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On the use of simple ratios between lactation curve coefficients
to describe parity effects on milk production
a ,
*
b cN.C. Friggens
, G.C. Emmans , R.F. Veerkamp
a
Department of Animal Health and Welfare, Danish Institute of Agricultural Sciences, Research Centre Foulum, P.O. Box 50,
DK-8830 Tjele, Denmark b
Animal Biology Division, Scottish Agricultural College, West Mains Road, Edinburgh EH9 3JG, United Kingdom c
Department of Animal Breeding and Genetics, ID-DLO, P.O. Box 65, 8200 AB Lelystad, The Netherlands Received 26 October 1998; received in revised form 9 April 1999; accepted 20 April 1999
Abstract
The objective of this study was to quantify how the pattern of milk production relative to time from calving is affected by parity in cows fed high quality rations. For this purpose two models; those of Emmans and Fisher (1986) and Dijkstra et al. (1997), were considered. Comparison with Wood’s (1967) function was made to evaluate their fitting ability. Daily records of milk yield from 40 cows fed a grass silage based, high concentrate total mixed ration were used. The cows had ad libitum access to food, they were milked twice daily. Each of the cows had milk yield records from calving to 240 days post calving in parities 1, 2 and 3. The model of Dijkstra et al. (1997) was found, on inspection, to be an alternative parameterization of the Emmans and Fisher model (1986). In the analyses, the Emmans and Fisher form was used: Yield5aU exp [2c(days
from calving)] where U5exp h2exp (G02b[days from calving])j. The Emmans and Fisher model (1986) performed marginally better than the Wood’s function (1967) in terms of percentage of variance accounted for and residual standard error. There was no significant effect of parity on G or b, which are the main parameters describing the evolution of0 lactation to peak. However, there was a highly significant effect (P,0.001) of parity on coefficients a, the scalar, and c, the decay coefficient. The coefficient values in parity 1 and 2 were found to be a constant proportion of the values in parity 3. Parity effects can therefore be described by simple ratios, offering the possibility of simplifying the inputs needed in models to describe potential milk production. 1999 Elsevier Science B.V. All rights reserved.
Keywords: Dairy cows; Lactation; Parity; Model; Milk yield
1. Introduction dairy cows, that is the performance the cow would
achieve if it were not limited by diet or environment, The ability to predict the potential performance of and thus calculate the energy required to support potential, is important for two reasons. If potential can be predicted, both relative to days in milk and
*Corresponding author. Tel.:145-8999-1555; fax:1
45-8999-across parities, then rations can be designed that
1500.
E-mail address: [email protected] (N.C. Friggens) capitalise on the genetic ability of the cow. Further,
in models that aim to predict the partition of The model of Neal and Thornley (1983) uses five nutrients between body reserves and milk product- input values to define initial state, and has 14 ion, the potential performance provides the biologi- parameters. The model of Dijkstra et al. (1997) is a cal basis from which to develop rules for nutrient description of the rates of profileration and death of partitioning in all feeding situations. The largest mammary cells in simpler, and more general, terms component of potential, in terms of energy require- than the model of Neal and Thornley (1983) and can
ments, is milk production. be applied to describe the lactation curve using four
It is clear that milk production varies with time, parameters. The model of Emmans and Fisher ´
both within and between lactations (Remond et al., (1986) also uses four parameters. In this model the 1997), though the major focus of studies to char- growth phase is described by a Gompertz function acterise the temporal patterns of milk production has (Gompertz, 1825; Winsor, 1932) and the declining been on the variation within lactations. There are a phase by a partial gamma decay first used for substantial number of existing lactation curve models lactation curves by Brody et al. (1923) and included of varying complexity (Nelder, 1966; Wood, 1967; in the well established model of Wood (1967). The Cobby and Le Du, 1978; Goodall and Sprevak, 1985; same form was used by France et al. (1985) to Grossman and Koops, 1988; Morant and Gnanasak- describe faecal marker excretion patterns.
