252 A
. Kettunen et al. Livestock Production Science 66 2000 251 –261
¨ yield, it approximates mean TD milk production
to differ from 1.0 Kettunen and Mantysaari, 1996. throughout lactation.
This finding supported the hypothesis that inclusion Attempts to improve the accuracy of BV estima-
of RR function in BV estimation of dairy cattle from tion and a need to provide more comprehensive
TD data is necessary. A random regression TD management information to farmers has stimulated
approach has been recently used for both estimation an increased interest in the use of original TD
of genetic parameters and genetic evaluation of measurements instead of aggregated lactation re-
production traits Jamrozik and Schaeffer, 1997; cords. Use of the TD approach allows a more
Kettunen et al., 1997; Jamrozik et al., 1997a. detailed statistical model to be developed, which can
Use of RR test-day models in variance component account for genetic and environmental variation
estimation has, however, turned out to be somewhat specific to individual TD yields. For the Finnish
problematic. Firstly, very high estimates of heritabili- dairy cattle BV estimation the greatest advantage
ty for daily milk yield have been reported Jamrozik afforded would be a more precise definition of the
and Schaeffer, 1997; Kettunen et al., 1997, and the contemporary group CG. The current animal model
pattern of estimates is contradictory to that estimated uses herd-calving year HY to describe CG. Further
with multitrait models Meyer et al. 1989; Pander et partitioning of HY according to calving season is not
al., 1992. Secondly, RR analysis has resulted in possible due to small herd sizes. Year of calving
antagonistic relationships between early and late leads into an illogical grouping of records, since a
lactation daily yields of protein Jamrozik and situation can arise, where cows produce for the
Schaeffer, 1997 and milk Kettunen et al., 1997. majority of their lactation under the same environ-
This is due to deficiencies in the definition of cow ment but are assigned to different classes according
permanent environmental effects. Inclusion of RR to HY. Furthermore, HY characterises long-term
function to describe PE effects can potentially im- effects of a particular calving year in a herd rather
prove the properties of the statistical model. Thirdly, than short-term variation due to management effects
when a logarithmic polynomial function was used, at the time of production. Since season of production
RR coefficients were found to be highly correlated: accounts for more environmental variation than herd-
additive term and second order polynomial 20.97, calving year-calving season HYS Swalve, 1995;
and first and second order logarithms 20.98, in ¨ ¨
Poso et al., 1996, the use of the TD approach, where particular Kettunen et al., 1997. An orthogonal
CG is defined as herd-test month HTM, improves polynomial function as RR sub-model could be used
the properties of the statistical model. Furthermore, to overcome problems of dependency between vari-
solutions of HTM effects can be utilised to improve ables.
herd management. The objective of this study was to estimate genetic
The genetic shape of the lactation curve can be parameters for first lactation TD milk yield of
modelled by fitting regression coefficients within an Finnish Ayrshire cows. Two RR models with tradi-
animal, commonly referred to as random regression tional consideration of PE effects were used: loga-
RR coefficients Schaeffer and Dekkers, 1994. rithmic polynomial and normalised third order ortho-
Additive genetic solutions are simply a set of BV gonal polynomial functions. In addition, the effect of
estimates for the RR coefficients Jamrozik et al., modelling PE covariance structure with RR func-
1997a. The product of these estimates and the days tion on genetic parameters was assessed. Finally,
in milk DIM dependent covariates give a BV of an results from RR analysis were evaluated by com-
animal for each TD yield. This allows the genetic parison with estimates derived from a multitrait
ranking of animals to vary at different stages of model MT by continuous covariance function.
lactation. In addition, differences between actual and expected production can be calculated to monitor the
management of individual herds and of individual
2. Material and methods
cows within a herd. Within a primiparous Finnish Ayrshire cow popu-
2.1. Data lation genetic interrelationships between TD milk
yields at different stages of lactation were estimated Data provided by the Agricultural Data Processing
A . Kettunen et al. Livestock Production Science 66 2000 251 –261
253 Table 1
For MT analysis, lactations were divided accord-
Summary statistics of test-day data
ing to DIM at test to give 16 traits Table 2. Traits
Herds 78
indicate TD milk production at a certain stage of
Herd years 707
lactation and are therefore referred to as interval milk
Herd-test months 8038
production IMP. Short intervals were used to
Cows 8351
distinguish critical changes of daily yields during the
Cows with records 6310
beginning of lactation 4–30 DIM i.e. IMPs 1–3.
