Animal Science 2000, 71: 197-207
© 2000 British Society of Animal Science
1357-7298/00/97460197$20·00
Appropriate mathematical models for describing the complete
lactation of dairy sheep
G. E. Pollott1 and E. Gootwine2
1Wye
College – University of London, Ashford, Kent TN25 5AH, UK
of Animal Reproduction, Agricultural Research Organisation, The Volcani Centre, PO Box 6, Bet Dagan
50250, Israel
2Department
Abstract
Despite milk being an important product from sheep, there are very few reports of milk production from the
complete lactation of dairy sheep. The Improved Awassi in Israel is kept under an intensive system of management
with lambs being weaned soon after birth. Records from one such flock were analysed to investigate the suitability of
various mathematical functions for describing milk yield from the complete lactations of dairy sheep. This included
a consideration of whether the functions could cope with short lactations, a characteristic of dairy sheep, and a
limited number of test-day records per lactation.
Four non-linear mathematical functions were investigated (Wood, Morant, Grossman and Pollott), two of which
could also be fitted in a linear and a linear weighted form (Wood and Morant). These functions were fitted to weekly
data from a ‘typical Awassi lactation curve’, represented by least squares means of daily milk yield from each week
of a 40-week lactation derived from an analysis of 25605 test day records. Characteristics of the lactation were
calculated from the functions, such as total milk yield, day and level of peak yield and persistency. These functions
were also fitted to 1416 individual lactation records of up to 10 test-day records per lactation. The value of the
functions was investigated using the residual mean square (RMS) of the fitted curve as an indicator of how well
each function described the lactation. Forms of these functions with a reduced number of parameters were also
investigated.
The non-linear functions always fitted the data with a lower RMS than their linear equivalent and the weighted
form of the linear functions always had a lower RMS than the unweighted form. Of the linear functions, Morant
fitted better than Wood. Of the non-linear functions Grossman, Morant and Pollott (additive and multiplicative)
fitted the data equally as well but better than Wood. The various functions predicted characteristics of the lactation
curve differently; the Wood functions tended to overestimate yield in early lactation and the Morant functions
underestimated peak yield.
No function was better suited to short lactations than another. However the three-parameter function of Morant,
Pollott multiplicative and Pollott additive were considered to be the most suitable for describing the complete
lactation of dairy sheep.
Keywords: Awassi, dairy sheep, lactation curve, milk recording.
Introduction
reported studies on the complete lactation of sheep
and most of them deal with ewes under
experimental conditions or of a non-dairy type
(Groenewald et al., 1995; Portolano et al., 1996). The
lack of studies on the compete lactation of dairy
sheep is partly due to the fact that in most dairy
sheep production systems, lambs are allowed to suck
for at least 30 days post lambing and milk recording
Milk is an important product from sheep,
particularly in the Mediterranean, Middle East and
eastern Europe and a great deal of effort has been put
into improving its production by both genetic and
non-genetic means (Barillet and Astruc, 1998). Whilst
understanding the factors that influence the progress
of lactation is of interest, only a few authors have
197
198
Pollott and Gootwine
starts only after weaning (International Committee
on Animal Recording, 1992; Barillet and Astruc,
1998). However, in some dairy sheep flocks operated
under intensive management, such as the Improved
Awassi flock of Kibbutz Ein Harod, Israel (Epstein,
1985), the common practice is to milk the ewes from
the start of the lactation. Lambs are removed from
their mothers at lambing into an artificial rearing
unit.
Fitting an appropriate mathematical model to
lactation curves is required in order to study the
environmental and biological variables that affect
milk
production.
In
addition,
appropriate
mathematical models may be used to predict future
milk yield for lactations currently in progress. Both
linear and non-linear methods of curve fitting have
been used to describe lactation curves, mainly in
dairy cattle recording. These have been extensively
reviewed by Masselin et al. (1987). The widely used
incomplete gamma function (Wood, 1967) is
convenient for computation as it becomes linear in its
loge form. However as Cobby and Le Du (1978) point
out, this function tends to overestimate yields in
early lactation and underestimates them in late
lactation; the use of a suitably weighted function
may overcome this inaccuracy (Guest, 1961). Morant
and Gnanasakthy (1989) addressed the problem of a
high level of correlation between the parameters of
the incomplete gamma function and suggested a four
parameter curve (modified polynomial), which can
also be used in a loge linear form, to overcome these
correlations. Further non-linear functions describing
lactation curves have been proposed by several
authors (Yadav et al., 1977; Keown et al., 1986; Ali and
Schaeffer, 1987; Wilmink, 1987; Grossman and Koops,
1988; Elston et al., 1989; Sherchand et al., 1992; Rook
et al., 1993; Pérochon et al., 1996; Dijkstra et al., 1997).
