Application of an autoregressive process to estimate genetic parameters and breeding values for daily milk yield in a tropical herd of Lucerna cattle and in United States Holstein herds.

Abstract

The objectives of this study were to estimate from test day records the genetic and environmental (co)variance components, correlations, and breeding values to increase genetic gain in milk yield of Lucerna and US Holstein cattle. The effects of repeated observations (within cow) were explained by first-order autoregressive processes within and across lactations using an animal model. Estimates of variance components and correlation coefficients between test days were obtained using derivative-free REML methodology. The autoregressive structure significantly reduced the model error component by disentangling the short-term environmental effects. The additional information and the more heterogeneous environmental variances between lactations in the multiple-lactation test day model than in the first lactation model provided substantially larger estimates of additive genetic variance (0.62 kg2 for Lucerna; 14.73 kg2 for Holstein), heritability (0.13 for Lucerna; 0.42 for Holstein), and individual genetic merit. Rank correlations of breeding values from multiple lactations and from first lactations ranged from 0.18 to 0.37 for females and from 0.73 to 0.89 for males, respectively. Consequently, more selection errors and less genetic gain would be expected from selection decisions based on an analysis of first lactation only, and greater accuracy would be achieved from multiple lactations. Results indicated that substantial genetic gain was possible for milk yield in the Lucerna herd (34 kg/yr). Estimates of genetic variance for Holsteins were larger than previously reported, which portends more rapid genetic progress in US herds also; under our assumptions, increases would be from 173 to 197 kg/yr.

Full text

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1998 J Dairy Sci 81:2738–2751
2738
Received May 8, 1997.
Accepted May 28, 1998.
Application of an Autoregressive Process
to Estimate Genetic Parameters and Breeding
Values for Daily Milk Yield in a Tropical Herd
of Lucerna Cattle and in United States Holstein Herds
J.G.V. CARVALHEIRA,* R. W. BLAKE,* E. J. POLLAK,*
R. L. QUAAS,* and C. V. DURAN-CASTRO†
*Department of Animal Science,
Cornell University, Ithaca, 14853 NY
†Universidad Nacional de Colombia and
Hacienda Lucerna, Calle 34 Norte #9N-195,
Cali, Valle, Colombia
ABSTRACT
The objectives of this study were to estimate from
test day records the genetic and environmental
(co)variance components, correlations, and breeding
values to increase genetic gain in milk yield of Lu-
cerna and US Holstein cattle. The effects of repeated
observations (within cow) were explained by first-
order autoregressive processes within and across lac-
tations using an animal model. Estimates of variance
components and correlation coefficients between test
days were obtained using derivative-free REML
methodology. The autoregressive structure signifi-
cantly reduced the model error component by disen-
tangling the short-term environmental effects. The
additional information and the more heterogeneous
environmental variances between lactations in the
multiple-lactation test day model than in the first
lactation model provided substantially larger esti-
mates of additive genetic variance (0.62 kg2for Lu-
cerna; 14.73 kg2for Holstein), heritability (0.13 for
Lucerna; 0.42 for Holstein), and individual genetic
merit. Rank correlations of breeding values from mul-
tiple lactations and from first lactations ranged from
0.18 to 0.37 for females and from 0.73 to 0.89 for
males, respectively. Consequently, more selection er-
rors and less genetic gain would be expected from
selection decisions based on an analysis of first lacta-
tion only, and greater accuracy would be achieved
from multiple lactations. Results indicated that sub-
stantial genetic gain was possible for milk yield in the
Lucerna herd (34 kg/yr). Estimates of genetic vari-
ance forHolsteins were larger
reported, which portends more rapid genetic progress
in US herds also; under our assumptions, increases
would be from 173 to 197 kg/yr.
thanpreviously
( K ey words: test day animal model, tropics, auto-
correlation, genetic evaluation)
Abbreviation key: E PA = expected producing abil-
ity, L TE = long-term environmental effect, ME =
mature equivalent, ST E = short-term environmental
effect, TD = test day.
INTRODUCTION
Many cattle in the tropics come from a gene pool
comprising multiple breeds for which the genetic ob-
jective capitalizes on breed differences in additive
merit for economically important traits. The forma-
tion of synthetic or composite breeds for these settings
is attractive as a simple alternative to complex strate-
gies of continuous crossbreeding. Studies of breeding
strategies in the tropics are scarce, especially for syn-
thetic breeds. Lucerna cattle, a synthetic dual-
purpose (milk and beef) Colombian breed that con-
tains about 40% Holstein genes (11), are a product of
this alternative strategy to align the genetic potential
for milk yield with the modest potentials for feed
nutrient supply, mainly from forages, in a tropical
environment.
More genetic information and more accurate esti-
mated breeding values may be achievable through the
analysis of test day ( TD) records of milk yield in-
stead of cumulative lactation totals for genetic evalu-
ation of potential parents. Improved information is
especially important in the tropics where data are
scarce and costly, thus severely restricting selection
gains. Harville (14) suggested a first-order auto-
regressive process to model the potentially variable
permanent environmental effect across repeated lac-
tations of an individual cow. Kachman and Everett
(15) included an autocorrelation structure for residu-
als of the TD records in a lactation. Van Tassel et al.
(31) generated 305-d cumulative totals from adjusted
TD milk yields using the hypothesis that TD records

