Lactation Curve Models Research Articles

Modelling temperature effects on milk production: a study on Holstein cows at a Japanese farm.

Milk yield and its composition vary according to individual cows as well as to a variety of different environment conditions, such as temperature. Previous studies suggest that heat exerts considerable negative effects on milk production and its composition, especially during summer months. We investigate the production and fat composition of milk from individual dairy cows and develop a modelling framework that investigates the effect of temperature by extending a traditional lactation curve model onto a more flexible statistical modelling framework, a generalised additive model (GAM). The GAM simultaneously copes with multiple different conditions (temperature, parity, days of lactation, etc.), and, importantly, their non-linear relationships. Our analysis of retrospective data suggests that individual cows respond differently to heat; cows producing relatively high quantities of milk tend to be particularly sensitive to heat. Our model also suggests that most dairy cows studied fall into three distinct cases that underpin the variation of the milk fat ratio by different mechanisms.Electronic supplementary materialThe online version of this article (doi:10.1186/2193-1801-3-129) contains supplementary material, which is available to authorized users.

Prediction of Fortnightly Test-Day Milk Yields Using Four Different Lactation Curve Models in Indian Murrah Buffalo.

The present investigation was carried out using data on 18871 fortnightly test day milk yield (FTDMY) records during first lactation of 961 Murrah buffaloes calved during 1977-2012 maintained in an organized farm at National Dairy Research Institute, Karnal. The least squares means of the FTDMY ranged from 2.35kg to 7.92kg. The relative efficiency of four lactation curve models via. Gamma-type function (GF), Exponential function (EF), Mixed log function (MLF) and polynomial regression function (PRF) were compared. PRF described the highest coefficient of determination (R2) (99.78%) and with least value (0.02kg) of root mean squares error (RMSE), whereas, least coefficient of determination (93.13%) was observed in Gamma-type function having maximum (0.1kg) RMSE value suggesting PRF being best mathematical model for prediction of FTDMYs in Murrah buffaloes.

Anadolu Mandalarında Farklı Laktasyon Eğrisi Modellerinin Karşılaştırılması

In this study, eight different lactation curve models were compared by using test day milk yield records belonging Anatolian Buffaloes raised in different Farm conditions between 2011 and 2013. To identify the best lactation curve models of Wood, Cobby and Le Du, Logaritmic Quadratic, Exponential, Parabolic exponential, Quadratic, Inverse Polynomial and Logaritmic Linear mathematical functions were used. The coefficient of determination (R2) and residual standard deviation (RSD) statistics were used for determination of best fitted model in lactation curve. Logaritmic Quadratic and Quadratic functions are the best goodness of fit model as having the highest R2 and lowest RSD coefficients. As a result, the parameters are estimated by logarithmic quadratic or quadratic models, for use in breeding programs will make an important contribution to research in this field.

Use of Lactation Curve Models for Prediction of Milk Yield in Sahiwal Cattle

A study was carried out on of 25499 weekly test day milk yield records of first lactation pertaining to 593 Sahiwal cows spread over 49 years located at National Dairy Research Institute, Karnal, India. The relative efficiency of three lactation curve models via . quadratic model, gamma type function and mixed log function were compared. Mixed log function described the best fit with highest coefficient of determination (97.9%) with root mean squares error (RMSE) of 0.025 kg which is followed by gamma type function. Least coefficient of determination (77.8%) was observed in quadratic model with maximum (0.081) RMSE. The peak yield of 8.08 kg in 8 th week of lactation was observed by mixed log function. The predicted first lactation 305 day milk yields by quadratic model, gamma type function and mixed log function were 1993.23± 42.09, 1991.51± 40.38 and 1993.23± 42.10 kg, respectively.