These methods tend to describe the lactation curve of
cattle better then the incomplete gamma function
although the residuals derived from the fitting the
functions are not always random with respect to time
(see for example Olori et al., 1999). Neal and
Thornley (1983) proposed a mechanistic model of
lactation based on the biology of the udder, whilst
Rook et al. (1993) and Dijkstra et al. (1997) used a
biological approach to lactation curve analysis.
Recently, Pollott (1999 and 2000) has developed a
series of functions based on the known biology of
milk secretion (Knight et al., 1998). The method fits
two logistic curves, one representing the increase in
milk production through lactation mediated by
secretory cell proliferation and differentiation, and
the other representing the decline in milk production
during lactation mainly due to programmed
secretory cell death (apoptosis).
The objective of this paper is to investigate the use of
a range of mathematical functions for describing
complete dairy sheep lactations, including an
investigation into the suitability of such functions to
cope with short lactations, a characteristic of dairy
sheep (Portolano et al., 1996).
Material and methods
The flock
The Improved Awassi flock of kibbutz Ein Harod
comprised about 1200 milking ewes. Its management
has been previously described by Gootwine et al.
(1995). Briefly, the flock was kept and given food
indoors throughout the year. Four 34-day mating
periods were scheduled annually. Ewe lambs were
mated for the first time between 8 and 10 months of
age. For each parity (lactation) the following
variables were obtained or calculated: litter size (LS),
lactation number (LN), lactation length (LL) and total
milk yield during the lactation (TMY). Ewes were
milked twice a day from the day of lambing until
their milk yield declined to about 0·5 l/day, or when
they had to be dried off for lambing. Milk yield was
recorded on the 15th day of each month for all
lactating ewes and the daily records used to estimate
TMY with the formula:
TMY = I1 M1 + Σ10r=2 Ir (Mr + Mr–1)/2
where I1 is the interval, in days, between lambing
and the first milk record, Ir the intervals between the
various monthly milk recordings and Mr is the daily
milk yield at the rth monthly recording. Test day
milking records were validated and stored using the
on-farm ‘Ewe and Me’ software (Gootwine et al.,
1994).
Poor performing ewes were usually culled after the
third lactation, otherwise culling occurred due to
poor health or age. Records were analysed from all
1366 ewes that lambed between the beginning of
1993 and the end of 1997. Altogether, there were
25605 daily milk yield records taken from 3512
lactations, with an average of seven records per
lactation. The lactations were grouped by lambing
date according to the month and year of lambing and
thus formed 60 monthly groups, which were used in
the analyses. Because of the confounding of lactation
number and age of ewe at lambing, only lactation
number is considered in the present study.
Preliminary analyses suggested this as a practical
alternative to either fitting age on its own or both age
and lactation number together.
Lactation curve fitting
In order to formulate the overall shape of the Awassi
lactation curve and to provide weekly milk records
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Modelling of complete lactations in sheep
for the curve fitting investigation, all individual daily
milk recordings were analysed by least-squares
methods for unbalanced data. The following model
was fitted to the data:
Mijklmnpqr = µ + Ei + LNj + Wk + MCl + WCm + MRn
+ LSp + Yq + eijklmnpqr
where Mijklmnpqr was a daily milk record from the Eith
ewe in its LNjth lactation in the Wkth week of that
lactation; MCl was the interval, in months, between
lambing and successful conception and WCm was the
interval, in weeks, between conception and date that
the milk record was taken; MRn was the month of the
Yq year of recording; LSp was the number of lambs
born to the ewe at the beginning of the lactation and
eijklmnpqr was the randomly distributed error term. The
model was fitted using the method of least squares to
fit general linear models for unbalanced data.
Computation was carried out using the general
linear model (GLM) procedure in the Statistical
Analysis Systems Institute (SAS, 1989).
The results from this analysis provided an estimate
of the average daily milk yield in weeks 1 to 40 of
lactation, forming what will be called a ‘typical
Awassi lactation curve’. Several linear, non-linear
and two-phase mathematical functions were then
chosen (Table 1) and their fit to the ‘typical Awassi
lactation curve’ was investigated. The Wood (1967)
function was chosen because it is widely used in
dairy recording. The Morant and Gnanasakthy (1989)
function was found to fit cattle data well (Pollott,
2000) and produces uncorrelated parameters. The
Grossman and Koops (1988) function is a multiphase model and the Pollott (2000) functions provide
parameters with a more immediate biological
interpretation than the other functions. In this paper
the use of the Morant function in its weighted form is
investigated for the first time. In all models Mn is the
daily milk yield at day n. The remaining unexplained
terms are parameters of each model.
The functions were fitted to the ‘typical Awassi
lactation curve’ data using an iterative non-linear
curve fitting procedure (procedure NLIN in SAS
(1989)). The parameters of each curve were estimated
using the least squares method and the
computational strategy of Marquardt was used to
search for the ‘best fit’ solution. The ‘best fit’ curve
was obtained for each lactation when there was a less
than 10-6 difference between the error sums of
squares in successive iterations.