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Journal of Dairy Science Vol. 81, No. 10, 1998
AUTOREGRESSIVE TEST-DAY ANIMAL MODEL
2739
TABLE 1. Number of records and means for 305-d mature equiva-
lent (ME ) milk yield (kilograms) and within-herd standard devia-
tion (S D) class for nine selected Holstein herds.
1Three herds per standard deviation (S D) class: low = <1644 kg
of phenotypic SD, medium = 1783 to 1898 kg of phenotypic SD, and
high = >2022 kg of phenotypic SD.
Year-season
adjusted
means for
Class1
RecordsME yield
(no.)
7266
9942
23,835
X
9281
9768
9461
SD
1508
1835
2040
Low
Medium
High
contain less residual variance and similar genetic
variance compared with the totals. Wade and Quaas
(35) were the first to demonstrate the methodology
for incorporating a first-order, autoregressive random
effect for year-season into the mixed model equations
of a sire model. More milk records per cow ≥10 daily
yields versus one cumulative total per lactation) are
expected to increase the number of unique contem-
porary groups of animal cohorts: the TD effect.
The use of TD models that accurately disentangle
the covariance structure of environmental effects
could increase the reliability of estimates of genetic
prediction. Gain would also increase because of the
reduction in the magnitude of residuals, thus increas-
ing the heritability of a single record or the mean of n
repeated records.
The primary objectives of this study were to com-
pare the effectiveness of alternative (co)variance
structures in TD models for partitioning random en-
vironmental variation to estimate the components of
genetic and environmental (co)variance and to in-
crease daily milk yield, especially for cattle in the
tropics. Secondary objectives were to estimate other
parameters for daily milk yield, including contrasting
Holstein herd environments in the northeastern US.
Steps included quantifying the biological responses in
different cattle systems, assessing the feasibility of
implementingthe model
gorithms in a multiple-herd scenario, and genetically
evaluating Lucerna cows and sires and alternative
genetic improvement programs for this breed.
andcomputational al-
MATERIALS AND METHODS
Tropical Climatology
and Lucerna Breed Formation
Data were provided under a collaborative research
agreement by the Lucerna Cattle Breeders Associa-
tion (ASOL UCE RNA) from the foundation herd, Ha-
cienda Lucerna (Bugalagrande, Valle del Cauca,
Colombia). At an altitude of 960 m above sea level,
the annual mean temperature at Hacienda Lucerna is
about 24°C, and mean temperature minima and max-
ima range from 19 to 29°C. The annual mean precipi-
tation of about 1100 mm is distributed in two rainy
seasons (April to May and October toDecember), and
relative humidity varies from 65 to 80%.
The Lucerna breed is the product of entrepreneur-
ship and a single-herd crossbreeding strategy fol-
lowed by inter se mating to form a new breed. Start-
ing in 1937, a crossbreeding program for milk was
initiated with US Holstein bulls with a local founda-
tion or Criollo breed, Harto Ân del Valle. Red Milking
Shorthorn bulls were introduced in 1951 for beef and
other characteristics. The herd has been closed to
outside breeding since 1958. Selection has been based
on milking performance (11). The resulting synthetic
dual-purpose breed is now mostly reddish in coat
color; purportedly, about 40% of its genes come from
Holsteins, about 30% come from Milking Shorthorns,
and about 30% come from Harto Ân del Valle cattle
(11). Artificial insemination using Lucerna semen
was implemented in 1980, and sire selection was
based primarily on the dams of sires pathway to
increasemilkyield. The
Agropecuario (Bogota Â, Colombia) recognized Lucerna
as a dual-purpose breed in 1983.
Instituto Colombiano
Data
The first TD measurement of milk yield for Lu-
cerna cows was taken either on the 1st or the 15th d
of the month following calving and thereafter at
monthly intervals. Data consisted of 30,256 TD
records for the first three lactations of 1819 Lucerna
cows calving from 1985 to 1995. The final pedigree
file consisted of 2538 animals, including these cows
and their ancestors.
Data from Holstein herds in the northeastern US
were from first, second, and third lactations of cows
calving from 1987 to 1995. Several studies (2, 9, 10,
19, 27, 29) showed that the genetic variance or
heritability of milk yield in Holstein cows especially
increased with the within-herd phenotypic variance of
mature equivalent ( ME )
American herd environments. Phenotypic means and
standard deviations for within-herd ME were ob-
tained after adjustment for the effects of year and
season of calving. Nine herds with similar mean
values for 305-d ME milk yields were chosen to
yield in US and Latin