Prediction of Fortnightly Test-Day Milk Yields Using Four Different Lactation Curve Models in Indian Murrah Buffalo

Prediction of Fortnightly Test-Day Milk Yields Using Four Different Lactation Curve Models in Indian Murrah Buffalo

Comparison of mathematical models applied to f1 dairy sheep lactations in organic farm and environmental factors affecting lactation curve parameter.

The objective of this study was to compare the goodness of fit of four lactation curve models: Wood’s Gamma model (WD), Wilmink (WL), and Pollott’s multiplicative two (POL2) and three parameters (POL3) and to determine the environmental factors affecting the complete lactation curve of F1 dairy sheep under organic management. A total of 5,382 weekly milk yields records from 150 ewes, under organic management were used. Residual mean square (RMS), determination coefficients (R2), and correlation (r) analysis were used as an indicator of goodness of fit for each model. WL model best fitted the lactation curves as indicated by the lower RMS values (0.019), followed by WD (0.023), POL2 (0.025) and POL3 (0.029). The four models provided total milk yield (TMY) estimations that were highly correlated (0.93 to 0.97) with observed TMY (89.9 kg). The four models under estimated peak yield (PY), whereas POL2 and POL3 gave nearer peak time lactation estimations. Ewes lambing in autumn had higher TMY and showed a typical curve shape. Higher TMY were recorded in second and third lambing. Season of lambing, number of lambing and type of lambing had a great influenced over TMY shaping the complete lactation curve of F1 dairy sheep. In general terms WL model showed the best fit to the F1 dairy sheep lactation curve under organic management.

Genetic variance in micro-environmental sensitivity for milk and milk quality in Walloon Holstein cattle

Animals that are robust to environmental changes are desirable in the current dairy industry. Genetic differences in micro-environmental sensitivity can be studied through heterogeneity of residual variance between animals. However, residual variance between animals is usually assumed to be homogeneous in traditional genetic evaluations. The aim of this study was to investigate genetic heterogeneity of residual variance by estimating variance components in residual variance for milk yield, somatic cell score, contents in milk (g/dL) of 2 groups of milk fatty acids (i.e., saturated and unsaturated fatty acids), and the content in milk of one individual fatty acid (i.e., oleic acid, C18:1 cis-9), for first-parity Holstein cows in the Walloon Region of Belgium. A total of 146,027 test-day records from 26,887 cows in 747 herds were available. All cows had at least 3 records and a known sire. These sires had at least 10 cows with records and each herd × test-day had at least 5 cows. The 5 traits were analyzed separately based on fixed lactation curve and random regression test-day models for the mean. Estimation of variance components was performed by running iteratively expectation maximization-REML algorithm by the implementation of double hierarchical generalized linear models. Based on fixed lactation curve test-day mean models, heritability for residual variances ranged between 1.01×10−3 and 4.17×10−3 for all traits. The genetic standard deviation in residual variance (i.e., approximately the genetic coefficient of variation of residual variance) ranged between 0.12 and 0.17. Therefore, some genetic variance in micro-environmental sensitivity existed in the Walloon Holstein dairy cattle for the 5 studied traits. The standard deviations due to herd × test-day and permanent environment in residual variance ranged between 0.36 and 0.45 for herd × test-day effect and between 0.55 and 0.97 for permanent environmental effect. Therefore, nongenetic effects also contributed substantially to micro-environmental sensitivity. Addition of random regressions to the mean model did not reduce heterogeneity in residual variance and that genetic heterogeneity of residual variance was not simply an effect of an incomplete mean model.

