N .C. Friggens et al. Livestock Production Science 62 1999 1 –13
5
standard errors S.E.s of each coefficient in the additional covariate in the analysis of variance for
curve fitting procedure were used to weight the parity effects of that coefficient parity data set.
analysis of variance for that coefficient by including
2
1 S.E. as a covariate. Given the uneven distribu- tion of parities across calving years Table 1 it was
3. Results and discussion
not possible to directly include calving year effects in the analysis of variance for parity effects. Quanti-
The results and discussion have been presented in fication of calving year effects is discussed in
an order which reflects the stepwise progress of the Section 2.4.
analyses rather than dealing with the objectives Further analyses were carried out on those co-
according to their priority. The order is: statistical efficients which were affected by parity. When
performance of the models examined and selection discussing these ratios, parity numbers are given as
of model for the parity analyses, effect of parity on subscripts to the relevant coefficient. For a given
lactation curve coefficients, and finally calving year coefficient, the ratio between the coefficient values
effects. for any pair of parities e.g., c c was calculated
1 3
within individual cows. The average value of the 3.1. Curve fitting
coefficient across the same pair of parities was also calculated within individual [e.g., c 1c 2]. The
The average lactation curves for each parity are
1 3
relationship between the 40 values of the ratio shown in Fig. 1. One cow was removed from the
between parities and the 40 values of the average of study because of chronic health problems in two out
the coefficient over the same two parities was of three of her lactations. An additional six lacta-
examined using linear regression analyses Minitab tions, out of 117, were discarded because of chronic
Version 9.1, 1992. health problems. Using the Emmans and Fisher
model 1986 the curve fitting procedure failed to 2.4. Calving year effects
converge in a further six cases, this was due to runs of missing data at the start of lactation for these
The effect of calving year on milk production was particular lactations. For the 105 cases where the
2
examined by selecting a second data set from the curve fitting procedure converged, the average R of
genotype by environment experiment Veerkamp et the curve fit was 84.1 min.545.0; first quartile5
al., 1994. The criteria for including cows in this 80.4; third quartile591.3; max.597.8 and the
second set were that; i they were being fed the high average residual standard deviation of the curve fit
concentrate TMR, ii they were in their third was 1.74 kg d min.50.76; first quartile51.38; third
lactation, and iii their third lactation was in one of quartile52.02; max.53.89 kg d. The residuals
the calving years included in the parity analysis. This from the curve fit of the average milk yield curve of
resulted in a data set of 18 795 records on 67 cows, third lactation cows Fig. 1 are shown for the
spread evenly across calving years, which included Emmans and Fisher model 1986 in Fig. 2. The
the third lactation cows that were in the parity correlations between the coefficients a, G , b and c,
analysis. The cows in this set were managed in of the Emmans and Fisher model 1986 are shown
exactly the same way as the cows in the parity data in Table 2. The relationship between coefficients a
set and the data were processed in the same way to and c of the Emmans and Fisher model 1986 is
derive lactation curve coefficients for the Emmans shown in Fig. 3.
and Fisher model 1986. Year effects were quan- For the 105 cases where curve fit statistics were
2
tified by analysis of variance with year included as a available for both models, the average R
of the fixed effect Genstat
E Version 5.3.2., 1994. In order Wood’s curve 1967 fit was 82.1 min.530.1, first
to examine if calving year would influence the parity quartile579.1, third quartile590.4, max.597.8
analysis, the average value of a given coefficient for and the average residual standard deviation of the
third lactation cows in each calving year derived Wood’s curve fit was 1.86 kg d min.50.82, first
from the calving year data set was used as an quartile51.42, third quartile52.15, max.53.78 kg
6 N
.C. Friggens et al. Livestock Production Science 62 1999 1 –13
Fig. 1. The relationship between average milk yield and time from calving for cows in parity 1, 2 and 3. The curve with the highest peak yield is parity 3, the curve with the lowest peak yield is parity 1.
