Short communication
Bayesian analysis of random regression models using B-splines
to model test-day milk yield of Holstein cattle in Brazil
A.B. Bignardi
a,n, L. El Faro
b, M.L. Santana Jr.
c, G.J.M. Rosa
d, V.L. Cardoso
b,
P.F. Machado
e, L.G. Albuquerque
a,fa
Departamento de Zootecnia, Faculdade de Ciˆencias Agra´rias e Veterina´rias, Universidade Estadual Paulista, Jaboticabal, SP, CEP 14884-900, Brazil bAgˆencia Paulista de Tecnologia dos Agronego´cios—APTA, Po´lo Regional Centro Leste, Ribeirao Preto, SP, CEP 14030-670, Brazil~
cGrupo de Melhoramento Animal de Mato Grosso (GMAT), Instituto de Ciˆencias Agra´rias e Tecnolo´gicas, Universidade Federal de Mato Grosso, Campus Rondono´polis, MT-270, Km 06,CEP 78735-001, Rondono´polis, MT, Brazil
d
Department of Dairy Science, University of Wisconsin, Madison, WI 53706, United States e
Departmento de Zootecnia, Universidade de Sao Paulo, Piracicaba, SP, CEP 13418-900, Brazil~ f
Conselho Nacional de Desenvolvimento Cientı´fico e Tecnologico (CNPq) and Instituto Nacional de Ciˆencia e Tecnologia - Ciˆencia Animal (INCT- CA), Vic-osa, MG CEP 36570-000, Brazil
a r t i c l e
i n f o
Article history: Received 27 March 2012 Received in revised form 2 August 2012 Accepted 10 September 2012 Keywords: Covariance functions Genetic parameters Segmented polynomials
a b s t r a c t
The objective of this paper is to model variations in test-day milk yields of first lactations of Holstein cows by RR using B-spline functions and Bayesian inference in order to fit adequate and parsimonious models for the estimation of genetic para-meters. They used 152,145 test day milk yield records from 7317 first lactations of Holstein cows. The model established in this study was additive, permanent environ-mental and residual random effects. In addition, contemporary group and linear and quadratic effects of the age of cow at calving were included as fixed effects. Authors modeled the average lactation curve of the population with a fourth-order orthogonal Legendre polynomial. They concluded that a cubic B-spline with seven random regression coefficients for both the additive genetic and permanent environment effects was to be the best according to residual mean square and residual variance estimates. Moreover they urged a lower order model (quadratic B-spline with seven random regression coefficients for both random effects) could be adopted because it yielded practically the same genetic parameter estimates with parsimony.
&2012 Elsevier B.V.
1. Introduction
Various studies have used random regression models (RR) based on Legendre polynomials to estimate genetic parameters for test-day milk yields (Arau´jo et al., 2006; Bignardi et al., 2009a; Cobuci et al., 2005; El Faro and
Albuquerque, 2003; El Faro et al., 2008). One major
disadvantage of these models is the need to use high-degree polynomials and the lack of uniformity in fitting the data along the curve. Alternatives to reduce the degree of polynomials are currently being studied. One such alternative is the use of spline functions, also called
segmented polynomials (Bohmanova et al., 2008;Meyer,
2005a;Silvestre et al., 2006).
Splines consisting of individual segments of low-degree polynomials joined at specific points, called knots (De Boor, 1978). Spline functions almost completely meet the requirements of an optimal function, i.e., they possess a simple biological interpretation, are easily estimated (linear in the parameters), and reduce multicollinearity Contents lists available atSciVerse ScienceDirect
journal homepage:www.elsevier.com/locate/livsci
Livestock Science
1871-1413&2012 Elsevier B.V.
http://dx.doi.org/10.1016/j.livsci.2012.09.010 n
Correspondence to: Departamento de Zootecnia, Faculdade de Ciˆencias Agra´rias e Veterina´rias, Universidade Estadual Paulista, Via de acesso Paulo Donato Castellane s/n, Departamento de Zootecnia, Pre´dio 2, Jaboticabal, SP, CEP 14884-900, Brazil.
E-mail address:[email protected] (A.B. Bignardi).
Open access under the Elsevier OA license.
