Detection of Different Shapes of Lactation Curve for Milk Yield in Dairy Cattle by Empirical Mathematical Models

Abstract

The study of relationships between mathematical properties of functions used to model lactation curves is usually limited to the evaluation of the goodness of fit. Problems related to the existence of different lactation curve shapes are usually neglected or solved drastically by considering shapes markedly different from the standard as biologically atypical. A deeper investigation could yield useful indications for developing technical tools aimed at modifying the lactation curve in a desirable fashion. Relationships between mathematical properties and lactation curve shapes were analyzed by fitting several common functions (Wood incomplete gamma, Wilmink's exponential, Ali and Schaeffer's polynomial regression, and fifth-order Legendre polynomials) to 229,518 test-day records belonging to 27,837 lactations of Italian Simmental cows. Among the best fits (adjusted r(2) higher than 0.75), the 3-parameter models (Wood and Wilmink) were able to detect 2 main groups of curve shape: standard and atypical. Five-parameter models (Ali and Schaeffer function and the Legendre polynomials) were able to recognize a larger number of curve shapes. The higher flexibility of 5-parameter models was accompanied by increased sensitivity to local random variation as evidenced by the bias in estimated test-day yields at the beginning and end of lactation (border effect). Meaning of parameters, range of their values and of their (co) variances are clearly different among groups of curves. Our results suggest that analysis based on comparisons between parameter values and (co)variances should be done carefully. Comparisons among parameter values and (co)variances could yield more robust, reliable, and easy to interpret results if performed within groups based on curve shape.

Analysis of lactation shapes in extended lactations

“…Results obtained in this study for individual lactations of standard length agree with previous reports for dairy cattle (Olori et al, 1999;Macciotta et al, 2005;Silvestre et al, 2006), with a better fit for polynomials (AS, LEG and SPL) compared with models with three parameters. No studies are currently available in the literature on fitting individual extended lactation curves.…”

Modeling lactation curves of Polish Holstein-Friesian cows. Part II: Prediction of 305-d lactation milk, fat and protein yields

“…When the first TD yield was at the end of the first month of lactation, the yields in the first month were extrapolated, and that might be why LEG3 and LEG4 very often modeled lactation curves without a peak, that is, as continuously decreasing curves. Macciotta et al (2005) confirmed that such atypical lactation curves were mainly a result of lack of records in the first days after calving. They also noticed that the occurrence of the curves without a peak arose from the peculiar combination of TD yields and their distribution along the entire lactation.…”

“…It should be added that Olori et al (1999) and Quinn et al (2005) considered weekly yields; what is more, Olori et al (1999) used only data on heifers from one herd, kept under uniform feeding and management conditions. Macciotta et al (2005) suggested that three-parameter models such as the WIL function ensured a superior fit to data with a single curvature. According to them, five-parameter functions like ALI and LEG4 are more flexible and permit description of a larger number of lactation curve shapes; that is why they were often used in random regression analyses of TD data to model individual deviations from an average fixed curve.…”

“…This confirmed the need for further research on the effect of data structure on the accuracy of modeling lactation curve and indirectly of 305-d yields using fitted models. Many authors have pointed out that the availability of TD records before peak yield is crucial for correct estimation of lactation curve shape (Macciotta et al, 2005;Strabel et al, 2005;Silvestre et al, 2006).…”

“…Among the most popular functions describing the course of lactation are those of Wood (1967), Ali and Schaeffer (1987), and Wilmink (1987). The functions differ mainly in the type of regression, the number of parameters, and the degree of relationship with the main characteristics of a typical lactation curve, such as peak yield, time at peak and persistency (Macciotta et al, 2005). With time, these functions have been replaced by orthogonal polynomials that model the lactation curve properly and tend to converge faster because of a lack of correlation among the coefficients.…”