r2 values from envfit in NMDS

Hello,

Please forgive my ignorance if this is really basic.

I am Using NMDS and envfit in the package vegan to determine the
environmental variables contributing to the differences between two
distinct groups of sites and to determine if the two groups of sites are in
fact distinct.

Envfit provides r2 values that, if I understand it correctly, quantify the
degree to which the environmental variables influence the distribution of
the sites (points) in the ordination space. They don't however reflect the
degree to which the variables affect the separation of the distinct site
groups.  An environmental variable may have a high r2 value but be
perpendicular to the axis of greatest group separation.

Is my understanding of the r2 values provided by envfit correct? Is there a
way to quantify the contribution of environmental variables to site group
separation in NMDS? Or should I separately use something like randomForest?

Kind of, though your explanation is back to front. The model fitted is

x_i = \beta_1 Ax_{ji} + \beta_2 Ax_{ki} + \varepsilon

i.e. we say the axis scores on axes j and k (Ax) affect the values of
the environment. And as such we reverse this silly statement and say
that if there is such a relationship (between axis scores and
environment) then the environment explains, to some degree, the
dissimilarity between sites.

Note that NMDS, like most other ordination methods, is focussed on
sites, not groups of sites. Hence no one ever claimed that it was
designed to best separate groups of sites. envfit simply fits vectors
into k-d ordination configurations; again it knows nothing of groups.

If we *had* a discriminant analysis method in vegan, which does set out
to best separate groups (under certain conditions), then that ordination
may be doing what you want but envfit would still not care about the
groups; it would simply project a vector into the resulting ordination
space.

If you want to discriminate between groups that you have defined a
priori, then a random forest is one of the machine learning tools that
might usefully be applied.

HTH

G

Gavin Simpson, PhD                          [t] +1 306 337 8863
Adjunct Professor, Department of Biology    [f] +1 306 337 2410
Institute of Environmental Change & Society [e] gavin.simpson at uregina.ca
523 Research and Innovation Centre          [tw] @ucfagls
University of Regina
Regina, SK S4S 0A2, Canada