Modeling lactation curves of Polish Holstein-Friesian cows. Part I: The accuracy of five lactation curve models

Modeling lactation curves of Polish Holstein-Friesian cows. Part I: The accuracy of five lactation curve models

Comparison of different lactation curve models in Karan Fries cattle of India

The first lactation weekly test day milk yield (WTDMY) records (57233) and monthly test day milk yield (MTDMY) records (11970) spread over a period of 23 years (1984– 2006) sired by 82 bulls of Karan Fries cattle were used to develop the best lactation curve model. The lactation curve parameters of Quadratic cum log model (QCLM), Gamma function (GF), Cobby Le Du model (CLDM), Polynomial regression function (PRF) and Multiphasic logistic function (MLF) were estimated. The average weekly test day milk yield was predicted with high degree of accuracy (R2 > 85%) by all the models with the maximum accuracy (R2 = 99.50%) obtained by polynomial regression function (PRF) and the least fit (R2 = 87.90%) was obtained by Gamma function (GF). However Quadratic cum log model (R2 = 99.20%) was almost equal to polynomial regression function and was better than Cobby Le Du model (R2 = 92.80%). The average root mean square error (RMSE) was found to be minimum with PRF (0.0121 Kg) followed by MLF (Triphasic). Thus the best fit model was polynomial regression function, which was better than other functions for prediction of first lactation WTDMY. The average monthly test day milk yield was predicted with high degree of accuracy (R2 > 75%) by all the models with the maximum accuracy (R2 > 99.41%) obtained by polynomial regression function (PRF) and the least fit was obtained with gamma function (R2 = 79.05%). However, quadratic-cum-log model (R2 = 99.28%) was almost equal to polynomial regression function and was better than CLDM (R2 = 89.50%). The average root mean square error (RMSE) was found to be minimum with PRF (0.0061 kg) followed by QCLM (0.0620). Thus the best fit model was polynomial regression function for prediction of both weekly and monthly test day milk yield of first lactation in Karan Fries cows.

Comparative evaluation of a new lactation curve model for pasture-based Holstein-Friesian dairy cows

Fourteen lactation models were fitted to average and individual cow lactation data from pasture-based dairy systems in the Australian states of Victoria and Tasmania. The models included a new “log-quadratic” model, and a major objective was to evaluate and compare the performance of this model with the other models. Nine empirical and 5 mechanistic models were first fitted to average test-day milk yield of Holstein-Friesian dairy cows using the nonlinear procedure in SAS. Two additional semiparametric models were fitted using a linear model in ASReml. To investigate the influence of days to first test-day and the number of test-days, 5 of the best-fitting models were then fitted to individual cow lactation data. Model goodness of fit was evaluated using criteria such as the residual mean square, the distribution of residuals, the correlation between actual and predicted values, and the Wald-Wolfowitz runs test. Goodness of fit was similar in all but one of the models in terms of fitting average lactation but they differed in their ability to predict individual lactations. In particular, the widely used incomplete gamma model most displayed this failing. The new log-quadratic model was robust in fitting average and individual lactations, and was less affected by sampled data and more parsimonious in having only 3 parameters, each of which lends itself to biological interpretation.

Comparison of different lactation curve models in Sahiwal cows

The present investigation was carried out on 12,859 fortnightly test-day milk yield records of first lactation pertaining to 643 Sahiwal cows spread over a period of 49 years (1961-2009), maintained at the National Dairy Research Institute, Karnal, India. The comparison of 3 lactation curve models, i.e. the quadratic model, gamma-type function, and mixed log function, was done using fortnightly test-day milk yield records. The mixed log function was the best fit, with the highest (88.75%) coefficient of determination (R2 value) and the lowest (0.08 kg) root mean square error (RMSE) value, whereas the lowest (56.93%) R2 value was observed in the quadratic model with maximum (0.15 kg) RMSE value. The closeness of fit of the mixed log function and gamma-type function with the observed lactation curve was almost of the same order. However, the gamma-type function gave a low peak yield, and therefore it is recommended that this function can give the best fit for low-yielding cows.