Test-day observations 63,331
Sires 1900
Intervals of 20 days described IMPs 4–6 31–90
Sires with daughters with records 1380
DIM and 13–16 271–350 DIM, whereas mid-
Mean of test day milk yield kg 20.3
lactation was divided into 30 day intervals to give IMPs 7–12 91–270 DIM.
Centre originated from the national milk recording database and comprised all TD measurements col-
2.2. Statistical model and methods lected since 1988. Incomplete data at the beginning
of the database and of short lactations were excluded 2.2.1. Multitrait model
to maximise the number of observations per animal. To estimate heritability for 16 IMPs and genetic
Consequently, TD milk yields produced from 4 to correlations between them, 37 trivariate REML runs
350 DIM of cows calving between April 1988 and were conducted. The same multitrait linear model
March 1996 were used. To attain an informative and was assumed for each IMP record to estimate
representative CG size, data from 78 herds with covariance components. For IMP in the tth interval
between 7 and 14 heifers calving each HY was used. the following model was used:
As a result, 63,331 TD milk records of 6310 y
5 age 1 DCC 1 YS 1 herd
thijkmno th
ti tj
tk
primiparous Finnish Ayrshire cows were obtained 1 b DIM
1 HY 1 a 1 e
Table 1. Pedigree information for animals without
t thijkmno
tm tn
thijkmno
records was traced back two generations. Cows with 1
records were daughters of 1380 sires, while after removing the non-informative animals from the
where y is the IMP record; age
is the fixed
thijklmno th
pedigree the total number of male and female effect of calving age; DCC
is the fixed effect of
ti
animals in the data was 1900 and 8351, respectively. days carried calf; YS is the fixed effect of calving
tj
year-season class; herd is the fixed effect of herd; b
tk t
is the regression coefficient of IMP on days in milk;
Table 2 ¯
Days in milk DIM at test, number of observations N , mean x
DIM is the covariate of days in milk at test;
thijkmno
and standard deviation S.D. of test-day milk production at
HY is the effect of herd-calving year, a
is the
tm tn
different stages of lactation
additive genetic effect of an animal n and e is
thijkmno
¯ Interval
DIM N
x S.D.
the residual associated with y . Calving age
thijkmno
divided into 7 groups indicated the age of the cow at
1 4–10
1428 18.8
4.1 2
11–20 2072
21.2 4.1
first calving. Five classes of DCC effect described
3 21–30
2091 22.4
4.0
the number of days from the last insemination to test.
4 31–50
4072 23.2
4.3
Pregnancy during the first 120 days was assumed to
5 51–70
4125 23.3
4.3
have no effect on milk production, and therefore
6 71–90
4138 23.2
4.4
DCC effect was excluded from models for 1–7
7 91–120
5950 22.5
4.5 8
121–150 5874
21.7 4.4
IMPs. Combining 9 calving years and 3 month
9 151–180
5820 20.9
4.4
classes within each year March–June, July–Sep-
10 181–210
5724 20.1
4.3
tember, October–February gave 26 calving year-
11 211–240
5571 19.2
4.4
season classes. Effects of HY, animal and residual
12 241–270
5408 17.9
4.4
were assumed to be random with zero means and
13 271–290
3426 16.7
4.5 14
291–310 2899
15.7 4.5
varHY5I H , vara5A G and
vare5
15 311–330
2171 15.3
4.6
I R , where I is an identity matrix, A is the
16 331–350
1560 14.7
4.5
additive genetic relationship matrix between animals,
254 A
. Kettunen et al. Livestock Production Science 66 2000 251 –261
H , G and R are 3 by 3 covariance matrices of calving month classes k X
5 1, X 5DIM c,
1p 2p
p 2
herd-year, animal and residual effect, respectively, X
5 DIM c , X
5 lnc DIM , X 5lnc
3p p
4p p
5p 2
for the three IMPs in question, and denotes the DIM , c 5305; HTM is the effect of test month
p m
direct product operator. The covariance components of production within herd, and e
is the
hijklmnop
were estimated with restricted maximum likelihood residual associated with y
. Classifications for
hijklmnop
REML method and the average information AI calving age and DCC effects were identical to those
algorithm. The AI matrix is the average of Fisher’s in 1. Herd-test month was used as CG to account
information and Newton–Raphson second derivative for short-term environmental variation associated
matrices Johnson and Thompson, 1995. with the time of production in a particular herd.
