Data
on test day (TD) milk yield of cows from a dualpurpose herd were analysed by a mixed
linear model in order to: choose the appropriate phenotypic (co)variance matrix structure,
analyse the existence of trends in variances and covariances over time, and obtain
lactation curves for Bos taurus x Bos indicus cows from an experimental
station of the Colegio de Postgraduados at Cardenas, Tabasco, Mexico. The data set was
comprised of 5566 test day milk yield records from 321 lactations (233
The
best (co)variance structure was then used to obtain lactation curves for the ¾
Recently, the analysis of repeated data such as test day (TD) milk yield records has increased. The TD models can be used to analyse individual TD records of cows rather than a cumulative lactation yield, which is currently used. In addition, TD models allow for: a more accurate estimation of fixed effects; the consideration of a different number of records from each lactation; the variation of fixed effects estimates across herds and lactation stages; the adjustment for different effects of sampling date (Stanton et al 1992; Swalve 2000) and the prediction of daily milk yields from a limited number of TD records (Pool and Meuwissen 1999). Furthermore, the shape of the lactation curve can be accommodated in the model to account for differences in lactation curves among cows and problems associated with persistency (Brotherstone et al 2000).
The analysis of repeated records requires special attention due to problems of correlated errors and heterogeneity in the error (co)variance matrix structure. The general mixed linear model methodology is able to address these issues by directly modelling the (co)variance structure among repeated measures (Little et al 1998). Such a property seems suited for dualpurpose milk production analyses involving environmental effects on daily milk yield which can vary markedly across lactation stages, mainly due to aspects of the dualpurpose management system. Ptak and Schaeffer (1993) used a TD model, which assumes that covariances between successive TD are equal to those between TD that are far apart. Garcia and Holmes (2001) found the mixed model suitable for the description of the lactation curve of dairy cows in pasturebased systems.
In this study, TD yields
of cows from a dualpurpose herd were analysed by mixed linear methodology in order to:
model the (co)variance structure; analyse the existence of trends in variances and
covariances over time and obtain the lactation curve for Bos taurus x Bos
indicus cows.
The data consisted of
lactations records collected from a population of dualpurpose, crossbred cows born in an
experimental station at the Colegio de Postgraduados at
The data set was comprised of 5566 test day milk yield records from 321 lactations cows (233 HZ and 88 HS lactations) collected between 1992 and 1999. Lactations with less than 8 TD records were discarded. Lactations up to 287 days in length were used and were divided into 21 intervals (DIM = days in milk) of 14 days each, starting from day 7 of lactation. Elimination of data was decided because of the few observations for lactations shorter than 105 days and longer than 287 days and to allow the statistical programs to converge. The number of lactations per genotype and parity are shown in Table 1. The average number of lactations per cow was 2.7. Repeated lactations were assumed to be uncorrelated, as previously suggested by other authors (Stanton et al 1992).
Table
1. Distribution of lactations per genotype and parity in a dualpurpose cattle system
in 

