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Revista mexicana de ciencias pecuarias

versão On-line ISSN 2448-6698versão impressa ISSN 2007-1124

Rev. mex. de cienc. pecuarias vol.7 no.2 Mérida Abr./Jun. 2016

 

Notas de investigación

Characterization of lactation curves in Siboney cattle using nonlinear mixed models

Alejandro Palacios-Espinosaa 

Joel Domínguez-Viverosb  * 

Yamariz Padrón-Quinteroc 

Manuel Rodríguez Castroc 

Felipe Alonso Rodríguez-Almeidab 

José Luis Espinoza-Villavicencioa 

Narciso Ysac Ávila-Serranod 

aDepartamento de Zootecnia, Universidad Autónoma de Baja California Sur. La Paz, Baja California Sur. México.

bFacultad de Zootecnia y Ecología, Universidad Autónoma de Chihuahua. Periférico Francisco R. Almada km 1, 31453, Chihuahua, Chihuahua, México; Tel (614)4341448.

cCentro de Investigaciones para el Mejoramiento Animal de la Ganadería Tropical. La Habana, Cuba.

dInstituto de Recursos, Universidad del Mar. Puerto Escondido, Oaxaca. México.


ABSTRACT

Nonlinear models (NLM) have been used to evaluate dairy cattle lactation curves (LC), and nonlinear mixed models are increasingly applied. The most appropriate model for estimating LC components in Siboney cattle was identified by applying a NLM for LC to 240 d; comparing a non-mixed NML vs a mixed NML; and estimating peak production (PMAX), total production at 240 d (PTOTAL) and days to maximum production (DPMAX). The database consisted of 15,324 records for daily milk production (kg) corresponding to first lactation in 2,809 Siboney cows (calving date in 2000-2012) in 28 herds in Cuba. Five NLM were evaluated: Wood, Wiltmink, Cobby, Brody and Sikka. Two SAS NLMIXED analyses were run: ANA1, including only the random effect of residuals; and ANA2, including a regression coefficient random effect plus residuals. Selection of the best fitting model was done based on average prediction error; prediction error variance; the Durbin Watson statistic; the determination coefficient (R2); the Akaike information criterion (AIC); and the Bayesian information criterion (BIC). The mixed NLM (ANA2) had the best fit: a) its R2 was higher (9 to 12 %); b) residual variance was notably reduced; and c) the AIC and BIC criteria exhibited a better fit. The model with the overall best fit was the ANA2 WOD, with estimates of 7.71 kg for PMAX, 40 d for DPMAX and 1,653 kg for PTOTAL.

Key words: Dairy cattle; Crossbreeding; Synthetic breed; Tropical livestock; Milk production; Nonlinear models

RESUMEN

En bovinos productores de leche la curva de lactancia (CL) se ha analizado con modelos no lineales (MNL), y recientemente se ha implementado la metodología de MNL mixtos. Los objetivos fueron: a) ajustar un MNL a la CL a 240 días; b) comparar MNL vs MNL mixtos; c) estimar producción al pico (PMAX) y total a 240 días (PTOTAL), así como días para alcanzar la producción máxima (DPMAX). Se analizaron 15,324 observaciones de producción diaria (kg), correspondientes a primera lactancia de 2,809 vacas Siboney, con fecha de parto de 2000 a 2012 en 28 hatos en Cuba. Se evaluaron cinco MNL: Wood, Wiltmink, Cobby, Brody y Sikka. Se realizaron dos análisis con el procedimiento NLMIXED de SAS: ANA1, incluyó sólo el efecto aleatorio de los residuales; y ANA2, incluyó el efecto aleatorio correspondiente a un coeficiente de regresión, más los residuales. La selección del modelo con mejor ajuste se realizó con: error de predicción promedio; varianza del error de predicción; estadístico Durbin Watson; coeficiente de determinación (R2); criterios de información Akaike (AIC) y Bayesiano (BIC). Los MNL mixtos mostraron el mejor ajuste con base en: a) los R2 del ANA2 fueron superiores, en un intervalo de 9 a 12 %; b) por la reducción en la varianza residual en el ANA2; y, c) los criterios AIC y BIC, mostraron mejor ajuste en los MNL del ANA2. El modelo de mejor ajuste en todo el estudio fue WOD del ANA2, con estimaciones para PMAX, DPMAX y PTOTAL de 7.71 kg, 40 días y 1,653 kg, respectivamente.

