We named our index for achieving desired weight gains at specific time points during growth while maximizing the total weight gain during growth (but excluding constrained gains at designated time points) the “maximum growth index.”

The maximum growth index (I_GEBV) was defined as IGEBV=∑j=0k-1bjGEBVj=GEBVa′L′b, where b_j is the index weight for the j^th order of the GEBV Legendre RR coefficient; GEBV_j is the GEBV for the j^th order of the Legendre polynomial RR coefficient (α_Lj) including a constant; GEBV_αL is a (k×1) column vector containing GEBV_j (j = 0,1,..,k−1) for Legendre polynomial RR coefficients (α_L); k is the number of Legendre coefficients, α_L = (α_L0, α_L1, …, α_Lk−1); and b is a (k×1) column vector of index weights derived to maximize correlation with total genetic value while satisfying restrictions. Legendre RR was fitted as a growth curve in this study. A previously reported index [1,2] for achieving a desired weight gain while minimizing selection intensity, rather than maximizing total weight gain throughout the growth process, is referred to as the “point gain-index.”

Desired genetic gains (ΔG_s) at s specific times during the fattening process can be described according to BLUP properties [17] as:

where S is an (s×k) matrix;

s is the total number of restrictions in the fattening process; t_i is the standardized time value to the interval between −1 to +1 for the i^th specific time in fattening process for the desired gains (i = 1,..,s); ϕ_j(t_i) is the j^th order of Legendre polynomial (j = 0,..,k−1) evaluated at t_i; ΔG_S is an (s×1) column vector;

ΔG_s = (ΔG_t1 ΔG_t2 . . ΔG_ts)'; ΔG_ti is the desired genetic gain for the i^th specific time during growth; ī is the intensity of selection; σ_IGEBV is the standard deviation of the maximum growth index (I_GEVB) and V_GEBVαL is a (k×k) (co)variance matrix of GEBV_αL.

The vector of the difference in Legendre polynomial coefficients between after and before selection (Δα_L is a [k×1] column vector, i. e., α_L after selection – α_L before selection) can be described according to BLUP properties [17] as

Furthermore, desired genetic gains (ΔG_s) at s specific times during the fattening process are described as ΔG_s = SΔα_L.

Total genetic value from the beginning to the end of the fattening process (i.e., from the 1^st through m^th specific times during the growth process) is described as G_L, GL=∑i=1mFiαL=FαL; i = 1, 2, …, m time point,

where F is a (1×k) row vector; and F_i,j = the j^th Legendre polynomial of covariate in the i^th specific time.

Let the breeding goal be to maximize the growth throughout the fattening process but excluding specific time points constrained to predetermined weight gains, subject to a restriction of SVGEBVαLbi¯σIGEBV=ΔGs associated with a prespecified value of ī. This is equivalent to maximizing the correlation (r) between I_GEBV and G_L subject to the constraint SVGEBVαLbi¯σIGEBV=ΔGs.

The method of Lagrange multipliers is widely used to find the maximum or minimum of a function when restrictions are imposed on the variables of the function. The method of Lagrange multipliers gives the following function (f):

where η is an (s×1) column vector of Lagrange multipliers with elements η_i (i = 1, 2, ..., s).

Because cov(GL, IGEBV′)=cov(FαL, IGEBV′)=cov(FαL,GEBVαL′b)=FVGEBVαLb, σIGEBV2=b′VGEBVαLb, σGL2=FKF′ and var(α_L) = K, the correlation (r_{IGEVB, GL}) between I_GEBV and G_L can be shown as:

Differentiating the function f with respect to b and equating the resulting partial derivatives to zeros results in the following equations:

, where I_k is a (k×k) identity matrix.

Differentiating the function f with respect to η and equating the resulting partial derivatives to zeros results in the following equations:

where, σIGEBV=b′VGEBVαLb.

Eqs. 2, 3 can be jointly expressed as the following system of equations:

The inverse of the coefficient matrix of Eq. 4 can be obtained through inversion by partitioning [18]. As a result, index weights (b) of I_GEBV can be obtained:

, where A=i¯σIGESVS′, B=1σIGEBV2bΔGs′, and C=i¯σIGESVSVGEBVαL.

