136 M
.H. Pool, T.H.E. Meuwissen Livestock Production Science 64 2000 133 –145
2
variance and scaled observations were calculated as required estimates for Vark and
s . For models
e
follows: C-LEGm and H-LEGm those parameters were
estimated using the selected data set and for model LEGm using the original data. The residual vari-
ˆ
9
y e
i q i q
2
ˆ ]]]
s 5
3 ance over DIM was assumed constant, for model
e
i q
n sc
i
C-LEGm and varied over cDIM for model H- LEGm. The model with the best order of fit, is the
]]
2
ˆ s
model that expects a covariance and a correlation
e
0 q
]]
y 5 y
4 structure which is equal, i.e., same shape as the
i q 11 i q
2
ˆ s
e
i q
œ
observed ones OBS and C-OBS. The second criterion, MSEP, investigates how well
2
ˆ where
s 5estimated residual variance for cDIM
missing records could be predicted. Therefore, differ-
e
i q
2
ent patterns of records were deleted from the original ˆ
class i and s
5overall residual variance in the
e
0 q
2
data set to obtain subsets with missing records, i.e., ˆ
model estimated in iteration q s
was re-esti-
e
0 q
2
part lactation records of different lengths were ˆ
mated in each iteration, as the mean of s
; y
e i q
i q
created. The deleted observations were predicted
and y
5vector with the current and updated
i q 1 1
using the information in the subset and compared to observations in the cDIM class i in iteration q,
the actually observed records see Pool and Meuwis-
ˆ respectively; e
5vector with residuals for cDIM
i q
sen, 1999. MSEP values were calculated as: class i in iteration q; n 5number of observation in
i
cDIM class i; sc5[N 2rankX] N 5scaling factor
s
i
accounting for the degrees of freedom used by the
2
ˆ MSEP 5
O
y 2 y s 5
i ij
ij i
model, where N 5the total number of test day
j 51
records and X5design matrix of the fixed effects one sc factor was used for all cDIM classes, because
where y 5missing record j in subset i, which was
ij
sc 5[n 2rankX ] n for cDIM class i could not be
ˆ known in the complete data set; y 5predicted value
i i
i i
ij
calculated since the design matrix X extends across
of missing record j in subset i, and s 5number of the
i
i
cDIM. missing records in subset i. The pattern in MSEP
was expected to decrease steadily over DIM if information accumulates, i.e., if part lactations be-
2.5. Comparison of models come longer. The model with the lowest and con-
tinuously decreasing MSEP pattern has the best fit. The criteria for the goodness of fit used were as in
Pool and Meuwissen 1999, namely: 1 a graphical comparison of the variance of the data [Vary], and
2 mean square errors of predictions of missing
3. Results
observations MSEP for the different TDMs. The first criterion compares the pattern of the
3.1. Variance estimates
Vary predicted by the model and that observed between the residuals when only the fixed effects of
Residual variances for milk yield estimated for the the model were fitted. Hence, observed variances
models LEGm, C-LEGm and H-LEGm are
were obtained without imposing a structure on Vary given in Table 1. Estimates decreased with the order
and were termed as OBS and C-OBS for the original of fit for all models but differences were small for
and reduced data set, respectively, for which correla- the higher orders of fit. Between models, estimates
tion structures were calculated based on weekly can not be compared because estimates for H-
classes of DIM. For all other models, Vary was LEGm were on a different scale as for C-LEGm
estimated by Eq. 2, where y includes a record for and LEGm. Variances and covariances were esti-
each day in the lactation trajectory 1–305 and mated and presented in the form of correlations
M .H. Pool, T.H.E. Meuwissen Livestock Production Science 64 2000 133 –145
137 Table 1
C-LEGm from 20.64 to 0.23 [r
and
k ,
2
1 7 6 7
Residual variances of daily milk yield kg estimated by the
r , respectively] and for H-LEGm from
k
model using Legendre polynomials with an order of fit m for
5 6 ,6 6
20.33 to 0.39 [r and r
, respectively]
LEGm [the reference model, based on the original data] and the
k k
,k
5 7 ,6 7 0 2
1 2
models C- and H-LEGm based on complete lactation records,
only results for the third- and fifth-order of fit are
respectively after a correction for heterogeneous variances over
presented in Table 2. Although correlations were in
days in milk classes within parity
general not very strong, they were not negligible.
Model LEGm
C-LEGm H-LEGm
Where C-LEGm estimated the strongest negative
LEG0 9.39
9.15 9.07
correlation between the higher order random regres-
LEG1 6.56
6.56 6.49
sion coefficients, H-LEGm estimated the strongest
LEG2 5.53
5.47 5.43
positive correlation between the lower order ran-
LEG3 4.96
4.85 4.84
dom regression coefficients.
