[R-sig-eco] NMDS axes scores

Mon Jan 11 13:13:24 CET 2016

On 11/01/16 12:08, Roman Luštrik wrote:
> 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?
Hadi & Ling is a good reference:

Hadi A, Ling R. Some cautionary notes on the use of principal components 
regression. Am Stat [Internet]. 1998 [cited 2012 Mar 21];52(1):15–9. 
Available from: http://www.jstor.org/stable/10.2307/2685559

Basically, if you have all your PCs in the model, you're OK (because 
that's just a rotation of the covariates anyway: in fact regression 
algorithms do their own rotation anyway). Similarly, with MDS it only 
makes sense if you include all of the axes.

Whilst I'm filling bandwidth, I'm not sure Jari's suggestion that you 
need the interaction term is correct. If a model is linear in axis1 and 
axis2, then any rotation is also linear, i.e. the transformation is c_1 
axis1 + c_2 axis2. So you only need Y ~ axis1 + axis2. Basically, it's 
still plane. The interaction adds a curve: if you want that you also 
need to include quadratic terms, i.e. Y ~ axis1 + axis2 + axis1^2 + 
axis2^2 + axis1:axis2.

Bob

> 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]]
>>>>
>>>> _______________________________________________
>>>> R-sig-ecology mailing list
>>>> R-sig-ecology at 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 at r-project.org
>>> https://stat.ethz.ch/mailman/listinfo/r-sig-ecology
>> _______________________________________________
>> R-sig-ecology mailing list
>> R-sig-ecology at r-project.org
>> https://stat.ethz.ch/mailman/listinfo/r-sig-ecology
>
>
>

-- 

Bob O'Hara

Biodiversity and Climate Research Centre
Senckenberganlage 25
D-60325 Frankfurt am Main,
Germany

Tel: +49 69 7542 1863
Mobile: +49 1515 888 5440
WWW:   http://www.bik-f.de/root/index.php?page_id=219
Blog: http://blogs.nature.com/boboh
Journal of Negative Results - EEB: www.jnr-eeb.org