Furthermore, the values of b can also be obtained directly by inversion of Eq. 4.

When the value of variance of the maximum growth index (σσIGEBV2) is very large compared to the values of elements of the (k×s) matrix (bΔG'_s), the term of 1σσIGEBV2bΔGs′(=B) in Eq. 4 can be neglected. In this situation,

Eq. 4 has three distinctive characteristics: 1) the coefficient matrix of Eq. 4 is not symmetric and nonlinear (in terms of b); 2) the coefficient matrix contains the selection intensity (ī), indicating that index coefficients vary depending on the value of (ī) as specified before selection; and 3) the coefficient matrix contains the unknown solution b, thus requiring an iterative approach to solve the equations. To start the iteration, it is logical to set the unknown b to the unrestricted growth index weights. The unrestricted growth index (Supplement 1) maximizes growth throughout the fattening process without restrictions on weight gains at prespecified time points during growth process. Because Eqs. 5, 6 is not a function of η, there is no need to assume the initial values for η. In an animal breeding context, selection intensity (ī) and the elements of ΔG_s are rather small constants. Eq. 4 has full rank (k+s). Therefore, iteration on Eq. 4 would converge to yield a unique solution. It is worth noting that the predetermined constrained weight gain should be biologically reasonable. It would be difficult to achieve restrictions when selection intensity is too small when the number of restrictions is large. A failure to achieve convergence in Eq. 4 is a good indication that the predetermined genetic gains at specific times during growth is more extreme than is genetically feasible and the selection intensity applied is too small. The iteration procedure is summarized as:

There are too many restrictions to satisfy the Eq. 3, i.e., SVGEBVαLbi¯σIGEBV=ΔGs, when the number of restrictions regarding desired gains (s) exceeds the number of RR coefficients fitted (k). In general, when difference in Legendre polynomial coefficients after and before selection is given (Δα_L, i.e., α_L after selection – α_L before selection), an equation (SΔα_L = ΔG_S) holds true without any restrictions on s and k, i.e., s>k, s<k, or s = k. On the other hand, the difference (Δα_L) can only be obtained uniquely when the number of restrictions (s) is equal to the number of Legendre polynomial coefficients fitted to the growth function (k). Since ΔαL=VGEBVαLbi¯σIGEBV from Eq. 1, SΔαL=SVGEBVαLbi¯σIGEBV=ΔGs. The terms of Δα_L and b are obtained uniquely when s = k. The term i¯σIGEBV in Eq. 3 must be 1 when s = k. Therefore, the term σ_IGEBV is a constant and cannot be differentiated with respect to b, even though σ_IGEBV is written as b′VGEBVαLb. In addition, when the number of desired gains during growth is equal to the number of RR coefficients fitted (s = k), the RR covariate matrix S becomes an (s×s) square matrix and inverse of S is described as S⁻¹. Therefore, b in Eq. 3 is described as:

And selection intensity (ī) required to achieve ΔG_s is b′VGEBVαLb.

Eq. 3 is simply an equation that satisfies the intended weight gains at specified time points without regard to maximizing genetic gain throughout the growth process. That is, when the number of desired weight gains during growth is equal to the number of fitted RR coefficients, then the maximum growth index will be an index that only meets the desired weight gain at a particular time point, rather than maximizing growth during growth process. Therefore, the selection intensity cannot be specified in advance to a particular value.

Additionally, the previously reported index weights which satisfy the desired weight gains specified in advance while minimizing selection intensity was shown in [1,2], i.e.,

Especially when s = k, b = S′(SV_GEBVαL S′)⁻¹ ΔG_s =

Eq. 8 is the same as Eq. 7. In the end, when s = k, the index becomes a just valid index to satisfy the desired gains specified in advance without minimizing selection intensity or maximizing growth in whole growth process. Therefore, when s<k, while achieving the specified amount of gains during growth, the developed index in this study can maximize growth in whole growth process and the previously reported index [1,2] can minimize selection intensity. For example, in quartic equation (k = 5), the number of desired gains should be 4 or less to maximize growth during the entire growth process while achieving pre-determined desired gains at specific time points, because constant is included to the number of RR coefficients fitted (k).