(Fuller, 1969). However, data fit will depend on the choice of the polynomial degrees and the sites where the knots will be inserted. One particular type of spline function, called B-spline, is preferred over other types of splines because of its excellent numerical properties (Eilers and Marx, 2010;Ruppert et al., 2003). This spline function is adequate to model random effects in mixed model ana-lyses and is also efficient in the estimation of covariance
functions (Rice and Wu, 2001). According to Misztal
(2006), RR models using spline functions tend to be highly affected by the distribution of the data, the degree of the functions, and especially by the choice of the number and position of knots. In the case of models using penalized
splines, Ruppert et al. (2003) recommend knots to be
placed at equidistant intervals. According to Meyer
(2005b), the use of a B-spline function, which is a kind of penalized spline, attenuates the influence of knot position on the estimates.Meyer (2005a), while modeling the growth curve of beef cattle by RR using B-spline functions, observed no differences in genetic parameter estimates between models with equidistant and non-equidistant knots.
The objective of the present study was to model variations in test-day milk yields of first lactations of Holstein cows by RR using B-spline functions and Baye-sian inference in order to fit adequate and parsimonious models which could be used for the genetic evaluation for test-day milk yield of Dairy Holsteins cattle in Brazil. 2. Material and methods
A total of 152,145 test-day milk yield records from 7317 first lactations of Holstein cows recorded from 1995 to 2003 were analyzed. The cows were descendants of 612 sires, were distributed over 93 herds located in the southeastern region of Brazil, with age at first calving ranging from 18 to 48 months. The data were obtained from the Herd Analyses and Milk Quality Program carried out by Clı´nica do Leite (Milk Quality Laboratory of Luiz de Queiroz Agriculture School, ESALQ-USP). Test-day milk yields were recorded from 305 days of lactation, with the first record being obtained up to 45 days after calving. Test-day milk yields were divided into weekly classes of days in milk, ranging from one to 44 classes. Lactations with at least five individual records were included. Con-temporary groups were formed by herd-test-day. Only data of animals with at least six animal-observations were kept. A total of 2539 contemporary groups were formed. The relationship matrix had a total of 17,688 animals. The structure of the data set after editing is summarized in Table 1.
The analyses were conducted using a single-trait RRM. The model included the direct additive, permanent envir-onmental and residual random effects. In addition, con-temporary group and linear and quadratic effects of the age of cow at calving were included as fixed effects. The average lactation curve of the population was modeled with a fourth-order orthogonal Legendre polynomial. The additive genetic and permanent environmental covar-iance functions were estimated by random regression on B-spline functions of days in milk. Residual variances
were modeled by step functions considering with six classes of variance (1, 2, 3, 4–6, 7–12 and 13–44 weeks) throughout.
The B-spline function can be defined recursively
(Meyer, 2005a), with basis functions of degree p¼0
assuming values of unity for all points in a given interval; otherwise, the value is zero. For thek-th interval given by knotsTkandTkþ1withTkrTkþ1.
Bk,0ðtÞ ¼
1 if TkrtoTkþ1 otherwise 0
High degree basis functions, forp40, are then deter-mined from the values of the lower degree basis functions and the width of the adjoining intervals between knots (Meyer, 2005a). According toDe Boor (2001), the function can be described as:
Bk,pðtÞ ¼ tTk TkþpTk Bk,p1ðtÞ þ Tkþpþ1t Tkþpþ1Tkþ1 Bkþ1,p1ðtÞ, Linear (L), quadratic (Q) and cubic (C) polynomials were considered for each individual segment, with basis functions of degreep¼1, 2 and 3, respectively. According
to Meyer (2005a), in random regression models using
B-spline functions as basis function, a type of penalized spline, the choice of knot position is less crucial.Mknots were chosen to divide the classes of days in milk into
m1 equally spaced intervals and the external knots are located in classes 1 and 44, i.e. at the extreme points, for all models. However, on the original scale of age at recording the knots were not located at equally spaced intervals since the division of the classes was not equally distant. Up to seven knots (six segments) were used for direct additive genetic and permanent environmental effects of the animal, with the same number of knots fitted to both effects. The number of random regression coefficients to model the trajectory of the linear, quadratic and cubic basis functions is given bym,mþ1 andmþ2, respectively.