Estimates from separate REML-runs were com- Two different RR sub-models were used to de-
piled to form G and R of order 16 by 16. This was scribe the BVs for the shape of lactation curves of
done by using the method of expanded part matrices individual cows RRSM. Initially, a logarithmic
¨ Mantysaari, 1999, which guarantees that the result
polynomial function was tested in 2, where u
qo
remains positive definite. To allow comparisons were regression coefficients of TD milk on DIM
between estimates of variance components from MT functions describing the shape of a lactation curve
and RR analyses, a fourth order continuous co- within an individual cow o, and covariates X
qp
variance function CF was fitted on G while for R a identical to those used to estimate average lactation
fourth order CF with measurement error was used curves Ali and Schaeffer, 1987. This model will
Kirkpatrick et al., 1994. Mean DIM was selected to subsequently be referred to as ASM. An alternative
represent DIM values of each interval. Then, co- RR sub-model was fitted in models 3 and 4,
variance functions were used to predict variance where r
were regression coefficients within a cow,
qo
components for individual test days. In our study, and covariates Z
DIM dependent terms of normal-
qp
variance components and genetic parameters derived ised third order orthogonal polynomial function
in this manner were used as standard for RR models. Snedecor and Cochran, 1980. Models 3 and 4
will subsequently be referred to as OPM and
PE1
OPM .
2.2.2. Random regression models
PE4
Similarly, two methods of accounting for the Covariance components for TD milk yield were
permanent environmental PE effect of an indi- estimated assuming the following linear RR models:
vidual cow were analysed. Models ASM and
5
OPM assumed a simple structure for within-cow
y 5 age 1 DCC 1 herd 1
O
b X
PE1 hijkmnop
h i
j qk
q
p
q 51
covariances, and thus pe described one common
no
PE effect associated with all TD yields of cow o. 1 HTM
m
Model OPM described a more complicated
PE4 5
covariance structure for PE effect; p were
qo
1
O
u X 1 pe
1 e 2
qo q
no hijkmnop
p
within-cow regression coefficients and covariates Z
q 51 qp
terms of normalised third orthogonal polynomial or
function. This model separated non-genetic within-
4
cow variation into a PE function and a residual term. 1
O
r Z 1 pe
1 e 3
qo q
no hijkmnop
A total of 18, 13 and 22 covariance components
p
q 51
were obtained by ASM, OPM and OPM
,
PE1 PE4
or respectively. Effects of HTM, RRSM, PE and re-
4 4
sidual were assumed random with zero means and
2
1
O
r Z 1
O
p Z 1 e
4 structure
of variances
varHTM5Is ,
qo q
qo q
hijkmnop HTM
p p
2 q 51
q 51
varRRSM5A G, vare5Is for all models,
e 2
var pe5Is for ASM and OPM
and var p5 where for all models y
is TD milk yield; age ,
PE PE1
hijklmnop h
I PE for OPM , where I is an identity matrix, A
DCC and herd are as described above; b are
PE4 i
j qk
is the matrix of additive genetic relationships among regression coefficients of TD milk on DIM functions
the animals, and denotes the direct product describing the shape of lactation curves within
A . Kettunen et al. Livestock Production Science 66 2000 251 –261
255 Table 3
operator. For ASM matrix G is a 5 by 5, and for
Estimates of variance components and heritability for daily milk
OPM and OPM
4 by 4 covariance matrix of
PE1 PE4
yields on selected test days derived by covariance function
the RR coefficients for animal effects. In addition,
2 2
2 2 a
2 b
DIM s
s s
s h
g HY
e P
PE denotes the 4 by 4 covariance matrix of the RR coefficients for p. Covariances between HTM,
5 2.68
0.62 10.54
13.84 0.20
25 2.68
0.68 8.73
12.08 0.23
RRSM, pe or p, and e were assumed zero. The
45 2.81
0.76 8.47
12.04 0.25
covariance components were estimated with re-
85 3.19
0.86 9.23
13.28 0.26
stricted maximum likelihood REML method and
125 3.45
0.89 9.50
12.85 0.27
the expectation–maximisation EM algorithm.
165 3.54
0.95 9.21
13.70 0.28
205 3.49
1.04 9.03
13.56 0.28
265 3.39
0.95 9.36
13.70 0.27
285 3.39
0.80 9.59
13.79 0.26
3. Results and discussion