Selection of a mathematical model to describe the lactation curves of Jersey cattle

Abstract. The extent of the usefulness of a lactation model depends on how well it succeeds in imitating the biological lactation process and how well it adjust for environmental and other factors that could influence production. Therefore, the objective of this study was to compare five different lactation curve models (Wood, Cobby and Le Du, Wilmink, Exponential and Parabolic Exponential model), and to find the best model that provided a good description of the lactation curve of Jersey cattle herd. Data used in this study were the first to seventh lactation official milk yield records from monthly recording of 3 630 lactations between 1984 and 2008 in the farm. The results showed that Wood model which has minimum residual standard deviation (3.562), maximum adjusted R2 value (91.6 %) and maximum persistency value (93.3 %) performed the best fit to the data and allowed a suitable description of the lactation curve. It was concluded that the Wood model provided accurate estimates of milk yield for all lactation numbers because this model was found to be more superior to the other models. Consequently, the usage of Wood model would provide some useful information on genetic improvement of the Jersey breed under pasture-based dry seasonal production systems.

Lactation curve models for estimating gene effects over a timeline

The effects of genes are commonly estimated using random regression models based on test-day data and only give a general gene effect. Alternatively, lactation curve models can be used to estimate biological and environmental effects, or to predict missing test-day data and perform breeding value estimation. This study combines lactation curve models and estimation of gene effects to represent gene effects in different stages of lactation. The lactation curve models used were based on the Wood, Wilmink, and Ali and Schaeffer models. A random regression test-day model was used to compare estimated gene effects with the results of commonly used models. The well-characterized DGAT1 gene with known effects on milk yield, milk fat, and milk protein production was chosen to test this new approach in a Holstein-Friesian dairy cattle population. The K232A polymorphism and the promoter VNTR (variable number of tandem repeats) of the DGAT1 gene were used. All lactation curve models predicted the production curves sufficiently. Nevertheless, for predicting genotype effects, the Wilmink curve indicated the closest fit to the data. This study shows that the characteristic gene effects for DGAT1 genotypes occur after lactation d 40, which might be explained by a link to other genes affecting metabolic traits. Furthermore, allele substitution effects of allele K of the K232A locus showed that the typical effect of low milk and protein yield is due mainly to a lower overall production level, whereas the higher fat and protein content is reached by increased production toward its peak and fat yield is increased because of a higher production after this peak. Predicting gene effects with production curves gives better insight into the timeline of gene effects. This can be used to form genetic groups, in addition to feeding groups, for managing livestock populations in a more effective way.

Selection of models of lactation curves to use in milk production simulation systems

The objective of this study was to select models of lactation curves with a better adjustment to the observed data in models of milk production simulation systems. A data base on 6,459 recordings of daily milk production was used. These data were obtained from monthly and fortnightly controls of milk between 2004 and 2007, from 472 lactations of animals from ten different milking cow herd farms. Based on rolling averages of milk production (MP-L/day) per cow, the ten herd farms were divided into low (L < 15), medium (15 <M < 20) and high (H > 20). Data were also divided according to the lactation numbers in first, second, third or greater. Eight lactation curve models commonly used in literature were compared. The models were individually adjusted for each lactation. The goodness of fit used for comparison of those models was the coefficient of determination, mean square error, mean square prediction error and the Bayesian information criterion. The values for the goodness of fit obtained in each model were compared by using 95% probability confidence interval. Wilmink (1987) model showed a better adjustment for cows of the first lactation numbers, whereas the Wood (1967) model showed a better adjustment for cows of the third or greater lactations numbers for the low milk production groups. Wood model showed a better adjustment for all the lactation numbers for the medium milk production group. Dijkstra (1997) model showed a better adjustment for all lactation numbers for the high milk production group. Despite of being more recent, the model by Pollott (2000), mechanist based and with a higher number of parameters, showed a good convergence for the used data.