After the parameters of all functions shown in Table
1 were estimated for the ‘typical Awassi lactation
curve’, each of the resulting equations was used to
estimate daily milk yield on the 4th day for each of
the 40 weeks. The residuals of these estimated values
were calculated using the ‘typical Awassi lactation
curve’ values and residual mean squares (RMS)
computed as a measure of goodness of fit using the
formula:
2
RMS = (ΣN
r=1 (Mrest – Mrtyp)) /N – Q
where Mrest and Mrtyp were the estimated and
‘typical’ values in each of the 40 lactation weeks,
Table 1 The four mathematical functions, and their variations, used to describe lactation
No. of
parameters
Name
Function
Wood (Wood, 1967)
Wood linear (Wood, 1967)
Morant
(model C4 in Morant and
Gnanasakthy, 1989)
Morant reparameterized
(see above)
Morant linear (see above)
Grossman (biphasic model of
Grossman and Koops (1988)
Pollott additive
(Pollott, 1999)†
Pollott multiplicative
Pollott, 1999)†
Mn = anbe–cn
ln(Mn) = ln(a) + b ln(n) –cn
Mn = exp(f + gn’ + hn’2 + i/n)
where n’ = (n – 150)/100
3
3
4
Mn = exp(f + gn’ (1 + kn’) + hn’2 + i/n)
where k was the regression of h on g.
ln(Mn) = f + gn’(1 + kn’) + hn’2 +i/n
Mn = Σ {aai bbi [1-tanh2 (bbi (n–cci))]} where i = 1 to 2
4
Mn = ((MSmax/(1 + ((1 – NO)/NO) exp(–GR n)))
– (MSLmax/(1 + ((1 – NOD)/NOD) exp(–DR n)))
Mn = ((MSmax/(1 + ((1 – NO)/NO) exp(–GR n)))
✕ (1/(1 + ((1 – NOD)/NOD) exp(–DR n)))
5
4
6
4
Weighted forms of both the Wood and Morant loge linear functions were investigated using the appropriate weights
suggested by Guest (1961), the square of Mn.
† MS and MSL are the milk secretion potential and loss of potential respectively, NO and NOD are the proportions of MS
and MSL achieved at parturition and GR and DR are the growth and death rate parameters of the two logistic curves. NO
replaced by 0·9999999 and n = (n–150) in all Pollott models (Pollott, 2000) reducing the number of parameters by one.
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Pollott and Gootwine
respectively, N was the number of daily milk records
in the lactation (in this case 40) and Q was the
number of parameters in the model.
Calculated characteristics of the curve
Peak yield (PY), day of peak yield (DP), calculated
total milk yield to a particular day (CTMY) and
persistency of the lactation (i.e. the rate of decline in
milk yield on day 150 of lactation) were calculated
for each function. In some cases the calculated values
were obtained by inspection and CTMY by
summation, after calculating the daily milk yield
values for each day of lactation. In other cases they
were derived using the various mathematical
functions as described in the original papers.
Completed lactations
In order to investigate the value of the various
mathematical functions for describing complete
sheep lactations with monthly milk records a subset
of data was selected which included 1416 complete
lactations. A lactation was assumed to be completed
when its last daily milk yield record reached
0·6 l/day. These lactations contained from four to 10
test-day records. Short lactations are a feature of
dairy sheep breeds and one objective of the analyses
reported here was to investigate the value of the
lactation curve functions when applied across the
full range of lactations encountered. The lactations
were grouped according to their TMY into low (total
milk yield <300 l), medium (300 to 600 l) or high
(>600 l) yielding lactations, and according to their
length into short, (<180 days), medium (180 to 270
days) or long (>270 days) lactations.
It is not feasible to fit a mathematical function with
five parameters to a lactation with four records. Thus
the functions were modified by reducing the number
of parameters used in the model, where appropriate.
Since the parameters dealing with the rising phase of
lactation are least accurately estimated, due to the
occurrence of a maximum of one milk record before
lactation peak, these parameters were most
commonly replaced in the following ways. The
Wood function was investigated using a constant for
b (Wood 2 model; the number indicates the number
of parameters remaining). The Morant functions
were fitted using the three-parameter version, with a
constant for i (Morant 3 model, after Williams, 1993).
Pollott additive models 4, 3 and 2 were derived by
replacing GR, MSL and NOD, successively, with a
constant. Pollott multiplicative models 3 and 2 were
derived by replacing GR and NOD, successively,
with a constant. Where a parameter was replaced by
a constant, the substitute value used was the
appropriate value from fitting the complete function
to the ‘typical Awassi lactation curve’. For example,
when using the Morant 3 function, i was replaced
with the value –0·5992 (Table 3).
