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1 Dipartimento di Scienze Zootecniche, Università di Sassari, Via De Nicola 9, 07100 Sassari, Italy
2 Italian Association of Simmental Breeders, Via Nievo 19, 33100 Udine, Italy
Corresponding author: N. P. P. Macciotta; e-mail: macciott{at}uniss.it.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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Key Words: lactation curve • lactation peak • test day • multivariate factor analysis
Abbreviation key: TD = test day
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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In spite of the great technical importance, there is no consensus on the most suitable measure of persistency and, in general, of main aspects of lactation-curve shape (Gengler, 1996; Jamrozik et al., 1998; Grossman et al., 1999; Rekaya et al., 2001). The modeling of lactation curves by analytical functions of time allows for estimating the increasing rate of production in the first part of lactation, the time at which lactation peak is attained, and the rate of decline of milk yield after the peak, as combinations of function parameters (France and Thornley, 1984). Such an approach is very efficient for averaging lactation curves of homogeneous groups of animals, but its usefulness becomes questionable when individual lactation patterns are fitted (Olori et al., 1999). The alternative methods based on combinations of test-day (TD) records taken at different stages of lactation, such as ratios between cumulated yields or measures of TD variation (Sölkner and Fuchs, 1987; Swalve, 1995), have been essentially aimed at expressing the shape of the lactation pattern by a single measure. However, they failed in characterizing persistency in a unique manner because they are not invariant with respect to the time period chosen (Rekaya et al., 2001). As a consequence, for example, genetic correlations between milk yield and persistency depend largely on what measure is used (Sölkner and Fuchs, 1987; Haile-Mariam et al., 2003). The major drawback of these empirical methods is the arbitrary assignment of relative weights to TD of different stages of lactation. This problem remains when EBV estimated by random regression TD models are combined in a genetic index of lactation persistency (Schaeffer and Dekkers, 1994). In fact, when relative weights of TD are estimated from the genetic (co)variance matrix of random regression coefficients (Togashi and Lin, 2003), a priori genetic changes for different lactation stages have to be assumed.
In this work, individual lactation curve shapes of Italian Simmental cows are analyzed by means of factor analysis of TD data. This multivariate methodology is characterized by both the power of synthesis of mathematical modeling and direct reference to a raw data structure. These features allow factor analysis to express the 2 main components of lactation-curve shape—i.e., the rate of increase of yield toward the lactation peak, and the persistency of lactation by 2 linear combinations of TD whose relative weights are objectively derived from the correlation structure of original data without previous assumptions.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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According to the factor analysis model, each of the m TD records can be represented as a linear combination of 2 common factors:
 | ([1]) |
where
| yi | = | TD records taken at different distance from calving,
| bij | = | factor coefficients (or loadings), i.e., correlations between the jth common factors and the ith TD record,
| Xl and Xp | = | latent common factors, and
| ei | = | random residual.
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The rationale of factor analysis is the modeling of the correlation matrix of TD records within lactation (S)
 | ([2]) |
where
| B | = | the matrix of the factor coefficients of model [1], and
|  | = | residual (co)variance matrix (McDonald, 1985; Morrison, 1976).
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The maximum-likelihood method estimates BB' by minimizing the residual matrix . Moreover, the factor rotation allows for the estimation of different B matrices that, with the constraint of keeping BB' invariant, simplify the factor structure and facilitate their interpretation in terms of relationships with the 2 main aspects of lactation curve shape. The scores of 2 common factors for each lactation can be then calculated as follows:
 | ([3]) |
where
| x' | = | [ l p],
| y' | = | row vector of standardized TD records, and
| (BB'+)-1*B | = | scoring coefficients.
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According to model [3], all TD records are considered in the calculation of the scores of both factors. Moreover, relative TD weights are not assigned a priori, but they are derived from the correlation structure of original data (BB'+) and from the correlations between factors and the original variables (B).
Data
Original data were milk TD yields recorded according to the A4 scheme (about 4 wk between 2 consecutive tests, respectively) in the period 1989 to 2002, supplied by the Italian Association of Simmental Cow Breeders. From these data, an archive of 5728 lactations of 4932 cows was extracted. The number of TD records per lactation was fixed at 7. Lactations with less than 7 TD records were discarded, whereas extra TD records for lactation with more than 7 records were deleted. Edits were also on DIM at which the first TD was recorded (<20), parity (1 to 6), lactation length (200 to 280 d), calving interval (300 to 600 d), and herd size (>15 cows).
The 7 TD records for each cow were regarded as different traits (MILK1, MILK2, . . . MILK7). Means and standard deviations for the 7 TD milk yields are reported in Table 1
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. To evaluate relationships between the 2 common factors Xl and Xp and the environmental effects known to influence the shape of the lactation curve, factor scores were analyzed with the following mixed linear model:
 | ([4]) |
where
| Y | = | Xl or Xp scores,
| PARi | = | fixed effect of the parity class (1, 2, . . . , 6),
| SEAj | = | fixed effect of calving season (1 = Jan–Feb, .., 6 = Nov–Dec),
| YEARk | = | fixed effect of year of calving (1 = 1989, ..., 11 = 1999),
| PRODl | = | covariable represented by the 305-d milk,
| Hm | = | random effect of the herd, and
| eijklmn | = | random residual.
