**ORIGINAL ARTICLES**

**Estimation of genetic parameters for test-day milk yield in first
calving buffaloes ^{¤}**

**Estimación de parámetros genéticos para producción de leche en el día de control en búfalas de primer
parto**

**Estimação de parâmetros genéticos para produção do leite no dia do controle em búfalas de primeiro
parto**

Naudin A Hurtado-Lugo^{1,2}, Zoot, PhD (c); Severino C de Sousa^{3}, Zoot, PhD; Rúsbel R Aspilcueta^{1}, Zoot,
PhD; Swammy Y Gutiérrez^{2},Tec Pec, Est Zoot; Mario F Cerón-Muñoz^{2*}, Zoot, PhD; Humberto Tonhati1,
Zoot, PhD.

* Corresponding author: Mario Fernando Cerón Muñoz. Facultad de Ciencias Agrarias. Universidad de Antioquia. Carrera 75 No. 65-87. Ciudadela de Robledo. Medellín, Colombia. Tel (574) 2199140. E-mail: cerongamma@gmail.com

1 Universidade Estadual Paulista (UNESP), Faculdade de Ciências Agrárias e Veterinárias 14884900 Jaboticabal, São Paulo, Brasil.

^{2} Grupo de Genética, Mejoramiento y Modelación Animal (GaMMA), Facultad de Ciencias Agrarias, Universidad de
Antioquia UdeA; Calle 70 No. 52-51, Medellín, Colombia.

^{3} Universidade Federal do Piauí (UFPI), Departamento de Zootecnia, Bom Jesus, Piauí, Brasil.

(Received: August 20, 2012; accepted: April 10, 2013)

**Summary**

**Background:** the milk yield records measured along lactation provide an example of repeated measures;
the random regression models are an appealing approach to model repeated measures and to estimate genetic
parameters. **Objective:** to estimate the genetic parameters by modeling the additive genetic and the residual
variance for test-day milk yield in first calving buffaloes. **Methods:** 3,986 test-day data from 1,246 first
lactations of crossbred buffalo daughters of 23 sires and 391 dams between 1997 and 2008 from five farms
were used. The model included the genetic and permanent environment additive as the random effect and
the contemporary group (year, month of test-day) and age at calving as covariable (linear) fixed effects. The
fixed (third order) and random (third to ninth order) regressions were obtained by Legendre polynomials.
The residual variances were modeled with a homogeneous structure and various heterogeneous classes.
The variance components were estimated using the WOMBAT statistical program (Meyer, 2006). **Results:** according to the likelihood ratio test, the best model included four variance classes, considering Legendre
polynomials of the fourth order for permanent environment and additive genetic effects. The heritabilities
estimates were low, varying from 0.0 to 0.14. The estimates of genetic correlations were high and positive
among PDC1 and PDC8, except for PCD9, which was negative. This indicates that for any of the selection criteria adopted, the indirect genetic gain is expected for all lactation curves, except for PCD9. **Conclusion:** heterogeneity of residual variances should be considered in models whose goal is to examine the alterations
of variances according to day of lactation.