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Journal of Dairy Science Vol. 81, No. 10, 1998
CARVALHEIRA ET AL.
2740
TABLE 2. Numbers of test day (T D) records, fixed effects, animals, and mean daily milk yield in eight data files for first and multiple
lactations for the Lucerna and the Holstein breeds.
1Three herds per standard deviation (S D) class: low = <1644 kg of phenotypic SD for 305-d mature equivalent (ME ) milk yield,
medium = 1783 to 1898 kg of phenotypic SD for 305-d ME milk yield, and high = >2022 kg of phenotypic SD for 305-d ME milk yield.
2First lactations only.
3First, second, and third lactations.
Holstein within-herd SD class1
Lucerna Low MediumHigh
Breed 12
1 + 2 + 33
11 + 2 + 311 + 2 + 311 + 2 + 3
Total TD records, no.
TD
Age, mo
DIM
Animals, no.
Cows with records, no.
Mean TD records per cow, no.
Mean daily yield, kg
13,998
120
28,808
129
14,552
208
32,526
217
22,194
300
46,956
301
56,567
282
123,113
283
15
297
8 10
99
8 159 15
295
4136
2926
16
9
33 97291
3126
2176
9999
2326
1584
2538
1819
2748
1845
3753
2569
9958
7213
10,935
8237
9
7.9
16
8.6
815
32.1
98 15
31.1 29.030.4 32.528.5
represent three classes of phenotypic standard devia-
tion (three herds per class; Table 1). The final
pedigree files consisted of 3126, 4136, and 10,935
animals for the low, medium, and high classes of
phenotypic standard deviation.
For all data, TD observations were deleted if a TD
class contained <4 observations or if DIM was >320 d.
Lactations were required to have at least 2 TD
records per cow. Lactation records were deleted 1) if
initiated by abortion, 2) if missing either the date of
birth or date of calving, 3) if first reported TD ex-
ceeded 75 DIM, 4) if TD intervals were >75 d, or 5) if
age at calving for Lucerna cows was not between 20
to 45 mofor first parity (20 to 36 mo for Holstein), 36
to 60 mofor second parity to permit a second lactation
for the older primiparous cows (28 to 48 mo for
Holstein), and 55 to 90 mo for third parity (40 to 72
mo for Holstein). (T he Lucerna management policy
was to cull cows that did not conceive within a year of
calving.) Other edits removed highly improbable lac-
tation or TD yields.
The Lucerna cows were grouped in 10 classes by
age at calving: 1 class each at ≤36 mo, 37 to 38 mo, 39
to 40 mo; 3 classes with 4-mo intervals from 41 to 52
mo; 3 classes with 6-mo intervals from 53 to 70 mo;
and 1 class at ≥71 mo of age. Holstein cows were
placed into 15 classes by age at calving: 8 bimonthly
classes from 20 to 36 mo, 6 classes with 4-mo inter-
vals from 37 to 60 mo, and cows at ≥61 mo of age.
Data from the Lucerna herd and from the Holstein
standard deviation classes each provided 2 data
filesÐ first lactations only and multiple lactations
(first three lactations). These 8 data files are sum-
marized in Table 2.
TD Models
A genetic TD model partitioning the random en-
vironmental component into short-term environmen-
tal ( ST E ) effects (following a first-order autoregres-
sive process within cow and lactation), long-term
environmental ( L TE ) effects (following a first-order
autoregressive process across lactations), and in-
dependent residuals was used to describe the data.
Two animal models differing by the presence (full
model) or absence (reduced model) of STE effects
were applied to analyze TD observations in data for
first lactation and multiple lactations. The full model
was as follows:
yijklmn = HTDi+ agej+ DIM(H )k( l)
+ am+ pm( l) + tn( ml) + eijklmn
where
yijklmn = amount of milk for an individual TD
sample from the cow m,
HTDi = fixed effect due to cows tested in the
same herd TD (H T D),
agej = fixed effect of same age at calving,
DIM(H )k( l) = fixed effect for contemporary cows
tested in the same lactation and DIM
within herd (H ),
am= random effect of animal,
pm( l) = random effect of LTE (only animals
with records contribute to this effect)
and assumed to follow a first-order
autoregressive process across lacta-
tions,