The functions were fitted to the 1416 individual
complete lactations using the iterative non-linear
curve fitting procedure, as described above. The RMS
was calculated for each function fitted to each
complete lactation. The RMS from fitting each
function to the lactations were then analysed for each
function in turn. The effect of milk yield group,
lactation length group and their interaction was
investigated using a least squares procedure fitting a
general linear model, as described above, with the
following model:
Rijk = TGi + LLj + TGi ✕ LLj + eijk
where Rijk was the residual mean square from fitting
a particular function to the 1416 lactation records,
TGi was the ith total milk yield group, LLj the jth
lactation length group, TGi ✕ LLj was their
interaction and eijk was the randomly distributed
error term. Least squares means, within an effect,
were compared and the paired differences between
levels within an effect were tested against a twotailed t distribution.
Differences between 14 functions in the accuracy
with which they fitted the 1416 lactations were
investigated using the following model:
Rijk = LRi + MODj + eij
where
MODj
and eij
means
test.
LRi was a lactation record (1416 in total),
was a model function (14 as shown in Table 4)
was the error term. Differences between the
were tested using a Duncan’s multiple range
Results
Daily milk yield
The pattern of daily milk yield in each week of
lactation derived from the least squares analysis (not
shown) of all daily milk yield data is shown in Figure
1. The change in daily milk yield throughout the
lactation shows the usual lactation curve pattern and
these least squares means (typical Awassi lactation
curve) were used in the next section to investigate
the value of different mathematical functions for
describing lactation yields.
Typical Awassi lactation curve
The equations derived from fitting the different
functions (Table 1) to the 40 weekly milk yield leastsquares means (‘typical Awassi lactation curve’;
Figure 1) are shown in Table 2. These equations were
used to estimate daily milk yield in the middle of
201
Modelling of complete lactations in sheep
0·4
4·0
0·2
Deviation (l)
3·0
2·5
0
–0·2
2·0
1·5
–0·4
1·0
–0·6
0·5
0
0
0
10
20
Week of lactation
30
40
Figure 1 Least-squares mean values of daily milk yield (l)
by week of lactation obtained from the analysis of daily
milk yield data.
each week of lactation. The RMS for each model
when its calculated daily milk yield values were
compared with those of the ‘typical Awassi lactation
curve’ are presented in Table 2. The residuals are
shown graphically in Figure 2 for the four linear
functions and in Figure 3 for the four non-linear
functions. The results show that the weighted
calculations always gave a better fit than their
unweighted counterparts and the Morant functions
always gave a better fit than the Wood functions for
the equivalent model. The non-linear Morant,
Grossman, Pollott additive and Pollott multiplicative
functions had similar residual mean squares and can
be considered to fit the ‘typical Awassi lactation
curve’ equally accurately.
The residuals from fitting the Wood and Morant
functions in their linear form showed a distinct
pattern whereas those from the Morant linear
weighted function (Figure 2) were evenly distributed
about zero. Thus, of the linear models, the linear
100
200
Day of lactation
0·3
0·2
0·1
0
–0·1
–0·2
–0·3
–0·4
0
100
200
Figure 3 The residuals derived from fitting the four nonlinear models (Wood, ●; Morant, ▲; Grossman,
; Pollott
additive,
) to the ‘typical Awassi lactation curve’.
weighted form of the Morant function described the
lactation curve most accurately. Of the non-linear
functions (Figure 3) only the Wood function had non-
Function
Typical Awassi lactation curve
Wood
Wood linear
Wood linear weighted
Morant
Morant linear
Morant linear weighted
Morant linear weighted
reparameterized
Grossman
M = 1·987 n0·2413 exp–(0·009075 n)
ln(M) = ln(1·6892) + 0·3156 ln(n) – 0·01062 n
ln(M) = ln(2·0530) + 0·2294 ln(n) – 0·00882 n
M = exp (0·6006 – 0·8147 n’ –0·2597 n’2 – 0·5992/n)
ln(M) = 0·5841 – 0·7969 n’ – 0·21546 n’2 – 0·8573/n
ln(M) = 0·6022 – 0·8081 n’ – 0·2543 n’2 – 0·6175/n
ln(M) = 0·6022 – 0·8081 n’ (1 + kn’)+ 0·2136 n’2 – 0·6175/n
using k = 0·5
M = (3·41 ✕ 0·074 ✕(1 – tanh2 (0·074 (n – 26·15))))
+ (470 ✕ 0·0066 ✕ (1 – tanh2 (0·0066 (n –33·3))))
M = (3·79/(1 + ((1 – 0·8)/0·8) exp (–0·1104 n)))
– (3·61/(1 + ((1 + 0·0831)/0·0831) exp (– 0·01725 n)))
M = (3·75/(1 + ((1 – 0·801)/0·801) exp (–0·1207 n)))
✕ (1/(1 + ((1 – 0·0712)/0·0712) exp (0·0184 n)))
Pollott multiplicative
300
Day of lactation
Table 2 The results of fitting the various functions to the ‘typical Awassi lactation curve’
Pollott additive
300
Figure 2 The residuals derived from fitting the four linear
models (Wood linear, ■; Wood linear weighted, ▲; Morant
linear
; Morant linear weighted
) to the ‘typical
Awassi lactation curve’.