Fig. 2. The residuals from non-linear regression of daily average milk yield of third lactation cows relative to days post calving using: d the Emmans and Fisher model 1986, and s Wood’s function 1967. The fitted curve Emmans and Fisher, 1986 is shown for reference
solid line.
d. The curve fitting procedure converged in all cases that the Wood’s function has a fixed zero intercept at
using the equation of Wood 1967 indicating that it calving and was thus not vulnerable to runs of
was more robust, in these terms, than the Emmans missing data in early lactation. Around peak yield
and Fisher model 1986. This is not surprising given and in mid lactation, there was a tendency for greater
N .C. Friggens et al. Livestock Production Science 62 1999 1 –13
7 Table 2
representative of potential. Rook et al. 1993 com-
The correlations between the coefficients of the Emmans and
pared a range of lactation curve models including
a
Fisher 1986 lactation curve model, calculated in two ways
one which was a form of the Emmans and Fisher
G b
a c
model. However, this study Rook et al., 1993 is not
G 0.625
20.117 20.063
directly relevant to the present study as the data used
b 0.424
20.686 20.612
were derived from cows which for the most part
a 0.098
20.269 0.937
were fed in a way likely to result in nutrition
c 0.196
20.199 0.750
distortions of lactation curves relative to potential
a
Within each of the 105 cow lactations where the curve fitting
milk production.
procedure converged, the correlation between coefficients was
In the study of Rook et al. 1993 their form of the
estimated in the curve fit. Each value above the diagonal is the
Emmans and Fisher model [Eq. 7A; Rook et al.,
average of the corresponding 105 correlations. Below the diag-
1993] did not perform well, ranking 11th out of 13
onal, each value is a single correlation, across cows and lactations, between the 105 values of the relevant two coefficients. There was
models on the basis of mean rank of residual mean
no significant effect of parity on the correlations.
squares. The Wood’s function 1967 ranked third out of the 13 models in their study Rook et al.,
deviations from zero in the residuals from the curve 1993. In contrast, in the present study the Emmans
fitting procedure when using the Wood’s function and Fisher model 1986 performed marginally
relative to the Emmans and Fisher model Fig. 2. better than the Wood’s function 1967 in terms of
On the basis of the curve fit statistics presented percentage of variance accounted for and residual
above there is relatively little to distinguish between standard error. In our view, the difference found
the models of Wood 1967 and Emmans and Fisher between the present study and that of Rook et al.
1986 but, as discussed in Section 1, the Emmans 1993 in the statistical performance of the Emmans
and Fisher model gives a better representation of the and Fisher model relative to Wood’s function
onset of lactation and is more easily interpretable in serves to emphasis the importance of using suitable
biological terms. data for testing models of potential production.
The above is, to our knowledge, the first published The original reason for comparing lactation curve
account of the ability of the Emmans and Fisher models in this study was to select for the subsequent
model to fit the milk yield of dairy cows whose parity analyses the better performing of two bio-
lactation curves can reasonably be assumed to be logically interpretable models Emmans and Fisher,
Fig. 3. The relationship between coefficients a and c of the Emmans and Fisher lactation curve model.
8 N
.C. Friggens et al. Livestock Production Science 62 1999 1 –13
1986; Dijkstra et al., 1997. These models were which is bounded between 0 and 1 when G tends to
derived from substantially different perspectives and 2` and `, respectively. The average value of G in
there appeared to be no obvious equivalence between the present study indicates that the milk producing
them. However, as shown in Section 2.3, the model system is approximately 40 developed at calving.
of Dijkstra et al. 1997 in fact reduces to the earlier The coefficient M in the equation of Dijkstra et al.
Emmans and Fisher model 1986. This applies to 1997 is equal to the product of the Emmans and
the equation of Dijkstra et al. 1997 for predicting Fisher milk yield scalar, a, and their degree of
milk yield [Eq. 11; Dijkstra et al., 1997] and to the maturity expression, exp [2exp G ]. Consequently,
equation for mammary growth post-partum [Eq. M
gives the milk yield at calving which in this 5b; Dijkstra et al., 1997] because their parameter T
study was calculated as kg d; 14.2, 20.5 and 21.2 is a constant within species. The comments do not
for parities 1, 2 and 3, respectively. The coefficient relate to the pre-partum mammary growth model
m in the equation of Dijkstra et al. 1997 is defined
T
Dijkstra et al., 1997. The fact that the same model as the rate of secretory cell proliferation at calving
arises from two distinctly different philosophies for Dijkstra et al., 1997 and thus relates to descriptions
describing the biological processes or mechanisms of mammary growth but does not easily translate to a
underlying milk production is reassuring. Indeed, one describable property of the milk production curve.
basis for ascribing generality to a functional form or For a given value of k,
m controls the amplitude of
T
theory is that it has arisen independently from the growth phase of the milk yield curve but said
different approaches or disciplines Von Bertalanffy, amplitude is expressed in a ratio with k and as an
1968. exponential multiplier for M [M ?exp
m k].
