Minimizing inferential bias in the theory and design of nutritional experiments through the application of the equilateral mixture triangle

Abstract

Animal nutritionists seek to understand how animals regulate the intake and balance of multiple nutrients, yet the design and analysis of such experiments are often limited by how nutrient spaces are represented. The geometric framework for nutrition (GFN) provides a powerful means to visualize nutrient interactions, but when more than two nutrients are considered, most empirical studies employ right-angled mixture triangles (RMTs). While convenient, this coordinate system can introduce visual artifacts that can bias inference. Here, we reintroduce the equilateral mixture triangle (EMT) as a complementary and biologically meaningful framework that preserves proportional relationships among nutrients and connects directly with the design principles of mixture experiments in engineering disciplines, where it is more commonly known as a simplex. The simplex provides a consistent bridge between theory and experimentation, allowing the same geometric representation to be used for choice and no-choice assays. For choice experiments, we extend GFN theory to show that animal feeding trajectories are bounded by the convex hull of available foods, show how it can scale to higher-dimensional nutrient systems, and develop statistical tests to distinguish between random choice and the defense of an intake target. For no-choice experiments, we demonstrate how simplex mixture designs can be integrated with response surface methodologies to model multivariate performance landscapes, identify optimal nutrient ratios, and avoid biologically unrealistic regions of nutrient space. For both types of experiments, we provide simulated case studies on how to design studies around the simplex. Together, these developments refine both the theoretical and experimental foundations of nutritional ecology, providing clear guidance for designing, visualizing, and interpreting multidimensional studies of nutrient regulation with diverse applications across ecological and agricultural disciplines.