**Key words**: biomodeling, genetic correlations, Legendre polynomials, random regression models.

**Resumen**

**Antecedentes:** los registros de producción de leche medidos a lo largo de la lactancia son un ejemplo
de medidas repetidas, los modelos de regresión aleatoria presentan un enfoque atractivo para modelar
medidas repetidas y para estimar parámetros genéticos. **Objetivo:** estimar parámetros genéticos a través de
la modelación de la varianza genética y residual para producción de leche en el día de control en búfalas de
primer parto. **Métodos:** fueron analizados 3986 controles de producción de leche en la primera lactancia de
1246 búfalas, hijas de 391 hembras y 23 toros, durante los años 1997 hasta 2008 en 5 fincas. El modelo incluyó
como efectos aleatorios genético aditivo y de ambiente permanente, como efectos fijos grupo contemporáneo
compuesto por mes, año de control y la covariable de la edad de la búfala al parto (lineal). Las regresiones
fijas (3er orden) y aleatorias (3er a 9no orden) fueron obtenidas mediante polinomios de Legendre. Las varianzas
residuales fueron modeladas con una estructura homogénea y varias clases heterogéneas. Los componentes
de varianza fueron estimados utilizando el programa WOMBAT. **Resultados:** de acuerdo con la prueba de
la razón de verosimilitud, el mejor modelo fue con 4 clases de varianzas residuales, siendo considerado
un polinomio de Legendre de cuarto orden para el efecto de ambiente permanente y genético aditivo. Las
heredabilidades fueron bajas, variando desde 0,00 hasta 0,14. Las correlaciones genéticas fueron altas y
positivas entre los PDC1 a PDC8, excepto en el PDC9 que fue negativo con respecto a los demás controles.
**Conclusiones:** es necesario considerar la heterogeneidad de varianzas residuales en los modelos estudiados,
con el fin de modelar los cambios en las variaciones respecto a los días en lactancia.

**Palabras clave:** biomodelación, correlación genética, modelos de regresión aleatoria, polinomios de
Legendre

**Resumo**

**Antecedentes:** os registros da produção do leite medidos ao longo da lactação, apresentam um exemplo
de medidas repetidas. Os modelos de regressão aleatória apresentam abordagem atraente para modelar
medidas repetidas e estimar parâmetros genéticos. **Objetivo:** estimar parámetros genéticos mediante a
modelação das variâncias genéticas e residual da produção do leite no dia do controle em búfalas de primeiro
parto. **Métodos:** foram analisados 3986 controles de produção de leite em primeiras lactações de 1246
búfalas, filhas de 391 fêmeas e 23 touros, entre 1997 e 2008 em 5 fazendas. No modelo foram incluídos
como efeitos aleatórios o genético aditivo e ambiente permanente, e como fixos o grupo contemporâneo
(mês e ano de controle) e a covariável a idade da búfala ao parto (Lineal). As regressões fixas (3° ordem) e
aleatórias (3° a 9° ordem) foram obtidas mediante polinômios ortogonais de Legendre. As variâncias residuais
foram modeladas mediante estruturas homogêneas e diferentes classes heterogêneas. Os componentes de
variância foram estimadas mediante o software WOMBAT. **Resultados:** de acordo com a prova da máxima
verossimilhança, o melhor modelo foi com 4 classes de variâncias residuais, sendo considerado polinômios
de Legendre de quarto ordem para o efeito de ambiente permanente e genético aditivo. As herdabilidades
foram baixas, variando desde 0,00 até 0,14. As correlações genéticas foram altas e positivas entre o PDC1 e
PDC8, a exceção do PDC9 que apresentou valores negativos com respeito aos outros controles. **Conclusões:**é necessário considerar heterogeneidade de variâncias nos modelos estudados, tentando modelar as mudanças
nas variações respeito aos dias em lactação.

**Palavras chave:** biomodelagem, correlação genética, modelos de regressão aleatória, polinômios de
Legendre.

**Introduction**

Random regression models (RRM) can be
applied to test-day milk yield as an alternative to
standard procedures used for the genetic evaluation
of longitudinal traits in dairy cattle. According to
Meyer (2006), RRM allow to fit random lactation
curves for each individual expressed as deviations
from the average lactation curve of the population
or of groups of individuals. RRM allows the
prediction of breeding values for the lactation curve
as a whole; in other words, for any desired point
on the curve (Kirkpatrick *et al.*, 1990). Moreover,
these models assume that the shape of the lactation
curve is influenced by different random effects, such
as genetic and permanent environmental effects.
In addition, RRM consider the deviation of the
lactation curve of animals in relation to the fixed
curve for subclasses of effects to which it belongs
(Araújo *et al.*, 2007).

