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1 National Agricultural Research Center for Hokkaido Region, Hitsujigaoka 1, Toyohiraku, Sapporo, Japan 0628555
2 Dairy and Swine Research and Development Centre, Agriculture and Agri-Food Canada
Corresponding author: K. Togashi; e-mail: tkenji{at}naro.affrc.go.jp.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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ratio of daily estimated breeding value (EBV)(r280/65 = D280/D65), 4) ratio of partial lactation EBV (P280/65 = D66~280/D5~65), and 5) differential daily EBV (d65–280 = D65 – D280), where Di refers to EBV at days in milk (DIM) i. Fundamental differences among these 5 selection criteria were interpreted conceptually with a graph. Persistency, defined as k = (
G65 – G280)/215, was the average daily rate of decline in selection gain from DIM 65 to 280, which is free from the effect of lactation milk on the rate of decline. Parameter k provides an objective measure of persistency, which increases when k < 0 and decreases when k > 0. Of the 5 selection criteria compared, d65–280 and P6 achieved greater persistency at the expense of genetic gain in lactation milk, whereas selection based on EBVL achieved the highest response in lactation milk, but was coupled with greatest decline in persistency. Selection on P280/65 or r280/65 improved both lactation milk and persistency and, thus, is recommended for simultaneous improvement of these 2 economically important traits. Further study of the relative economic values of persistency and lactation milk in order to combine both traits into an index for selection decision is warranted.
Key Words: persistency • lactation milk • selection criterion • test day model
Abbreviation key: RR = random regression
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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Genetic correlations between first lactation yield and various measures of persistency are moderate and favorable (Swalve and Gengler, 1999; Jakobsen et al., 2002), suggesting that it is possible to improve lactation yield and persistency simultaneously. Danell (1982) combined the individual test-day yields into an index to study the possibility of changing the shape of the lactation curve. Ferris et al. (1985) combined yield and the parameters of Wood’s lactation curve (Wood, 1967) with some arbitrary weighting factors to select for lactation curve and yield. Persistency and peak yield vary by countries and feeding systems (Zwald et al., 2001), suggesting that the most profitable lactation curve could vary by feeding systems or by countries. Easing stress caused by negative energy balance in early lactation would raise fertility (Butler and Smith, 1989; Senatore et al., 1996; Loeffler et al., 1999; de Vries et al., 1999). Redistributing the peak yield toward late lactation (i.e., improved persistency) would reduce stress and metabolic disorders in early lactation so that the cow will have higher fertility, thus raising lifetime production.
Index selection based on stage EBV and index selection based on RR coefficients have been presented as 2 equivalent procedures for simultaneous improvement of lactation milk and persistency (Lin and Togashi, 2002; Togashi and Lin, 2003). Both procedures were developed through restricting genetic gains to certain specified lactation stages to achieve the desired curve. The primary objective of this study is to compare the effectiveness of 5 selection criteria in improving total yield and persistency in dairy cows. The secondary objective is to present a procedure for converting the genetic covariance matrices estimated under different production periods (e.g., 305-d vs. 335-d production period).
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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A Legendre polynomial function of the fifth degree (6 covariates including 0 order) was used to fit the lactation curve in the framework of a RR test day model.
The formula used to standardize DIM (ranging from –1 to 1) under a 335-d production period is
 | ([1]) |
Similarly, the standardization of DIM under a 305-d production period is
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Let matrix G* be the genetic covariance matrix of daily yields from DIM 5 to 305 under a 335-d production system. Then G* corresponds to the (301 x 301) upper-left submatrix of a (331 x 331) genetic covariance matrix of daily yields from DIM 5 to 335:
 | ([3]) |
where 
* = a 301 x 6 matrix containing quintic Legendre polynomials with DIM standardized according to Eq. [1] and 
genetic covariance matrix of RR coefficients for 335-d production periods.
Similarly, the genetic covariance matrix of daily yields under a 305-d production period (G) can be written as
 | ([4]) |
where = a 301 x 6 matrix containing quintic Legendre polynomials with DIM standardized according to Eq. [2] and K6 = a 6 x 6 genetic covariance matrix of RR coefficients for a 305-d production period.
Premultiplying and postmultiplying Eq. [4] by ' and , respectively, leads to
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Re-arranging this formula leads to K6 = (')–1'G(')–1. It is expected that G = G* = *K*6*'. Therefore,
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and subsequently,
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Equation (5) was used to convert the genetic covariance matrix of RR coefficients from a 335-d (Pool and Meuwissen, 2001) to 305-d production period in this study.
Selection Criteria Compared
Selection based on lactation EBV (EBVL).
