Lactation Persistency as a Component Trait of the Selection Index and Increase in Reliability by Using Single Nucleotide Polymorphism in Net Merit Defined as the First Five Lactation Milk Yields and Herd Life

Derivation of survival matrix

We divided the life cycle of a dairy cow into 5 stages: stage 1 refers to the period from birth to the end of the first lactation; stage 2 is from the end of the first lactation to the end of the second lactation; and so forth. The 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 r₀₁ 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., r₁₂ = 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 r₀₁, r_12, r_23, r_34, and r₄₅ 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

Genetic lifetime milk yield (H₅) is defined as follows:

where G_i and w_i are the genetic value and relative economic weight of the i^th lactation milk yield, respectively (i = 1 to 5). H₅ is genetic lifetime milk yield derived by using only milk yield for the first five lactations.

The discounted expressions (Hill, 1971; Weller et al., 1994) are as follows:

where d_j is the appropriate discounting factor, for which the subscript 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 = (w₁, w₂, w₃, w₄, w₅)) reflect the actual average of each lactation milk yield, so that w₅ is w₅=AM_5×5S_5×120d with the vector d = (d₁, … d₁₂₀)′ and the diagonal matrix AM_5×5 whose diagonal elements are the averages of the i^th lactation milk ( Mi¯, i = 1 to 5). The discounted total milk yield during the given period of 120 months from the original investment is described as w5′1.

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

The relative economic weights (in genetic standard deviations) of total milk yield (the sum of the first five lactation milk yields) and HL were studied as: wmilk*:wHL*=1:0.389, where wmilk* and wHL*.. are the relative economic weights (in genetic standard deviations) for the first five lactation milk yields and HL, respectively. The ratio of wmilk*:wHL*=1:0.389 is equivalent to the ratio of 7.2:2.8 currently used in Japan and is in close agreement with the reported value of 7.0:2.5 (Burnside et al., 1984; Allaire and Gibson, 1992; Weller and Ron, 1992; Dekkers, 1994). Net merit ( Hnet merit*), defined in terms of the first five lactation milk yields and HL in genetic standard deviations, is as follows:

where wia (i = 1 to 5) is the relative economic weight for the i^th lactation milk yield described as wia=wi/(∑i=15wi); ∑i=15wia=1, w6ais the relative economic weight for HL,

and σGHL is the genetic standard deviation of HL. On the other hand, wHL*/wmilk*=2.87.2=w6aσGHLσH5, so that net merit (H_{net merit}) in units of actual values of milk (kg) and HL (days) is as follows:

Comparison of selection indices in terms of maximizing net merit

Lactation milk yield and persistency during the first one, two, or three parities are taken as the component traits of selection indices (I₁–I₆) to maximize net merit (H_{net merit}) 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 (I₇), comprising the same components as those for net merit, is set as a basis of comparison:

where M_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.

Here we use the index I₆ = b₁M₁+b₂M₂+b₃M₃+b₄P₁+b₅P₂+b₆P₃ as an example to demonstrate the derivation. In matrix notation, the index can be written as,

where and b6′=(b1b2b3b4b5b6) and g6′=(M1M2M3P1P2P3).

As indicated earlier, the goal is to maximize the genetic superiority of animals selected as breeding candidates in terms of net merit (H_{net merit}). The covariance matrix between I₆ and H_{net merit} is as follows:

where w′=(w1aw2aw3aw4aw5aw6a) and

The variance of I₆ is as follows:

where GP_i is the genetic value of the i^th lactation persistency, rGi is the accuracy of the i^th lactation milk yield EBV, and rGPi is the accuracy of the i^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.

The genetic superiority of selected bulls as breeding candidates or the genetic difference between candidate bulls and all the bulls due to selection on I₆ for net merit (H_{net merit}) is as follows:

where σI6 is the standard deviation of the index I₆ and i¯ is the selection intensity.

And selection accuracy is as follows:

where G is the genetic covariance matrix of the first five lactation milk yields and HL. The genetic superiorities of selected bulls as breeding candidates in the first five lactation milk yields (ΔG₁,ΔG₂,…,ΔG₅) and HL(ΔG_HL) are shown as follows:

We compared the seven selection indices in terms of net merit, selection accuracy, genetic superiorities over the total of the first through fifth lactation milk yields ( ∑i=15ΔGi) and herd life (Δ_HL). Selection intensity was set to unity ( i¯=1) for all selection criteria compared in the current study.