Deviation (l)
Daily milk yield (l)
3·5
Residual mean
square (l2)
0·0615
0·1645
0·0395
0·0023
0·0179
0·0033
0·0033
0·00165
0·00159
0·00161
202
Pollott and Gootwine
Table 3 The estimated characteristics of the overall lactation
curves using the four linear and four non-linear functions
Milk yield to Peak
day 280 (l) yield (l)
Typical Awassi
curve
Wood
Wood linear
Wood linear
weighted
Morant
Morant linear
Morant linear
weighted
Grossman
Pollott additive
Pollott
multiplicative
Day of
peak
yield
Persistency
(g/day)
543
546
545
3·34
3·44
3·59
27
27
30
15·3
12·7
14·2
550
542
541
3·45
3·29
3·36
26
21
19
12·5
14·7
14·1
543
542
543
3·29
3·35
3·29
20
27
24
14·6
15·4
15·4
542
3·31
25
15·3
random residuals. Thus the non-linear Morant,
Grossman and Pollott functions were as good as each
other.
Calculated values. Four characteristics of the overall
lactation curve were estimated using the various
functions: milk yield to day 280, maximum daily
milk yield, the day on which it occurred and
persistency at day 150. These are shown in Table 3
for the ‘typical Awassi lactation curve’ and the
various functions. The ‘typical Awassi lactation
curve’ had a total milk yield to day 280 of 543 l with
a peak of 3·34 l on day 27 and a decline in milk
production on day 150 of 15·3 g/day. All the
functions estimated total milk yield to within 3 l of
the ‘typical Awassi lactation curve’, with the
exception of the Wood linear weighted function
which gave an overestimate by 7 l. Peak yield was
accurately estimated by all functions except those
involving the Wood function, which tended to
overestimate it. Estimates of the day of peak lactation
were all within 3 days of the ‘typical Awassi lactation
curve’ value, with the exception of the Morant
functions which underestimated the day of peak
yield by an average of 7 days. The Grossman and
Pollott models estimated persistency in agreement
with the ‘typical’ values but all other models
underestimated it.
Completed lactations
The least-squares analyses of RMS from fitting the
various functions to the 1416 completed lactations
are summarized in Table 4. From the mean RMS
values in Table 4 the Wood models showed the
poorest fit, although the Wood 3 had a lower mean
RMS than some of the other models (Pollott
multiplicative 2, Pollott additive 4). The Morant 3
and 4 functions had a lower mean RMS than all other
models. Of the Pollott additive models the three- and
Table 4 A summary of analyses of variance investigating the effect of total milk yield group (TMY) and lactation length group (LL) on
the residual mean square of fitting various 2, 3, 4, 5 and 6-parameter functions to complete lactations
Significance of
Function and
no. of
parameters
Wood 3
Wood 2
Wood linear weighted 3
Morant 4
Morant 3
Morant linear weighted 3
Grossman 6
Pollott additive 5
Pollott additive 4
Pollott additive 3
Pollott additive 2
Pollott multiplicative 4
Pollott multiplicative 3
Pollott multiplicative 2
Mean RMS and
s.e. (l2)†
0·1366D ±
0·1531C ±
0·1735B ±
0·0803H ±
0·1020G ±
0·1187F ±
0·1486C ±
0·1260EF ±
0·1576C ±
0·1253F ±
0·1366D ±
0·1346DE ±
0·1189F ±
0·1845A ±
0·004134
0·004195
0·006568
0·002485
0·003089
0·004485
0·008874
0·004617
0·005902
0·004147
0·003785
0·005023
0·003634
0·004601
TMY
LL and
TMY ✕ LL
**
***
*
**
*
*
*
*
***
*
**
***
*
***
Mean residual mean square
Low
TMY (l2)
Medium
TMY (l2)
0·0486a
0·0500a
0·0602a
0·0316a
0·0407a
0·0465a
0·1068a
0·0621a
0·0636a
0·0518a
0·0526a
0·0628a
0·0504a
0·0586a
0·1340b
0·1547b
0·1808b
0·0764b
0·1006b
0·1245b
0·1298a
0·1171a
0·1797b
0·1302b
0·1441b
0·1482b
0·1269b
0·1915b
High
TMY (l2)
0·2169b
0·2949c
0·2764ab
0·1304b
0·1352b
0·1542ab
0·1785a
0·1655a
0·1758ab
0·1992b
0·2915c
0·1706ab
0·1921b
0·4368c
a,b,c TMY group means in the same row with the same superscript are not significantly different.
