envfit and NMDS

Howdy folks,

This is the second time this week we have this issue. There are two (or three) separate points:

(1) You should not correlate environmental variables with axes in any ordination method. This applies to PCoA, PCA, CA or anything else just as well as to NMDS. You can see this by fitting the vectors: they are rarely parallel to the axes. Even in CCA/RDA, the vectors for constraints are rarely parallel to the axes.

(2) The ordination space in NMDS is metric. The non-metric part is the monotonic (non-metric) regression from metric ordination space to observed dissimilarities. The observed dissimilarities between sampling units ("plots", "sites") need not be metric, but they can be semimetric or non-metric, but the ordination space derived from them is metric. 

(3) As a separate issue, it is often better to use fitted surfaces than fitted vectors. Fitted vectors are appropriate when the fitted surface is a plane (first degree linear trend surface). This is rarely the case, and this applies to all ordination methods: the fitted surfaces in CA, PCoA or PCA are usually just as non-planar as in NMDS.

For point 2: Look at the stressplot(<NMDS-result>). Here horizontal axis gives Euclidean distances in NMDS space -- these are metric. The vertical axis gives the observed dissimilarities -- these can be anything. The fit lines gives the monotonic regression -- this is non-metric. 

With vegan::metaMDS() the ordination space is not only metric, but it is strictly Euclidean. We do and we can rotate the ordination space.

As a historic note, the vector fitting code for vegan was based on a Bell Labs document that describes vector fitting for their NMDS (KYST software). The Bell folks invented NMDS, and they regarded vector fitting suitable for NMDS from the very beginning. That is, form 1960s.

Cheers, Jari Oksanen