}_{1}is defined as the proportion of genes from individual i that are present in individuals in generation t

_{2}deriving by descent from individual i, where (t

_{2}–t

_{1}) →∞ [5]. That is, after several generations, the genetic contributions of ancestors stabilize and become equal for all descendants, i.e., the ultimate proportional contribution of an ancestor to its descendants is reached.

**1′ = (1 1 1 1), N**is a 4×4 diagonal matrix containing the number of selected parents for element (i, i) as N

_{i,i}, N

_{1,1}is the number of sires in SS and is referred to as

*N*

*, N*

_{SS}_{2,2}is the number of sires in SD and is referred to as

*N*

*, N*

_{SD}_{3,3}is the number of dams in DS and is referred to as

*N*

*, and N*

_{DS}_{4,4}is the number of dams in DD and is referred to as

*N*

*. In addition,*

_{DD}*u*

*is the expected lifetime long-term genetic contribution of individual i in category SS conditional on its selective advantage (which in mass selection is the genomically enhanced breeding value [GEBV]), and*

_{i,SS}*u*

*,*

_{i,SD}*u*

*, and*

_{i,DS}*u*

*are the expected lifetime long-term genetic contributions of individual i in categories SD, DS, and DD, respectively. Furthermore,*

_{i,DD}**δ**= (δ

*δ*

_{SS}*δ*

_{SD}*δ*

_{DS}*), where δ*

_{DD}*is the correction factor for deviations of the variance of family size from independent Poisson variances in the selected offspring from sires in SS; δ*

_{SS}*, δ*

_{SD}*, and δ*

_{DS}*are corrections for deviations of the variance of the family size from independent Poisson variances in the selected offspring from parents in SD, DS, and DD, respectively.*

_{DD}*S*

*) and in SD (*

_{i,SS}*S*

*) in the linear model is:*

_{i,SD}*A*

*is the breeding value of sire i in SS or SD,*

_{i,SS}*Ā*

*is the average breeding value of dams mated to the ith sire in SS and SD, respectively; the dams mated to the ith sire in SS belong to the DS category, and the dams mated to the ith sire in SD belong to the DD category; and*

_{i,DS and DD}*Ā*

*,*

_{i,SS}*Ā*

*,*

_{i,SD}*Ā*

*, and*

_{i,DS}*Ā*

*, are the average breeding values of the individuals in the SS, SD, DS, and DD categories.*

_{i,DD}*S*

*) and in DD (*

_{i,DS}*S*

*) in the linear model is:*

_{i,DD}*A*

*is the breeding value of dam i in DS and DD, respectively;*

_{i,DS and DD}*A*

*is the breeding value of a sire mated to the ith dam in DS and DD, respectively; the sires mated to the ith dam in DS belong to the SS category; and the sires mated to the ith dam in DD belong to the SD category.*

_{i,SS and SD}*u*

*) are predicted by linear regression on the selective advantage. That is,*

_{i,SS,SD,DS,or DD}*α*

*is the expected contribution of an average parent in*

_{x}*x*, and

*β*

*is the regression coefficient of the contribution of i on its selective advantage (*

_{x}*S*

*). In addition,*

_{i,x}*α*

*can be obtained according to Woolliams et al [9]:*

_{x}**G**is a 4×4 matrix representing the parental origin of genes of selected offspring in the order of SS, SD, DS, and DD category, i.e., representing rows offspring and columns parental categories. That is,

**α′N**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).

##### (1)

*β*

*is inversely proportional to the number of parents (that is,*

_{x}**I**

_{4}is a 4×4 identity matrix,

**Π**is a 4×4 matrix of regression coefficients with

*π*

*being the regression coefficient of*

_{xy}*S*

*of a selected offspring i of category*

_{i,x}*x*(SS, SD, DS, DD) on

*s*

*of its parent j of category*

_{j,y}*y*(SS, SD, DS, DD). For example,

*π*

*is the regression coefficient of*

_{SD,SS}*S*

*of a selected offspring i of SD on*

_{i,SD}*S*

*of its parent j of SS. Given that SS is the sires to breed sons category, we have non-zero elements,*