Values in the mean RMS column with the same superscript are not significantly different.
† 1416 records used for two- and three-parameter models (224 Low, 820 Medium, 372 High); 1383 for four-parameter models
(193 Low, 818 Medium, 372 High); 1220 for five-parameter models (90 Low, 759 Medium, 371 High); 1033 for six-parameter
models (30 Low, 638 Medium, 365 High).
Modelling of complete lactations in sheep
five-parameter versions were equally as good and of
the multiplicative models the three-parameter model
had the lowest RMS.
The results in Table 4 for the Wood 3 model indicate
that there were significant differences (P < 0·01)
between the mean RMS for the three TMY groups
but not to the LL groups or the interaction between
TMY and LL. The model fitted the Low TMY group
of records more accurately (mean RMS 0·0486) than
the medium and high TMY (mean RMS 0·1340 and
0·2169, respectively). The same general pattern can
be seen for all other models in Table 4, with the
exception of the five- and six-parameter models. Due
to low numbers of records in the low TMY group the
5/6 parameter models were similar in their RMS
values for all three TMY groups. The Morant
functions always gave the lowest RMS of all the
functions within the TMY groups. Lactation length,
and its interaction with TMY, was only important in
the two-parameter Pollott functions, which tended to
fit the high yielding lactations very poorly.
Discussion
In this paper the use of various mathematical
functions for describing the complete lactation curve
of the Improved Awassi dairy ewes were
investigated. Four basic functions were considered.
Two of the basic functions can be calculated using
linear multiple regression methods and three were
used with a reduced number of parameters,
depending on the assumptions made. The degree of
fit of the various mathematical functions were tested
using two data sets: a ‘typical Awassi lactation curve’
comprised of 40 daily records obtained for each week
of the lactation, starting from the week of lambing,
and 1416 completed lactations with different
numbers of monthly test-day records.
Although the lactation of sheep has been extensively
reported, studies on the complete lactation are rare
due to the common practice in dairy ewes of
allowing the lambs to suck for at least a month after
lambing. The suckling period coincides with the
rising phase and peak of lactation, making them
impossible to estimate. Post-weaning milk
production is thus entirely during the declining
phase of lactation and can be adequately described
by a simple linear regression model. It is not
surprising, therefore, to find few reports of work
dealing with the complete lactation of sheep and in
such cases to find authors relying heavily on research
in cattle, a species in which weaning commonly
occurs within a few days of parturition.
Portolano et al. (1996) used the Wood function to
describe the complete lactations of 92 Comisana ewes
203
but gave no indication of how accurately the model
fitted the data. Groenewald et al. (1995) studied 63
lactations of Merino sheep, a non-dairy breed and
compared the Wood, Grossman and Morant
functions for describing the first 16 weeks of
lactation based on 10 records. They found the
Grossman model to have the best fit (mean RMS
0·0317 kg2) followed by Morant (0·0341 kg2) and
Wood (0·0461 kg2).
Linear functions
The two linear functions compared here were the
incomplete gamma function as described by Wood
(1967), with modifications suggested by Cobby and
Le Du (1978) and the modified polynomial,
suggested by Morant and Gnanasakthy (1989) after
comparing six possible alternatives. Both functions
were fitted in a logarithmic form in order to achieve
linearity. The method of weighting the Morant
function using the square of the daily milk yield was
investigated here for the first time.
When fitting the two linear functions, in both their
unweighted and weighted form, to the ‘typical
Awassi lactation curve’ (Table 2), the findings of
Cobby and Le Du (1978), Williams (1993) and Morant
and Gnanasakthy (1989) were found to apply to
sheep lactations as well. The original gamma
function of Wood (1967), used in its loge form, tended
to overestimate the yields in early lactation and
underestimated them in late lactation (Figure 2). The
distribution of residuals from the Wood linear
weighted function was not random with respect to
day of lactation (Figure 2). The use of the Wood
function weighted by the square of the daily milk
yield (Cobby and Le Du, 1978) improved the fit in
early lactation, compared with the unweighted linear
Wood model, but tended to overestimate daily milk
yield in late lactation. Nevertheless the Wood linear
weighted function reduced the RMS of the estimated
daily milk yields in weeks 1 to 40 from 0·1645 l2
(Wood linear) to 0·0395 l2. The total milk yield to the
end of the 40th week was overestimated by Wood
linear weighted (550 v. 543 l) but day of peak yield
was estimated almost exactly (day 26) and peak yield
was estimated at 3·45 l compared with the original
value of 3·34 l.
The Morant linear function fitted the weekly leastsquares means better than Wood linear weighted
function, reducing the RMS from 0·0395 to 0·0179 l2
(Table 2). This was mainly due to a better fit in mid
and late lactation because, in early lactation the
Morant linear function underestimated yields.
