### INTRODUCTION

### MATERIALS AND METHODS

### Derivation of survival matrix

**S**

_{i,j}then is derived, in which the row (i) represents the lactation number and the column (j) represents the age (in months) at the end of the i

^{th}lactation. Thus, matrix

**S**represents the age distribution of a cow population. Currently in Japan, the average age at first calving is 25 months, and the calving interval is 14 months (MAFF, 2011). Let

*r*

_{01}be the survival rate from birth to the end of first lactation or the end of stage 1. Let

*r*

_{ij}be the survival rate from the end of the i

^{th}lactation or the end of the i

^{th}stage to the end of the j

^{th}lactation or the end of the j

^{th}stage, e.g.,

*r*

_{12}= survival rate from the end of the first lactation or the end of the first stage to the end of the second lactation or the end of the second stage. The survival rate (

*r*

_{ij}) is determined by a cow being alive and getting pregnant from the end of the i

^{th}lactation to the end of the j

^{th}lactation. The survival rates of

*r*

_{01},

*r*

_{12,}

*r*

_{23,}

*r*

_{34,}and

*r*

_{45}currently in Japan are 0.6900, 0.7773, 0.7525, 0.5936, and 0.5464, respectively (Hagiya et al., 2010). The survival matrix starts when a female calf is born. The investment period is set at 120 months. The elements of

**S**

_{i j (i = 1, 2, 3, 4, 5 stage; j = 1,2, …, 120 months)}are derived as described by Weller et al. (1984). Increased survival rates are set as t times as large as the current survival rate (t = 1.05, 1.1, 1.15, 1.2, and no loss rate of 100%).

### The relative economic weights of the first five lactation milk yields

_{5}) is defined as follows:

_{i}and w

_{i}are the genetic value and relative economic weight of the i

^{th}lactation milk yield, respectively (

*i*= 1 to 5). H

_{5}is genetic lifetime milk yield derived by using only milk yield for the first five lactations.

*is the appropriate discounting factor, for which the subscript*

_{j}*j*is the time interval (in months) from the original investment to mean trait expression (

*j*= 1, … , 120 months), and δ is the discounted rate per month %, e.g., δ = 0/12%, 1/12%, or 4/12% per month. The relative economic weights for the first five lactation milk yields (

**w**′

_{5}= (w

_{1}, w

_{2}, w

_{3}, w

_{4}, w

_{5})) reflect the actual average of each lactation milk yield, so that

**w**

_{5}is

**w**with the vector

_{5}=AM_{5×5}S_{5×120}d**d**= (d

_{1}, … d

_{120})′ and the diagonal matrix

**AM**

_{5×5}whose diagonal elements are the averages of the i

^{th}lactation milk (

*i*= 1 to 5). The discounted total milk yield during the given period of 120 months from the original investment is described as

### Net merit defined in terms of the first five lactation milk yields and HL

*i*= 1 to 5) is the relative economic weight for the i

^{th}lactation milk yield described as

*H*) in units of actual values of milk (kg) and HL (days) is as follows:

_{net merit}### Comparison of selection indices in terms of maximizing net merit

_{1}–I

_{6}) to maximize net merit (

*H*) and to clarify the effects of adding lactation persistency as a component trait of selection index to selection index consisting of only milk yields. Selection index (I

_{net merit}_{7}), comprising the same components as those for net merit, is set as a basis of comparison:

_{i}, P

_{i}, and hl are the EBV for the i

^{th}lactation milk yield, i

^{th}lactation persistency, and HL, respectively and b

_{i}is the selection index coefficient for the i

^{th}component trait of selection index. Lactation persistency is defined as the difference in EBVs for 240 DIM and 60 DIM (Togashi et al, 2008). The selection index is derived as described by Togashi et al. (2011), in which differences between the reliabilities of the EBV of the component traits of the selection index are taken into consideration.