_{j,SS}*π*

*and*

_{SS,SS}*π*

*, in*

_{SD,SS}**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,

*π*

*and*

_{DS,SD}*π*

*, in*

_{DD,SD}**Π**as elements (3,2) and (4,2), respectively. Because DS is the dams to breed sons category, we have non-zero elements,

*π*

*and*

_{SS,DS}*π*

*, in*

_{SD,DS}**Π**as elements (1,3) and (2,3), respectively. And given that DD is the dams to breed daughters category, we have non-zero elements,

*π*

*and*

_{DS,DD}*π*

*, in*

_{DD,DD}**Π**as elements (3,4) and (4,4), respectively.

**Λ**is a 4×4 matrix of regression coefficients, with

*λ*

*being the regression coefficient of the number of selected offspring of category*

_{xy}*x*on

*S*

*of its parent j of category*

_{j,y}*y*. In the same way as

**Π**, we have non-zero elements,

*λ*

*and*

_{SS,SS}*λ*

*,*

_{SD,SS}*λ*

*and*

_{DS,SD}*λ*

*,*

_{DD,SD}*λ*

*and*

_{SS,DS}*λ*

*, and*

_{SD,DS}*λ*

*and*

_{DS,DD}*λ*

*in*

_{DD,DD}**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,

**Π**and

**λ**were calculated from Woolliams et al [9] and Bijma and Woolliams [11], as outlined in Appendices A and B.

*E*denotes the expectation with respect to the selective advantage,

*E*(

*Ā*

*–*

_{DS}*Ā*

*) = (*

_{DD}*i*

*–*

_{DS}*i*

*)*

_{DD}*σ*

*,*

_{A,f}*N*

*,*

_{SS}*N*

*,*

_{SD}*N*

*, and*

_{DS}*N*

*), since the term of*

_{DD}*k*

*,*

_{SS}*k*

*,*

_{SD}*k*

*, and*

_{DS}*k*

*are variance reduction coefficients for offspring selection in SS, SD, DS, and DD, respectively. Note that covariances of mates between SS and SD and between 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.*

_{DD}### Correction of E (ΔF) from Poisson variances

*, δ*

_{SS}*, δ*

_{SD}*, and δ*

_{DS}*, can be approximated by Woolliams and Bijma [10].*

_{DD}**Δ**

*V*

_{SS}**, Δ**

*V*

_{SD}**, Δ**

*V**, and*

_{DS}**Δ**

*V**, are 4×4 matrices which are variances of selected family size deviated from Poisson variance by applying binomial distribution to the family size from the parents of SS, SD, DS, and DD, respectively, and*

_{DD}*s*

*is the selective advantage of parents in category*

_{x}*x*(SS, SD, DS, DD). Elements of

**Δ**

*V**are shown in Appendix C.*

_{SS,SD,DS,or DD}### Example applications of the formula

*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

*a*

_{1}

*g*

_{1}+

*a*

_{2}

*g*

_{2}, where

*g*

_{1}is the true genetic value for trait i,

*a*

*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*

_{i}*a*

_{1}

*g*

_{1}+

*a*

_{2}

*g*

_{2}, 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

*A*

*. Similarly, the breeding goal value (*

_{i,SS}*H*) of sire i in SS can be expressed as

*H*

*; note that the formula that we developed in Methods can be applied not only to breeding value (*

_{i,SS}*A*) but also to breeding goal value (

*H*).

*I*) was performed as

*I*=

*a*

_{1}

*GEBV*

_{1}+

*a*

_{1}

*GEBV*

_{2}, because GEBVs are assumed to be derived from multiple-trait 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.

*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.

### RESULTS AND DISCUSSION

### Rates of inbreeding

*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.

*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).

*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

^{−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.