T
The choice of which parameterisation to use In both the model of Emmans and Fisher 1986
[Emmans and Fisher 1986 or Dijkstra et al. 1997] and the equation of Dijkstra et al. 1997 the rate of
is dependent upon how well the parameter interpreta- decline in milk yield is described by a single
tions relate to the purpose for which the model is to coefficient, called c and
l, respectively. In terms of be used. The mathematical equivalence between the
biological interpretation this has a clear advantage two parmeterisations is given in Section 2.3. In
over the Wood’s function 1967 where the rate of biological terms, the roles of the different parameters
decline is controlled by two parameters. There is, can be summarised thus.
however, a good argument for introducing a second The scaling coefficient, a, in the Emmans and
parameter which relates to the declining phase of Fisher model 1986 can be seen as the main
lactation, to describe the effect of pregnancy in coefficient by which differences between cows in
depressing concurrent milk production Hooper, milk yield potential would be expressed Congleton,
1923; Coulon et al., 1995. Such modifiers have been Jr. and Everett, 1980. It does not, however, directly
proposed Coulon et al., 1995 and indeed parameter give milk yield at any time in lactation and has no
b in the Wood’s function 1967 may provide the direct equivalence to any one parameter in the
flexibility to account for the depression in milk yield equation of Dijkstra et al. 1997. The coefficient
due to the extent of pregnancy in late lactation. which allows scaling in the equation of Dijkstra et al.
However, in order to be biologically sensible, a 1997 is M .
parameter to describe the effect of pregnancy should The growth of the milk producing system is
clearly be related to the date of conception of the calf determined by G and b in the Emmans and Fisher
and be independent of the stage of lactation of the model 1986, and by
m and k in the equation of cow.
T
Dijkstra et al. 1997. The rate at which milk yield increases to peak is described by a single coefficient
3.2. Parity effects in both models, b Emmans and Fisher, 1986 and k
Dijkstra et al. 1997. The coefficient G in the
The results concerning both parity effects and Emmans and Fisher model 1986 quantifies the
calving year relate only to the Emmans and Fisher degree of maturity of the milk producing system at
model 1986. The average values of the lactation calving through the expression exp [2exp G ]
curve coefficients in parities 1, 2 and 3 are presented
N .C. Friggens et al. Livestock Production Science 62 1999 1 –13
9 Table 3
creased indicating that the rate of decline in milk
The average values of the coefficients of the Emmans and Fisher
yield post-peak was progressively steeper with in-
1986 lactation curve model for parities 1, 2 and 3
creasing parity. Similar significant effects of parity
Parity Coefficient
on Wood’s function 1967 have been frequently
G b
a c
reported Rao and Sundaresan, 1979; Congleton Jr. and Everett, 1980; Wood, 1980; Rowlands et al.,
1 20.206
0.0694 32.1
0.00218
1982; Yadav and Sharma, 1985; Collins-Lusweti,
2 20.245
0.0916 44.9
0.00322 3
20.089 0.0888
52.8 0.00393
1991. However, and in contrast to the above results, as the coefficients of Wood’s function are not
a
SED 0.072
0.0118 1.7
0.00023
mutually exclusive in describing the underlying
b
P ns
ns
biology they are all affected by parity.
a
Standard error of the difference.
The finding that only two of the lactation curve
b
Significance of the effect of parity: ns indicates P .0.05,
coefficients were affected by parity suggests that a
indicates P ,0.001.
simplification in the inputs needed to generate lacta- tion curves for different parities may be possible.