LEG4 4.61
4.50 4.43
LEG5 4.38
4.23 4.17
LEG6 4.20
4.09 4.05
3.2. Modeled versus observed covariances
LEG7 4.07
3.99 3.93
The covariances of milk yields expected by each model [LEGm, C-LEGm and H-LEGm] were
calculated from the covariance matrix of random between the random regression coefficients k and k
i j
regression coefficients k and the residual vari-
for the order of fit m [r and i ±j] in Table 2
jm k
, k
im jm
ance for each DIM Eq. 2. In Figs. 1 and 2 the and are comparable between the models. Values
variances observed in the data and expected are showed differences between the models LEGm,
presented for the reference model [LEGm, charts in C-LEGm and H-LEGm and varied also slightly
the upper right corner, described by Pool and with the order of fit estimated up to a seventh-order
Meuwissen 1999] and C-LEGm main graphs. of fit. Correlations ranged for LEGm from 20.51
The shape of the variance curve expected for the to 0.52 [r
and r , respectively], for
k ,k
k ,k
0 0 1 0
5 6 6 6
Table 2 Correlations between random regression coefficients using a third- and fifth-order fit Legendre polynomial expected by the model LEGm,
C-LEGm and H-LEGm i.e., for the reference model, based on complete lactation records, respectively, after a correction for heterogeneity of variances of milk yield over days in milk classes within parity
a
Model k
k k
k k
k k
k
0m 1m
2m 0m
1m 2m
3m 4m
LEGm k
0.2879 k
0.2291
1 3 1 5
k 20.1705
0.0119 k
20.2952 20.0279
2 3 2 5
k 0.1756
20.1040 20.0740
k 0.1246
20.1185 20.1524
3 3 3 5
k 20.3441
20.0356 20.0757
20.2653
4 5
k 0.1838
20.1835 20.0472
20.1663 20.4297
5 5
C-LEGm k
0.1106 k
0.1227
1 3 1 5
k 20.2851
20.0282 k
20.2893 20.0178
2 3 2 5
k 0.1504
20.1537 20.3757
k 0.1903
20.1116 20.3326
3 3 3 5
k 20.3188
20.0387 20.0348
20.2942
4 5
k 0.1821
20.1569 20.0231
20.1451 20.5366
5 5
H-LEGm k
0.3779 k
0.3567
1 3 1 5
k 20.1193
0.2248 k
20.0567 0.2205
2 3 2 5
k 0.2916
0.0490 20.0657
k 0.2948
0.0758 20.0436
3 3 3 5
k 20.1629
20.0482 20.0316
20.0451
4 5
k 0.1052
20.0487 0.0570
20.0668 20.1875
5 5 a
k 5random regression coefficient j in model LEGm using Legendre polynomials with an order of fit m.
jm
138 M
.H. Pool, T.H.E. Meuwissen Livestock Production Science 64 2000 133 –145
Fig. 1. Variances of milk yield observed by a test day model based on complete lactation records model C-OBS and expected for the order of fit 0, 1 and 3 using Legendre polynomials [C-LEGm, based on the selected data], compared with the variances observed versus expected
by the reference model [OBS and LEGm, based on the original data, chart in upper right corner].
different orders of fit should be compared to the pattern expected was sufficiently accurate for model
observed ones, i.e., OBS in the upper graphs and C-LEG4, where the reference model needed
C-OBS in the main graphs. In general, the goodness LEG5. The increase at the end of the lactation
of fit increased with the order of fit for both models. period for the reference model [LEG4] was not
However, the variances expected by C-LEGm i.e., observed for C-LEG4 for which random regression
using only complete lactation records of at least 280 coefficient estimates from the reduced data set were
days long approximated the observed variances used. For H-LEGm the data was scaled such that
better than model LEGm did. Fig. 3 shows the the observed variance equals the variance that was
expected variances of the records that were corrected expected by the model, which means that the choice
for heterogeneous variances [H-LEGm]. The pat- of the model did no longer depend on the expected
terns of the curves in Fig. 3 are different from those covariance matrix, but on the correlation matrix.
of the other models [LEGm and C-LEGm] and not comparable because the variances of the cor-
3.3. Expected versus observed correlation rected records were scaled towards the variances
structures expected by the TDM. Variance estimates for all
H-LEGm, except H-LEG0, which was constant, The expected and observed correlation structures
increased towards the end of the lactation trajectory, for C-LEGm and the reference model, LEGm, are
starting halfway the lactation period, with the most presented in Fig. 4. The correlation structure ob-
rapid increase for the higher orders of fit. served in the data changed slightly towards the end
Based on complete lactation records the variance of the lactation period, showing a tendency for
M .H. Pool, T.H.E. Meuwissen Livestock Production Science 64 2000 133 –145
139
Fig. 2. Variances of milk yield observed by a test day model based on complete lactation records model C-OBS and expected for the order of fit 4, 5 and 7 using Legendre polynomials [C-LEGm, based on the selected data], compared with the variances observed versus expected
by the reference model [OBS and LEGm, based on the original data, chart in upper right corner].
