Comparison of goodness of fit in lactation curve modeling in Holstein dairy cows

Introduction: Lactation curves are essential tools for understanding milk production patterns in dairy cattle, providing critical insights for herd management and genetic improvement. Modeling these curves accurately is vital to predict yield, optimize feeding strategies, and enhance overall productivity. Among the various approaches, non-linear models such as Wood, Wilmink, Dhanoa, Cappio-Borlino, and Cobby & Le Du have been widely used to describe the lactation kinetics in Holstein cattle. However, comparative evaluations of these models' performance under similar conditions remain limited. This study aimed to assess and compare the goodness-of-fit of these five established models to identify the most suitable for characterizing lactation curves in Holstein cows, ensuring more precise and reliable production forecasting. Additionally, the possibility of utilizing the Dynamic Ordinary Least Squares (DOLS) method instead of the traditional nonlinear least squares (LS) approach for the best-fitted model was investigated to explore potential improvements in estimation accuracy.
Materials and methods: This study utilized milk production records from a dairy farm in Tehran Province, Iran, under the supervision of the National Animal Breeding Center. The dataset included 1050 test-day records from the first three lactation periods (1^st to 3^rd), spanning 305 days in milk (DIM). Only cows with a minimum of 10 valid test-day records and lactations exceeding 200 days were considered. Lactation curves were fitted using five non-linear models, Wood, Wilmink, Dhanoa, Cappio-Borlino, and Cobby & Le Du, implemented in R software (packages nlme and minpack.lm). Model performance was assessed using goodness-of-fit criteria, including the coefficient of determination (R²), root mean square error (RMSE), and Akaike Information Criterion (AIC). Upon identifying the best-performing model, the DOLS method was applied to evaluate its potential for improving parameter estimation accuracy. This analysis was conducted using records from the first three lactations of eight Holstein dairy cows.
Results and discussion: Comparative analysis of the five non-linear models revealed the Wood model as the most statistically suitable for characterizing Holstein lactation curves. While effective, the Wood model demonstrated limitations under traditional LS estimation, particularly in accurately predicting peak yield magnitude and timing. These shortcomings stemmed from the LS method's sensitivity to initial parameter values and its inability to properly synchronize estimated events with observed farm data. Implementation of the DOLS method substantially enhanced model performance. The DOLS-adapted Wood model showed improved goodness-of-fit metrics and, crucially, provided more biologically accurate estimates of key lactation events, including initial yield, peak timing, and peak magnitude. These estimates closely matched actual farm records, demonstrating superior predictive capability compared to the LS approach. The study confirmed established biological patterns: third-lactation cows exhibited higher initial yields (α parameter) and greater persistence (lower c parameter values) compared to first-lactation animals. These findings align with known metabolic adaptations in mature cows and previous reports of lactation curve dynamics. Visual and statistical comparisons revealed distinct advantages of the DOLS approach. While both methods achieved reasonable goodness-of-fit, the DOLS-generated curve showed better alignment with critical lactation points and lower standard errors, particularly for peak-related parameters. This temporal calibration is essential for practical applications, as even models with high R² values may yield misleading recommendations if event timing is inaccurate. The results substantiate theoretical advantages of DOLS in production function estimation, demonstrating its capacity to generate properly calibrated, differentiable concave functions that faithfully represent biological reality while maintaining statistical rigor.
Conclusions: This study demonstrates that the DOLS method offers substantial improvements over traditional nonlinear LS for modeling Holstein lactation curves using Wood’s equation. The DOLS approach effectively addresses critical limitations of conventional nonlinear estimation, particularly its dependence on manual parameter initialization and its tendency for temporal miscalibration. By providing parameter estimates that align more closely with biological reality, DOLS enables superior prediction of key lactation traits, including peak yield and persistence, crucial factors for genetic selection and herd management decisions. The methodological advantages of DOLS extend beyond statistical performance to practical implementation; its ability to estimate marginal productivity within the actual lactation timeline while preserving biological plausibility makes it especially valuable for dairy management applications. These findings suggest that adopting DOLS in breeding programs and farm management systems could enhance both the accuracy and economic outcomes of production decisions. While this research presents a novel application of DOLS to lactation modeling, certain limitations merit consideration. The relatively small sample size, the absence of formal cointegration testing, and unexamined econometric assumptions, such as stationarity and residual independence, highlight the need for validation with larger and more diverse datasets. Future studies should examine DOLS performance across different breeds, management systems, and lactation stages while incorporating appropriate econometric validation procedures. Nevertheless, this work establishes DOLS as a promising alternative to conventional nonlinear approaches, offering both theoretical rigor and practical utility for lactation curve analysis in dairy science.