Palabras clave: Bovinos leche; Cruzamiento; Raza sintética; Ganadería tropical; Modelos no lineales

Genetic improvement aimed at increasing milk production in warm tropical regions is based, among other strategies, on crosses between adapted breeds (BI; Bos indicus) and specialized breeds from temperate and cold climes (B7; Bos taurus). Different degrees of crossbreeding provide different advantages1. Beginning in the 1960s, crossbreeding began between the Holstein and Zebu breeds in Cuba with the aim of developing genotypes apt for milk production and adapted to adverse tropical conditions. Two synthetic breeds resulted2,3: Siboney (5/8 Holstein + 3/8 Zebu) and Mambí (3/4 Holstein + 1/4 Zebu). The Siboney breed was created as part of the National Genetic Improvement Program to produce genotypes less demanding than Holstein that could attain adequate production potential under tropical conditions with feeding based mainly on grazing4,5.

The lactation curve (LC) is used to analyze dairy cow milk production in three periods with four main components: 1) initial production; 2) increased production or ascending phase; 3) maximum production or peak phase; and 4) descending rate or reduced production, called persistence6,7,8. Nonlinear models (NLM) have been used to characterize and analyze LCs. In this approach, regression coefficients are interpreted in association with LC stages, or help to derive other indicators that describe the LC9,10. Adjusting NLM has been done using nonlinear regression analysis involving iterative methods such as those of Gauss, Marquardt and Newton, among others11. Statistically analyzing milk production through lactation measurements involves repeated measurements over time. This generates a variance/covariance structure that can be analyzed using mixed models12,13, with positive effects in NLM fit analyses. Nonlinear regression analyses, including random effects related to the regression coefficients, are run using a different iterative method (e.g. Firo, Gauss, Hardy and Isamp)14,15 than used in NLMs with only fixed effects.

Understanding LC is important for dairy cattle management, feed and genetic improvement programs16. Analysis of LC is done at 305 d lactation in intensive or corral production systems with specialized breeds such as Holstein. However, the environmental effects and handling conditions in tropical production systems shorten cow productive life, and as a consequence the definition, analysis and characterization of lactation is done over shorter intervals, from 210 to 240 d17,18,19. In an effort to characterize and analyze the Siboney cattle LC, the present study objectives were to a) select a NLM that fits the Siboney LC; b) compare nonlinear regression procedures for the only fixed effects NLM versus the mixed NLM; and c) use the NLM with the best fit to estimate LC components or indicators.

The data analyzed in this study were from the data base of the Research Center for Genetic Improvement of Tropical Livestock (Centro de Investigaciones para el Mejoramiento Animal de la Ganadería Tropical), a division of the Ministry of Agriculture of the Republic of Cuba. A total of 15,324 daily milk production (kg) records were analyzed, in an interval of 1 to 240 d, for first lactations in 2,809 Siboney cows that had given birth between January 2000 and December 2012. These data were collected from 28 herds located on the island of Cuba (20 and 23° N, 74 and 85° W) and managed using a grazing system. The island experiences two clear seasons, a rainy season from May to October during which 70 to 80 % of rainfall occurs, and a dry season from November to April. Mean annual temperature is 23.1 °C and average relative humidity ranges from 60 to 70 % in the day to 89 to 90 % at night20.

Five NLMs were evaluated for their effectiveness in analyzing and characterizing LC7,9,21:

  1. Wood (WOD; yt= β1*(tβ2)*(exp(-β3*t)) + εij)

  2. Wiltmink (WIL; yt= β1 + β2*t+ β3*(exp(-0.05*t))+ εij)

  3. Cobby (COB; yt= β1- β2*t-1*(exp(-β3*t)) + εij)

  4. Brody (BRO; yt= β1*(exp(-β2*t))- β1*(exp(-β3*t))+ εij)

  5. Sikka (SIK; yt= β1*(exp((β2*t)-(β3*t2))) + εij)

Where: yt is milk production (kg) on day t (1 to 240); β1, β2 and β3 are each model's regression coefficients; and εij are random effects, associated with errors or residuals, of j-th production on the i-th day of lactation. Four LC indicators were estimated using each model's β values: lactation initial production (IP); days to maximum production (DPMAX); peak or maximum production (PMAX); and total or accumulated production (PTOTAL) at 240 d lactation 9,10,22.