somewhat lower correlations between days early in Legendre polynomial seemed to fit the observed
lactation and higher between days late in lactation structure in the data accurately. The fit of C-LEG4
for the reduced data set compared to the original data was better than C-LEG3 because stacked areas
set C-OBS and OBS, respectively. The observed showed a slightly odd bend between days in the
correlation structure shows the overall pattern clear- middle of the lactation, especially for the lower
ly, although the observed correlations were not correlation areas, which was generated probably by
smoothed. Observed correlations were calculated for the increased variances expected at the end of the
weekly classes of DIM so that number of observa- lactation see Fig. 1. Compared to C-LEG5 the
tions was large enough for each class given the size goodness of fit of C-LEG4 was more smoothed,
of the data set. Reducing the class length for DIM suggesting that C-LEG4 would be less sensitive to
and a larger data set is expected to smooth the uncertainties in the data. Based on the expected
observed correlations more. covariance and correlation pattern Figs. 2 and 4
In general the correlation structure was modeled C-LEG4 and C-LEG5 were almost identical
better for the higher orders of fit for both models, suggesting that the order of fit, i.e., the number of
C-LEGm and LEGm. Estimated correlations parameters to be estimated per animal could be
ranged from almost unity for successive days to reduced by one to a fourth-order of fit.
almost zero for days far apart. The correlation The correlation structures expected by the model
structures expected with model C-LEGm seemed to after correction for heterogeneity of variances [H-
fit the observed one better as those from model LEGm] are presented in Fig. 5. The scaling of the
LEGm. For model C-LEGm a fourth-order of fit covariance matrix in the data towards the expected
140 M
.H. Pool, T.H.E. Meuwissen Livestock Production Science 64 2000 133 –145
Fig. 3. Variances of milk yield expected by a test day model using Legendre polynomials after correction for heterogeneity of variance for milk yield over cDIM classes for an order of fit of 0, 1, 3 and 5 [i.e., H-LEGm, based on complete lactation records].
one resulted in a better fit of the observed correlation accumulated compared to the reference model
structure C-OBS in Fig. 4 compared to model [LEGm]. Comparing different orders of fit for C-
C-LEGm, especially for days late in the lactation LEGm the MSEP-pattern improved up to C-
[e.g., the observed stacked correlation range 0.8–0.9 LEG4 and was stable for higher orders of fit Fig.
after day 210 observed in the data was expected by 6, right chart.
H-LEG3, but not by model C-LEG3]. Overall, A correction for heterogeneity of variance im-
H-LEG3, two-orders of fit lower than the reference proved the MSEP-pattern further, especially for the
model [LEG5], seemed to yield a good fit of the lower orders of fit, up to LEG3 Fig. 7, left chart.
correlation structure observed in the reduced data. The MSEP for H-LEGm improved especially when
part lactation records of 80 days and longer had to be 3.4. MSEP of missing observations
extrapolated implicitly by the TDM. The unexpected increase of MSEP, when part lactation records of 100
In Fig. 6 left chart the MSEP is given for several days or longer were used, was almost stabilized
subsets with part lactations of different lengths for completely for the lower orders of fit in model
the reference model [LEGm] and for the model H-LEG3 compared to C-LEGm Fig. 7, left
C-LEGm. The MSEP was expected to decrease chart. For the higher order of fits [H-LEG2 and
continuously if information accumulates, i.e., if part higher] MSEP-patterns were stable Fig. 7, right
lactations becomes longer. For the lower orders of fit chart.
up to a third-order the MSEP decreased with lacta- Overall, the results for MSEP showed that includ-
tion length for part lactations up to a length of 80 ing information about test day records at the end of
DIM but increased thereafter. Overall, model C- the lactation period was important in order to get
LEGm was more consistent when information accurately estimated model parameters, i.e., lactation
M .H
. P
ool ,
T .H
.E .
Meuwissen
Livestock Production
Science 64
2000 133
– 145
141 Fig. 4. Correlation structures for milk yield among days in milk observed in the data OBS and C-OBS and those expected by a test day model using Legendre polynomials
with different orders of fit m based on the selected data with lactation records of at least 280 days long [C-LEGm], compared to those for the reference model [LEGm], based on the original data set. Stacked areas on top, from left to right, are OBS lower triangle and C-OBS upper triangle; LEG1 lower triangle and C-LEG1 upper
triangle. Stacked areas on bottom, from left to right, are LEG3 lower triangle and C-LEG3 upper triangle; LEG4 lower triangle and C-LEG4 upper triangle; LEG5 lower triangle and C-LEG5 upper triangle.
142 M
.H. Pool, T.H.E. Meuwissen Livestock Production Science 64 2000 133 –145
Fig. 5. Correlation structures for milk yield among days in milk expected by a test day model using Legendre polynomials with different orders of fit m after correction for heterogeneity of variances of milk yields over classes for days in milk [H-LEGm]. Stacked areas, from
left to right, are H-LEG2 upper triangle and H-LEG3 lower triangle; H-LEG4 upper triangle and H-LEG5 lower triangle.
records with information over the whole lactation parameters to be estimated per animal without
period should be used. Further, the unexpected significantly reducing the goodness of fit for the
increase of MSEP probably generated due to unequal expected variance–covariance matrix using Legendre
weighting of the data points was improved clearly polynomials in a random regression TDM. The
after the correction for heterogeneity of variance. goodness of fit was measured here by MSEP and a
graphical comparison of the observed versus ex- pected variance and correlation structures. Differ-
4. Discussion ences in goodness of fit could be compared by