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Journal of Dairy Science Vol. 81, No. 10, 1998
AUTOREGRESSIVE TEST-DAY ANIMAL MODEL
2741
tn( ml) = random effect of STE nested within
cow and lactation and assumed to
follow a first-order autoregressive
process, and
eijklmn = random residual term.
In matrix notation, the full model can be represented
as
y = Xb + Za + Mp + Qt + e,
where y ∼N(X b, V), b = unknown vector of fixed
effects, which, with known X, defines the mean; a, p,
and t = vectors representing the random effects of
animal, LTE, and STE associated with records in y by
Z, M, and Q respectively; e = vector of residuals; and
V = the (co)variance matrix. The expectations and
(co)variances for this model are
∼N


y
a
p
t
e




;


Xb
0
0
0
0




V
symm
ZG
G
MJ
0
J
QS
0
0
S
R
0
0
0
R




and then (e.g., for l = 1, 2, 3 lactations)
V = ZGZ′ + MJ M +(QlS1Ql′) + R ,
∑
l = 1
3
G =
,Aqisa
2
J=
⊗
sp
2


1
symm
rp
1
rp
rp
1
2


Iq2
S =
,


S1
S2
S3


S1=diag
,{F }m(l)st1

,INse
2
diag{F }m(l)=
diag,


1 rt1
1
symm
rt1
rt1
...
2
¼
...
rt1
n±2
rt1
1
rt1
rt1
...
rt1
rt1
1
n±1
n±2
2


m(l)
R =
2
where A = numerator relationship matrix,
genetic variance component, J
matrix following a first-order autoregressive process
across the three lactations,
nent,= LTE autocorrelation coefficient (for the
rp
single lactation analysis J reduces to
(co)variance matrix following a first-order auto-
regressive process within lactations reflecting the as-
sumption of independence of this effect between lacta-
tions, F = autocorrelated block diagonal matrix cor-
responding to the cow m with n TD samples in
lactation 1,= STE variance component 1,
st1
corresponding autocorrelation coefficient (for single
lactation analysis, Sreduces to
= the
sa
2
= LTE (co)variance
= LTE variance compo-
sp
2
), S = STEIsp
2
=
2
rt1
, R= diag{F}mst
= residual variance
2
residual (co)variance matrix,
component, q1
= number of animals having records, N = the totalq2
number of records in the analysis, I = identity matrix,
and ⊗ = Kronecker tensor product.
The reduced model (ignoring the STE effect) had
the same assumptions as the full model, except that
LTE is the only random environmental effect measur-
ing potential correlations that were due to the cow
both within and between lactations.
The parameters were estimated using derivative-
free REML (3, 28). Likelihood functions were max-
imized by the multivariate simplex (polytope) al-
gorithm as described by Nelder and Mead (24). Con-
vergence was assumed when the variance of log-
likelihood functions of the polytope was <10±8, and log
likelihoods after restarting did not differ (up to the
fourth decimal place). Models were compared by
likelihood ratio tests (L RT, x2) with 2 df for the
analysis of first lactations and 6 df for the analysis of
multiple lactations (5).
se
2
= number of animals being evaluated,
RESULTS AND DISCUSSION
Full Model with STE Effect
Variance components and parameter estimates
from analyses of first and multiple lactations for the
Lucerna herd and for Holstein standard deviation
classes are in Tables 3 and 4. The log-likelihood
values, ±2log(£), which maximized the likelihood
functions for each model, are also given in the tables.
Universally large differences between the ±2log(£) of
the full and reduced models clearly indicated that the
fit to these data was best for the full model. Differ-