Two analyses were run with the NLMIXED procedure in the SAS program, using the Gauss iterative method13,23: 1) a NLM without random effects related to the regression coefficients and only the random effect (e) of the residuals (ANA1); and 2) a mixed NLM in which the random effect of each model's β1 was added to the residuals (ANA2). The non-correlated random effects of εij and β1 fit the assumptions of a normal distribution, with a mean equal to zero and variances of σ2e and σ2β1, respectively. The random effects of the regression coefficients β2 and β3 were not included because their variance components were very low magnitude and tended towards zero. Selection of the best fit model was done based on six criteria24,25,26:

  1. Average prediction error [APE=(i=1nepi-pmipmi*100)/n]

  2. Prediction error variance: [PEV=i=1npmi-epi2/n];

  3. Durbin Watson statistic [DW=21-ñ; ρ=t=2n(et-et-1)2t=1net2]

  4. Determination coefficient [R2= (1 - (ssr/tss))];

  5. Akaike information criterion [AIC= -2*Log lik+2k];

  6. Bayesian information criterion [BIC = -2*Log lik +log(n)*k]

Where pmi is the production measured on the i-th day of lactation; epi is the estimated production for the i-th day of lactation; n is the total number of observations; ssr is the sum of squares of residuals; tss is the total sum of squares; Log lik is the likelihood function logarithm; and k is the number of model parameters. The APE analyzes the relationship between measured and estimated production for the i-th day of lactation, and, based on the combination of "+" and "-" signs, the NLM is shown to either over- (+) or underestimate (-) the predictions. For APE, PEV, AIC and BIC, the model with the lowest value is considered that with the best fit. In contrast, for R2 the model with the highest value is considered to have the best fit. The DW statistic analyzes autocorrelations in the errors, using three categories: a) 2<DW<4 indicates a negative autocorrelation; b) 0<DW<2 indicates no autocorrelation; and c) DW≤0 indicates a positive autocorrelation.

No differences in APE, PEV and DW were present between the two analyses run to select the model with the best fit (Table 1). Based on APE and the DW statistic, all the NLM predictions tended to underestimate the analyzed information, with no autocorrelation between residuals. The mixed NLM exhibited the best fit based on the three results in Table 1: a) the residual variances of ANA2 were lower (up to 82 % in the SIK and WOD models) than those estimated in ANA1; b) consequently, the ANA2 R2 values (97 % on average) were higher than those produced by ANA1 (87 % in all), in an interval of 9 to 12 %; and c) the AIC and BIC criteria congointly exhibited the best fit in ANA2. As a function of the AIC, BIC and R2 criteria, the best fit within ANA2 was with the WOD model, followed by the SIK and BRO models. With the ANA2 results, estimations were made for IP, DPMAX, PMAX and PTOTAL (Table 1).

Table 1 Nonlinear model fit in the characterization of lactation curves in Siboney cattle 

Item WOD WIL COB BRO SIK
Nonlinear model fit, not including random effects (ANA1)
β1 5.89 8.46 8.12 8.18 7.54
β2 0.0996 -0.0117 0.00953 0.00135 0.00040
β3 0.00245 -1.9026 0.2676 0.2678 0.0000073
σ2e 7.27 7.28 7.31 7.32 7.29
AIC 73916 73912 73985 73997 73947
BIC 73946 73942 74016 74028 73977
R2 87.3 87.4 87.2 87.1 87.3
APE -15.7 -15.6 -16.3 -16.2 -15.9
PEV 7.28 7.28 7.33 7.33 7.29
DW 1.44 1.42 1.42 1.44 1.43
Nonlinear model fit, including β1 random effect (ANA2)
β1 6.09 8.45 8.11 8.29 7.67
β2 0.0912 -0.0116 0.00951 0.00147 0.000089
β3 0.00241 -1.9064 0.2677 0.2334 0.0000062
1.31 2.75 2.78 1.32 1.30
σ2β1 4.42 4.52 4.52 8.11 7.03
AIC 73698 73906 73980 73767 73732
BIC 73736 73944 74018 73805 73770
R2 98.3 95.2 95.1 97.7 97.3
APE -15.7 -15.5 -15.8 -15.8 -15.7
PEV 7.28 7.27 7.31 7.32 7.30
DW 1.41 1.41 1.40 1.41 1.40
Lactation curve components
IP 6.08 6.64 1.90 1.71 7.66
DPMAX 40 42 20 20 28
PMAX 7.71 7.74 8.03 8.04 7.60
PTOTAL 1653 1655 1667 1664 1660

