Which results of the standard test for community-weighted mean approach are too optimistic?

Aims
The community-weighted mean (CWM) approach is used to analyse the relationship between species attributes (traits, Ellenberg-type indicator values) and sample attributes (environmental variables, richness) via the community matrix. It has recently been shown to suffer from inflated Type I error rate if tested by a standard test and the results of many published studies are probably affected. I review the current knowledge about this problem, and clarify which studies are likely affected and by how much.

Methods
I suggest classifying hypotheses commonly tested by CWM approach into three categories, which differ in the formulation of the null hypothesis. I use simulated and real data to show how the Type I error rate of the standard test is affected by data characteristics.

Results
The CWM approach with the standard test returns a correct Type I error rate for hypotheses assuming a link between species attributes and composition (Category A). However, for hypotheses assuming a link between composition and sample attributes (Category B) or not assuming any link (Category C), the standard test is inflated, and alternative tests are needed to control for this. The inflation of standard tests for Category C is negatively related to the compositional β-diversity, and positively to the strength of the composition–sample attributes relationship and data set sample size. These results apply to CWM analyses with extrinsic sample attributes (not derived from the compositional matrix). CWM analysis with intrinsic sample attributes (derived from the composition, such as species richness) is a case of spurious correlation and can be tested using a column-based (modified) permutation test.

Conclusions
The concept of three hypothesis categories offers a simple tool to evaluate which hypothesis has been tested and whether the results have correct or inflated Type I error rate. In the case of inflated results, the level of inflation can be estimated from the data characteristics.