Estimation of the maximum safe level of feed ingredients by spline or broken-line nonlinear regression models

The use of non-linear regression models in the analysis of biological data has led to advances in poultry nutrition. Spline or broken-line nonlinear regression models are commonly used to estimate nutritional requirements. One particular application of broken-line models is estimating the maximum safe level (MSL) of feed ingredients beyond which the ingredients become toxic, resulting in reduced performance. The objectives of this study were to evaluate the effectiveness of broken-line models (broken-line linear or BLL; and broken-line quadratic or BLQ) in estimating the MSL; to identify the most efficient design of feeding trials by finding the optimal number of ingredient levels and replications; and to re-estimate the MSL of various test ingredients reported in the nutrition literature for comparison purposes. The Maximum Ingredient level Optimization Workbook (MIOW) was developed to simulate a series of experiments and estimate the MSL and the corresponding descriptive statistics (SD, SE, CI, and R2). The results showed that the broken-line models provided good estimates of the MSL (small SE and high R2) with the BLL model producing higher MSL values as compared to the BLQ model. Increasing the number of experimental replications or ingredient levels (independently of each other) reduced the SE of the MSL with diminishing returns. The SE of the MSL was reduced with increasing the size (total pens) of the simulated experiments by increasing either the number of replications or levels or both. The evaluation of MSLs reported in the existing literature revealed that the multiple range procedure used to determine the MSL in several reports can both overestimate and underestimate the MSL compared to the results obtained by the broken-line models. The results suggest that the broken-line linear models can be used in lieu of the multiple range test to estimate the MSL of feed ingredients along with the corresponding descriptive statistics, such as the SE of the MSL.