### INTRODUCTION

### MATERIALS AND METHODS

### Point gain index to achieve the desired gains at the specific times in fattening process with minimum selection intensity

*I*

*) is described as*

_{p}*b*

*is the index weight for the jth order of Legendre polynomial coefficient,*

_{j}*GEBV*

*is GEBV of the jth order of Legendre polynomial coefficient, and k is the order of Legendre polynomial function. In matrix notation,*

_{j}*I*

*=*

_{p}

*b′GEBV**, where*

_{αL}**is a (k+1) vector of index weights and**

*b*

*GEBV**is a (k+1) column vector containing*

_{αL}*GEBVj*(

*j*= 0,1,..,k) for RR coefficients (

**). Hereafter**

*αL**I*

*refers to the point gain index.*

_{p}

*G**) at s specific times in the fattening process are described as*

_{s}##### (1)

*G*

*is the desired genetic gain for the ith specific time during growth,*

_{ti}**is a**

*S**s*×(

*k*+1) matrix,

*s*is the total number of restrictions in the fattening process,

*t*

*is the age standardized for the ith specific time in fattening process for the desired gains (*

_{i}*i*=

*1*,..,

*s*),

*ϕ*

*(*

_{j}*t*

*) is the jth order of Legendre polynomial (*

_{i}*j*=

*0*,..,

*k*) evaluated at age

*t*

*standardized, and Δ*

_{i}

*α**= (Δ*

_{L}*α*

_{L}_{0}, Δ

*α*

_{L}_{1}, .., Δ

*α*

_{L}_{k})′, where Δ

*α*

_{L}_{i}is the difference in ith Legendre polynomial coefficient (

*α*

_{L}_{i}) before and after selection, i.e., Δ

*α*

_{L}_{i}=

*α*

_{L}_{i}after selection and −

*α*

_{L}_{i}before selection.

*α**) can be described according to BLUP properties [20] as*

_{L}

*α**is a [(*

_{L}*k*+1)×1] vector of true genetic Legendre coefficients,

*V*

_{GEBV}_{αL}is the (co)variance matrix of

*GEBV*_{αL},

*ῑ*is the intensity of selection, and

*σ*

_{I}_{p}is the standard deviation of the point gain index (

*I*

*). The selection intensity (*

_{p}*ῑ*) required to achieve Δ

*α**can be obtained by setting*

_{L}*ῑ*=

*σ*

_{I}_{p}. Therefore,

*σ*

_{g}_{–}

_{αL}_{i,j}is the genetic covariance for the ith order and jth order of Legendre coefficients.

*α**so that the index constructed based on Δ*

_{L}

*G**has a minimum variance, with the restriction that the vector of expected genetic gains at specific*

_{s}*s*points during the fattening process (

**= [**

*λ*

*λ*_{1}

*λ*_{2}…..

*λ**] is a vector of Lagrange multipliers.*

_{s}*f*with respect to Δ

*α**equal to zero leads to*

_{L}*f*with respect to

**equal to zero leads to**

*λ*

*α**in equation (4) would lead to the minimum selection intensity and satisfy the constraints of the expected genetic gains being equal to the desired genetic gains.*

_{L}

*α**in (5) has to satisfy the pre-specified gains as shown in (1), i.e.,*

_{L}**Δ**

*S*

*α**= Δ*

_{L}

*G**. It can be proved as*

_{s}**) for the selection index based on Legendre polynomial coefficients are shown as**

*b**k*+1), i.e., s≤

*k*+1.

*k*+1 =

*s*,

*S*is a square matrix, such that

*α**that satisfy the same vector of desired gains would have different selection intensities. This indicates that given a set of desired genetic gains, the solution to achieve the fixed set of genetic gains is not unique.*

_{L}### Eigenvector index to achieve desired gains at specific time points in the fattening process by using the minimum selection intensity

*I*

*) using n eigenvectors of the additive genetic RR covariance matrix is described as*

_{e}

*ɛ**is a ith eigenvector of the additive genetic RR covariance matrix, i.e., covariance matrix of the additive genetic Legendre polynomial coefficients (*

_{i}*i*= 1,2,.., n), and the elements of

*ɛ**are shown as*

_{i}*e*

*is the jth element of*

_{ij}

*ɛ**.*

_{i}

*V**is a (n×n) matrix and is shown as*

_{E}

*G**) during the fattening process are described as*

_{s}

*G**is a (*

_{s}*s*×1) vector of the true genetic values at s specific times during the fattening process, Δ

*G*

*is the genetic gain at the ith time point in the fattening process, and*

_{ti}*σ*

_{I}_{e}is the standard deviation of the eigenvector index (

*I*

*). In addition,*

_{e}*cov*(

*G**,*

_{s}*I*

*) can be described according to BLUP properties [20] as*

_{e}*ῑ*) required to achieve Δ

*G**can be obtained by setting*

_{s}*ῑ*=

*σ*

_{I}_{e}. Therefore,

**) to choose a vector**

*λ***so that the index constructed based on Δ**

*b*

*G**has a minimum variance, with the restriction that the vector of expected genetic gains is equal to the vector of desired genetic gains. The function to be minimized is*

