Statistical basis and outputs of stable isotope mixing models: Comment on Fry (2013)

Fry (2013; Mar Ecol Prog Ser 472:1−13) reviewed approaches to solving under - determined stable isotope mixing systems, and presented a novel approach based on graphi - cal summaries. He inaccurately characterized the statistics and interpretation of outputs from IsoSource and more recent Bayesian mixing model tools (e.g. SIAR, MixSIR), however, and as an alternative promoted an approach — not based on likelihood methods — that uses graphing and 2 new metrics for tracking source contributions to a mixture. Fry's approach does not provide statisti- cal probability densities associated with source contribution parameter estimates, has little appli- cability to complex mixing systems such as hierarchical models, and relies on the subjective inter- pretation of graphing products. We clarify the analytic theory underlying common mixing model approaches and provide an analysis of the 4-source, 2-tracer underdetermined mixing system ex- ample in Fry (2013), using both a Bayesian mixing model and Fry's graphical analysis and summary metrics. We demonstrate that properly interpreted Bayesian approaches yield distribu- tions of parameter estimates that can reflect multi-modality, covariance and parameter uncertainty.