in Table 3. There were no significant effects of parity This depends upon the relationship between the
on coefficients b and G Table 3. This indicates coefficients of the Emmans and Fisher model 1986
that any effects of parity on both the degree of across parities. The data used in the study of parity
maturity of the milk producing system G and the effects were chosen so as to allow this relationship to
rate at which milk production increases to peak b be examined. For each cow, the ratio in the value of
are small relative to the variation present. Thus, coefficient a between parities 1 and 2, 1 and 3, and 2
when generating potential milk yield curves for and 3 was calculated. The same ratios were also
different parities in general prediction models, a calculated for coefficient c. The averages, across
single average value across parities of both b and G cows, of these ratios are presented in Table 4. In
can be used. However, there were highly significant order to test whether the values of these ratios were
effects of parity on coefficients a and c P ,0.001; independent of the size of the coefficient, regression
Table 3. The lactation curve scalar, a, increased analyses between the ratios and the average values of
indicating that potential yields increased with in- the coefficient were carried out. Examples of the
creasing parity. The decay coefficient, c, also in- relationship between the ratio and the average value
Table 4 The average values of the ratios, calculated within individuals, between parities for coefficients a and c of the Emmans and Fisher model
a
1986
b
Ratio between parities a a
a a a a
c c c c
c c
1 2
1 3
2 3
1 2
1 3
2 3
Mean 0.73
0.63 0.88
0.69 0.57
0.87
c
S.E.M. 0.029
0.032 0.030
0.067 0.054
0.048
d
Residual SD 0.163
0.177 0.168
0.348 0.276
0.276
2 d
R 4.6
0.0 3.7
17.4 10.3
0.0
d
Slope coefficient 0.0075
20.0014 20.0059
168 86
226
d
S.E. of slope 0.00474
0.00488 0.00401
60.3 42.0
38.9
d
P ns
ns ns
ns ns
a
Summary statistics are also presented for linear regressions, across cows, of the ratios e.g., a a on the averages of the coefficients in
1 2
the same ratio [e.g., a 1a 2].
1 2
b
Parities are indicated by the subscript numbers in the ratios.
c
Standard error of the mean.
d 2
Summary statistics from the regression analyses: R is the percentage of variance accounted for by the regression, adjusted for degrees of freedom; P is the significance of the slope coefficient, ns indicates P .0.05, indicates P .0.01.
10 N
.C. Friggens et al. Livestock Production Science 62 1999 1 –13
Fig. 4. The ratio of the value of coefficient a Emmans and Fisher, 1986 in parity 1 a and parity 3 a shown relative to the average
1 3
value of the coefficient across these two parities [a 1a 2]. Both the ratios and the averages were calculated within cows. The line shows
1 3
the average value of the ratio.
of the coefficient are shown for coefficient a in Fig. general means to generate potential milk yield curves
4 and coefficient c in Fig. 5. With one exception, the for different parities from information relating to one
slopes of all the regressions were not significantly parity only. Accounting for the proportion of the
different from zero and consequently the proportion variation due to the general effect of parity, in-
of variance accounted for by the regressions was dependent of coefficient size, is also important
very low Table 4. This indicates that for both because it allows true individual variation to be
coefficient a and coefficient c the ratios were in- quantified and exploited Taylor, 1985. However,
dependent of the average size of the coefficient. The this represents an issue subsequent to and outside the
exception was the ratio between parities 1 and 2 of scope of the present study.
coefficient c which is shown in Fig. 5. The regres- Further, if an assumption is made about the
sion equation of the relationship between the ratio relationship between coefficients a and c Emmans
and the average value of c across these two parities and Fisher, 1986 across cows within the chosen
was: c c 50.2471168[c 1c 2]. However, the reference parity then potential milk yield curves can
1 2
1 2
regression accounted for only 17.4 of the total be generated for all parities from one single value
variation. such as desired or potential 305 day yield. There is
There was a significant slope in only one out of evidence to suggest that the main difference between
six regressions. Given that this relationship had a high and low producing cows of equal parity in the
correlation coefficient of 0.42, it does not provide shape of the lactation curve is in the scaling parame-
reasonable grounds for rejecting the simpler finding ter, a, Congleton Jr. and Everett, 1980. Thus, for
that the ratios between parities in coefficients a and c the purpose of predicting potential lactation curves, it
were independent of the average size of the coeffi- may be reasonable to assume that coefficient c is a
cient. Thus, in the present study, the differences constant within parity, independent of a. An alter-
between parity in the shape of the lactation curve native assumption could make use of the relation
were accounted for, across cows, by simple stable between a and c shown in Fig. 3 which can be
ratios in the coefficients a and c. This offers a described by linear regression as: a 525.015918c
N .C. Friggens et al. Livestock Production Science 62 1999 1 –13
11
Fig. 5. The ratio of the value of coefficient c Emmans and Fisher, 1986 in parity 1 c and parity 2 c shown relative to the average
1 2
value of the coefficient across these two parities [c 1c 2]. Both the ratios and the averages were calculated within cows. The line shows
1 2
the average value of the ratio.
2
Table 5
R 555.8, residual SD58.04 kg. This may be a
The average values of the coefficients of the Emmans and Fisher
reasonable assumption for the purposes of prediction
model, derived from a control data set of third parity cows, for
but it clearly is due, in large part, to the form of the
each calving year included in the parity analysis
lactation model and the associated correlations in the
a
Calving year Coefficient
coefficient estimates Table 2.
