Peer Review #2 of "A critical issue in model-based inference for studying trait-based community assembly and a solution (v0.2)"

Statistical testing of trait-environment association from data is a challenge as there is no common unit of observation: the trait is observed on species, the environment on sites and the mediating abundance on species-site combinations.A number of correlation-based methods, such as the community weighted trait means method (CWM), the fourth-corner correlation method and the multivariate method RLQ, have been proposed to estimate such trait-environment associations.In these methods, valid statistical testing proceeds by performing two separate resampling tests, one site-based and the other species-based and by assessing significance by the largest of the two p-values (the p max test).Recently, regression-based methods using generalized linear models (GLM) have been proposed as a promising alternative with statistical inference via site-based resampling.We investigated the performance of this new approach along with approaches that mimicked the p max test using GLM instead of fourth-corner.By simulation using models with additional random variation in the species response to the environment, for example, due to unobserved traits or environmental variables, the site-based resampling tests using GLM are shown to have severely inflated type I error, of up to 90%, when the nominal level is set as 5%.In addition, predictive modelling of such data using site-based cross-validation very often identified trait-environment interactions that had no predictive value.The problem that we identify is not an "omitted variable bias" problem as it occurs even when the additional random variation is independent of the observed trait and environment data.Instead, it is a problem of ignoring a random effect.In the same simulations, the GLM-based p max test controlled the type I error in all models proposed so far in this context, but still gave slightly inflated error in more complex models that included both missing (but important) traits and missing (but important) environmental variables.For screening the importance of single trait-environment combinations, the fourth-corner test is shown to give almost the same results as the GLM-based tests in far less computing time.

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