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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, Pearson and partial correlations are reported, the latter measuring relationships among each pair of variables controlling possible effects of other variables. A marked reduction of partial correlations in comparison with Pearson correlation can be noticed. An objective measure of such a reduction is the Kaiser measure of sampling adequacy that was about 0.91 in this study, higher than the 0.80 usually assumed as the minimum threshold for the suitability of the data set for the factor analysis (Cerny and Kaiser, 1977).
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According to equation [2], the correlation matrix of original variables can be decomposed as follows (symm = symmetric)
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The goodness of fit of the factor model is confirmed by the values of (second matrix of the right term of the equation). Off diagonal elements are the residual correlations, i.e., the part of the original correlations that have not been reconstructed by the factor model, whereas diagonal elements are the uniqueness of variables, i.e., the specific variance of each original variable plus the random error (Enevoldsen et al., 1996). Values of all elements, especially of those off diagonal, are very close to zero, thus evidencing the suitability of the factor model.
Factors Xl and Xp are mainly correlated with the first 3 TD and the last 3 TD of the lactation, respectively (Table 3). Because the maximum yield occurs within the first 3 TD, Xl scores can be considered as an index of the increasing rate of milk yield in the first phase of lactation. On the other hand, Xp scores can be proposed as a measure of the inverse of the rate of decrease of milk yield in the second part of lactation (persistency).
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using scoring coefficients, i.e., the relative weights to be assigned to each standardized TD record, reported in Table 4. All TD records are used in the calculation of the 2 factors, even if with different relative weights and signs. As an example, the calculation of Xl and Xp scores for a cow is reported. Original values (16.6, 19, 16.8, 14, 16.2, 15, and 12.4 kg for MILK1, MILK2, MILK3, MILK4, MILK5, MILK6, and MILK7, respectively) are standardized using the overall means and standard deviations (Table 1).
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Correlations among Xl and Xp scores and the total 305-d milk yield were 0.65 and 0.73, respectively, indicating that an increase of either the rate of increase to lactation peak or the lactation persistency, or both, results in a higher 305-d milk. The low correlation between Xp and Xl scores (0.11) can be understood by examining the crossed frequency distribution of factor scores (Table 5). Actually, because factor scores are normal standardized variables, most of cows have values of both Xl and Xp scores around the mean, but the distribution of the factor scores within each of the levels of the other shows values in almost all the cells. It can be observed that, in the classes of average values of Xl ("-1 0" or "0 1"), there are 578 cows out of 4162 that have values of Xp higher than one. Thus about 14% of cows with a medium peak level show high persistency of lactation.
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, 2, and 3, where average lactation patterns of groups of animals classified in different positions of Table 5 are shown. Lactation patterns, reported in Figure 1, show that an increase of both the 2 factors results in a parallel shift of the whole curve to higher values, i.e., in an increase of the total lactation yield, whereas increasing values of one factor for a constant value of the other result in patterns characterized by a progressively higher rate of increase to lactation peak or persistency (Figures 2 and 3, respectively). On the basis of these results, Xl and Xp scores can be proposed as criteria for detecting cows with the more suitable lactation-curve shape at a given 305-d yield level.
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and 7. The 2 variables considered were particularly affected by parity and calving season (P < 0.0001), as evidenced by the change in signs of least squares means across levels of each factor. Calving year also had a significant effect, even if to a lesser extent (P < 0.01), as evidenced by the absence of changes in sign of least squares means across years.
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). Spring calvings showed the highest peak value due to the favorable environmental conditions (pastures, climate). Finally, a slight, even if irregular, tendency of peak factor to increase across years can be observed. This result could be ascribed to the effect of the selection program carried out on this dual-purpose breed.
First-parity cows showed the highest values of persistency that tends to decrease in older cows (Table 7), in agreement with previous figures obtained for cattle (Shanks et al., 1981; Sölkner and Fuchs, 1987). The higher persistency of younger animals is usually explained with the maturation process, which is still in progress in young animals and that counteracts the normal decline in milk yield in the second part of lactation (Stanton et al., 1992). Cows calving in fall and winter were the most persistent, probably because they had the end of lactation in the spring, i.e., when environmental conditions are better. This result agrees with previous findings on persistency calculated with the mean standard deviation of TD yields along the 305-d lactation (Sölkner and Fuchs, 1987). The lowest persistency has been observed for cows calving in spring. No detectable trends of persistency were evidenced across years of calving.
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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The 2 latent factors allow for the separation of individual lactation curves characterized by different shapes. At the phenotypic level, Xl and Xp are lowly correlated, whereas each of them is highly related to the 305-d milk. If confirmed at a genetic level, such a result could be of great interest for the genetic improvement of lactation yield by selecting animals on the basis of persistency.
Results of the mixed linear model analysis highlight relationships among Xl and Xp and some environmental factors known to affect lactation-curve shape, such as calving season and parity.
Finally, a major limitation of the multivariate methodology, i.e., the requirement of a fixed number of observation for each subject, can be overtaken by combining factor analysis with one of the several extrapolation techniques proposed in the literature.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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Received for publication June 9, 2003. Accepted for publication October 6, 2003.
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REFERENCES |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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= -1 ÷ 0; 
= 0 ÷ 1; 
= 1 ÷ 2; –*– = >2). (Xl = latent common factor related to the lactation peak; Xp = latent common factor related to the lactation persistency). Points are plotted for the average day in milk on each test day.
