The aim of the work reported here was to investigate the appropriateness of a sinusoidal function by applying it to model the cumulative lactation curves for milk yield and composition in primiparous Holstein cows, and to compare it with three conventional growth models (linear, Richards and Morgan). Data used in this study were 911 144 test-day records for milk, fat and protein yields, which were recorded on 834 dairy herds from 2000 to 2011 by the Animal Breeding Centre and Promotion of Animal Products of Iran. Each function was fitted to the test-day production records using appropriate procedures in SAS (PROC REG for the linear model and PROC NLIN for the Richards, Morgan and sinusoidal equations) and the parameters were estimated. The models were tested for goodness of fit using adjusted coefficient of determination $\lpar {R_{{\rm adj}}^2 } \rpar $, root mean square error (RMSE), Akaike's information criterion (AIC) and the Bayesian information criterion (BIC). $R_{{\rm adj}}^2 $ values were generally high (>0.999), implying suitable fits to the data, and showed little differences among the models for cumulative yields. The sinusoidal equation provided the lowest values of RMSE, AIC and BIC, and therefore the best fit to the lactation curve for cumulative milk, fat and protein yields. The linear model gave the poorest fit to the cumulative lactation curve for all production traits. The current results show that classical growth functions can be fitted accurately to cumulative lactation curves for production traits, but the new sinusoidal equation introduced herein, by providing best goodness of fit, can be considered a useful alternative to conventional models in dairy research.

中文翻译：

---

引入正弦方程来描述荷斯坦奶牛累积产奶量和成分的泌乳曲线

本文报告的工作目的是通过将正弦函数应用于初产荷斯坦奶牛产奶量和成分的累积泌乳曲线建模来研究正弦函数的适用性，并将其与三种常规生长模型（线性、理查兹和摩根）进行比较）。本研究中使用的数据是 911 144 个测试日的牛奶、脂肪和蛋白质产量记录，这些记录由伊朗动物育种中心和动物产品推广中心从 2000 年到 2011 年记录在 834 个奶牛群中。使用 SAS 中的适当程序（PROC REG 用于线性模型，PROC NLIN 用于理查兹、摩根和正弦方程）将每个函数拟合到测试日生产记录，并估计参数。使用调整后的决定系数测试模型的拟合优度$\lpar {R_{{\rm adj}}^2 } \rpar $、均方根误差 (RMSE)、Akaike 信息准则 (AIC) 和贝叶斯信息准则 (BIC)。$R_{{\rm adj}}^2 $值普遍较高（>0.999），表明数据拟合合适，并且模型之间的累积产量差异很小。正弦方程提供了 RMSE、AIC 和 BIC 的最低值，因此最适合累积牛奶、脂肪和蛋白质产量的泌乳曲线。线性模型对所有生产性状的累积泌乳曲线的拟合最差。目前的结果表明，经典的生长函数可以准确地拟合到生产性状的累积泌乳曲线，但本文引入的新正弦方程通过提供最佳拟合优度，可以被认为是乳品研究中传统模型的有用替代方案。