Parity 
¾ 
½ 
1 
87 
26 
2 
57 
20 
3 
38 
16 
4 
25 
11 
5 
17 
10 
6 
9 
5 
Data were classified according to the following factors: year of calving (19921999), season of calving (dry, rainy, and windy and rainy), parity (1, 2, 3–4, 5 or greater), genotype (HZ and HS) and 21 DIM intervals. Season of calving was grouped based on temperature and rainfall of the region as: dry (February to May), rainy (June to September) and windy and rainy (October to January).
Data were analysed with the following linear model
y_{ijklmno} = M + A_{i} + E_{j} + P_{k} + G_{l} + DIM_{m} + L(G)_{ln} + e_{ijklmno}
where y_{ijklmno} = test day milk yield of lactation, M = overall mean; A_{i} = fixed effect of year of calving, E_{j} = fixed effect of season of calving, P_{k} = fixed effect of parity, G_{l} = fixed effect of genotype, DIM_{m} = fixed effect of DIM interval, L(G)_{ln} = random effect of lactation within genotype; and e_{ijklmno} = random residual.
Three different (co)variance structures were fitted to the TD data: (1) a structure known as compound symmetry (CS) which uses the same variance and correlations for all pairs of measures on the same animal. It represents the most commonly used (co)variance structure in phenotypic analysis of repeated measures (Stanton et al (1992) due to the advantage of only two parameters to be estimated: the between lactation and residual components of variance; (2) The second and third structures were considered in order to take account of two main aspects of correlations between repeated measures. Firstly, two measures can be correlated simply because they share a common contribution from the same lactation. Secondly, measures on the same lactation closer in time are usually more correlated than measures further apart in time. The second structure uses a model that calculates covariances and correlations for each pair of measures on the same animal, known as unstructured (UN). The third structure was a (co)variance structure that combined the CS structure with a firstorder autoregressive process [CS + AR(1)].
The goodness of fit of the
(co)variance models was assessed by comparing values of REML Log likelihood, Akaike’s
Information Criterion (AIC) and Schwartz Bayesian Criterion (SBC). Larger values for the parameter
criteria indicates a better fit of the model. The best (co)variance structure
was then used to obtain generalized least squares
estimates for TD milk yields, which were then plotted against DIM to represent lactation
curves for Holstein x Zebu and Holstein x Sahiwal cows. All analyses were performed using the
mixed procedure of SAS (1995).
The CS structure produced variance, covariance and correlation values of 630, 309 and 0.49, respectively. (Co)variances and correlations between TD at different DIM intervals estimated for UN and CS + AR(1) are reported in Tables 2 and 3, respectively. Correlation values between repeated measures showed a clear decreasing pattern with time for the CS + AR(1) structure. This means that environmental effects in the dual purpose system are important and also that estimation of total milk production based on partial lactations may not be as efficient as desired. In the CS + AR(1) structure, covariance and correlation values were practically zero after the 10^{th} DIM interval. Decreasing trend in variances, covariances and correlations, are recognized in the UN matrix case, although they are partially hidden.
Table 2. Matrix of variances, covariances and correlations (along, above and below the diagonal, respectively) for milk yield (kg^{2}) using a matrix structure that makes no assumptions regarding equal variances and correlations 

