Implications of regression bias for multi-element isotope analysis for environmental remediation

Measuring changes in the stable isotope ratios of multiple elements (e.g. Δδ¹³C, Δδ³⁷Cl, and Δδ²H) during the (bio)transformation of environmental contaminants has provided new insights into reaction mechanisms and tools to optimize remediation efforts. Dual-isotope analysis, wherein changes in one isotopic system are plotted against another (to derive an interpretational parameter expressed as Λ), is a key tool in multi-element isotopic assessment. To date, most dual-isotope analyses use ordinary linear regression (OLR) for the calculation, which can be subject to regression attenuation and thus an inherent artifact that depresses slope values, expressed as Λ. Here, a series of Monte Carlo simulations were constructed to represent common data conditions and variations within dual-isotope data to test the degree of bias when deriving Λ using OLR compared to an alternative regression technique, the York method. The degree of bias was quantified compared to the modeled or "true" Λ value. For all simulations, the York method provided the least bias in slope estimates (<1%) over all data conditions tested. In contrast, OLR produced unbiased estimates only under a limited set of conditions, which was validated through a mathematical model proof. Both the mathematical model and simulations show that bias of at least 5% in OLR occurs when the extent of enrichment in the x-variable (X_M) is equal to or less than ≈15 times the 1σ precision in the isotope measurement (σ_X), for both Cl/C and C/H plots. The results give practitioners tools to evaluate whether bias is present in data and to estimate the extent to which this negatively impacts the interpretations and predictions of remediation potential for new and previously published datasets. This study demonstrates that integration of such robust statistical tools is essential for dual-isotope interpretations widely used in contaminant hydrogeology but relevant to other disciplines including environmental chemistry and ecology.