Eight models were tested using different combinations of B-spline functions (BS) of weeks of lactation for additive genetic and permanent environmental effects. Therefore, models were identified as follows: ‘‘BSX k’’, where X corresponds to L, Q or C, i.e., the degree of the
polynomial segment, and k are the specific numbers of
random regression coefficients for direct additive genetic and permanent environmental effects.
The matrix presentation of the single-trait random regression model is given byy¼XbþZaþWcþe, wherey
is the vector ofNobservations measured inNdanimals; Table 1
Summary of data structure.
Information Statistics
Number of records 152,145
Number of animals with records 7317
Number of sires 612
Number of dams 1858
Number of contemporary groups 2539
Overall mean (kg) 27.45
bis the vector of the systematic effects and fixed
regres-sion coefficients; a is the vector of additive genetic
random regression coefficients;cis the vector of
perma-nent environmental random regression coefficients;e is
the random vector of error effects, andX,ZandWare the incidence matrices of fixed effects and additive genetic and permanent environmental random effects.
The following assumptions are made: y9b, a, c,
s
2el, :::,
s
2 efNMVðXbþZaþWc, RÞ, with R¼diags
2 el , the likelihood function, in that l¼1, 2,..., f, where f is the number of residual classes;bpconstant;a9KaNMV(0,G), with G¼AKa, where A is the numerator relationshipmatrix between animals andKais the matrix containing
(co)variances between additive genetic random regres-sion coefficients;c9KcNMV(0,C), withC¼IKc, whereIis
an identity matrix, Kc is the matrix containing
(co)var-iances between permanent environmental random
reg-ression coefficients, and NMVrefers to a multivariate
normal distribution;Ka9va, S2aW1 va, va S2a
;Kc9vc,
S2cW1 vc, vc S2c
, where va, S2a and vc, S2c are interpreted as degrees of belief and priori values for (co)variances of the additive genetic and permanent environmental regression coefficients, respectively, and
W1is an inverted Wishart distribution.
s
2el9vel, s2el
X2
vel, vel S2el
, where vel, S2el are interpreted as degrees of belief and priori values for residual variance, andX2
is a scaled inverse chi-square distribution. The covariance matrices of additive genetic, perma-nent environment random regression coefficients and residual variances of all competing RRM were estimated using the GIBBS3F90 (Misztal et al., 2002). This program generates Markov chains for the parameters of a random regression model by Gibbs sampling. For each analysis the length of the chain was set to 250,000. The burn-in period was 25,000 iterations, higher than the minimum burn-in
required according toRaftery and Lewis (1992) method.
Convergence was tested using the criteria proposed by
Heidelberger and Welch (1983)and Geweke (1992)
cri-teria. The software R, with some routines of the package Bayesian Output Analysis (BOA), was used to calculated Geweke’s and Heidelberger and Welch’s statistics (Smith, 1997). Using a thin the output by 25, inferences were performed on the remaining 9000 samples. The highest posterior density interval was determined for each para-meter of the model with 95% confidence.
The deviance information criterion (DIC) was used as criteria to choose the most adequate model. The DIC introduced bySpiegelhalter et al. (2002)combines both goodness of fit and degree of parameterization of the model and can be calculated easily from the samples generated by a Markov Chain Monte Carlo simulation. The DIC was defined bySpiegelhalter et al. (2002)as:DIC¼ Dð
y
Þ þpD¼2Dðy
ÞDðy
Þ, where Dðy
Þ was the posterior expectation of the Bayesian deviance fDðy
Þ ¼Ey9y½Dðy
Þ andDðy
Þ ¼ 2logpðy9y
Þgrepresented a measure of the fit of the model, and pD was the effective number ofpara-meters representing penalty for increasing model com-plexity defined aspD¼Dð
y
ÞDðy
Þ, wherey
is the vector of parameters of the model,Dðy
Þwas the Bayesian devianceevaluated at the posterior mean of the parameters. Models having smaller DIC are favored because it indicates a better goodness-of-fit associated with a lower degree of complexity. 3. Results and discussion
The deviance information criterion (DIC) obtained for
all models are summarized inTable 2. The models
con-taining a smaller number of parameters presented the worst fit according to criteria. Problems of convergence of the iterative process were observed for models BSQ8 and BSC8. An increase in the number of knots in the models resulted in lower DIC values, irrespective of the order of the B-spline (linear, quadratic or cubic). Comparing mod-els containing the same number of parameters, those of cubic order and containing a smaller number of segments (BSC7) were always superior according to DIC.
