The importance of being random! Taking full account of random effects in nonlinear sigmoid hierarchical Bayesian models reveals the relationship between deadwood and the species richness of saproxylic beetles

Hierarchical models are used to study the relationship between a response variable and a predictor in structured data. Random effects are meant to capture the structured part of variability among groups of observations. In ecology, random effects are usually incorporated into the intercept. Their application to the other parameters of the curve, especially in nonlinear curves, has been understudied. However, applying random effects to different parameters of the function is of interest, as it allows us to account for variations in the shape of the relationship over groups of observations. Our study was based on Bayesian models linking the local quantity of deadwood to the local species richness of saproxylic beetles in French forests. Our hypothesis was that it was important to account for inter-forest variations of the relationship to better fit the data. Since a sigmoidal curve seemed adapted to studying this relationship from an ecological point of view, we paid special attention to commonly used sigmoidal functions, but also included two new ones for biogeography originating from ecophysiology (one sigmoid with estimated asymptotes and one with estimated asymptotes allowing asymmetry). We applied various settings of random effects to these different mean functions. We compared, evaluated and interpreted the models and results according to several criteria (WAIC, comparison of significance of the difference in terms of LOOic, goodness-of-fit p-values and magnitude of the effect). We first found that models without random effects were systematically the worst and that the best model was not necessarily the one with random effect incorporated into the intercept, as is usually done in ecology. Secondly, we found that, in most cases, for a given mean function, the best model had several random effects, and the model with the most random effects performed nearly as well as the best models. Furthermore, the inclusion of random effects revealed statistically significant relationships between deadwood volume and species richness. Thirdly, we revealed a complementarity between the different assessment criteria, each one giving important information for the selection and interpretation of the models. In conclusion, future forest biodiversity management studies should incorporate random effects into the modeling framework so that more robust conclusions can be made about the relationships, based on complementary post-fitting analysis criteria.