Given that the purpose of this study was to develop an index rather than to apply Legendre RR to a growth curve, we assumed the birth weight; the body weights at 5, 81, 127, 128, and 130 weeks of age during the fattening process; the Legendre coefficients in the absence of selection (Supplement 2); and the genetic covariance matrix of Legendre polynomial coefficients (Supplement 3) in the same way as previously [1,2], which was based on the fattening process in Japanese Black steers [19,20].

The reliability of GEBV (j = 0, 1, ... , k − 1) for the j^th order of Legendre RR coefficients was assumed to be 0.7. In the current study, quartic Legendre polynomials were assumed as done previously [1,2]. Because Japanese Black steers are slaughtered at approximately 30 months of age [21], we fitted a growth curve to 130 weeks of age. We assumed that the growth curve in the absence of selection is similar to the curve derived from fitting a Gompertz growth curve [19]. Instead, we fitted a RR model for that curve to develop a selection index based on RR coefficients. With our selection index, the main goals of breeding are to have a smaller birth weight than at pre-selection and to reach the 130-week weight earlier. We set constraints on body weights in relation to the breeding goals and four examples of desired weight gains, as we did previously [2]. These examples reflect differences in how specific constraints on the amount of weight gain at specific times during growth are set. The four examples are shown in Table 1, that is,

Example 1: birth weight (−2.5 kg);

Example 2: birth weight (−2.5 kg) and weight at 128 weeks (+4.6 kg);

Example 3: birth weight (−2.5 kg), weight at 66 weeks (+2.8 kg), and weight at 128 weeks (+4.6 kg); and

Example 4: birth weight (−2.5 kg), weight at 43 weeks (+1.4 kg), weight at 87 weeks (+3.9 kg), and weight at 128 weeks (+4.6 kg).

The number of constraints in these examples (s = 1, 2, 3, or 4) is less than the number of quartic Legendre RR coefficients fitted including a constant (k = 5). Therefore, these constraints on desired body weight gains are achieved and the amount of growth is maximized throughout the growth process with the exception of the desired weight gain at pre-specified points.

Table 2 shows the index coefficients and total weight gains at convergence for selection intensities of 0.5 and 1.0 for constraint Examples 1, 2, 3, and 4. Convergence was reached earlier as the number of constraints on desired weight gains decreased. For example, when the intensity of selection was equal to 1.0, the number of rounds of iteration at convergence was 5, 6, 12, and 12 for 1, 2, 3, and 4 constraints, respectively. In addition, when the intensity of selection was 0.5, the number of rounds of iteration at convergence was 5, 9, 57, and 40 for 1, 2, 3, and 4 constraints, respectively. Therefore, convergence occurred later as the selection intensity decreased. The prespecified desired genetic weight gains (ΔG_S) and the weight gains achieved with the maximum growth index were consistent for all the constraint Examples 1, 2, 3, and 4.

The total weight gain throughout growth (excluding gains at constrained weeks of age) decreased as the number of constraints on desired gains increased, indicating that the fewer the constraints, the more diverse the growth curve and the greater the potential for a greater total weight gain. Conversely, the growth curve is more controlled when the number of constraints increases. As a result, less constraint increases the likelihood of diverse growth curves and greater weight gain over the entire growth period. Even when the selection intensity increased from 0.5 to 1.0, total gain during growth (excluding gains at constrained weeks of age) did not double, because the intensity of selection is not independent of the index coefficients. The index coefficients vary depending on the specified selection intensity, even though the constraints on weight gain might be same; that is, different selection intensities result in different indices, even when the desired weight gain is the same.

We then compared the total weight gain throughout the growth process and weight gains at specific weeks of age between the maximum growth index, the point-gain index, and the unrestricted growth index. Table 3 shows the results for constraint Examples 1 and 2, and Table 4 shows the results for constraint Examples 3 and 4, with the specific conditions of the four Examples shown in Table 1. The numbers of constraint Examples 1, 2, 3, and 4 refer to the numbers of constraints on weight gain throughout the growth process. We applied the same selection intensity for all three indices; the value of the selection intensity was determined by using the point-gain index that achieved the desired weight gain at the lowest selection intensity. Overall, the selection intensity was relatively low, i.e., 0.0713, 0.1264, 0.4179, and 0.3965 for constraint Examples 1, 2, 3, and 4, respectively.