The phenotypic variances estimated with the two models (BSQ7 and BSC7) were high in the first two weeks of lactation, declined until mid-lactation, and again increased after week 20 until the end of lactation (Fig. 1). These estimates were very close for both models. The genetic variances estimated with the models were also higher during the first two weeks of lactation, declined until mid-lactation, and again increased after week 25. A similar finding has been reported byKettunen
et al. (2000) and Rekaya et al. (1999) and in Brazil by
Cobuci et al. (2005)for Holstein cattle using RR models and different functions. Lower genetic variability was observed during peak lactation (weeks 18 and 19). Addi-tive genetic variances were always lower than the other variances throughout lactation. The same trends have
been reported for Holstein cattle by Jamrozik and
Schaeffer (2002)andOlori et al. (1999).
The permanent environmental variances estimated with the two models were practically constant until week 37 of lactation, increasing slightly thereafter until the end of lactation. In contrast, Cobuci et al. (2005), using the Wilmink function, andCosta et al. (2008), using Legendre polynomials, reported higher estimates at the end of lactation for Brazilian Holstein cattle. When compared to the results reported byBignardi et al. (2009b), using the same data set but applying the restricted maximum likelihood method and RR on Legendre polynomials for
Table 2
Models, number of parameters (np) in the model, position of knots (pk), number of segments (ns), deviance information criterion (DIC).
Modelsa np pk ns DIC (1) BSL5 36 1-12-23-34-44 4 838,00 (2) BSL6 48 1-10-19-26-35-44 5 833,64 (3) BSQ6 48 1-12-23-34-44 4 832,53 (4) BSQ7 62 1-10-19-26-35-44 5 830,62 (5) BSQ8 78 1-9-16-23-30-37-44 6 858,33 (6) BSC6 48 1-16-30-44 3 832,64 (7) BSC7 62 1-12-23-34-44 4 830,28 (8) BSC8 78 1-10-19-26-35-44 5 858,05 a
BSXk, where X corresponds to L (linear), Q (quadratic) or C (cubic), i.e., the degree of the polynomial segment, and k are the specific numbers of random regression coefficients for direct additive genetic and permanent environmental effects.
genetic effects and a stationary parametric correlation
function associated with a seventh-order variance
function for permanent environmental effects, it can be concluded that models BSQ7 and BSC7 provided lower permanent environmental variances practically through-out the period of lactation. The residual variance esti-mates obtained with model BSC7 (35.04, 19.15, 14.61,
11.90, 9.65 and 11.27 kg2) were lower than those
obtained with model BSQ7 (40.42, 19.07, 14.57, 12.27, 9.59 and 11.20 kg2), especially at the beginning of lacta-tion. Similar results have been reported for Holstein cattle byBignardi et al. (2009a)andOlori et al. (1999)that were
using RR on Legendre polynomials and by Bohmanova
et al. (2008)using a linear spline function. According to Lo´pez-Romero et al. (2003), the heterogeneity of residual variances is related to the stage of lactation and a set of factors not specified in the model, such as characteristics of the dry season, stage of pregnancy and body condition at calving, contributing to increase residual variance at the extremes of the lactation curve.