Studies on dairy cattle using RRM are based
on orthogonal functions to model the (co)variance
structure for random genetic effects and residual
classes (Strabel and Misztal, 1999). Among these
functions are orthogonal Legendre polynomials
of different orders, which are used to analyze
patterns of genetic variation and are influenced by
environmental or management conditions across
the curve. In general, the reliability of heritabilities
obtained with RRM is greater than that obtained
with finite dimensional models, indicating that
RRM allow the detection of a greater genetic
variability across lactation (Brotherstone *et al.*,
2000; Pool *et al.*, 2000). The goal of the present
study was to estimate the genetic parameters by
modeling the additive genetic and the residual
variance for test-day milk yield in first calving
buffaloes.

**Materials and Methods**

*Location and animals*

The information was collected in five farms
located in the province of Cordoba. The province of
Cordoba is located in Colombia's northeast, north of
the western mountain range between the geographic
coordinates 09°26'16''N to 07°22'05''N and
74°47'43''W to 76°30'01''W. Its climate is warm
tropical, where the average annual rainfall ranges
from 1300 mm in the coastal zone to approximately
3000 to 4000 mm in the upper area of the Sinu
and San Jorge Rivers (Ballesteros *et al.*, 2006);
rainfall pattern is unimodal with a rainy season
between May and November and a dry season
from December to April (Corporación Autónoma
Regional de los Valles del Sinú y San Jorge, 2005).
Crossbred animals with high percentage of Brazilian
and Bulgarian Murrah breeds were used. Animals
grazed on native grass fields mixed with Brachiaria
Spp grass and some forage legumes; females were
milked once a day.

*Study design*

This study used 3986 test-day milk yield records (TDMY) from 1,246 females during their first lactation. These females were descendants of 23 sires and 391 dams born between 1997 and 2008. Buffalo cows younger than 22 months and older than 48 months at calving were removed from the study. TDMY was estimated between 5 and 270 days after calving and was divided into intervals of 30 days, corresponding to nine test-day records (TDMY1 to TDMY9) for first lactation cows. Only cows that had their first test-day record before 45 days after calving were considered in the analyses. TDMY was divided into nine classes (1 to 9) of days in lactation.

*Statistical analysis*

The characteristic of milk production was analyzed by means of single trait random regression models. All models included as random effect: the additive genetic, permanent environment and residuals effects of the animal; and as fixed effects: the contemporary group (year, month of test-day) and age at calving as covariable (linear). The fixed (third order) and random (third to ninth order) regressions were obtained by Legendre polynomials. The residual variances were modeled with a homogeneous structure and heterogeneous classes. The variance components were estimated using WOMBAT program (Meyer, 2006). In matrix form, the model can be represented by:

y = Xb + Za + Wp + e

Where:

*y* = vector of observations.

*b* = vector of fixed effects (CG, (co)variate age
of dam at calving, and average curve of the
population).

*a* = vector of solutions for additive genetic random
regression coefficients.

*ap* = vector of solutions for permanent
environmental random regression coefficients.

*e* = vector of different residual effects.

*X, Z, W* = incidence matrices for fixed and random
additive genetic and permanent environmental
effects, respectively.

The dimension of vector *a* consists of *k _{a}* x

*N*coefficients, where:

_{a}*k _{a}* corresponds to the order of the polynomial.

*N _{a}* is the number of animals in the relationship
matrix.

The dimension of vector ap consists of *k _{ap}* x

*N*coefficients, where:

_{d}
*k _{ap}* corresponds to the order of the polynomial and

*N*is the number of animals with records.

_{d}For analysis, it was assumed that the records were distributed with mean Xβ. The following assumptions were made for additive genetic, permanent environmental and residual effects:

Where:

*K _{a}* and

*K*are covariance matrices between additive genetic and permanent environmental random regression coefficients, respectively.

_{ap}*A* is the relationship matrix between individuals.

*I _{Nd}* is the identity matrix of dimension

*N*.

_{d}is the Kronecker product between matrices.

R is a block diagonal matrix containing residual variances. It was assumed that the residuals are independent.