_{s}*f*with respect to

**equal to zero leads to**

*b**f*with respect to

**equal to zero leads to**

*λ***) are shown as**

*b*### Stage gain index to achieve the desired gains at specific stages during the fattening process by using the minimum selection intensity

*I*

*) based on RR coefficients to achieve the desired genetic gains at specific growth stages by using the lowest possible selection intensity. The stage gain index (*

_{s}*I*

*) is described in the same way as the point gain index (*

_{s}*I*

*), that is*

_{p}*I*

*),*

_{p}

*SΔα**= Δ*

_{L}

*G**.*

_{s}*I*

*), the vector Δ*

_{s}

*G**is the desired genetic gains for s stages, and*

_{s}*G*

*is the desired genetic gain for the jth stage in the fattening process. Note that Δ*

_{sj}*G*

*is not the jth specific time point during growth but the jth specific stage during the fattening process. Therefore,*

_{sj}**does not correspond to a specific point but to a specific stage during the fattening process. That is,**

*S***is described as**

*S**ϕ*

*(*

_{i}*t*) is the ith order of Legendre polynomial evaluated at week

*t*standardized, and

*m*

*and*

_{j}*n*

*are the first and last week of age of the jth stage, respectively. In this study, the fattening process was measured in units of weeks of age; accordingly stage was divided into units of weeks of age. Note that the only difference between the point gain and stage gain indices is the definition of*

_{j}*S*. The point gain index and stage gain index correspond to the genetic gains for specific points and specific stages, respectively; all other equations are completely the same between these two indices. Therefore, as in the previous section on the point gain index (

*I*

*), the difference in Legendre polynomial coefficients (Δ*

_{p}

*α**) (*

_{L}*α*after selection –

*α*before selection) in the stage gain index (

*I*

*) can be described as*

_{s}*I*

*) is shown as*

_{s}*k*+1), i.e.,

*s*≤

*k*+1.

### Numerical example

*k*= 4) were assumed as done previously [26,27,13]. Japanese Black steers are slaughtered at approximately 30 months of age [28], so we fitted a growth curve to 130 weeks of age, corresponding to 30 months of age. We assumed that the growth curve before selection was similar to the curve from [24], who fitted a Gompertz growth curve. Instead, we fitted a RR model for that curve to develop a selection index based on RR coefficients. Gompertz growth curve depends on three parameters, i.e., A, B, and K are the asymptotic weight, growth starting point, and maturity rate of the growth curve, respectively. The three parameters, A, B, and K are used from [24] such as 768, 3.4, and 0.03, respectively. Weekly body weights from birth through 130 weeks of age were estimated from Gompertz growth curve by using the three parameters [24]. Covariates for Legendre RR curve are shown as

**in Supplementary file. RR coefficients before selection are estimated from weekly body weights from birth through 130 weeks of age and covariates for Legendre RR curve. The desired curve was derived according to the breeding goal that the birth weight was less than before selection and that the 130-week-old weight was achieved earlier than before selection. Birth weight; body weight at 5, 81, 127, 128, and 130 weeks of age during the fattening process; and the Legendre coefficients before selection and those of the desired curve are shown (Table 2).**

*ϕ**k*+1, i.e., 5. We could provide a maximum of 5 index traits as products between the eigenvector and Legendre coefficients expressed in GEBV, that is,

*GEBV*

*(*

_{j}*j*= 0, 1,..,

*k*) for the jth order of Legendre polynomial coefficients was assumed to be 0.7. From the point that inaccurate estimation of population parameters could bias estimates of theoretical gains, in addition to the reliability of 0.7, we also added reliability of 0.5 and 0.6 to the point gain selection index in which the four target ages were 0, 42, 86, and 130 weeks and the respective desired gains were −2.5, 15.7, 16.6, and 5.4 kg.

### RESULTS AND DISCUSSION

### Gains achieved by using the point gain, stage gain, and eigenvector indices

*k*+1 = 5) is greater than the number of restrictions or desired gains at the specific points (4) or stages (3), the index with the lowest selection intensity is uniquely selected. Thus, the point gain and eigenvector indices each achieved a 2.5 kg lower birth weight. Moreover, these indices achieved the same body weight at 130 weeks of age as before selection (720.4 kg) but approximately 2 weeks earlier (Table 4).