Evaluation of five lactation curve models fitted for fat:protein ratio of milk and daily energy balance

Selection for milk yield increases the metabolic load of dairy cows. The fat:protein ratio of milk (FPR) could serve as a measure of the energy balance status and might be used as a selection criterion to improve metabolic stability. The fit of different fixed and random regression models describing FPR and daily energy balance was tested to establish appropriate models for further genetic analyses. In addition, the relationship between both traits was evaluated for the best fitting model. Data were collected on a dairy research farm running a bull dam performance test. Energy balance was calculated using information on milk yield, feed intake per day, and live weight. Weekly FPR measurements were available. Three data sets were created containing records of 577 primiparous cows with observations from lactation d 11 to 180 as well as records of 613 primiparous cows and 96 multiparous cows with observations from lactation d 11 to 305. Five well-established parametric functions of days in milk (Ali and Schaeffer, Guo and Swalve, Wilmink, Legendre polynomials of third and fourth degree) were chosen for modeling the lactation curves. Evaluation of goodness of fit was based on the corrected Akaike information criterion, the Bayesian information criterion, correlation between the real observation and the estimated value, and on inspection of the residuals plotted against days in milk. The best model was chosen for estimation of correlations between both traits at different lactation stages. Random regression models were superior compared with the fixed regression models. In general, the Ali and Schaeffer function appeared most suitable for modeling both the fixed and the random regression part of the mixed model. The FPR is greatest in the initial lactation period when energy deficit is most pronounced. Energy balance stabilizes at the same point as the decrease in FPR stops. The inverted patterns indicate a causal relationship between the 2 traits. A common pattern was also observed for repeatabilities of both traits, with repeatabilities being largest at the beginning of lactation. Additionally, correlations between cow effects were closest at the beginning of lactation (rc=−0.43). The results support the hypothesis that FPR can serve as a suitable indicator for energy status, at least during the most metabolically stressful stage of lactation.

Analysis of water intake and dry matter intake using different lactation curve models

The objective was to evaluate 6 different lactation curve models for daily water and dry matter intake. Data originated from the Futterkamp dairy research farm of the Chamber of Agriculture of Schleswig-Holstein in Germany. A data set of about 23,000 observations from 193 Holstein cows was used. Average daily water and dry matter intake were 82.3 and 19.8kg, respectively. The basic linear mixed model included the fixed effects of parity and test-day within feeding group. Additionally, 6 different functions were tested for the fixed effect of lactation curve and the individual (random) effect of cow-lactation curve. Furthermore, the autocorrelation between repeated measures was modeled with the spatial (power) covariance structure. Model fit was evaluated by the likelihood ratio test, Akaike's and Bayesian information criteria, and the analysis of mean residual at different days in milk. The Ali and Schaeffer function was best suited for modeling the fixed lactation curve for both traits. A Legendre polynomial of order 4 delivered the best model fit for the random effect of cow-lactation. Applying the error covariance structure led to a significantly better model fit and indicated that repeated measures were autocorrelated. Generally, the best information criteria values were yielded by the most complex model using the Ali and Schaeffer function and Legendre polynomial of order 4 to model the average lactation and cow-specific lactation curves, respectively, with inclusion of the spatial (power) error covariance structure. This model is recommended for the analysis of water and dry matter intake including missing observations to obtain estimation of correct statistical inference and valid variance components.

Comparison of different lactation curve models in Holstein cows raised on a farm in the south-eastern Anatolia region

Abstract. The objective of this study was to compare the goodness of fit of seven mathematical models (including the gamma function, the exponential model, the mixed log model, the inverse quadratic polynomial model and their various modifications) on daily milk yield records. The criteria used to compare models were mean R2, root mean squared errors (RMSE) and difference between actual and predicted lactation milk yields. The effect of lactation number on curve parameters was significant for models with three parameters. Third lactation cows had the highest intercept post-calving, greatest incline between calving and peak milk yield and greatest decline between peak milk yield and end of lactation. Latest peak production occurred in first lactation for all models, while third lactation cows had the earliest day of peak production. The R2 values ranged between 0.590 and 0.650 for first lactation, between 0.703 and 0.773 for second lactation and between 0.686 and 0.824 for third lactation, depending on the model fitted. The root mean squared error values of different models varied between 1.748 kg and 2.556 kg for first parity cows, between 2.133 kg and 3.284 kg for second parity cows and between 2.342 kg and 7.898 kg for third parity cows. Lactation milk yield deviations of Ali and Schaeffer, Wilmink and Guo and Swalve Models were close to zero for all lactations. Ali and Schaeffer Model had the highest R2 for all lactations and also yielded smallest RMSE and actual and predicted lactation milk yield differences. Wilmink and Guo and Swalve Models gave better fit than other three parameter models.