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Journal of Dairy Science Vol. 81, No. 10, 1998
CARVALHEIRA ET AL.
2742
TABLE 3. Estimates of variance components (kilograms per day)2for animal (
2
s Ãe
either including or ignoring a first-order autoregressive process for short-term environmental (ST E ) effects on daily milk yield in first
lactations from a herd of Lucerna cows and from each of the Holstein standard deviation (S D) classes.
), long-term environment (), short-term temporary
s Ãa
2
s Ãp
2
environment (STE ,), residual ( ), phenotype ( ), heritabilities (), and autocorrelations (r Ã) using test day animal models
s Ãt
2
s ÃPHENO
2
hÃ2
1Three herds per SD class: low = <1644 kg of phenotypic SD for 305-d mature equivalent (ME ) milk yield, medium = 1783 to 1898 kg of
phenotypic SD for 305-d ME milk yield, and high = >2022 kg of phenotypic SD for 305-d ME milk yield.
2Model including the STE effect.
3Model ignoring the STE effect.
Holstein within-herd SD class1
Parameter
estimate
Lucerna LowMedium High
Include2
Ignore3
Include IgnoreInclude IgnoreIncludeIgnore
s Ãa
s Ãp
2
0.561.017.187.456.22 6.118.598.94
2
<0.00 1.28 8.60 9.7211.89 13.18 14.8016.59
s Ãt
r Ãt
s Ãe
s ÃPHENO
hÃ2
±2log(£)
2
3.24
0.83
. . .
. . .
8.96
0.57
. . .
. . .
10.14
0.61
. . .
. . .
10.49
0.58
. . .
. . .
2
1.052.485.2412.40 5.5013.427.46 15.71
2
4.84
0.11
4.77
0.21
29.98
0.24
29.56
0.25
33.74
0.18
32.71
0.19
41.34
0.21
41.24
0.22
28,505.40 30,510.4354,562.0355,574.30 84,621.29 86,532.82228,031.92231,494.89
ences between the log-likelihood values were of multi-
ple orders of magnitude (e.g., chi-squared critical
values for P = 0.0005 with 2 and 6 df, respectively, are
15.2 and 24.1; the smallest chi-squared value in this
study is 1012). The full model was unequivocally
superior for every analysis. Consequently, compari-
sons of results from the two models are primarily to
emphasize the opportunity losses in genetic gain from
insufficiently disentangling the environmental effects
from the genetic ones.
Variance Components, Autocorrelations,
and Heritabilities of Daily Yield
in First Lactation
The TD records from Lucerna cows were highly
correlated (0.83), and most of the nongenetic vari-
ance caused by cow was explained by the STE effect.
Correlations between TD records in each of the three
Holstein classes for phenotypic standard deviation,
although smaller than for the Lucerna classes, rang-
ing from 0.51 to 0.61, were similar to the coefficient of
0.58 used in the northeast TD evaluations (12).
The autocorrelation structure was clearly effective
in removing STE influences that, if ignored, remained
partly confounded with the genetic component, espe-
cially in Lucerna data. This confounding caused the
genetic component of variance to be overestimated in
the reduced model in three of four data files. The
estimated genetic variance from the full model for
daily milk yield in primiparous Lucerna cows was
45% smaller than the biased estimate from the
reduced model (Table 3), which could mean less
genetic progress if the STE effect were ignored, espe-
cially in the dual-purpose Lucerna herd.
Family ties in the Lucerna data were fewer than
anticipated; 45% of records was from cows with
unknown sires. The resulting sparse relationship
coefficient matrix probably constrained the estimated
heritability (hÃ2= 0.11) because of partial confound-
ing of genotype and environment. More relationship
ties for all Holstein data files probably helped to
capture relatively more of the genetic variation and
gave heritability estimates from full and reduced
models that ranged from 0.18 to 0.25 (Table 3).
Genetic relationships among the cattle being evalu-
ated are the only data permitting separation of the
genetic component from the nongenetic components in
the cow effect. Meyer et al. (21) reported that the
heritability of milk yield was underestimated for Aus-
tralian Black and White cows from sire misidentifica-
tion, which produced a less informative structure of
genetic covariance. Insufficiently dense relationship
matrices may predispose smaller estimates of genetic
variance (4).
Estimates of the phenotypic variance in the single
lactation analysis were nearly the same from full and