_{6}= b

_{1}M

_{1}+b

_{2}M

_{2}+b

_{3}M

_{3}+b

_{4}P

_{1}+b

_{5}P

_{2}+b

_{6}P

_{3}as an example to demonstrate the derivation. In matrix notation, the index can be written as,

*H*). The covariance matrix between I

_{net merit}_{6}and

*H*is as follows:

_{net merit}_{6}is as follows:

*GP*is the genetic value of the i

_{i}^{th}lactation persistency,

^{th}lactation milk yield EBV, and

^{th}lactation persistency EBV. The reliabilities of the EBV of selection traits of candidate bulls and the genetic parameters were derived from Hagiya et al. (2010) and Togashi et al. (2008). The square of the accuracy is reliability. The average reliability is used in this study.

_{6}for net merit (

*H*) is as follows:

_{net merit}_{6}and

*G*

_{1},Δ

*G*

_{2},…,Δ

*G*

_{5}) and HL(Δ

*G*) are shown as follows:

_{HL}_{HL}). Selection intensity was set to unity (

### Application of SNP

### RESULTS AND DISCUSSION

### Effects of survival and discounted rates on the relative economic weights of the first five lactation milk yields

**S**

*and shows the effects of both survival and discounted rates on the relative economic weights of the first five lactation milk yields, the pooled weights of the third, fourth, and fifth lactation milk yields, and these economic weights under a 100% survival rate. The relative economic weights of the second, third, fourth, and fifth lactation milk yields (w*

_{i j}_{2}, w

_{3}, w

_{4}, and w

_{5}) were expressed as ratios of the first lactation milk (w

_{1}) by setting w

_{1}=1. The relative economic weights of w

_{2}through w

_{5}increased as the survival rate increased from t = 1 to t = 1.2 for all discounted rates. For a given survival rate, the economic weights of w

_{2}through w

_{5}decreased as the discounted rate increased because the time to reach the second through fifth lactations is longer than to reach the first lactation.

_{1}>w

_{2}>w

_{3}>w

_{4}>w

_{5}). This effect occurred because survival rate decreases as parity increases, as mentioned earlier, i.e., the survival rates of

*r*

_{01},

*r*

_{12},

*r*

_{23},

*r*

_{34,}and

*r*

_{45}are 0.6900, 0.7773, 0.7525, 0.5936, and 0.5464, respectively (Hagiya et al., 2010). The decrease in economic weight from the third to fifth parity was greater than that from the first to third parity because survival rate decreased more from the third to fifth parity than from the first to third parity. The economic weight of the fourth and fifth lactation milk yields increased with the survival rate.

_{2}and w

_{3}), particularly under the current survival rate (t = 1). Overestimation of w

_{2}and w

_{3}decreased as the survival rate increased. In contrast, when the survival rate was achieved without loss (100%), the relative economic weight of the third lactation was greater than that of the second lactation, which in turn was greater than that of first lactation (w

_{3}>

*w*

_{2}> w

_{1}) under discounted rates of 0/12% and 1/12% per month. This result occurs because the average yield of the third lactation milk is greater than that of the second lactation milk, which in turn is greater than that of first lactation milk

### Comparison of selection indices in terms of maximizing net merit

*(*I

_{1}to I

_{7}) for selected bulls as breeding candidates under the current survival rate and a discounted rate of 0 are shown in Table 2. Net merit and selection accuracy increased by adding lactation persistency as a component trait to selection index comprising only milk yields. That is, net merit and selection accuracy during the first; first and second; and first, second, and third parity components was larger in I

_{2}(b

_{1}M

_{1}+b

_{2}P

_{1}) than in I

_{1}(M

_{1}), in I

_{4}(b

_{1}M

_{1}+b

_{2}M

_{2}+b

_{3}P

_{1}+b

_{4}P

_{2}) than in I

_{3}(b

_{1}M

_{1}+b

_{2}M

_{2}), and in I

_{6}(b

_{1}M

_{1}+b

_{2}M

_{2}+b

_{3}M

_{3}+b

_{4}P

_{1}+b

_{5}P

_{2}+b

_{6}P

_{3}) than in I

_{5}(b

_{1}M

_{1}+b

_{2}M

_{2}+b

_{2}M

_{3}), respectively. In addition, the genetic superiority of HL in first and second parity and in first, second, and third parity components was larger in I

