A Mathematical Explanation for Why Ratio-Based Isotopic Analyses are Commonly Misleading: Theory

Abstract Stable mass isotopic ratios (such as ¹³C:¹²C, ¹⁵N:¹⁴N, ¹⁸O:¹⁶O ⁸⁷Sr:⁸⁶Sr and ³⁴S:³²S) are used to interpret archaeological, climate change, ecological, geological, and physiological studies. Most isotopic reports evaluate changes in observed isotopic ratios or ratio-based expressions over time or among treatments. To address concerns that ratios or ratio-based expressions may not produce conclusions that support known physiological or ecological principles, source (isotopic ratio of the material being added or lost) analyses are proposed as an alternative to statistically analysing observed isotopic ratios. Mathematically defined relationships between observed ratios, backgrounds (isotopic ratio of a system before any loses or gains), sources and total element concentrations as well as denominator vs. numerator relationships are presented. These mathematical relationships suggest that source-based approaches often produce conclusions that differ from ratio-based evaluations. Total element concentrations, necessary for source analyses, are presented in less than half of isotopic publications. Without evaluating total element, relative background and source ratios cannot be determined. Even, when total element data is available, researchers rarely conduct source analyses. This is unfortunate because determining sources solves most interpretive issues. Our goal is to advocate better methods when analyzing isotopic ratios in the thousands of mass isotope publications annually produced.