thy, 1989) and a number of studies which have Both the Emmans and Fisher model and the model examined the performance of different models in of Dijkstra et al. (1997) have two clear advantages terms of their ability to fit milk yield data (Rowlands over the simpler, three-parameter, Wood’s function
´
et al., 1982; Rook et al., 1993; Perochon et al., 1996) (1967). The disadvantages of the Wood’s function
b
or give accurate predictions of future yield from part arise from the use of a power function, t , to describe records (Wilmink, 1987; Jones, 1997). However, for the growth phase of lactation. When t50, the the purpose of predicting potential a high degree of function predicts zero yield. The data used by Wood flexibility of curve shape and ability to fit environ- (1967) were weekly records which may have made mentally or nutritionally induced distortions of the the zero intercept seem likely. However, it is clear underlying curve are not valuable attributes. The key from daily records that cows can have appreciable attributes of a useful model to describe potential are milk yields at calving as would be expected from the that it is biologically interpretable and provides the finding that the secretory potential of the mammary simplest sufficient description of the potential yield gland is well developed at parturition (Knight and
b
curve. Both of these attributes are important when it Wilde, 1993). Further, t does not have an comes to subsequently developing modifiers to the assymptote and so continues to have an effect, potential curve. Development of models to incorpo- implying continued growth of the gland, throughout rate, in a logical and general manner, effects such as lactation. Under normal conditions there is no evi-those of parity, pregnancy, limiting nutrition and dence for this (Knight and Wilde, 1993), and from environment depends upon a clear association be- the point of view of interpreting the model, the decay tween model parameters and the underlying bio- phase is not simply described but is the consequence
b
logical phenomena (Dijkstra et al. 1997). The pro- of two parameters. By replacing the t term with an cess of extending such a model to accommodate asymptotic function which can have a positive parity effects provided the impetus for this study. intercept, both the Emmans and Fisher model (1986) The set of lactation curve models which explicitly and the model of Dijkstra et al. (1997) overcome aim to reflect the underlying biology is a small one these two problems albeit at the cost of an extra (Neal and Thornley, 1983; Emmans and Fisher, parameter.
1986; Dijkstra et al., 1997) though it can be argued Although they are derived from very different sets that any model which comprises explicit growth and of theoretical assumptions, the models of Emmans decay phases is biologically interpretable (Wood, and Fisher (1986) and Dijksta et al. (1997) would 1977). All of these models, either implicitly (Dijk- both appear to satisfy the fundamental criteria for use stra et al., 1997) or explicitly (Neal and Thornley, as biologically interpretable models to describe 1983; Emmans and Fisher, 1986) relate to potential potential milk production. However, neither of these
Table 1
forms, has been characterised in terms of statistical
The distribution of lactations according to calving year and parity
performance. Because these models are descriptions
a Parity Calving year
of potential, the appropriate data against which to
evaluate them are; milk yields from cows fed in a 1990 1991 1992 1993 1994 1995
manner likely to result in lactation curves whose
1 8 7 13 12 0 0
shapes are characteristic of potential milk product- 2 0 8 7 13 12 0
ion. The data used in the present study provided the 3 0 0 8 7 13 12
opportunity for such an evaluation.
Total 8 15 28 32 25 12
The first objective of this study was to evaluate the
a
two models (Emmans and Fisher, 1986; Dijkstra et The calving year started on the 1st September in any given year.
al., 1997) in order to determine which should be used in addressing the main objective of this study.
The ability of these models to fit milk yield data was 1994) were not considered in the present study. The assessed relative to Wood’s function (1967). cows were managed in three groups for feeding The second, and main, objective of this study was according to days in lactation; less than 100 days to quantify the effect of parity on the coefficients of (early), between 100 and 200 days (mid), and greater a simple, biologically based, model of the lactation than 200 days (late). The proportions in total dry curve and, if possible, to develop simple relation- matter (DM) of silage, brewers’ grains and concen-ships between parities in the lactation curve co- trate in the TMRs offered to early, mid and late efficients thus simplifying the inputs necessary for a lactation cows were; 40:5:55, 50:5:45 and 60:5:35 general model of lactational performance. respectively. The feeding system was designed to achieve, over a full lactation, average proportions of silage, brewers’ grains and concentrate of 50:5:45 in total DM. The average DM content (oven drying at
2. Materials and methods 808C for 24 h) of the early and mid TMRs was 364
and 334 g / kg fresh, respectively. The average crude 2.1. Animals, feeding and management protein content (Association of Official Analytical Chemists, 1990) of the early and mid TMRs was 178 Data were obtained from 40 Holstein–Friesian and 176 g / kg DM, respectively. The average cows, each of which had daily milk yield records for metabolisable energy content, estimated from stan-first, second and third lactations. The cows were dard ME values for the ration ingredients, of the housed and managed at the Scottish Agricultural early and mid TMRs was 12.2 and 11.9 MJ ME / kg College / University of Edinburgh Langhill Dairy DM, respectively. The average NDF content (Rober-Research Centre, Edinburgh, Scotland (longitude; 38 tson and Van Soest, 1977) of the early and mid 109W, latitude; 558529N) as part of a genotype by TMRs was 372 and 374 g / kg DM, respectively. environment study (Veerkamp et al., 1994). Calving Further details of the composition of the TMRs used occurred between September and January in any in the genotype by environment experiment, and of given year. The cows were kept indoors in conven- other performance measures made, have been re-tional cubicle housing from calving through to the ported by Veerkamp et al. (1994).
