wascores() for metaMDS?

Hi sig-ecology!

Here comes a probably stupid question... I am looking for smart ways to 
include information about underlying variables in MDS plots. In other 
words, after having computed an ordination with isoMDS or metaMDS from a 
community table, I would like to add something like species 
coefficients/loadings as vectors to the plot of sites. As no species 
coefficients exist in this case, the best I could come up with so far is 
simply vectors calculated from correlation coefficients of the 
individual species with the site scores (on two MDS axes).
The function metaMDS allows to compute "species scores" using the 
function wascores().... I have now pondered for 2 days how these scores 
are calculated and what their precise meaning would be.

An individual taxon's "species score" is computed as the weighted
average of the "site scores", weights being the abundance of that taxon
in each site. It is the abundance weighted centroid of all the samples
in which the species occurs. The motivation for this is that in CA,
species scores are weighted averages of site scores that are themselves
weighted averages of species scores and so on in the Two-way algorithm
of Mark Hill - not that vegan computes the CA solution that way in cca()
- so it is an analogous approach to computing species scores for nMDS.

Would these 
species scores be appropriate to show as vectors in the MDS?

Not as vectors, as that implies directionality or increasing abundance
and there is no reason to assume that the abundance of a given taxon
will increase linearly or even monotonically in a given direction across
the nMDS plot.

Although I hesitate to call it that, the species score computed as the
weighted average of the site scores, is an optima (of nMDS site scores)
and thus abundance declines as one moves away from the point. So in this
sense, you display the species scores in the same manner as on a CA or
CCA plot, as a point, instead of the vector in PCA/RDA. However, the
decline in CA is uniform in any direction (fitted not actual abundance),
i.e. in 2-D the species score is the point at the top of a 2-D
bell-shaped surface as this is the implied response model in CA. With
nMDS there is no reason to assume this is the case.

For one or two taxa, you could just project a surface of actual
abundances using ordisurf() or you could just use the points as you
would in a CA diagram, more or less. The problem with the surface
approach is that you can only show a couple of species at most on a
single ordination plot.

ordisurf would likely be the best option for most extra data you wish to
impose on to the nMDS plot, again for the reason that the relationship
between nMDS axes and the variable of interest need not be a simple
linear or monotonic surface.

HTH

G

Thanks for any answer...

Gabriel Singer

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