### Comparison of the number of index traits in eigenvector selection

**ɛ**

_{1}**′GEBV**

**and**

_{αL}**ɛ**

_{2}**′GEBV**

**. In the same way, the first, second, and third eigenvectors are treated as index traits when the eigenvector number is 3 (Table 5), that is, the three index traits are**

_{αL}**ɛ**

_{1}**′GEBV**

**,**

_{αL}**ɛ**

_{2}**′GEBV**

**, and**

_{αL}**ɛ**

_{3}**′GEBV**

**, and so on for all four conditions.**

_{αL}### Selection intensity and Legendre coefficients at different target weight gains

*α**), and index weights for stage, point gain, and eigenvector indices with different desired increases (listed as restriction values), are shown (Table 6). The number of stages in the stage gain index is one (Table 6), i.e., the entire fattening period is a single stage. The genetic gain for the entire fattening period was 1,490 kg, which we calculated from the Legendre function (Table 2), that is, (*

_{L}

*α**), and index weights calculated based on the restriction value of 1,490 were 1.49 times greater than those calculated by using the restriction value of 1,000. In the point gain index in which the four target ages were 0, 42, 86, and 130 weeks, the respective desired gains were −2.5, 15.7, 16.6, and 5.4 kg. However, the desired gains changed to −1, 6.28, 6.64, and 2.16 kg when the desired increase in birth weight was represented by −1.*

_{L}

*α**), and index weights calculated by using a birth weight restriction value of −2.5 were 2.5 times larger than those calculated based on a birth weight restriction value of −1. The relationship between these indices is that the directions of the Legendre coefficients (Δ*

_{L}

*α**) and index weights are multiplied by a constant only. Therefore, the results (Table 6) show that genetic gain can be expressed as actual weight gain or as any scaled value and that these indices are essentially the same.*

_{L}### Effects of point gain index selection on body weights throughout the fattening process

*k*+1≥

*s*). In contrast, when

*k*+1<

*s*, there are too many restrictions (

*s*) regarding desired gains (Δ

*G**) to satisfy the condition that*

_{s}**Δ**

*S*

*α**= Δ*

_{L}

*G**.*

_{s}

*α**), and selection intensity in reliability of GEBV (0.7,0.6, and 0.5) are shown in Table 7. The absolute value of index coefficients increased as reliability of GEBV decreased from 0.7 to 0.5. Selection intensity increased with decreasing reliability of GEBV, since variance of index increases with an increase in index coefficients and variance is the square of selection intensity (*

_{L}*ῑ*=

*σ*

_{I}_{p}). The increase in GEBV prediction error associated with the decrease in reliability may have necessitated a slightly higher selection intensity to achieve the intended gain. However, difference in RR coefficients (after selection – before selection, Δ

*α**) was almost the same despite the difference in reliability of GEBV. As a natural result, the amount of weight gain at each age from birth through 130 weeks was almost the same regardless of the difference in reliability of GEBV (not shown). Of course, the intended weight gain was achieved at all three reliability values of GEBV. In this study, Δ*

_{L}

*α**was expressed as (5), i.e., Δ*

_{L}

*α**=*

_{L}

*V*

_{GEBV}_{αL}

*S***′**(

*SV*

_{GEBV}_{αL}

*S***′**)

**Δ**

^{−1}

*G**. The terms of*

_{s}

*V*

_{GEBV}_{αL}

*S***′**and (

*SV*

_{GEBV}_{αL}

*S***′**)

**could have canceled out the effects on**

^{−1}

*V*

_{GEBV}_{αL}, thereby reducing the effect of genetic parameter (

*V*

_{GEBV}_{αL}) on Δ

*α**. This study minimized index variance (*

_{L}

*V*

_{GEBV}_{αL}) onΔ

*α**could have been decreased. Almost the same weight gain during growth process could have been emerged regardless of the difference in reliability of GEBV, because expected responses to selection for body weight in cattle are increase in weight at age of selection but also substantial increases in weight at all other ages [4]. Further research would be necessary to clarify the effect of inaccurate genetic parameter on expected genetic gain. More sophisticated procedures would be necessary to estimate genetic parameters for GEBV [31].*

_{L}**Δ**

*S*

*α**= Δ*

_{L}

*G**, to*

_{s}**Δ**

*S′S*

*α**=*

_{L}**Δ**

*S′*

*G**, and Δ*

_{s}

*α**= (*

_{L}**S′S**)

^{−1}**S′**Δ

*G**. The difference in RR coefficients (Δ*

_{s}

*α**= (*

_{L}**S′S**)

^{−1}**S′**Δ

*G**) should be described to satisfy the desired gains. However, the difference in RR coefficients fails to achieve the desired gains (Δ*

_{s}

*G**), because*

_{s}**Δ**

*S*

*α**=*

_{L}**(**

*S*

*S***′**

**)**

*S*

^{−1}

*S***′**Δ

*G**≠ Δ*

_{s}

*G**. The approach of [8] results in an index with minimum selection intensity only when the number of desired gains is equal to the number of RR coefficients fitted. In contrast, Δ*

_{s}

*α**satisfies the equation of*

_{L}

*S***′**

**Δ**

*S*

*α**=*

_{L}

*S***′**Δ

*G**for any number of desired gains and any order of fitted RR coefficients. Therefore, RR coefficients obtained by adding Δ*

_{s}

*α**= (*

_{L}**S′S**)

^{−1}**S′**Δ

*G**to the RR coefficients before selection is an option for obtaining new growth curve coefficients. However, the new growth curve needs to be checked to confirm that it at least nearly meets the breeder’s intention. This verification is necessary because this approach will not yield the desired gains unless the number of desired gains is equal to the order of the fitted RR coefficients.*

_{s}