[R-sig-eco] distances in NMDS ordination space

[Jari Oksanen]

Hi Kate,

I think we should use the name Euclidean NMDS for the kind of NMDS we have in vegan, because its ordination space is strictly Euclidean. There is nothing non-metric in the ordination space. What is non-metric is the transformation of observed dissimilarities to optimize the fit to the Euclidean ordination space. The central term in the squared stress function is

SUM (theta(d) - delta)^2

where delta are the Euclidean distances among points in the ordination, d are the observed community dissimilarities among sampling units, and theta() is a non-metric monotone function to transform d. Sum of squared differences is -- by definition -- squared Euclidean distance, and hence our kind of NMDS is Euclidean. 

The isolated axes are not meaningful *because* the space is Euclidean. Euclidean space is invariant under rotation: you can rotate axes and distances among points does not change, and you can rotate the axes and the configuration of points does not change. Any direction of axes is just as good.

In vegan, the default is to rotate axes to principal components so that first dimension is longest. However, you can also rotate a dimension parallel to an environmental variable using function MDSrotate. These rotations do not change the stress, configuration or distances among points. Such a rotated dimension can be more meaningful and there is some justification in using that as a variable in some other analysis.

To repeat: vegan NMDS is Euclidean NMDS and the NMDS ordination space is Euclidean. Because it is Euclidean, it is rotation-invariant and any rotation is equally good. Therefore axes do not have a natural orientation in Euclidean space. The only thing that is non-metric is the transformation of community dissimilarities. That non-metric transformation is made to optimize the goodness of fit to Euclidean ordination space.

Cheers, Jari Oksanen

On 16/07/2015, at 22:19 PM, Kate Boersma wrote:

> Hi all.
> 
> I have a methodological question regarding non-metric multidimensional scaling. This is not specific to R. Feel free to refer me to another venue/resource if there is one more appropriate to my question.
> 
> Correct me if I'm wrong: NMDS axes are non-metric, which is why NMDS frequently makes sense for community data, but it also means that distances in NMDS ordination space cannot be interpreted simplistically as they can in eigenvalue-based methods like PCA. This is why it is inadvisable (meaningless) to use NMDS axes as response variables in a linear modeling framework (e.g., with environmental variables as predictors).
> 
> My question is this: Does that mean that it is also inadvisable to use distances among points in ordination space as response variables?
> 
> My (potentially flawed) understanding: While the coordinates may not make sense in isolation, they should be meaningful relative to each other. In a 2D ordination, if communities A & B are closer together in ordination space than communities C & D, that means they have more similar species compositions. Therefore, I should be able to predict the distance between points in a linear modeling framework.
> 
> Alternately, I could use the actual distances among communities from my dissimilarity matrix with a method like db-RDA. But I used NMDS over RDA or CCA for a reason. It seems more straightforward to use the distances from my NMDS ordination instead of generating new coordinates from a PCoA to fit an RDA framework (as in db-RDA)... but this logic only works if NMDS distances are informative.
> 
> Are these comparable analyses? If not, why not?
> 
> I'd love your opinions.
> 
> Thank you,
> Kate
> 
> -- 
> Kate Boersma, PhD
> Department of Biology
> University of San Diego
> 5998 Alcala Park
> San Diego CA 92110
> kateboersma at gmail.com
> http://www.oregonstate.edu/~boersmak/
> 
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