Application of SNP

Currently in Japan, the reliabilities of EBV among selected bulls as breeding candidates for milk yield of the first five lactations are 0.816, 0.708, 0.597, 0.550, and 0.500, respectively; those for persistency of the first three lactations are 0.689, 0.612, and 0.493, respectively; and that for HL is 0.570 (Hagiya et al., 2010). The reliabilities of first-lactation traits are higher than those of second or third lactation traits, because collection of records is more rapid for first-lactation compared with second- and third-lactation traits. Using SNP to improve reliability is more important for second- or third-lactation traits than first-lactation traits, due to this difference in data collection time. Therefore, we considered three scenarios: increasing the reliabilities of second-and third-lactation traits and of HL by 10% while maintaining the reliabilities of first-lactation traits at their current levels; increasing the reliabilities of second- and third-lactation traits and of HL by 20% while maintaining the reliabilities of first-lactation traits at their current levels, and increasing the reliabilities of second- and third-lactation traits and of HL by 20% and the reliabilities of first-lactation traits by 10%.

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

The relative economic weights for the first five lactation milk yields in terms of actual milk yield (kg) are shown in Table 1, which is based on survival matrix S_{i j} 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 (w₂, w₃, w₄, and w₅) were expressed as ratios of the first lactation milk (w₁) by setting w₁ =1. The relative economic weights of w₂ through w₅ 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 w₂ through w₅ decreased as the discounted rate increased because the time to reach the second through fifth lactations is longer than to reach the first lactation.

Table 1

Relative economic weights of the first five lactation milk yields

Among the 9 combinations determined at all levels of discounted rates (0/12%, 1/12%, and 4/12% per month) and survival rates (t = 1.0, 1.05, and1.1; t times as large as the current survival rate), the relative economic weight of the first lactation was greater than that of the second lactation, which in turn was greater than that of third lactation and so forth (w₁>w₂>w₃>w₄>w₅). This effect occurred because survival rate decreases as parity increases, as mentioned earlier, i.e., the survival rates of r₀₁, r₁₂, r₂₃, r_34, and r₄₅ 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.

The relative economic weight of the pooled third, fourth, and fifth lactation milk yields was higher than that of first or second lactation milk yield, especially at low discounted rates and high survival rates. This result indicates that, compared with first- or second-lactation milk yield, the third-, fourth-, and fifth-lactation milk yields play important roles in the genetic improvement of lifetime performance, especially under low discounted and high survival rates. As an example, when the survival rate improved by 20% (t = 1.2) under a discounted rate of 0, the relative economic weight of the pooled third-, fourth-, and fifth-lactation milk yields was about two times greater than that of the first parity.

Currently in Japan, the first three lactation milk yields are given equal economic weights. This assignment overestimates the relative economic weight of the second and third lactations (w₂ and w₃), particularly under the current survival rate (t = 1). Overestimation of w₂ and w₃ 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 (w₃ > w₂ > w₁) 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 (M3¯>M2¯>M¯1).

The value of w5′1 (the total discounted yields of the first through fifth lactation milks) during the 120 months from the original investment is shown in Table 1 as a ratio of the total discounted yields of the first through fifth lactation milks under the current survival rate. The total discounted yields of the first through fifth lactation milks during 120 months increased with increasing survival rate. For example, under a discounted rate of 0, when survival rate improved by 20% (t = 1.2), the total discounted yields of the first through fifth lactation milks increased by 1.57 times compared with that under the current survival rate (t = 1). Therefore, it is important to improve survival rate through both genetics and management.

Comparison of selection indices in terms of maximizing net merit

The selection accuracy and genetic superiorities of net merit, the total yield of the first through fifth lactation milks and HL derived by using selection indices (I₁ to I₇) 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 I₂ (b₁M₁+b₂P₁) than in I₁ (M₁), in I₄ (b₁M₁+b₂M₂+b₃P₁+b₄P₂) than in I₃ (b₁M₁+b₂M₂), and in I₆ (b₁M₁+b₂M₂+b₃M₃+b₄P₁+b₅P₂+b₆P₃) than in I₅ (b₁M₁+b₂M₂+b₂M₃), respectively. In addition, the genetic superiority of HL in first and second parity and in first, second, and third parity components was larger in I₄ (b₁M₁+b₂M₂+b₃P₁+b₄P₂) than in I₃ (b₁M₁+b₂M₂) and in I₆ (b₁M₁+b₂M₂+b₃M₃+b₄P₁+b₅P₂+b₆P₃) than in I₅ (b₁M₁+ b₂M₂+b₂M₃), respectively. In particular, the genetic superiority of HL was 45.4 days from I₆ compared with 39.0 days from I₅. 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 I₃ (b₁M₁+b₂M₂) and I₄ (b₁M₁+b₂M₂+b₃P₁+b₄P₂) and between I₅ (b₁M₁+b₂M₂+b₂M₃) and I₆ (b₁M₁+b₂M₂+b₃M₃+b₄P₁+b₅P₂+b₆P₃), 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.