Current selection under RR test day model is based on lactation EBV, i.e., EBVL = 1'
, where 1 is a summing vector, is the Legendre polynomial matrix of a whole lactation, and is a vector of RR coefficients. The correlated response in DIM i (Gi, i = 5, 6, ... 305) to selection on EBVL is
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where bDi•EBVL is the regression coefficient of EBV at DIMi (Di) on lactation EBV (EBVL), 
the intensity of selection, and 
EBVL the standard deviation of EBVL.
The daily correlated responses to selection on EBVL can be expressed in a matrix form:
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where L is a 301 x 1 vector of daily responses from DIM 5 to 305, G is the 301 x 301 genetic covariance matrix of daily yields from DIM 5 to 305, and 
. The total lactation response is GT•EBVL = 1'L.
Selection based on differential EBV.
The difference between EBV at 2 DIM is defined as d65–280 = D65 – D280, where 65 is DIM at peak based on the report of Pool and Meuwissen (2001). Because D65 is greater than D280, the smaller the difference (d65–280), the greater the persistency. When selection is on d65–280, the correlated response in DIM j is
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In matrix form, the daily correlated responses from DIM 5 to 305 to selection on d65–280 are
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where 
is a 301 x 1 vector containing daily responses from DIM 5 to 305, and G65 and G280 are the columns of the genetic covariance matrix of daily yields (G) corresponding to DIM 65 and 280, respectively. Note that the first column of G (i.e., G1) contains the (co)variances between DIM 5 and each day of the lactation; so, G65 and G280 are columns 61 and 276 of G, respectively. Total correlated response to selection on d65–280 is equal to GT•d65–280 = 1'd.
Selection based on ratio of daily EBV.
According to Taylor series expansion (Mood et al., 1987), a ratio (say r = x/y) can be approximated as follows:
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It follows that 
and 
Let Di be the daily EBV of DIM i. The ratio of D280 to D65 is r280/65 = D280/D65. By this definition, a large ratio is desirable for persistency. When selection is based on r280/65, the total correlated response (GT•r280/65) is
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where
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and
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Note that the linear index derived to maximize the genetic value of 
is identical to that developed by Lin (1980) to maximize the genetic value of 
Lin (1980) presented a linear index approach to sidestep the problems of estimating the genetic parameters of a ratio and to maximize the response in a ratio.
Selection based on ratio of partial lactation EBV.
The ratio of partial lactation EBV is defined as P280/65 = D66~280/D5~65, where D66~280 is the cumulative EBV from DIM 66 to 280 with the symbol "~" denoting the range. On the basis of Taylor series expansion of a ratio as shown previously,
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where
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Correlated response in DIM j to selection on P280/65 is Gj = Cov(P280/65, Dj)(i/P280/65). Total response in lactation is 
.
Selection based on P6.
Jakobsen et al. (2002) defined 
with a large value of P5, indicating a high persistency. In contrast to their report, the initial results of this study indicated that selection should be for a small P5 rather than a large P5 to improve persistency. This study defined P6 as 
, with a large value of P6 indicating a high persistency. The DIM at peak was around 65 (Pool and Meuwissen, 2001). Correlated response in DIM j to selection on P6 is 
where 
. Total response to selection on P6 is 
.
Basis for Comparison of Selection Criteria
The rate of decline between 2 consecutive DIM after selection is
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where k0 is the rate of decline before selection and k1 is the genetic change in rate of decline because of selection.
In this study, persistency (k) is defined as the average daily rate of decline in genetic gain from DIM 65 (peak) to 280:
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By this definition, the value of k is not a function of EBV and, thus, is free from the effect of lactation milk on the rate of decline. Therefore, parameter k provides an objective measure of persistency. Note that parameter k is in inverse relationship to persistency; the smaller the value k, the greater the improvement of persistency.
Let m be the midpoint between DIM 65 (peak) and 280. When Gm~280 > G65~m, persistency improves. If Gm~280 = G65~m, persistency remains unchanged. If Gm~280 < G65~m, persistency deteriorates. Total lactation response, genetic response for each stage of the lactation, and average daily genetic gains were used to assess the improvement of persistency associated with each selection criterion. Selection intensity () is set at 0.1 for all criteria to compare selection responses. Genetic covariance matrix of daily yields from DIM 5 to 305 (G) was calculated as G = K', where K is a 6 x 6 genetic covariance matrix of RR coefficients and is a 301 x 6 Legendre polynomial matrix. Matrix K* of a 335-d production system (Pool and Meuwissen, 2001) was converted into matrix K of a 305-d production system for this study according to conversion Eq. [5].
