Diagnosing underdetermination in stable isotope mixing models

Stable isotope mixing models (SIMMs) provide a powerful methodology for quantifying relative contributions of several sources to a mixture. They are widely used in the fields of ecology, geology, and archaeology. Although SIMMs have been rapidly evolved in the Bayesian framework, the underdetermination of mixing space remains problematic, i.e., the estimated relative contributions are incompletely identifiable. Here we propose a statistical method to quantitatively diagnose underdetermination in Bayesian SIMMs, and demonstrate the applications of our method (named beta -dependent SIMM) using two motivated examples. Using a simulation example, we showed that the proposed method can rigorously quantify the expected underdetermination (i.e., intervals of beta -dependent posterior) of relative contributions. Moreover, the application to the published field data highlighted two problematic aspects of the underdetermination: 1) ordinary SIMMs was difficult to quantify underdetermination of each source, and 2) the marginal posterior median was not necessarily consistent with the joint posterior peak in the case of underdetermination. Our study theoretically and numerically confirmed that beta -dependent SIMMs provide a useful diagnostic tool for the underdetermined mixing problem. In addition to ordinary SIMMs, we recommend reporting the results of beta -dependent SIMMs to obtain a biologically feasible and sound interpretation from stable isotope data.

Automated summary

This study introduces a statistical method to address the issue of uncertainty in Bayesian SIMMs and demonstrates its application using two examples. The results show that beta-dependent SIMMs can rigorously quantify uncertainty and serve as a useful diagnostic tool for the mixing problem.