The heritability estimates obtained with the two models showed the same trend throughout most of lactation (Fig. 2). However, some differences in the magnitudes of the esti-mates were observed in almost the entire period of lactation. After a decline at the beginning of lactation, heritabilities
increased and the highest estimates were observed around the 31th to 37th weekly test-day milk yields. Greater oscillations in the heritability estimates were observed by
Bignardi et al. (2009a) when fitting Legendre polynomial
regressions and Bignardi et al. (2011) fitting parametric functions and B-spline linear using restricted maximum likelihood estimation to the same data set. This agrees with
Meyer (2005a) who studied the growth of beef cattle and
found that B-spline functions are less susceptible to the problems of erratic estimates at the extremes of the curve frequently encountered with Legendre polynomials. This lower susceptibility is due to the lower degree of polynomials in the individual segments and to the better control of the global influence of individual observations. For that reason, it can be inferred that the use of B-splines analysis test-day milk yield is appropriate, because these functions are indi-cated for data with large variations with time, what is observed in this case. The posterior means of heritability ranged from 0.21 (week 21) to 0.31 (week 34) for model BSQ7 and from 0.19 (week 22) to 0.33 (week 34) for model BSC7. Similar estimates have been reported for Holstein cattle byWhite et al. (1999)using a cubic spline function and byArau´jo et al. (2006),Bignardi et al. (2009a,b, 2011), Cobuci et al. (2005)andCosta et al. (2008)using RR models with different functions fitted to test-day milk yield records 0 20 40 60 80 100 120 σ 2 p 0 20 40 60 80 100 120 σ 2 p 0 20 40 60 80 100 120 σ 2 a 0 20 40 60 80 100 120 σ 2 a 0 20 40 60 80 100 120 1 7 13 19 25 31 37 43 σ 2 pe weeks 0 20 40 60 80 100 120 1 7 13 19 25 31 37 43 σ 2 pe weeks
Fig. 1.Posterior means () of phenotypic (s2
p), genetic (s2a) and permanent environmental variances (s2pe) and approximate 95% highest posterior
in Brazil. However, Silvestre et al. (2005) obtained lower heritability estimates with RR models using a cubic spline function.
The model BSQ7 provided a smaller credibility interval than model BSC7 for the estimates of all variance compo-nents and for the heritability estimates at the beginning and end of lactation, suggesting a higher confidence of the estimates obtained with model BSQ7 during these periods (Figs. 1 and 2). This finding indicates that model BSQ7 is more efficient in fitting milk yields at the extremes of the lactation curve, periods that are known to be more difficult to model.
Higher phenotypic correlations, around 0.70, were observed between adjacent test-day milk yields, especially during mid-lactation. Estimated genetic correlation coeffi-cients were found higher than phenotypic correlation coef-ficients. Adjacent milk yields obtained during mid-lactation presented genetic correlation estimates close to one. These results agree with those reported for Holstein cattle byOlori
et al. (1999) and Rekaya et al. (1999) and in Brazil by
Bignardi et al. (2009a)andCobuci et al. (2005). In addition, negative genetic correlations between the initial and final weeks of lactation were obtained with both models, but their frequency was lower for model BSQ7 when compared to model BSC7 and other models to the same data set but using restricted maximum likelihood and others functions (Bignardi et al., 2011). These negative correlations might be attributed to the difficulty in modeling milk yields at the beginning and at the end of lactation. At the beginning of lactation, the cow still suffers from the stress of calving and also presents an energy deficit. Negative genetic correlations were also estimated for Holstein cattle by RR using different
functions by Brotherstone et al. (2000), Jamrozik and
Schaeffer (1997), Kettunen et al. (2000) and Olori et al. (1999) and in Brazil by Bignardi et al. (2009a,b, 2011), Cobuci et al. (2005)andCosta et al. (2008). With respect to permanent environmental correlations between milk yields lower estimates were observed with increasing interval between test days. In general, the estimates of permanent environmental correlations were of lower magnitude than the genetic correlations. The phenotypic, genetic and per-manent environmental correlations obtained with models BSQ7 and BSC7 were similar.
4. Conclusion
In tropical regions like Brazil, B-spline random regres-sion models using Bayesian inference are indicated for
Dairy Holstein breed milk yield genetic evaluation. Although a cubic B-spline with seven random regression coefficients for both the additive genetic and permanent environment effects was found to be the best according to DIC criteria, a lower order model (quadratic B-spline with seven random regression coefficients for both random effects) could be adopted because it yielded practically the same genetic parameter estimates with parsimony. In addition, a posteriori estimates for the quadratic B-spline model had smaller credibility intervals at the extremes of lactation curve, suggesting higher reliability of the esti-mates provided by the model.
Conflict of interest statement
The authors declare that they have no conflict of interest.
Acknowledgments
This work was funded by the Fundac-ao de Amparo~ a
Pesquisa do Estado de Sao Paulo (FAPESP) and the~
Coordenac-ao de Aperfeic~ -oamento de Pessoal de Nı´vel Superior (Capes), Brazil.
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