β1, β2 and β3= regression coefficients for evaluated nonlinear models; σ2e= residual variance; σ2β1= variance corresponding to bi random effect; AIC= Akaike information criterion; BIC= Bayesian information criterion; R2= determination coefficient (%); APE= average prediction error; PEV= prediction error variance; DW= Durbin Watson statistic; IP= lactation initial production; DPMAX= days to maximum or peak production; PMAX= maximum or peak production; PTOTAL= cumulative total production at 240 d lactation. Models: WOD: Wood; WIL: Wiltmink; COB: Cobby; BRO: Brody; SIK: Sikka.

The WOD NLM is widely seen to have the best statistical fit when using different selection and hierarchization criteria; it is the most popular model to describe dairy cow LC. Its regression coefficients are associated with different LC components and allow derivation of other partial or total production indicators over time7,8,27. In one study, the WOD NLM was found to be the most apt of sixteen different NLM in LC for six cross genotypes using different BT:BI proportions21. This model was also found to be apt, with an ideal fit, for modeling milk production in beef cattle in different sampling schemes28,29.

In the NLM, β1 is associated with lactation initial production, β2 with the ascending phase or peak production, and β3 with descending rate or persistence. The combination of signs in β2 and β3 showed the WOD and WIL models to fit four unimodal curve types30: 1) standard or typical curve, WOD + β2,- β3 and WIL - β2,- β3; 2) continuously descending curve (atypical), WOD -β2,-β3 and WIL +β2,-β3; 3) inverted standard curve (U), WOD - β2,+ β3 and WIL + β2,+ β3; and 4) continuously increasing curve: WOD + β2,+ β3 and WIL - β2,+ β3. The standard LC based on the WIL model and the continuously increasing LC based on the WOD model exhibited similar estimations for PMAX and PTOTAL (Figure 1), but differed notably in their estimations of IP and DPMAX (Table 1). In contrast to the high β1 estimates they produced, both the COB and BRO models generated underestimates for initial lactation points such as IP (1.8 kg) and DPMAX (20 d). This may indicate that these models have certain problems estimating the early portions of the LC.

Figure 1 Siboney cattle lactation curves adjusted to 240 days based on the Wiltmink (a; WIL) and Wood (b; WOD) models 

Average PMAX was 7.8 kg and average PTOTAL was 1,660 kg, levels higher than previously reported for Siboney cattle5,31,32, and other BI x BT genotypes19. However, they are lower than reported in Mambí cattle33,34.

Dairy cattle production systems under warm tropical conditions depend on factors linked to the animal, environment, feeding and production technology. Variability in production at the animal level can be attributed to genotype as a function of the BI:BT proportion in a given individual35,36. At the technology level, milk production is affected by the cow/calf interaction and nursing management18. In tropical systems, feeding is largely based on grazing of forage grasses, occasionally complemented by forage legumes37. Factors involved in the environment level include prevalence of internal and external parasites, stress caused by high temperatures and rainfall, and forage quality and availability1,38. In the present results, the Wood model had the best fit for characterizing the lactation curve in Siboney cattle at 240 d. The curve was one of continuous increase with a peak production of 7.71 kg at 40 d and a cumulative total production of 1,653 kg. The nonlinear mixed models exhibited the best fits based on the Akaike and Bayesian information criteria, and reduced residual variance, with increases of 11 % in the determination coefficient.

ACKNOWLEDGEMENTS

Thanks are due to the Centro de Investigaciones para el Mejoramiento Genético de la Ganadería Tropical (CIMAGT) of the Republic of Cuba for providing the data on which this study is based.

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Received: August 01, 2014; Accepted: September 17, 2014

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