Estimation of the non-linear parameter in Generalised Diversity-Interactions models is unaffected by change in structure of the interaction terms

Research over the past few decades has assumed the richness (number of species) to be the primary driver of the biodiversity and ecosystem function (BEF) relationship. However, biodiversity is multi-dimensional, and richness alone does not capture all its attributes. Diversity-Interactions modelling is a regression-based framework that models the biodiversity and ecosystem function relationship by incorporating species-specific effects along with the interactions between species. The species interactions in a Diversity-Interactions model can take several different forms ranging in complexity from a single interaction term (assuming all pairs of species interact in the same way) to many interaction terms (e.g. assuming a separate interaction for all pairs of species). The specification of the interactions may also include a non-linear parameter ({\theta}) as an exponent to the product of the species proportions to allow for non-linear relationship between the response and species interactions, giving rise to Generalized Diversity-Interactions modelling. The structure of the interaction terms describes the underlying biological processes and thus the selection of a correct structure is important. The inclusion of {\theta} introduces complexity to the model selection process: we could (a) select the interaction structure first by assuming {\theta}=1 and then estimate {\theta}, or (b) estimate {\theta} first and then select the appropriate interaction structure by fixing {\theta}, or (c) test for {\theta} and its inclusion for each interaction structure. It is also unknown whether {\theta} is robust to changes in the underlying interaction structure. Using a simulation study, we test the robustness of {\theta} and compare multiple model selection approaches to identify an optimal and computationally efficient model selection procedure for Generalized Diversity-Interactions models.