Glossary

Formulas

  • Mean :

  • Variance :

  • Standardization :

  • Correlation :



Formulas with weights

  • Weighted mean : if weights sum to 1

  • Weighted variance : if weights sum to 1

  • Weighted standardization :

  • Weighted correlation :

  • Weighted covariance :



2016 : Peres-Neto

Community Weighted Mean (CWM)

CWM (for one community site) =

with :

  • xi = trait value of species i
  • wi = abundance of species i
  • n = total number of species


Community Weighted Mean correlation : correlation between CWM values and environmental values (non-weighted by species’ abundances) for all community sites


Species Niche Centroid (SNC)

SNC (for one species) =

with :

  • xi = environmental value of site i
  • wi = abundance of species in site i
  • n = total number of community sites


Species Niche Centroid correlation : correlation between SNC values and trait values (non-weighted by species’ abundances) for all species


Fourth-corner correlation

  • correlation (weighted standardized trait, weighted standardized environment)
  • weighted covariance (CWM, weighted standardized environment)
  • weighted covariance (weighted standardized trait, SNC)
  • weighted regression slope (CWM of weighted standardized trait, weighted standardized environment)
    considering the correlations among environmental variables, on the contrary to RLQ


CWM, SNC correlations 4th-corner correlation
divide by mean variation divide by total trait variation


Maximum (4th-corner correlation) :

  • reached if species distributions are perfectly ordered across columns (species) and rows (sites)
  • squared root of 1st eigen value of correspondance analysis of species distribution matrix (sites x species)


Chessel fourth-corner correlation : 4th-corner correlation / max(4th-corner correlation)


Decomposition of trait variation

CWM / SNC are based on weighted averages : ignore trait variation among species within communities
(i.e. within-community component)

Usually, CWM is used to assess among-community component, and average trait distance (e.g. MPD) to asses within-community component. But they are not additive : they do not sum up to total variation.

New solution for within-community component :

(for one community site) = alpha i *

with :

  • alpha i = abundance weight of site i

  • lij = abundance of species j in site i

  • SUM lij = abundance of species in site i

  • tj = trait value of species j

  • ci = CWM value of site i




Citations

  • Peres-Neto P R, Dray S and Braak C J F ter. 2017. Linking Trait Variation to the Environment: Critical Issues with Community-Weighted Mean Correlation Resolved by the Fourth-Corner Approach. Ecography 40 (7): 806–16. https://doi.org/10.1111/ecog.02302