The random additive genetic and permanent
environmental effects were modeled using third- to
sixth-order Legendre polynomials. A homogeneous
(R) and a heterogenous (r) structure with nine
classes (each month was considered to be a different
class) were adopted for residual variances. In
addition, heterogeneous variances with smaller
classes were also considered, which were divided
according to similarity between variances based
on the nine residual classes. Thus, test-day records
for milk yield were divided into four classes (1, 2
to 4, 5 to 8, and 9). The general RRM is given as
*LEGk _{a}*.

*k*_

_{ap}*R*or

*LEGk*, where:

_{a}.k_{ap}_r*k _{a}* and

*k*correspond to the order of the covariance function for additive genetic and permanent environmental effects, respectively.

_{ap}*R* or *r* is the structure of residual variances.

The covariance functions were estimated by the restricted maximum likelihood method using the WOMBAT statistical program (Meyer, 2006).

We assumed independence of the residuals.
The citation of the RRM follows the pattern of
*LEGk _{a}.k_{ap}_r*, referring to the order of the covariance
function for the additive genetic effects (

*k*), permanent environmental effects (

_{a}*k*) and the residual variances structure (

_{ap}*r*). For example, the LEG4,6_4 model denotes an analysis fitting a fourth and sixth order Legendre polynomial for the additive genetic and the permanent environmental effects, respectively, and the residual variances modeled with a step function with four classes.

The different models were compared using
the logarithm of the likelihood function (log L),
the likelihood ratio test (LRT) at 1% probability,
restricted maximum likelihood Akaike (AIC) and
Schwarz Bayesian (BIC) information criteria, and
examination of variances and correlations estimated
for the traits. The information criteria can be
described as: *AIC* = –2log*L* + 2*p* and *BIC* = –2log*L* + *p*log(*N–r(X)*, where:

*p* is the number of parameters estimated.

*N* is the number of data.

*r(X)* is the rank of the incidence matrix of fixed
effects in the model.

*log L* is the logarithm of the restricted maximum
likelihood function (Wolfinger, 1993).

**Results**

The phenotypic mean of TDMY was 3.78 ± 0.29 kg (Table 1). Milk production was 3.69 ± 0.09 kg at the beginning of the lactation curve and 4.01 ± 0.52 at peak lactation, declining thereafter to 2.25 ± 0.68 kg.

For the homogenous models (hom3,3_1), random additive genetic and permanent environmental effects were kept constant, assuming a homogenous residual variance structure (hom). For the heterogeneous models (het3,3_9, het3,3_4, het3,6_4, het4,4_4, het4,5_4, and het4,6_4), the random effects of residual variance were divided into four classes of heterogeneous variance (het).

The heritabilities obtained by the models ranged
from 0.00 to 0.14 (Figure 1). In general, the highest
heritabilities were observed between TDMY2 and
TDMY5 (0.09 to 0.14) and the lowest estimates
on TDMY8 and TDMY9 (0.00 to 0.03). For all
seven models tested, additive genetic, permanent
environmental and phenotypic variances were
greater at the beginning of lactation and lower at
the conclusion, ranging from 0.38 kg^{2} to 0.04 kg^{2},
from 1.65 kg^{2} to 0.20 kg^{2}, and from 1.98 kg^{2} to
0.53 kg^{2}, respectively. Residual variances were low
(0.001 kg^{2}) at the beginning of lactation and high
(1.95 kg^{2}) at the conclusion.

In general, the heterogeneous models presented lower log likelihood function (Log L), Akaike information criterion (AIC) and Schwarz Bayesian information criterion (BIC) values (Table 2). The BIC, which penalizes models with a larger number of parameters more rigorously, selected het3,6_4 and het4,4_4 as the best models for milk production. As a consequence, these models would be more adequate to describe the TDMY variation. Table 3 shows the eigenvalues associated with the matrix of random regression coefficients for additive genetic and permanent environmental effects obtained with the two best models (het3,6_4 and het4,4_4) for TDMY according to the BIC.