Graphic Depiction of Different Selection Criteria
The interrelationship among differential daily EBV (d65–280 = D65 – D280), ratio of daily EBV (r280/65 = D280/D65), ratio of partial lactation EBV (P280/65 = D66~280/D5~65) and 
was depicted graphically in Figure 1
. Criterion d65–280 refers to the difference in height between c and e (Figure 1). Therefore, selecting for a smaller difference between c and e tends to reduce the triangle cef, as the length between DIM 65 and 280 is a constant. Criterion r280/65 can be linearly approximated as D280 – (
280/65)D65 (Lin, 1980). Let g be the value of (280/65)D65 on DIM 65 in Figure 1. The ratio of r280/65 would improve persistency as long as the height of e (i.e., f) increases at a faster rate than the height of g. Thus, selecting for a larger ratio of r280/65 would reduce the difference in height between g and e in Figure 1, thus implying selection for a smaller area of the rectangle gefh.
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Criterion d65–280 assigns equal weights (1:1) between D280 and D65. Criterion r280/65 can be linearly expressed as D280 – (280/65)D65 (Lin, 1980), indicating that the weighting factors between D280 and D65 are 1:280/65. Because 280/65 is <1, criterion r280/65 places greater weight on D280 than on D65. Implicitly, r280/65 aims to improve lactation milk and persistency mainly by increasing D280. Similarly, criterion P280/65 can be linearly represented as 
. As shown in Figure 1
, D5~65 is equal to the area of mnac, and D66~280 is equal to (abfc – D65). Because 
is >1, criterion P280/65 puts greater weight on D5~65 than on D66~280 to increase the value of P280/65.
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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. The same principle applies to the development of conversion formulas between any 2 different production periods. Once K6 and 
are converted, the genetic covariance matrix of daily yields can be computed subsequently according to Eq. [3] and [4]. Matrix (Pool and Meuwissen, 2001), under a 335-d production system, was converted to K6, which in turn was used to compute genetic covariance matrix of daily yields (G) under a 305-d system. This G matrix was found to be identical to the 301 x 301 upper-left submatrix of the 331 x 331 full genetic covariance matrix of daily yields, thus confirming the validity of conversion between K6 and . Although the G matrix on a 305-d basis can be extracted directly from that on a 335-d basis, the conversion formula is needed to convert the G matrix from a 305-d basis to a 335-d basis for comparison purposes. However, the K matrix of RR coefficients is needed for conversion between 305-d and 335-d systems because the K matrix on a 305-d basis is not a subset of that on a 335-d basis.
Comparison of Selection Criteria
Persistency is affected by the lactation yield genetically (Jamrozik et al., 1998; Swalve and Gengler, 1999; Jakobsen et al., 2002). For the purpose of comparing the effectiveness of different selection criteria in improving persistency, this study defined the persistency parameter (k) as the average daily rate of decline in genetic gain from DIM 65 (peak) to DIM 280. Given a fixed amount of cumulative genetic gain from DIM 65 to 280, a selection criterion that yields a smaller value of k will achieve a greater improvement in persistency. Persistency improves when k is negative (i.e., G65 < G280), deteriorates when k is positive (i.e., G65 > G280), and remains unchanged when k is equal to 0 (i.e., G65 = G280). The value k was defined independently of the lactation yield before selection and is, thus, free from the effect of lactation milk on the rate of decline. Therefore, parameter k provides an objective measure of persistency. The average daily rate of decline in genetic gain from DIM 65 to 280 was divided into 3 intervals: DIM 65 to 89, DIM 90 to 210, and DIM 211 to 280. Genetic responses in persistency and lactation milk from the 5 selection criteria are given in Table 1.
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, conventional selection based on EBVL was expected to raise the lactation milk by 67 kg EBV and to decrease persistency because of the positive value of k. Although selection for a large value of 
increased persistency, it incurred the greatest loss of lactation milk by 26.53 kg EBV (Table 1). The improvement in persistency caused by P6 is in agreement with graphic interpretation of P6 in Figure 1. The report of Jakobsen et al. (2002) about P5 was misleading because selection should be for a lower value of P5 rather than a higher value for improvement of persistency.
Of the 5 selection criteria studied, criteria d65–280 (i.e., D65 – D280) and P6 resulted in greater improvement in persistency mainly because both criteria are more related to the definition of persistency k = (G65 – G280)/215 than were the other three criteria compared. However, both d65–280 and P6 incurred greater loss in lactation milk by 13.4 and 26.5 kg EBV, respectively (Table 1). This result is in agreement with the graphic depiction of selection criteria in the previous section. Therefore, both criteria are not a viable approach for simultaneous improvement of persistency and lactation milk. Dekkers et al. (1996) recommended the use of differential yield between DIM 60 and 280 for genetic evaluation for persistency of lactation because it was less correlated with 305-d yield.