Table 2

Genetic superiorities of selected bulls as breeding candidates based on selection indices compared (I₁ to I₇)1 under the current survival and zero discounted rates

The genetic superiority of the first five lactation milk yields was highest for I₂ (b₁M₁+b₂P₁) among the six indices (I₁ to I₆), and followed by I₁ (M₁). 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, M₂, M₃ and P₁, P₂, P₃, respectively.

The order of genetic superiority of HL was I₆ (b₁M₁+b₂M₂+b₃M₃+b₄P₁+b₅P₂+b₆P₆)>I5 (b₁M₁+b₂M₂+b₂M₃) >I₄ (b₁M₁+b₂M₂+b₃P₁+b₄P₄)>I₃ (b₁M₁+b₂M₂)>I₁ (M₁) ≅ I₂ (b₁M₁+b₂P₁). I₁ and I₂ 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.

Dairy producers’ decisions regarding culling cows reflect various factors such as milk yield, reproduction problems, mastitis, lameness, and labor costs. All of these factors can change over time. Functional traits (such as resistance to mastitis and lameness, fertility, calving ease, and longevity) are defined as the characteristics of an animal that increase production efficiency through reduced input costs rather than increased output of products (Groen et al., 1998). High milk production during the first lactation decreases the efficacy of the immune system in the udder and leads to damage that persists throughout subsequent lactations (Rupp et al., 2000). The genetic superiority of HL due to index I₅ (b₁M₁+b₂M₂+b₃M₃) or I₆ (b₁M₁+b₂M₂+b₃M₃+b₄P₁+b₅P₂+b₆P₃) was about 2.5 times greater than that due to I₄ (b₁M₁+b₂M₂+b₃P₁+b₄P₂). 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.

The component traits of selection index (I₇) are identical to those that define net merit components. This similarity means that net merit and selection accuracy due to selection index (I₇) are maximal in improving net merit (H_{net merit}). Net merit due to I₆ (b₁M₁+b₂M₂+b₃M₃+b₄P₁+b₅P₂+b₆P₆) was 99.4% of that of selection index (I₇). Net merit, selection accuracy, and HL based on I₆ were the largest among indices (I₁ to I₆).The first five lactation milk yields based on I₆ was the largest among indices (I₃ to I₆). The first five lactation milk yields based on I₁ or I₂ were larger than those based on I₃ through I_6. However, HL based on I₁ or I₂ was nearly 0. Furthermore, collecting data for HL is time-consuming and costly (Smith and Quaas, 1984). Therefore, selection index (I₆), 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

The effects of survival and discounted rates on selection accuracy, net merit, HL, and total milk yield during the first through fifth lactations according to selection index I₆ 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).

Table 3

Net merit, selection accuracy, and genetic superiorities of selected bulls as breeding candidates for first through fifth lactation milk yields, and for herd life based on selection index (I₆)1 at increased survival and discounted rates

Given the current survival rate (t = 1) and a discounted rate of 0% per month, the genetic superiority of selected bulls as breeding candidates in terms of total milk yield for the first through fifth lactations was 2,717.9 kg, according to I₆. 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.

Under the current survival rate (t = 1) and a discounted rate of 0%, genetic superiority in terms of HL was 45.4 days, according to I₆. 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 I₆ 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 I₆ 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

The net merit, selection accuracy, and genetic superiorities associated for selected bulls as breeding candidates based on selection index (I₆) 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).

Table 4

Net merit, selection accuracy, and genetic superiorities of selected bulls as breeding candidates for first through fifth lactation milk yields, and for herd life based on selection index (I₆)1 at increased survival and discount rates when reliability of EBV is increased by SNP2

In all alternative scenarios tested, applying SNP increased the first five lactation milk yields compared with these traits in the current scenario without SNP, and this increase became more pronounced as reliability increased (third scenario). Similarly, adding SNP data increased HL in all the scenarios (particularly the second) compared with that of the current scenario without SNP. The longer HL that occurred during the second scenario compared with that which occurred during the third scenario, where the reliabilities of second- and third-lactation traits and HL were increased by 20% while those of first-lactation traits were increased by 10%, reflects the genetic correlation between HL and the first-, second-, and third-lactation milk yields (−0.006, 0.118, and 0.257, respectively). Therefore, applying SNP to increase the reliabilities of EBV of second-and third-lactation traits and HL while maintaining those of first-lactation traits at their current levels increased net merit and HL by maximizing the increased economic importance of later lactation traits resulting from increased survival rates, by reducing the effect of first-lactation milk yield on HL, and by augmenting the effects of the second-and third-lactation milk yields on HL.