Desired gains were achieved for both the maximum growth index and point-gain indices. Although the unrestricted growth index yielded the highest total gain among the three indices compared, this index failed to achieve the desired gains, because it aims only to achieve the maximum weight gain throughout the growth process. Total weight gain throughout the growth phase was negative for constraint Example 1, where only birth weight was constrained (−2.5 kg), and it was −103.6 and −127.0 for the maximum growth index and point-gain index, respectively. The maximum growth index maximized the total weight gain throughout the growth process and achieved the desired weight gains at specified time points. However, the total weight gain was negative even for the maximum growth index (Table 3), because the selection intensity was 0.0713 and therefore almost zero. In contrast, with the unrestricted growth index, which maximizes weight gain throughout the entire growth period without the restriction of reducing the birth weight by 2.5 kg, the birth weight increased by 0.88 kg, and the overall weight gain during growth was 369 kg even though the selection intensity was only 0.0713.

In constraint Examples 2, 3, and 4, the total weight gain was greater for the maximum growth index than for the point-gain index. However, when the number of constraints was 3 or 4, this tendency was less evident. As mentioned earlier, this is because the growth curve is less diverse and more controlled with a large number of constraints than with a small number of constraints.

Genetic gains from the maximum growth index for constraint Examples 1 and 2 are shown for selection intensities of 0.5 and 1.0 (Figure 1). For constraint Example 1 (constraint on birth weight only [−2.5 kg]), for selection intensities 0.5 and 1.0, genetic gain increased with age toward 77 weeks and decreased gradually thereafter to the end of fattening. For constraint Example 2 (i.e., constraints on weight at birth [−2.5 kg] and 128 weeks [4.6 kg]), genetic gain increased with age toward 65 and 63 weeks for selection intensities of 0.5 and 1.0, respectively. After it peaked, genetic gain decreased toward the end of fattening, and the decreasing trend was steeper in constraint Example 2 than in Example 1. This difference occurred because genetic gain at 128 weeks was constrained to 4.6 kg in constraint Example 2, whereas constraint Example 1 has no restriction on genetic gain other than the decrease at birth. Although genetic gain was greater at a selection intensity of 1.0 than at 0.5, the trend in genetic gain along the growth trajectory was not parallel between these selection intensities. As mentioned earlier, this lack of parallelism is because the selection index is different for each selection intensity. The genetic gains for constraint Example 3 (constraints on birth weight [−2.5 kg], weight at 66 weeks [20.2 kg], and weight at 128 weeks [4.6 kg]) derived from the maximum growth index for selection intensities of 0.5 and 1.0 are shown in Supplement 4. Similarly, the genetic gains for constraint Example 4 (constraints on birth weight [−2.5 kg], weight at 43 weeks [15.7 kg], weight at 87 weeks [16.6 kg] and weight at 128 weeks [4.6 kg]) derived from the maximum growth index for selection intensities of 0.5 and 1.0 are shown in Supplement 5. The number of constraints on weight gain during growth is 3 and 4 in Supplements 4, 5, respectively. Weight gain between the constrained ages showed an upward convex curve because the amount of increase due to the upward convex curve is larger than the linear increase. In other words, the index that we developed maximizes the amount of weight gain over the entire growth period, so that the amount of weight gain between the constrained ages creates an upward convex curve.

As mentioned earlier, genetic weight gain at age t can be described as a linear combination of the Legendre polynomial at age t standardized and the difference in Legendre RR coefficients between after and before selection. This can be written in matrix form as SΔα_L = ΔG_S. Similarly, an index to achieve the intended differences in the Gompertz growth curve can be constructed by using the intended differences in the parameters A, B, and K, which are asymptotic weight, growth starting point, and maturity rate, respectively (Supplement 6). However, the genetic weight gain at age t cannot be expressed as a linear combination of the polynomial evaluated at age t and the differences in the parameters of the Gompertz growth curve between after and before selection. Therefore, using the RR growth curve and expressing the genetic weight gain at age t as a linear combination of the polynomial evaluated at age t and the differences in the parameters of the Legendre growth curve between after and before selection made it possible to maximize the amount of genetic gain throughout the growth process and concurrently achieve the desirable genetic gain at a particular growth point.