The response in persistency to selection based on the ratio D280/D65 is negative, indicating an improvement of persistency. In addition, this criterion resulted in a gain of 37 kg EBV in lactation milk (Table 1). The increase in lactation milk is because a relatively larger weighting factor was placed on the increase in D280 than on the decrease D65, as graphically depicted earlier (1: 280/65). Selection on the ratio of partial lactation EBV (D66~280/D5~65) increased persistency and increased lactation milk by 34 kg EBV. Note that the smaller the parameter k, the greater the increase in persistency. As shown in Table 1, all of the 5 selection criteria had a smaller genetic gain per day in stage DIM 65 to 89 than in stage DIM 90 to 210, which in turn had a smaller gain than in stage DIM 211 to 280. This indicates that persistency was mainly achieved in stage DIM 65 to 89, and, thus, more weights need to be placed on stages DIM 90 to 210 and DIM 211 to 280 to further reduce the slope of the lactation curve.
The correlated gains on DIM 5, 30, 60, 65, 90, 120, 180, 240, 280, and 305 are given in Table 2 and Figure 2. The daily genetic gains on DIM 5 through 220 caused by selection on D65–D280 were negative and then became positive from DIM 220 onward. In contrast, selection based on the ratio of daily EBV (r280/65) increased daily EBV throughout the whole lactation, particularly toward the end of lactation. Daily genetic response to selection on ratio of partial lactation EBV (P280/65) was negative from DIM 5 to 34 and then increased steadily from DIM 35 to 200 and tapered off from DIM 200 onward (Figure 2). Similar trend in daily response was observed for criterion r280/65, except that daily response increased steadily toward the end of lactation (Figure 2). This difference is due to the fact that P280/65 aims to reduce EBV from d 5 to 65 and increase EBV from d 66 to 280, whereas r280/65 is designed to reduce D65 and increase D280 so that EBV continues to increase beyond DIM 280 as correlated response. This response confirms the above graphic depiction that P280/65 places greater weight on D5~65 than on D66~280, whereas r280/65 places greater weight on D280 than on D65 to increase the respective value of the selection criteria. The responses in lactation milk caused by r280/65 and P280/65 are similar (36.8 vs. 34.2 kg EBV). The responses in persistency (k) from r280/65 and P280/65 are –0.00064 and –0.00053, respectively (Table 1). Criterion r280/65 produced a slightly better persistency than P280/65. Criterion P280/65 is most effective in decreasing EBV in early lactation and maintaining a rather constant gain in EBV in mid and late lactation. Criterion EBVL yielded greater daily responses than the other 4 election criteria (d65–280, r280/65, P280/65, and P6) during the lactation (Figure 2).
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. Average daily genetic gains among 4 stages (DIM 5 to 65, DIM 66 to 170, DIM 171 to 280, and DIM 281 to 305) are shown in Table 3. Average daily genetic gain caused by d65–280 was negative in stage DIM 5 to 65 and then increased steadily with advancing DIM of the lactation. This means an improvement of persistency by selection on d65–280 because average daily genetic gain in the latter stage must be larger than that of the former stage to raise persistency. Criterion r280/65 consistently increases the average daily genetic gain to the end of lactation, whereas criterion P280/65 increases the average daily genetic gain up to stage DIM 171 to 280 and then decreases slightly beyond DIM 280 (Table 3; Figure 2). The response pattern exhibited by both r280/65 and P280/65 indicates that both criteria did result in a better persistency. This is in agreement with graphic interpretation of criterion P280/65, which aims to reduce the area of mnac before the peak and increase the area of abfc beyond the peak in Figure 1. Improved persistency could result in higher milk yield in the last stage of lactation. Keeping high yield toward the end of lactation may not be directly related to long calving interval. Even if relatively high yield is observed in late lactation, cows can be dried for next calving. Daily genetic gains caused by selection on P280/65 decreased slightly in the last stage of lactation (stage DIM 281 ~ 305). Further study is needed regarding how to strike a balance between improved persistency and optimum calving interval. The daily genetic loss in early lactation from P280/65 would alleviate stress in early lactation. Average daily response to selection on EBVL decreased from peak period to the end of lactation, meaning a decrease in persistency (Table 3; Figure 2). Criterion P6 decreased the area of cef (Figure 1) to improve persistency by reducing the genetic gain at the peak (DIM 65) by 0.17 kg EBV and by increasing genetic gain at DIM 280 by 0.01 kg EBV (Table 2). P6 showed similar trend in persistency to d65–280 (Figure 2), but the former incurred greater genetic loss in lactation milk than the latter.
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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Received for publication March 13, 2003. Accepted for publication December 20, 2003.
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REFERENCES |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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, x = r280/65(D280/D65) and — = P280/65 (D66~d280/D5~65).