To fit or not to fit: evaluating stable isotope mixing models using simulated mixing polygons

Abstract

Summary1. Stable isotope analysis is often used to identify the relative contributions of various food resources to a consumer's diet. Some Bayesian isotopic mixing models now incorporate uncertainty in the isotopic signatures of consumers, sources and trophic enrichment factors (e.g. SIAR, MixSIR). This had made model outputs more comprehensive, but at the expense of simple model evaluation, and there is no quantitative method for determining whether a proposed mixing model is likely to explain the isotopic signatures of all consumers, before the model is run. 2. Earlier linear mixing models (e.g. IsoSource) are easier to evaluate, such that if a consumer's isotopic signature is outside the mixing polygon bounding the proposed dietary sources, then mass balance cannot be established and there is no logical solution. This can be used to identify consumers for exclusion or to reject a model outright. This point-in-polygon assumption is not inherent in the Bayesian mixing models, because the source data are distributions not average values, and these models will quantify source contributions even when the solution is very unlikely. 3. We use a Monte Carlo simulation of mixing polygons to apply the point-in-polygon assumption to these models. Convex hulls ('mixing polygons') are iterated using the distributions of the proposed dietary sources and trophic enrichment factors, and the proportion of polygons that have a solution (i.e. that satisfy point-in-polygon) is calculated. This proportion can be interpreted as the frequentist probability that the proposed mixing model can calculate source contributions to explain a consumer's isotopic signature. The mixing polygon simulation is visualised with a mixing region, which is calculated by testing a grid of values for point-in-polygon. 4. The simulation method enables users to quantitatively explore assumptions of stable isotope analysis in mixing models incorporating uncertainty, for both two-and three-isotope systems. It provides a quantitative basis for model rejection, for consumer exclusion (those outside the 95% mixing region) and for the correction of trophic enrichment factors. The simulation is demonstrated using a two-isotope study ( 15 N, 13 C) of an Australian freshwater food web.

Resource polygon geometry predicts Bayesian stable isotope mixing model bias

“…Bayesian mixing models can also be biased towards the prior generalist assumption when fractionation-corrected consumers fall outside of the resource polygon (Parnell et al 2010, Smith et al 2013. Consumers falling outside of the resource poly gon may indicate that either the potential food sources or the consumer trophic enrichment has been misrepresented.…”

“…Consumers falling outside of the resource poly gon may indicate that either the potential food sources or the consumer trophic enrichment has been misrepresented. Smith et al (2013) developed a creative Monte Carlo based approach to quantitatively test whether the point-in-polygon assumption is met and thus whether a Bayesian mixing model analysis is even warranted. In cases where the consumer does not fall within the resource polygon, Bayesian mixing models will provide misleading results.…”

Periphyton support for littoral secondary production in a highly humic boreal lake

“…We used trophic fractionation factors of 2.52 ± 2.5 for δ 15 N and 0.47 ± 1.23 for δ 13 C according to Vander Zanden and Rasmussen (2001). Before the SIAR model was run, we evaluated model uncertainty for 2 y with Monte Carlo simulations for mixing polygons defined by the putative food sources (Smith et al 2013) to resolve whether consumer values lay within the 95% mixing region. We ran Bayesian SIAR mixing models and Monte Carlo simulations of mixing polygons in R (version 3.2.3; R Project for Statistical Computing, Vienna, Austria).…”

Full Issue

“…Default SIAR MCMC parameters (iterations ¼ 2 3 10 5 , burning ¼ 5 3 10 4 , thinning ¼ 15) were used for modelling. No trophic enrichment factor has been Smith et al (2013) using the packages sp and splancs in program R to make an a priori assessment of the goodness-of-fit of the data to the mixing model. No sampled birds were excluded as all were within the 0.05 contour (Appendix Figure 4) indicating that a viable mathematical solution for the mixing model existed (Smith et al 2013).…”

“…Mixing region displaying the probability of the data fitting the MCMC mixing model followingSmith et al (2013). Contours depict the probability of a viable solution existing with the outermost contour representing a probability of 0.05 and the remaining contours increasing from 0.1 to 0.7 in 0.1 increments.…”