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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., (© 2016 Poultry Science Association Inc.)

Nutritional requirements and responses of all organisms are estimated using various models representing the response to different dietary levels of the nutrient in question. To help nutritionists design experiments for estimating responses and requirements, we developed a simulation workbook using Microsoft Excel. The objective of the present study was to demonstrate the influence of different numbers of nutrient levels, ranges of nutrient levels and replications per nutrient level on the estimates of requirements based on common nutritional response models. The user provides estimates of the shape of the response curve, requirements and other parameters and observation to observation variation. The Excel workbook then produces 1-1000 randomly simulated responses based on the given response curve and estimates the standard errors of the requirement (and other parameters) from different models as an indication of the expected power of the experiment. Interpretations are based on the assumption that the smaller the standard error of the requirement, the more powerful the experiment. The user can see the potential effects of using one or more subjects, different nutrient levels, etc., on the expected outcome of future experiments. From a theoretical perspective, each organism should have some enzyme-catalysed reaction whose rate is limited by the availability of some limiting nutrient. The response to the limiting nutrient should therefore be similar to enzyme kinetics. In conclusion, the workbook eliminates some of the guesswork involved in designing experiments and determining the minimum number of subjects needed to achieve desired outcomes.

Nutritional requirements are typically estimated based on feeding trials with animals or birds offered several amounts of the critical nutrient(s). A nutrient response function is then fitted to data from the feeding trials. Modern computer techniques allow for a variety of functional forms to be used as nutrient response functions. However, the performance of these models is almost undistinguishable from a purely statistical perspective. This paper approaches the issue of determining nutrient requirements from an economic prospective. Crude protein amounts that would maximize profits were calculated for combinations of corn, soybean meal, and live broilers prices using several nutrient response models fitted to technical data from a trial with several balanced CP amounts fed to broiler chickens. Under certain combinations of input prices, differences between the models were between 1.5 and 3.0% CP. No model consistently predicted the greatest or least CP amounts or net profits, emphasizing that the (tangential) slopes of the models change at different rates over the range of nutrient (CP) amounts studied. Models providing adequate statistical fits to research data do not necessarily provide functions that are clearly most appropriate for maximizing producer profits.

1. Research papers use a variety of methods for evaluating experiments designed to determine nutritional requirements of poultry. Growth trials result in a set of ordered pairs of data. Often, point-by-point comparisons are made between treatments using analysis of variance. This approach ignores that response variables (body weight, feed efficiency, bone ash, etc.) are continuous rather than discrete. Point-by-point analyses harvest much less than the total amount of information from the data. Regression models are more effective at gleaning information from data, but the concept of "requirements" is poorly defined by many regression models. 2. Response data from a study of the lysine requirements of young broilers was used to compare methods of determining requirements. In this study, multiple range tests were compared with quadratic polynomials (QP), broken line models with linear (BLL) or quadratic (BLQ) ascending portions, the saturation kinetics model (SK) a logistic model (LM) and a compartmental (CM) model. 3. The sum of total residuals squared was used to compare the models. The SK and LM were the best fit models, followed by the CM, BLL, BLQ, and QP models. A plot of the residuals versus nutrient intake showed clearly that the BLQ and SK models fitted the data best in the important region where the ascending portion meets the plateau. 4. The BLQ model clearly defines the technical concept of nutritional requirements as typically defined by nutritionists. However, the SK, LM and CM models better depict the relationship typically defined by economists as the "law of diminishing marginal productivity". The SK model was used to demonstrate how the law of diminishing marginal productivity can be applied to poultry nutrition, and how the "most economical feeding level" may replace the concept of "requirements".

A variety of models have been proposed to fit nutritional input-output response data. The models are typically nonlinear; therefore, fitting the models usually requires sophisticated statistical software and training to use it. An alternative tool for fitting nutritional response models was developed by using widely available and easier-to-use Microsoft Excel software. The tool, implemented as an Excel workbook (NRM.xls), allows simultaneous fitting and side-by-side comparisons of several popular models. This study compared the results produced by the tool we developed and PROC NLIN of SAS. The models compared were the broken line (ascending linear and quadratic segments), saturation kinetics, 4-parameter logistics, sigmoidal, and exponential models. The NRM.xls workbook provided results nearly identical to those of PROC NLIN. Furthermore, the workbook successfully fit several models that failed to converge in PROC NLIN. Two data sets were used as examples to compare fits by the different models. The results suggest that no particular nonlinear model is necessarily best for all nutritional response data.