Estimating continuous body size-based shifts in δ¹⁵N–δ¹³C space using multivariate hierarchical models

Abstract

Stable isotopes (δ¹⁵N and δ¹³C) offer one representation of an individual’s trophic niche and are important tools for elucidating ecological patterns and testing a diversity of hypotheses. Because δ¹⁵N and δ¹³C values are often obtained from the same sample, they compose a bivariate response that researchers commonly analyze using multivariate statistical methods. However, stable isotope data sets often exhibit hierarchical structure whereby samples may be clustered or grouped at multiple levels either as an artifact of sampling design or due to structure inherent in the sampled population (e.g., samples from individuals grouped according to life history stages, social groups, ages, or sizes classes). Ignoring such structure can result in overly optimistic confidence intervals and heighten the risk of observing significant differences where none exist. To address these issues, we suggest researchers utilize multivariate hierarchical models, which are a simple extension of univariate hierarchical methods. The models account for potential dependencies between δ¹⁵N and δ¹³C values, permit valid predictions of shifts in δ¹⁵N–δ¹³C space related to predictor variables, provide more accurate estimates of parameter uncertainty, and improved inferences on coefficients that correspond to groups with small to moderate quantities of data. We demonstrate advantages of multivariate hierarchical models by examining size-dependent shifts in δ¹⁵N–δ¹³C space in outmigrating post-smolt Chinook salmon sampled from an estuarine habitat. Given the prevalence of complex structure in ecological stable isotope data sets, multivariate hierarchical models should hold considerable value to food web and stable isotope ecologists.