The models of lower order are sufficient to capture all the milk production variation, with a fourth coefficient for the additive part being associated with a zero eigenvalue, unlike the permanent environmental effect. For the two models, the fourth regression coefficient for additive genetic variance, but not for permanent environmental variance, was associated with an eigenvalue of zero. The first eigenvalue explained more than 70% of the variation in milk yield data. The variability in additive genetic and permanent environmental effects was primarily explained by the first eigenvalues (> 92%).

The genetic variances obtained with the selected
models (het3,6_4 and het4,4_4) were greater at
the beginning of lactation, with the highest values
being observed on test days 3 and 1, respectively
(Figure 2). The permanent environmental variances
obtained with models het3,6_4 and het4,4_4
showed a similar trend, with values ranging from
1.20 kg^{2} to 0.49 kg^{2} and from 1.29 kg^{2} to 0.49 kg^{2},
respectively. The phenotypic variances obtained
with models het4,4_4 and het3,6_4 were greater at
the beginning of lactation, with maximum values
on test day 1 (1.37 kg^{2} and 1.45 kg2, respectively)
(Figure 2). Conversely, residual variances ranged
from 0.06 kg^{2} to 0.36 kg^{2} and from 0.03 kg^{2} to
0.48 kg^{2}, respectively.

For the selected models, the correlations between regression coefficients for additive genetic effects ranged from -0.82 to 1.0 across lactation, with the observation of high and positive genetic correlations between consecutive test days (Tables 4 and 5), and low and positive correlations between more distant test days. For the two models, negative genetic correlations were observed between TDMY9 and the other test days, with high correlations (-0.82) at the beginning of lactation and low correlations (-0.01) at the end of lactation (Tables 4 and 5).

High and positive phenotypic correlations were observed between adjacent test days and low and negative correlations between more distant test days, ranging from -0.17 to 0.82 and from -0.10 to 0.81 for models het3,6_4 and het4,4_4, respectively (Tables 4 and 5).

**Discussion**

TDMY followed the typical shape of the
lactation curve in dairy buffaloes (Araújo et al,.
2007; Hurtado-Lugo et al,. 2005). The models
considering homogenous residual variances across
lactation presented higher AIC and BIC values and
consequent problems in the estimation of TDMY
(Table 2). The results indicated that variances differ
across lactation and residual variances therefore
need to be modeled using heterogeneous variance
structures (Table 2). Comparison of the models
based on step functions showed that models
including classes of residual variances provided
better estimates than those considering homogeneity
of variances. In addition, superparameterized
models (*n*) were penalized rigorously by the BIC
criterion when compared to the AIC criterion
(Table 2). The difference in the estimates between
models based on the AIC and BIC selection criteria
might be related to the small number of test-day
observations. In general, the heterogeneous models
presented higher log L values (Table 2). In the study
of Araújo *et al.* (2007), second order models were
more efficient when the number of residual classes
was high. However, fourth order polynomial models
with four residual classes showed the highest log
L value. In this investigation, models het3,6_4
and het4,4_4, presented the lowest log L, AIC and
BIC values and were therefore the most efficient in
describing the variation in milk production across
the lactation curve.

According to Pool *et al.* (2000), the shape of the
lactation curve can be modeled with sufficiently
high precision using a third order Legendre
polynomial for the additive genetic component
and a fourth order polynomial for permanent
environmental effects. On the other hand, Lopez-
Romero and Carabaño (2003) suggested low order
Legendre polynomials for additive and permanent
environmental variances to be the most adequate to
model milk production in first lactation cows. The
eigenvalues obtained in the present study suggest
that the dimension of random effects can be reduced
without the loss of information, disagreeing with
the log L and AIC values for additive genetic and
permanent environmental effects and agreeing with
the BIC value for additive genetic effects. However, according to Legarra *et al.* (2004), the reduction
of the dimension of random effects as a result
of eigenvalues close to zero is not indicated in all
cases since the adoption of this criterion may lead to
simplistic and inadequate models.

The heritability estimates obtained with the models tested ranged from 0.00 to 0.14. All models, except for those selected (het3,6_4 and het4,4_4), presented estimation problems at the end of the lactation curve (Figure 1). As a consequence of this fact, Meyer (2005) suggested that models using high order polynomials are more flexible and are able to model variances on a continuous scale. However, these polynomials tend to place greater emphasis on the extremes of the trajectory, causing estimation problems at these points.