_{4}(b

_{1}M

_{1}+b

_{2}M

_{2}+b

_{3}P

_{1}+b

_{4}P

_{2}) than in I

_{3}(b

_{1}M

_{1}+b

_{2}M

_{2}) and in I

_{6}(b

_{1}M

_{1}+b

_{2}M

_{2}+b

_{3}M

_{3}+b

_{4}P

_{1}+b

_{5}P

_{2}+b

_{6}P

_{3}) than in I

_{5}(b

_{1}M

_{1}+ b

_{2}M

_{2}+b

_{2}M

_{3}), respectively. In particular, the genetic superiority of HL was 45.4 days from I

_{6}compared with 39.0 days from I

_{5}. The genetic superiority of the total yield of the first through fifth lactation milks in the first and second and first, second, and third parity components were equivalent between I

_{3}(b

_{1}M

_{1}+b

_{2}M

_{2}) and I

_{4}(b

_{1}M

_{1}+b

_{2}M

_{2}+b

_{3}P

_{1}+b

_{4}P

_{2}) and between I

_{5}(b

_{1}M

_{1}+b

_{2}M

_{2}+b

_{2}M

_{3}) and I

_{6}(b

_{1}M

_{1}+b

_{2}M

_{2}+b

_{3}M

_{3}+b

_{4}P

_{1}+b

_{5}P

_{2}+b

_{6}P

_{3}), respectively. Therefore lactation persistency during the second and (especially) third parity contributed to increasing net merit, in which HL was increased while the first five lactation milk yields were maintained at the same levels derived from the selection index consisting of only milk yields.

_{2}(b

_{1}M

_{1}+b

_{2}P

_{1}) among the six indices (I

_{1}to I

_{6}), and followed by I

_{1}(M

_{1}). These results reflect the high genetic correlations between the first-lactation milk yield and second- through fifth-lactation milk yields (r

_{G}= 0.759 to 0.974), the moderate genetic correlation between lactation persistency during the first parity with the first five lactation milk yields (r

_{G}= 0.42 to 0.53), and that the reliabilities of the EBVs for milk yield and lactation persistency rank (in descending order) as M

_{1,}M

_{2}, M

_{3}and P

_{1}, P

_{2}, P

_{3}, respectively.

_{6}(b

_{1}M

_{1}+b

_{2}M

_{2}+b

_{3}M

_{3}+b

_{4}P

_{1}+b

_{5}P

_{2}+b

_{6}P

_{6})>I5 (b

_{1}M

_{1}+b

_{2}M

_{2}+b

_{2}M

_{3}) >I

_{4}(b

_{1}M

_{1}+b

_{2}M

_{2}+b

_{3}P

_{1}+b

_{4}P

_{4})>I

_{3}(b

_{1}M

_{1}+b

_{2}M

_{2})>I

_{1}(M

_{1})

**≅**I

_{2}(b

_{1}M

_{1}+b

_{2}P

_{1}). I

_{1}and I

_{2}yielded the smallest ΔH

_{HL}, due to the low genetic correlations between HL and first-lactation milk yield (−0.006) and persistency (0.09). Tsuruta et al. (2004) found that the genetic correlation between productive life and milk yield declined from positive to 0 over the years from 1979 to 1993, mainly in response to changes in producers’ culling practices.

_{5}(b

_{1}M

_{1}+b

_{2}M

_{2}+b

_{3}M

_{3}) or I

_{6}(b

_{1}M

_{1}+b

_{2}M

_{2}+b

_{3}M

_{3}+b

_{4}P

_{1}+b

_{5}P

_{2}+b

_{6}P

_{3}) was about 2.5 times greater than that due to I

_{4}(b

_{1}M

_{1}+b

_{2}M

_{2}+b

_{3}P

_{1}+b

_{4}P

_{2}). These results indicate that milk yield during and persistency of the third lactation play important roles in determining the duration of HL, because third-lactation milk yield has a higher genetic correlation with HL (r

_{G}= 0.257) than does either the first-(r

_{G}= −0.006) or second- (r

_{G}= 0.118) lactation milk yields

_{,}and third-lactation persistency has a higher genetic correlation with HL (r

_{G}= 0.210) than does either the first-(r

_{G}= 0.090) or second- (r

_{G}= 0.180) lactation persistency. The milk yield or persistency of the lactation closest to the culling stage of a cow is much more important than that far back from the culling stage in extending the HL.