Consequently, peak yield was estimated by the
Morant linear function as 3·36 l on day 19, 8 days
earlier than the ‘actual’ value. The Morant linear
204
Pollott and Gootwine
function predicted total yield almost exactly. The use
of the weighted Morant linear function improved the
accuracy of fit to a residual mean square of 0·0033 l2,
accurately predicted total yield but underestimated
both the day and level of maximum yield. The
distribution of residuals from the Morant linear
weighted function was random with respect to day
of lactation (Figure 2). The Morant linear weighted
function would thus appear to be better than the
Wood linear weighted function for describing the
overall lactation curve of the typical Improved
Awassi sheep, both in terms of its accuracy of fit and
distribution of residuals.
Non-linear functions
The non-linear functions always fitted the ‘typical
Awassi lactation curve’ better than their linear
equivalents (Table 2). The Morant, Grossman and
Pollott functions had similar RMS (0·00159 to
0·0023 l2) which were all lower than that of the Wood
function (0·0165 l2). The residuals derived from
fitting the four non-linear models to the overall data
are shown in Figure 3. In its non-linear form, the
Wood function overestimated the milk yield in early
and late lactation and underestimated it in mid
lactation. The other three functions showed evenly
distributed residuals about the curves. This is
reflected in the estimates of the ‘typical Awassi
lactation curve’ characteristics shown in Table 3. The
Wood function overestimated peak yield (3·44 l) but
predicted day of peak yield well; the Morant
function underestimated both day and level of peak
yield and the Pollott functions underestimated peak
yield but predicted the day of peak yield closely.
In terms of their lower RMS values, the lack of
association of their residuals with day of lactation
and their ability to estimate TMY accurately, the
functions of Morant, Grossman and Pollott were all
similar and better than the Wood function, when
used in their non-linear form. However, the Morant
model predicted the characteristics of the peak yield
less accurately than the Grossman or Pollott
functions.
Using the functions on monthly recorded data
In contrast to the ‘typical Awassi lactation curve’,
which had 40 weekly least squares means, farm data
usually has a small number of monthly records. Any
suitable function must be able to cope with the
characteristics of farm data. These include lactations
with no apparent peak, a wide range of lactation
lengths and lactations with very different shapes, as
well as the short lactations, characteristic of dairy
sheep.
Comparing the functions used on lactations with monthly
records. The results of fitting the original functions
(Table 1) to the 1416 complete lactations are shown in
Table 4. These results indicate that the Morant
function fitted the 1416 lactations with the lowest
mean RMS (0·0803 l2). The Grossman function had
the highest mean RMS (0·1486 l2). The Wood and
Pollott multiplicative models were similar (0·1366
and 0·1346 l2) and intermediate, with the Pollott
additive having a lower mean RMS (0·1260).
Generally, reducing the number of parameters in a
model reduced the accuracy of the model. The
exceptions to this were the three-parameter Pollott
models, which had the lowest mean RMS within the
additive and multiplicative functions. The improved
fit of the three-parameter Pollott models was due to
the fact that the three parameters estimated all
related to the declining phase of the curve, and this
was the phase with the most data points. Often the
pre-peak phase had only one, or even no points, and
hence was difficult to estimate accurately.
Lactation characteristics and the fit of the functions. The
results in Table 4 indicate that all functions fitted
lactations with low yields better than those with
medium and high yields. The Grossman and Pollott
additive five functions were the only exception to
this, probably due to the low number of short
lactations in the low TMY group when fitting a fiveor six-parameter model. Since the lactations used in
these analyses were complete lactations, a lactation
with a high yield had either a higher peak or a
greater length than a low yielding lactation, or a
combination of both. In all but two of the functions
there was no effect of lactation length on RMS. The
lack of fit of the functions to the high yielding ewes
may have been due to the either the inability of the
functions to describe a more pronounced peak
accurately or higher yielding animals having a more
variable daily milk yield. It is not possible to explore
these two possibilities with the current data set but a
study of lactations based on weekly records would
help to address this question.
Only in a few cases was the model a better fit to the
medium compared with the high-yielding groups
(Wood 2, Pollott additive 2, Pollott multiplicative 2).
Lactation length did not affect the fit of the models,
with the exception of the two-parameter models,
which would probably not be used due to their poor
fit to high yielding lactations.
There is no need to choose a particular function to
deal with short lactations in sheep (Table 4). The best
functions were as equally effective as each other at
all levels of yield. The Morant functions clearly have
205
the lowest residual mean squares, particularly in the
high yielding group. The three-parameter Pollott
functions had the lowest RMS of the remaining
models studied. These results agree to some extent
with those of Groenewald et al. (1995) in that the
Wood models fitted the data least accurately.
However, in our analyses the Grossman model was
less accurate than the Morant model.
