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 <weis...@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]] >> >> _______________________________________________ >> R-sig-ecology mailing list >> R-sig-ecology@r-project.org >> https://stat.ethz.ch/mailman/listinfo/r-sig-ecology > > > -- > > ------------------------------ > Pokud je tento e-mail součástí obchodního jednání, Přírodovědecká fakulta > Univerzity Karlovy v Praze: > a) si vyhrazuje právo jednání kdykoliv ukončit a to i bez uvedení důvodu, > b) stanovuje, že smlouva musí mít písemnou formu, > c) vylučuje přijetí nabídky s dodatkem či odchylkou, > d) stanovuje, že smlouva je uzavřena teprve výslovným dosažením shody na > všech náležitostech smlouvy. > > _______________________________________________ > R-sig-ecology mailing list > R-sig-ecology@r-project.org > https://stat.ethz.ch/mailman/listinfo/r-sig-ecology _______________________________________________ R-sig-ecology mailing list R-sig-ecology@r-project.org https://stat.ethz.ch/mailman/listinfo/r-sig-ecology