_{7}) are identical to those that define net merit components. This similarity means that net merit and selection accuracy due to selection index (I

_{7}) are maximal in improving net merit (

*H*). Net merit due to I

_{net merit}_{6}(b

_{1}M

_{1}+b

_{2}M

_{2}+b

_{3}M

_{3}+b

_{4}P

_{1}+b

_{5}P

_{2}+b

_{6}P

_{6}) was 99.4% of that of selection index (I

_{7}). Net merit, selection accuracy, and HL based on I

_{6}were the largest among indices (I

_{1}to I

_{6}).The first five lactation milk yields based on I

_{6}was the largest among indices (I

_{3}to I

_{6}). The first five lactation milk yields based on I

_{1}or I

_{2}were larger than those based on I

_{3}through I

_{6.}However, HL based on I

_{1}or I

_{2}was nearly 0. Furthermore, collecting data for HL is time-consuming and costly (Smith and Quaas, 1984). Therefore, selection index (I

_{6}), which includes total milk yield and persistency of the first three lactations, as modified by the trait-specific EBVs, is a practical and favorable means of improving lifetime milk yield in the absence of data on HL.

### Effects of survival and discounted rates on selection accuracy, HL, and first through fifth lactation milk yields

_{6}are shown in Table 3. As the discounted rate increased, net merit and selection accuracy increased but HL decreased because increased discounted rate puts greater relative economic weight on earlier rather than later lactations (Table 1) and because the reliabilities of the associated EBVs are higher for earlier rather than later lactation milk yields. Net merit and selection accuracy decreased slightly and HL increased when survival rate increased because increased survival rate puts greater economic weight on later compared with earlier lactations (Table 1).

_{6}. Given the same survival rate (t = 1), when the discounted rate increased from 0% to 4/12% per month, the genetic superiority in terms of total milk yield for the first through fifth lactations increased to 2,721.6 kg

**,**due to the increased reliabilities of the EBVs of the earlier lactation milk yields. Similarly, given the same discounted rate (0% per month), when the survival rate increased from t = 1.0 to t = 1.2, the genetic superiority in terms of total milk yield for the first through fifth lactations decreased from 2,717.9 to 2,699.3 kg, due to the decreased reliabilities of the EBVs and increased relative economic weights of later lactations.

_{6}. Under the same survival rate (t = 1), when the discounted rate increased from 0% to 4/12% per month, genetic superiority in terms of HL due to I

_{6}decreased from 45.4 to 44.9 days (only approximately 1.0%). When the survival rate increased by 20% (t = 1.2) under the same discounted rate (0% per month), genetic superiority in terms of HL due to I

_{6}increased from 45.4 to 47.9 days (approximately 5%). This result indicates that increasing the discounted rate slightly shortens HL whereas increasing the survival rate greatly prolongs HL.

### Applying SNP to maximize the economic weights of later lactation milk yields

_{6}) are shown in Table 4, for which SNP was applied to increase the reliabilities of the trait-specific EBVs. Net merit and selection accuracy decreased with increasing survival rate with or without input from SNP. In the first scenario we tested, the ratio of net merit at 20% increased survival rate to that of the current survival rate was 0.9823 to 0.9830 without applying SNP but 0.9863 to 0.986 with SNP. In particular, the net merit ratio based on the second scenario, in which the reliabilities of second- and third-lactation traits and HL were increased by 20% while those of first-lactation traits were maintained at the current levels, was relatively high, i.e., 0.9890 to 0.9897. Similarly, the ratio of selection accuracy at 20% increased survival rate to that of the current survival rate was 0.9801 to 0.9802 without SNP but 0.9833 to 0.9845 with SNP, with that of the ratio based on the second scenario being particularly high, i.e., 0.9861 to 0.9873. Therefore, using SNP reduced the decreases in net merit and selection accuracy that otherwise were accompanied by the decrease in reliabilities of later lactation traits. In all three scenarios we tested, application of SNP regarding later lactation traits and HL increased net merit and selection accuracy. This trend becomes more evident as the difference in reliability between second- and third-lactation traits and first lactation traits becomes larger (as in the second scenario we tested).

### CONCLUSION

_{G}= −0.006), and by augmenting the effects of second- and third-lactation milk yields on HL (r

_{G}= 0.118 and 0.257, respectively).