Abstract
Abstract Statistical analysis of trait–environment association is challenging owing to the lack of a common observation unit: Community‐weighted mean regression (CWMr) uses site points, multilevel models focus on species points, and the fourth‐corner correlation uses all species‐site combinations. This situation invites the development of new methods capable of using all observation levels. To this end, new multilevel and weighted averaging‐based regression methods are proposed. Compared to existing methods, the new multilevel method, called MLM3, has additional site‐related random effects; they represent the unknowns in the environment that interact with the trait. The new weighted averaging method combines site‐level CWMr with a species‐level regression of Species Niche Centroids on to the trait. Because species can vary enormously in frequency and abundance giving diversity variation among sites, the regressions are weighted by Hill's effective number (N2) of occurrences of each species and the N2‐diversity of a site, and are subsequently combined in a sequential test procedure known as the max test. Using the test statistics of these new methods, the permutation‐based max test provides strong statistical evidence for trait–environment association in a plant community dataset, where existing methods show weak evidence. In simulations, the existing multilevel model showed bias and type I error inflation, whereas MLM3 did not. Out of the weighted averaging‐based regression methods, the N2‐weighted version best controlled the type I error rate. MLM3 was superior to the weighted averaging‐based methods with up to 30% more power. Both methods can be extended (a) to account for phylogeny and spatial autocorrelation and (b) to select functional traits and environmental variables from a greater set of variables.
Highlights
Functional traits are often useful in going beyond species-based understanding of community assembly processes (McGill et al 2006; Ackerly & Cornwell 2007; Miller, Damschen & Ives2018b)
Each regression is modified by weighting each site point by the effective number of species in the site and each species point by the effective number of occurrences of the species, where effective number is defined as the Hill number show some Type I error rate inflation, each test is carried out by randomisation using the ANOVA F-value of the regression as test statistic
Note that the permutation-based max test using the fourth-corner correlation is precisely equivalent to the permutation-based max test using the F-value of Community Weighted MeanRegression (CWM)- and Species Niche Centroids (SNC)-based regressions weighted by the sites and species totals, respectively. (Hill 1973; ter Braak & Verdonschot 1995)
Summary
In the model-based analysis proposed by Jamil et al (2013), i.e. the second multilevel model (MLM2) described in a recent paper by Miller et al (2018b), there are as many points as there are species (m). The fourth-corner correlation, a weighted correlation between trait and environmental variable, can be displayed as a plot of n×m points corresponding to all species-site combinations (Figure 1c). The n points so calculated do not play a role in the statistical inference, i.e. the assessment of the trait-environment association
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