Parameters values
In some cases, using the various functions resulted in
parameter estimates being outside their expected
range (e.g. negative values of b in the Wood
function). In these cases there was not enough
information in the data to allow the estimation of the
rising phase parameters. This may be because the
first milk record was taken about or after the peak, as
is quite possible with monthly data, or simply
because there is not enough differentiation between
pre- and post-peak yields. This was seen with the
results from the individual curve fitting using the
Wood function. A number of negative b values were
obtained from these analyses. In all cases the first
test-day record was the highest in the lactation but
this was not the complete answer because other
records also had their first value as the highest.
Calculated values and parameters
The various functions used for describing lactation
curves have been discussed in terms of their
accuracy of fit to the data and the credibility of the
parameter values. However, when using the results
of curve fitting it is important to use a function that
provides reliable estimates of the required
characteristics of the lactation. This includes
information on the peak yield, persistency and
predictions of the future course of the lactation, and
hence total milk yield. In addition it may be
important to use a function that provides parameter
values that have a meaningful biological
interpretation.
For the Wood function negative b and/or c values are
a problem. Not only are they outside the expected
range but the calculations of DP, PY and CTMY are
not feasible (Wood, 1967).
The Morant function is a second order polynomial
which relies on one of its maxima falling within the
normal length of the lactation period in order to
describe the curve realistically. In some cases the
fitted Morant curve can decline in early lactation. It
then turns up to the peak and then follows the
expected pattern for a lactation curve. This can result
in extremely high values of CTMY. The use of the
standard value for i, resulting in the Morant 3 model,
Daily milk yield (l)
Modelling of complete lactations in sheep
4·0
3·5
3·0
2·5
2·0
1·5
1·0
0·5
0
0
100
200
300
Day of lactation
Figure 4 The two phases of the overall lactation curve
using the Pollott additive model showing actual milk yield
(
), cell proliferation and differentiation phase curve
(▲), apoptosis curve (✖) and calculated milk yield (■).
corrects this problem but the function
underestimates yield in early lactation.
still
Using the Pollott functions to describe lactation
The Pollott functions have been specifically designed
to describe lactation based on the known biology of
milk production (Pollott, 2000). Figure 4 shows the
relationship between the two logistic curves and the
overall lactation curve using the Pollott additive
function. The phase describing cell proliferation,
differentiation and the increase in cell activity
reaches its upper limit soon after parturition. In the
‘typical Awassi lactation curve’ the daily milk
secretion potential was 3·79 l and the increase in milk
production was 24 g/day mid way between the start
and the peak of lactation. The relative increase in
milk production per day was 0·1104. The peak yield
of 3·31 l occurred on day 24 of the lactation. The daily
milk secretion loss at the start of lactation was 0·3 l
and the maximum daily secretion loss was 3·61 l. The
rate of loss of milk secretion midway between the
peak and end of lactation was 15 g/day and the
relative rate of milk loss was 0·01725.
Pollott (2000) discussed the effects of assuming that
there is no milk secretion loss until after lambing. In
this case, the second logistic curve of the model is set
to start at 0 and alters several values accordingly:
The daily milk secretion potential is 3·49 l and the
increase in milk production is 23 g/day. The daily
milk secretion loss is 3·31 l and the reduction in milk
production is 14 g/day mid way between the peak
and the end of lactation.
Since sheep lactations tend to be left to continue until
daily milk yield almost ceases, then the upper limit
of both logistic curves nearly coincide within the
lactation period. This contrasts with the situation in
many cattle herds where lactation is stopped
prematurely (Pollott, 2000).
206
Pollott and Gootwine
Conclusions
There are a number of factors which need to be
considered when choosing the optimal function to
describe the lactation of sheep. While the key factor
is the accuracy of the fit of the function, the
possibility of calculating characteristics of the curve
and the interpretation of the curve’s parameters is as
important. Of the four functions investigated in this
paper, the Grossman, Morant and Pollott models
improve on the widely used Wood model. The
Morant and Pollott functions outperform the
Grossman function in a number of ways; accuracy of
fit, use for calculating characteristics of the lactation
and interpretation of the parameters.
The Morant function appears to be robust and
flexible, giving a good fit in both its linear and nonlinear form. The three-parameter version of the
model can be used with little loss of accuracy and its
weighted linear form may be the best for use with
on-farm recording schemes. Its main drawback
relates to underestimation of the day of peak yield
and difficulties in calculation of CTMY, which can be
inaccurate due to the shape of the curve in early
lactation. However, the use of the three-parameter
form overcomes the latter problem.
The methods of Pollott (2000) are the most accurate
method when using weekly records of milk yield but
are less accurate than the Morant function with
monthly data. However, they have one advantage in
that they provide parameters which have a biological
interpretation and may thus be preferred in certain
situations. The Pollott additive 5 provides estimates
of all the parameters with the most accuracy. Both
the Pollott additive 3 and the Pollott multiplicative 3
forms of the lactation curve give the best fit to the
monthly data and result in credible parameter values
in all cases.
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