## ter Braak C., Dray S., Peres-Neto P. (2017).
*A critical issue in model-based inference for studying
trait-based community assembly and a solution*.
PeerJ, 5:e2885.
ISSN 2167-8359.
10.7717/peerj.2885.

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 pmax 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 pmax test using GLM
instead of fourth-corner. By simulation using models with
additional random variation in the species response to the
environment, 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 pmax 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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