Days in milking intervals 


1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
1 
685.2 
427.5 
436.5 
403.4 
418.5 
305.4 
317.5 
263.9 
238.9 
243.2 
238.8 
249.4 
209.3 
190.5 
214.8 
144.8 
211.3 
2 
0.635 
660.6 
503.0 
462.7 
422.3 
294.0 
337.3 
286.4 
239.5 
240.8 
252.6 
277.3 
219.6 
195.6 
193.9 
158.3 
186.6 
3 
0.574 
0.673 
845.4 
534.8 
541.4 
375.3 
375.2 
324.8 
284.1 
283.5 
265.7 
243.2 
251.9 
203.5 
251.3 
198.2 
207.8 
4 
0.555 
0.648 
0.662 
771.6 
566.6 
416.3 
488.5 
405.3 
350.6 
322.5 
349.9 
323.6 
287.9 
253.0 
262.9 
231.7 
244.4 
5 
0.577 
0.593 
0.672 
0.736 
767.8 
488.3 
449.6 
375.6 
370.6 
319.7 
318.8 
313.1 
323.4 
306.3 
297.5 
280.5 
290.2 
6 
0.472 
0.463 
0.523 
0.609 
0.714 
609.8 
746.2 
456.5 
455.4 
398.5 
384.0 
374.7 
300.7 
291.1 
247.3 
251.4 
236.1 
7 
0.444 
0.480 
0.472 
0.534 
0.645 
0.666 
456.5 
619.4 
420.2 
404.3 
355.5 
326.5 
348.7 
330.8 
314.6 
306.7 
256.2 
8 
0.405 
0.448 
0.449 
0.486 
0.578 
0.611 
0.671 
420.2 
622.5 
405.5 
390.6 
352.4 
333.8 
289.0 
268.7 
295.1 
271.5 
9 
0.366 
0.373 
0.392 
0.421 
0.507 
0.601 
0.668 
0.677 
405.5 
617.9 
401.8 
372.1 
369.8 
327.2 
287.7 
285.4 
265.4 
10 
0.374 
0.377 
0.392 
0.407 
0.468 
0.521 
0.587 
0.653 
0.654 
401.8 
536.5 
406.9 
364.1 
312.4 
272.9 
266.8 
238.1 
11 
0.394 
0.424 
0.394 
0.446 
0.545 
0.557 
0.607 
0.617 
0.676 
0.698 
406.9 
677.8 
367.7 
332.3 
312.0 
282.0 
258.2 
12 
0.366 
0.414 
0.321 
0.394 
0.449 
0.487 
0.527 
0.504 
0.542 
0.575 
0.675 
400.3 
400.3 
373.4 
333.2 
297.1 
263.0 
13 
0.333 
0.356 
0.361 
0.431 
0.486 
0.507 
0.531 
0.558 
0.617 
0.610 
0.661 
0.640 
577.2 
412.2 
366.2 
345.2 
293.1 
14 
0.307 
0.321 
0.295 
0.384 
0.466 
0.497 
0.511 
0.490 
0.553 
0.530 
0.605 
0.605 
0.724 
561.9 
399.6 
389.0 
340.6 
15 
0.354 
0.326 
0.374 
0.409 
0.464 
0.433 
0.498 
0.467 
0.498 
0.474 
0.582 
0.553 
0.659 
0.729 
535.3 
397.4 
341.8 
16 
0.240 
0.267 
0.295 
0.361 
0.438 
0.441 
0.486 
0.514 
0.496 
0.465 
0.527 
0.494 
0.622 
0.711 
0.744 
532.8 
391.1 
17 
0.334 
0.300 
0.296 
0.364 
0.434 
0.395 
0.388 
0.452 
0.441 
0.397 
0.462 
0.418 
0.505 
0.595 
0.612 
0.702 
583.0 
Table 3. Matrix of variances, covariances and correlations (along, above and below the diagonal, respectively) for milk yield (kg^{2}) using a (Co)variance matrix that combined the compound symmetry (CS) structure with a firstorder autoregressive process AR(1) 