G b
a c
3.3. Calving year effects
1990 0.456
0.0967 51.4
0.00399 1991
0.079 0.0993
53.2 0.00374
1992 0.074
0.0784 74.1
0.00566
The data set used for the parity analysis was
1993 20.177
0.0961 53.1
0.00341
chosen so as to be able to obtain estimates of the
1994 0.089
0.0932 60.3
0.00479
ratios between parities for lactation curve coefficients
1995 20.041
0.0872 52.2
0.00366
that were not biased by variation between cows; the
Overall mean 0.0608
0.0918 57.1
0.00417
same cows were present in each parity. The inevit-
b
SED 0.1817
0.03056 14.13
0.00023
able consequence of this approach was that there was
c
P ns
ns ns
ns
an uneven distribution of parities across calving
a
The calving year started on the 1st September in any given
years with more heifers present in the early years and
year.
more third lactation cows in the later years Table 1.
b
Standard error of the difference.
Thus, the design of the experiment was vulnerable to
c
Significance of the effect of year: ns indicates P .0.05.
both systematic effects of calving year on lactational performance and extreme effects in the first and last
years. To quantify whether such calving year effects were no significant effects of calving year on any of
existed a second data set was constructed, as de- the four coefficients. Further, the average values of a
scribed in Section 2.4. The average values of the given coefficient for each calving year Table 5
coefficients of the Emmans and Fisher model 1986 were used as a covariate in the analysis of variance
for each calving year are presented in Table 5. There for the effect of parity on the same coefficient in the
12 N
.C. Friggens et al. Livestock Production Science 62 1999 1 –13
parity data set. In all cases, inclusion of the calving Emmans and Fisher model 1986 was, if anything,
year effect covariate had no significant effect on the marginally better than the Wood’s function 1967.
analysis for parity effects. These results indicate that Parity had significant effects on only two of the
the analysis of the effects of parity on the lactation coefficients of the Emmans and Fisher model 1986,
curve coefficients and the calculation of the ratios a and c. The effect of parity was adequately de-
between parities for the curve coefficients were not scribed by simple ratios between any pair of parities
affected by the uneven distribution of parities across for a given coefficient. Thus, the coefficient values in
calving years. parity 1 and 2 were found to be a constant proportion
of the values in parity 3. This allows the general effect of parity to be accounted for, both in models
3.4. Further considerations to predict potential milk yield and in analyses to
characterise variation between individual cows. The main aim of this study, to provide a simple
means by which to generate potential lactation curves for different parities from limited information,
Acknowledgements
has been achieved. This has been done using data from cows managed within one system, in one herd.
The technical assistance of the staff at the Langhill Thus the results presented here can be seen as a first
Dairy Research Centre Edinburgh, Scotland, UK is step, an important subsequent step would be to
˚ gratefully acknowledged. Inge Riis Korsgard of the
examine these relationships in other herds which Biometry and Genetics Department, Danish Institute
have been managed and fed in a manner likely to of Agricultural Sciences provided valuable statistical
result in milk yield curves whose shapes are charac- help for which we are grateful. This study was
teristic of potential milk production. We have been funded by the Ministry of Agriculture Fisheries and
careful to present our study in terms of describing Food Consortium DS04; R
UMINT
project. The data potential, the reason for this being that both the aims
used were collected as part of a project funded by the and the model used in the study were not designed to
Scottish Office
Agriculture, Environment
and be able to accommodate the effects of sub-optimal
Fisheries Department, the Milk Marketing Board for feeding on milk production. Models to do this clearly
England and Wales, the Holstein Friesian Society for require to consider the balance of nutrient inputs and
Great Britain and Ireland, and the Ministry of outputs as well as the potential of the animal
Agriculture Fisheries and Food. Oldham and Emmans, 1989. We therefore chose to
study cows in an experiment in which it could be reasonably assumed that the resulting lactation
curves reflected potential. However, no single experi-
References
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Brody, S., Ragsdale, A.C., Turner, C.W., 1923. The rate of decline
this study will be taken up by others and further
of milk secretion with the advance of the period of lactation. J.
developed.
Gen. Physiol. 5, 441–444. Cobby, J.M., Le Du, Y.L.P., 1978. On fitting curves to lactation
data. Anim. Prod. 26, 127–133. Collins-Lusweti, E., 1991. Lactation curves of Holstein–Friesian
4. Conclusions