Greater heritabilities at the beginning and at the
end of lactation have been reported for dairy cattle
(Brotherstone *et al.*, 2000). The same trend at the
beginning of lactation was observed in the present
study. In contrast, heritabilities were lower at the
end of the lactation curve. This finding might be due
to the fact that buffalo farming is a recent activity in
Colombia and that production data are limited due
to the small number of animals and zootechnical
records. Araújo *et al.* (2007), who studied milk
production in buffaloes as a function of age using
RRM with low order Legendre polynomials,
estimated heritabilities of 0.08 to 0.40. These
estimates are higher than those obtained in the
present study.

The estimates obtained with model het4,4_4
presented higher AIC and BIC values than those
obtained with model het3,6_4. However, the latter
model included a larger number of parameters (*n*).
Therefore, het4,4_4 was the best model. This model
was selected based on the fact that less complex
models (*n*) are easier to interpret from a biological
standpoint. On the other hand, this fact indicates
the need to include a larger number of zootechnical
data from the herd. According to Meyer (2005),
Legendre polynomials place greater emphasis on
observations at the extremes of the lactation curve,
generating estimation problems at these points.
This fact may explain the difficulty in modeling
TDMY since cows are affected by postpartum
stress at the beginning of lactation, as well as by a
negative energy balance. In addition, the number of
production records is low at the end of lactation.

The phenotypic variances obtained with the two
models showed a similar variation across lactation
(Figure 2), with the observation of higher variances
at the beginning of lactation and peak values on the
first test day. Kettunen *et al.* (2000) reported higher
phenotypic variances at the beginning and at the end
of lactation for first lactating Holstein cows.

Using the models selected, the genetic correlations presented their greatest differences between TDMY6 and TDMY8 (Tables 4 and 5). The unexpected negative genetic correlations between TDMY9 and the other test days might be due to the lack of fit of polynomials at the end of lactation. In addition, the small number of zootechnical records from TDMY9 may cause estimation problems at these points.

Negative genetic correlations at the beginning
and at the end of lactation have been reported
for dairy cattle using RRM. These results might
be explained by postpartum stress and the small
number of records at the end of lactation (4,13). The
genetic correlations obtained with model het3,6_4
differ from those reported by Araújo *et al.* (2007),
suggesting that the genetic correlations between
milk yields were close to unity in Murrah buffaloes.

The phenotypic correlations varied, with the
observation of higher estimates between test days
at the beginning of lactation and lower estimates
between more distant test days. Similar results have
been reported by Hurtado-Lugo *et al.* (2005), who
used finite dimensional models and found higher
phenotypic correlations between adjacent test
days (0.64 to 0.84) and lower correlations between
more distant test days (0.01 to 0.30). Kettunen
*et al.* (2000) estimated phenotypic correlations
of 0.20 to 0.74, with the highest estimates being
observed between adjacent test days and the lowest
correlations between test days at the beginning and
at the end of lactation.

In conclusion, the present results demonstrate the need to consider heterogeneity of residual variances in the models tested in order to describe variations in days of lactation. The function with four residual variance classes was the best to model milk production across lactation. However, all model selection criteria used indicated het3,6_4 to be the best model (lowest AIC and BIC values), whereas model het4,4_4 was the most parsimonious because of its lower complexity at the time of biological interpretation.

**Notas**

**Acknowledgements**

This research was financially supported by the Colombian Ministry of Agriculture and Rural Development, Colombian Cattle Breeders Federation (FEDEGAN), Fundação de Apoio à Pesquisa do Estado de São Paulo (FAPESP-Process N º 2009/53773-1), the University of Antioquia, CODI/UdeA sustainability 2011-2012 and the Colombian Buffalo Breeders Association. Project title: ''Consolidation of the registration system in buffaloes and milk control, and Impact on production and improvement of Colombian herds''.

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