The daily gain based on the maximum growth index constrained only by birth weight (−2.5 kg) is shown in Figure 2. Before selection, daily gain peaked at 41 weeks of age (1.19 kg). For the maximum growth index at a selection intensity of 1.0, daily gain peaked at 39 weeks of age—earlier than before selection—and the daily gain at the peak was 1.31 kg, which was greater than before selection. This tendency was more pronounced at a selection intensity of 1.0 than at 0.5. The daily gain from the maximum growth index at selection intensities 0.5 and 1.0 with constraints on birth weight (−2.5 kg) and 128-week weight (4.6 kg) is shown in Figure 3. In the same way as in Figure 2, where constraints were set on birth weight only, daily gain peaked earlier than before selection, and the daily gain at peak was greater than before selection. For example, under a selection intensity of 1.0, the maximum daily gain (+1.30 kg) occurred at 37 weeks of age, compared with a maximum daily gain of +1.19 kg at 41 weeks of age before selection. Furthermore, the rate of decline in daily gain after the peak was steeper after selection than before.

Daily gain from the maximum growth index for constraint Examples 3 and 4 at a selection intensity of 1.0 is shown in Supplement 7. In the same way as for Figures 2, 3, maximum daily gain peaked earlier and higher with the maximum growth index selection than before selection. Daily gain after the peak decreased more steeply after selection than before selection. For the maximum growth index as shown in Figures 2, 3 and Supplement 7, birth weight was constrained to decrease by 2.5 kg. Given that the maximum average daily gain in late-maturing pigs peaked 10 days later than in early maturing pigs [22], an earlier growth peak after selection than before appears to be associated with precocious maturation. A maximum growth index in which birth weight is constrained to decrease might lead to transition to a precocious growth curve.

As mentioned earlier, to achieve the desired weight gain, the maximum number of constraints on weight at given time points must be less than the number of RR coefficients fitted including a constant (s<k). Therefore, the maximum growth index we developed can be easily applied in breeding practice because, in the case of a single constraint, such as reduced birth weight, any Legendre RR, whether quadratic, cubic, or quartic, can maximize weight gain during growth.

By developing a maximum growth index that suppresses birth weight by 2.5 kg and maximizes growth during the fattening period, we achieved a lower birth weight and heavier final weight than before selection (Figure 4); that is, the index we developed makes it possible to reach the pre-selection final weight 8 weeks earlier and to increase the post-selection final weight by 17 kg compared with that at pre-selection (737.7 kg vs. 720.4 kg), even when selection intensity is set to 0.5 only. At a selection intensity of 0.5, approximately 70% of the population is selected, and only 30% is culled, therefore suggesting weak selection. Increase in inbreeding is not that serious at a selection intensity of 0.5. Therefore, even at a low selection intensity of 0.5, the maximum growth index enables a large amount of growth. The property that the maximum growth index maximizes growth over the entire growth period reduces the need for high selection intensity required to achieve large amounts of weight gain. However, what is noteworthy is the selection intensity in the previously reported point-gain index was 0.127 when the number of constraints was two (i.e., constraints on weight at birth [−2.5 kg] and 128 weeks [4.6 kg]) [1]. That selection intensity of 0.127 was very low. Even at a very low selection intensity, the previously reported point-gain index made it possible to achieve desirable weight gains at birth and at 128 weeks, with a total weight gain throughout the growth period reaching 223 kg [1]. In addition, the lower the selection intensity, the more likely the selection goal will be achieved. That characteristics of the point-gain index helps maintain genetic diversity. Therefore, when low selection intensity is prioritized from a long-term perspective of genetic diversity, a point-gain index is also necessary. Which method to use—suppressing inbreeding or maximizing overall growth—is the breeder’s decision, considering the characteristics of the maximum growth index and the point-gain index. The selection index procedure that we developed might easily be applied to other longitudinal data such as growth curves in plants and egg-production curves in poultry. Although this study uses GEBVs of the RR coefficients, EBV or phenotype itself [2] of the RR coefficients can be treated as selection index traits of the maximum growth index. EBV coefficients can be applied simply substituting GEBV for EBV in EBV maximum growth index. Phenotypic maximum growth index is written in Supplement 8.