Days in milking interval 


1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
1 
356.44 
155.25 
67.62 
29.45 
12.83 
5.59 
2.43 
1.06 
0.46 
0.20 
0.09 
0.04 
0.02 
2 
0.436 
356.44 
155.25 
67.62 
29.45 
12.83 
5.59 
2.43 
1.06 
0.46 
0.20 
0.09 
0.04 
3 
0.190 
0.436 
356.44 
155.25 
67.62 
29.45 
12.83 
5.59 
2.43 
1.06 
0.46 
0.20 
0.09 
4 
0.083 
0.190 
0.436 
356.44 
155.25 
67.62 
29.45 
12.83 
5.59 
2.43 
1.06 
0.46 
0.20 
5 
0.036 
0.083 
0.190 
0.436 
356.44 
155.25 
67.62 
29.45 
12.83 
5.59 
2.43 
1.06 
0.46 
6 
0.016 
0.036 
0.083 
0.190 
0.436 
356.44 
155.25 
67.62 
29.45 
12.83 
5.59 
2.43 
1.06 
7 
0.007 
0.016 
0.036 
0.083 
0.190 
0.436 
356.44 
155.25 
67.62 
29.45 
12.83 
5.59 
2.43 
8 
0.003 
0.007 
0.016 
0.036 
0.083 
0.190 
0.436 
356.44 
155.25 
67.62 
29.45 
12.83 
5.59 
9 
0.001 
0.003 
0.007 
0.016 
0.036 
0.083 
0.190 
0.436 
356.44 
155.25 
67.62 
29.45 
12.83 
10 
0 
0.001 
0.003 
0.007 
0.016 
0.036 
0.083 
0.190 
0.436 
356.44 
155.25 
67.62 
29.45 
11 
0 
0 
0.001 
0.003 
0.007 
0.016 
0.036 
0.083 
0.190 
0.436 
356.44 
155.25 
67.62 
12 
0 
0 
0 
0.001 
0.003 
0.007 
0.016 
0.036 
0.083 
0.190 
0.436 
356.44 
155.25 
13 
0 
0 
0 
0 
0.001 
0.003 
0.007 
0.016 
0.036 
0.083 
0.190 
0.436 
356.44 
The autoregressive (p)
parameter value for the CS + AR(1) (co)variance structure was 0.44. This value is smaller
than the repeatability value for milk yield (0.65) reported for a dualpurpose cattle
system (HernándezReyes et al 2000) but similar to that notified by De Alba and Kennedy
(994) in purebred and crossbred Criollo cattle in
Goodness of fit criteria (Table 4) based on REML and AIC showed that the UN (co)variance structure was more appropriate for TD data analysis. However, the SBC criteria indicated that the CS + AR(1) structure was better in fitting TD records. Nevertheless, there were no marked differences among (co)variance structures. There are two major potential problems with using the unstructured covariance. One, it requires estimation of a large number of variance and covariance parameters and can lead to severe computational problems, especially for unbalanced data. Two, it does not exploit existence of trends in variances and covariances over time, and thus often results in erratic patterns of standard error estimates (Little et al 1998).
Table 4. Goodness of fit statistics for an unstructured (UN), a compound symmetry (CS) and a CS with a first order autoregressive process [CS + AR(1)] (co)variance structures for test day milk yields of a dual purpose herd of dairy cattle in Tabasco, Mexico 

UN 
CS 
CS + AR(1) 

REML 
23679 
24403 
23968 
AIC 
23889 
24405 
23972 
SBC 
24584 
24412 
23985 
REML= Restricted Maximum likelihood Log likelihood, AIC = Akaike’s Information Criterion; SBC = Schwartz Bayesian Criterion. 
The two components of (co)variance that the CS + AR(1) model disentangles for milk yield around the respective average lactation curve (the compound symmetry and the autoregressive components), can be regarded as effects of different random factors. The constant component given by the CS part can be related to factors affecting the whole lactation, such as genetic merit of the animal and permanent environmental factors. On the other hand, the AR(1) part suggests, as mentioned before, the existence of an appreciable component of covariation that decreases rapidly with increasing DIM intervals, which can be seen as an indication of important environmental factors that result in short term effects, which are lost within a few DIM intervals.
Average lactation curves for the HZ and HS estimated by imposing the CS + AR(1) structure to the model are reported in Figure 1. The HZ had higher TD yields than the HS cows. They show the typical ascending phase of milk yield curves. Peak milk yield was reached at 5 weeks of production. However, these shapes are different to those reported by Madalena et al (1979) for HZ cows in Brazil, who used Wood models to fit curves, and who found an earlier peak (at week one) than in this study.
Figure 1. Lactation curves of 3/4
in
The use of suitable (co)variance
matrices represents a fundamental point to correctly analyse repeated measure traits like
weekly or monthly milk yields or other economically important traits. This study, however, did not show
a clear benefit of using any of the (co)variance structures. Nevertheless the CS + AR(1)
structure was able to point out some essential features of the behaviour of individual
lactations around the average curve. In particular, it highlighted the differences between
factors that cause constant and short term covariation, and the relative importance that
such components can have on milk yield. The results of this study are also of technical
interest because they
seem to
indicate that a biweekly interval
between two consecutive records is a valid option for describing the lactation curve of
dualpurpose cattle.
The CS + AR(1) matrix structure is recommended for the analysis of TD milk yield data from dual purpose cattle. The mixed model methodology described well the lactation curve of Bos taurus x Bos indicus cows under the management conditions of this study.
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Received 20 December 2001