An Algebraic Model from Mitscherlich Function to Describe Lactation Curves of Dairy Animals

Prasad, S. 2007. An algebraic model from Mitscherlich function to describe lactation curves of dairy animals. J. Appl. Anim. Res., 31: 121–123. The Mitscherlich function [Yt = a (1—b e−ct)] is generally used as growth model. A lactation curve model Yt= α (1—b e−ct) -dt + ϵtwas derived from this function by subtracting the linear function (Y—dt) without intercept. The curve attains the peak position after time tm= [ln (abc/d)]/c since parturition. This model was compared with four parameter models by fitting them on average weekly milk production of crossbred cattle and graded Murrah buffaloes. The proposed model was found to be the best for describing the lactation curves and also performed better than Mitscherlich x Exponential model.

Modeling the lactation curve for test-day milk yield in Murciano-Granadina goats

Test-day yield measured at intervals of 4 weeks was used to investigate the mathematical description of early stages of lactation and entire lactation curves in dairy goats under field conditions. Characteristics of the lactation curve (maximum yield and time of maximum yield) were calculated separately by parity groups (first, second and third or greater). Test-day data from 733 Murciano-Granadina goats were fitted with five published empirical standard lactation curve models including variations of the incomplete gamma function of Wood, a linear decline and a diphasic model. The average milk yield was predicted with a high degree of accuracy ( R 2>0.98) by all models for all parities. Peak yield took place at about 4.4 weeks of lactation for first parity and 3.2 weeks for second parity and predicted peak yield ranged from 1.6 to 1.74 kg per day for first parity and from 2.34 to 2.57 kg per day for second parity. For parity 3 or greater, a decline phase with no peak was observed and predicted peak yield at the second week of lactation ranged from 2.66 to 2.79 kg per day. The residual showed low fluctuation around zero for the diphasic model and lower mean absolute deviation, indicating a more satisfactory description of the lactation curve. Of the five models, the diphasic model resulted in the best goodness of fit and gave a good description of the Murciano-Granadina lactation curve for commercial dairy herds when test-day yield from Official Milk Control is monitored.

Classification and Longitudinal Examination of Callused Teat Ends in Dairy Cows

To examine the development of teat end callosity thickness and roughness in early lactation and to quantify cow factors of interest, a system to classify teat end condition was developed. A distinction was made between rough and smooth rings around the teat orifice. In addition, a classification of the degree of callosity was developed. Kappa coefficients for the repeatability of scoring by this classification system by different workers were 0.71 for teat end callosity thickness and 0.86 for teat end callosity roughness. The teat end callosity classification system was used for a longitudinal study with 40 cows during the first 14 wk of lactation. Models were built to predict teat end callosity thickness and roughness, machine-on time, and milk yield. For the response variables, teat end callosity thickness, machine-on time, and milk yield, the consecutive measurements appeared to follow a lactation curve model with a subject-specific general slope and intercept. Teat end callosity increased rapidly the first 8 wk. Cow factors such as days in milk, parity, machine-on time, and teat end shape were associated with the degree of teat end callosity, and the probability of the callosity ring to become rough. Teat end callosity thickness did not decrease within the 14-wk trial period for most teats. Pointed or round teat ends showed more callus than inverted teat ends. Longer machine-on time resulted in a higher probability of the callosity ring to become rough. Rear teats showed less callosity than front teats in this study.