following July. The distribution of lactations
accord-ing to calvaccord-ing year and parity is shown in Table 1. 2.2. Data processing The cows were milked twice daily and received 0.5
kg of a standard dairy concentrate in the milking The full data set for the evaluation of the effect of
parlour at each milking. parity on lactation curve coefficients comprised
days post-calving were used as it has been shown on inspection of the equation of Dijkstra et al. it that the concurrent pregnancy in lactating dairy cows became apparent that it was an alternative form of causes a depression in milk yield in the last 18 weeks the model proposed by Emmans and Fisher (1986), of pregnancy (Hooper, 1923; Coulon et al., 1995). as shown below:
Estimation of the lactation curve coefficients, and the
letmT/k5q
effect of parity on them, was thus unaffected by
pregnancy and drying off. Extremely deviant points substituting q in the Dijkstra equation gives: were excluded from the data in a two step process
dY / dt5M ?exp qf 2q?exp (2kt)2ltg
using a spline fitting procedure with 6 degrees of 0
freedom (GenstatE Version 5.3.2., 1994). First, the
spline was fitted to the raw data for each lactation dY / dt5M0?exp ( q)?expf2q?exp (2kt)g
curve and individual milk yields which deviated
?exp(2lt)
from the spline by 65 standard deviations were excluded from the data. Second, the spline was fitted
dY / dt5M0?exp ( q) to the data set which excluded deviant values from
the first spline fitting. Individual milk yields which ?exph2exp ln ( q)f g?exp (2kt)j?exp (2lt)
deviated from the second spline by 65 standard
deviations were also excluded from the data. This dY / dt5M ?exp ( q)?exph2exp ln ( q)f 2ktg j
0
two step process resulted in the exclusion of, on
?exp (2lt)
average, 0.98 daily milk yields per lactation, the maximum number of data points excluded in any one
This is the Emmans and Fisher (1986) form cow lactation was 6 (out of 232).
where:
2.3. Lactation curve coefficients and statistical M0?exp ( q)5a, ln ( q)5G , k0 5b andl 5c analyses
Given that the equation of Dijkstra et al. (1997) is To determine whether the Dijkstra et al. (1997) or another form of the Emmans and Fisher model Emmans and Fisher (1986) model should be used in (1986) it could not provide an alternative model addressing the main objective of this study, the first against which to compare the curve fit of the objective of this study was to compare them in terms Emmans and Fisher model. For this purpose, the of their ability to fit data. The two equations are function of Wood (1967) was used:
shown below: calving, and a, b and c are the coefficients of the dY / dt5M0?exphmTf12exp (2kt) /kg 2ltj
equation of Wood (1967). The non-linear curve (Dijkstra et al., 1997)
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 calculated1 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., (c11c ) / 2]. The3 The average lactation curves for each parity are 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,0
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
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).
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
G0 b a c model. However, this study (Rook et al., 1993) is not
G0 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,
1986; Dijkstra et al., 1997). These models were which is bounded between 0 and 1 (when G tends to0
derived from substantially different perspectives and 2`and`, respectively). The average value of G in0
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.0
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,0
equation for mammary growth post-partum [Eq. M0 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 mTin the equation of Dijkstra et al. (1997) is defined (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, mTcontrols the amplitude of 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 [M0 0?exp (mT/k)].
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 .0 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 Fisher0 and be independent of the stage of lactation of the model (1986), and by mT and k in the equation of cow.
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
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
G0 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 indicates0 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 the0 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 G0 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 /a1 2 a /a1 3 a /a2 3 c /c1 2 c /c1 3 c /c2 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 in1 2 the same ratio [e.g., (a11a ) / 2].2
b
Parities are indicated by the subscript numbers in the ratios. c
Standard error of the mean.
d 2
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 average1 3 value of the coefficient across these two parities [(a11a ) / 2]. Both the ratios and the averages were calculated within cows. The line shows3 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 /c1 250.2471168[(c11c ) / 2]. However, the2 reference parity then potential milk yield curves can 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
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 average1 2 value of the coefficient across these two parities [(c11c ) / 2]. Both the ratios and the averages were calculated within cows. The line shows2 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).
G0 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
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; RUMINT 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
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![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 averagevalue of the coefficient across these two parities [(13a 1a )/2]](https://thumb-ap.123doks.com/thumbv2/123dok/1052331.933353/10.612.108.439.86.306/coefcient-emmans-fisher-parity-relative-averagevalue-coefcient-parities.webp)
