CCA / RDA

Constrained ordination arranges samples only along directions that the environmental variables can explain: points are samples, small green points are species (only the furthest from the origin are labelled), and red arrows are environmental variables — longer arrows at a smaller angle to an axis are more strongly correlated with it.

CCA assumes unimodal species responses along gradients (suited to community data spanning long gradients); RDA assumes linear responses, and community data should be Hellinger-transformed first.

The constrained proportion is the share of community variation explained by the environmental variables. The permutation test shuffles the rows of the environmental table to judge whether that share exceeds chance (P = (1 + number of permutations at least as extreme) / (1 + permutations)).

More variables always explain more; with nearly as many variables as samples the result is meaningless, so choose a few well-justified variables. Interpret arrows with care when variables are strongly correlated (collinear).

Scaling 1 is suited to relationships among samples, scaling 2 to relationships between species and environmental variables. Arrows are rescaled to the range of the sample scores, so read their direction and relative length only.

Method

ter Braak C. J. F. (1986) Ecology 67:1167–1179 (CCA); Legendre & Legendre (2012) Numerical Ecology §11.1–11.2; Legendre, Oksanen & ter Braak (2011) Methods Ecol Evol 2:269–277 (permutation tests). Eigenvalues and sample, species and biplot scores match scikit-bio rda and cca; pseudo-F can be recomputed from the constrained proportion, and permutation P values agree with an independent implementation within Monte Carlo error.

Data size

Up to 2,000 samples; the permutation test takes time proportional to samples × species × permutations.