[R-sig-eco] NMDS axes scores

[Roman Luštrik]

Thank you Jari for an, as always, insightful email. It has been a gut
feeling of mine for quite some time that using PCA scores as independent
variables is at least little wrong but never found any reference to
substantiate it. I would like to use this opportunity to ask you or other
readers if there are any (critical) references available regarding this
usage of PCA scores?

Cheers,
Roman

On Mon, Jan 11, 2016 at 11:11 AM, Jari Oksanen <jari.oksanen at oulu.fi> wrote:

> Contrary to common misbelief, NMDS ordination space is **metric**. In
> vegan, the ordination space (= the ordination result) is even guaranteed to
> be Euclidean (in isoMDS it can be Minkowski, but this is not allowed with
> vegan). What is non-metric is the regression from observed dissimilarities
> to the Euclidean distances in ordination space. The reason why we do not
> recommend using NMDS axes as independent beasts is that NMDS tries to
> preserve the *distances* among points. Any orthogonal rotation (= turning
> of ordination space) will change scores along rotated axes, but retain the
> distances among points. The vegan NMDS result is rotated to principal
> components, but still you should avoid thinking that this makes dimensions
> independent from each other, although the first maximizes the dispersion of
> points and axes are orthogonal (non-correlated).
>
> PCA ordination is Euclidean in the same way as NMDS. The difference to
> NMDS are that (1) only Euclidean distances among sampling units can be used
> in PCA (in NMDS you can use any adequate dissimilarity), and (2) the
> mapping is linear (instead of non-metric) from observed dissimilarities to
> Euclidean dissimilarities. Try function stressplot() in vegan to see what
> this means — it is available both for NMDS and rda (PCA) results.  CA is
> similar to PCA except that it is based on weighted Euclidean distances. I
> won’t go into mathematical details, but you can see ?wcmdscale in vegan to
> see how to get CA as a weighted Euclidean ordination of Chi-square
> transformed data.
>
> PCA and CA have some ordering criteria for their axis and therefore some
> people have used axes from those as independent beasts. I think this is
> dubious, too, but people do it all the time. The PCA/CA also define a
> multivariate space, and taking only one axis as an independent object
> sounds strange, in particular if you take something else than the first
> axes.
>
> So what to do with NMDS axes? If you take all NMDS axes and their
> interactions in a regression of type ~ axis1 + axis2 + axis1:axis2 then
> this is equal to fitting a linear trend surface, and the interaction term
> axis1:axis2 takes care that the result is invariant under rotation of NMDS
> space. Function ordisurf() in vegan gives further ideas how to fit surfaces
> to NMDS *space* (instead of simple axis). Also, if you think that some
> direction in NMDS (not necessarily parallel to the axes) is good and you
> have an indicator variable for that, you can use MDSrotate() function in
> vegan to rotate your solution to that direction and then take that rotated
> axis as your explanatory variable.
>
> HTH, Jari Oksanen
>
> > On 11 Jan 2016, at 10:38 am, Martin Weiser <weiser2 at natur.cuni.cz>
> wrote:
> >
> > Hi Conny,
> >
> > AFAIK NMDS is *non-metric* and represents distances among objects, not
> > gradients along axes (known or unknown): distances along axes are
> > stretched as needed locally (NMDS works with rank order), even order of
> > the elements along axes does not tell anything. NMDS is great if you
> > want to say: Object A resembles object C more than it resembles object
> > B, even though C and B are quite similar.
> > Try this: run NMDS several times, aim for different number of axes (e.g.
> > 1,2,3,5,10) and note the scores of the objects along the first one.  You
> > *may* get the same thing.
> >
> > If you need scores of the objects in the ordination, use something with
> > well defined metrics and axes, e.g. PCA, CA.
> >
> > HTH,
> > Martin
> >
> > On 9.1.2016 05:41, Conny wrote:
> >> Hi all,
> >>
> >>
> >>
> >> it has been frequently pointed out in this group, that NMDS axes scores
> >> shouldn't be used individually for further analysis.
> >>
> >> I therefore would like to include both of my NMDS site scores as a
> response
> >> into a GLM model simultaneously.  Unfortunately, I couldn't find any
> advice
> >> on how to actually do this. I found a  couple of papers using NMDS
> scores in
> >> GLMs, but they all seem to use them individually, fitting separate
> models to
> >> each of the ordination axes.
> >>
> >>
> >>
> >> I'm a bit at a loss here and any advice is very much appreciated,
> >>
> >> Conny
> >>
> >>
> >>      [[alternative HTML version deleted]]
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