Hi Jari,

Thank you for your prompt reply.  Gilbert and Bennett 2010 is very
interesting.  It seems that there are substantial drawbacks to these
techniques.  I think I may use these techniques only as supplementary and
may decide not to try to use them to correct for spatial autocorrelation in
the main manuscript... apparently there is not a good way to do this, I
guess.

Assuming I decide to investigate these techniques anyway, I wanted to
clarify a few things.

1. So the pcnm() function in vegan automatically removes the vectors
associated with negative eigenvalues, returning only those with positive
eigenvalues?

2.  It is perplexing to me that I find a significant spatial autocorrelation
using a Mantel test, but do not find a significant global test using all
pcnm vectors (and a 105 m threshold).  I'm not sure what to make of this.

3.  Lets assume I ended up with a significant global test and used
the ordiR2step and capscale to select significant pcnm axes.  As I
understand it, I could then use varpart and further capscale analsyses
including evironmental factors and signficant pcnm axes.  However, I was a
little unsure on what you were saying about adonis.  Is it innapropriate to
use the significant pcnm axes and environmental factors in an adonis
analysis with multivariate community response data?  (e.g.,
>adonis(ResponseData ~ pcnmFinal + env1 + env2 + env3, EnvData, permutations
= 4999, method = "bray")

Thanks!!

Kevin

On Sat, Nov 6, 2010 at 1:27 AM, Jari Oksanen <jari.oksanen@oulu.fi> wrote:

> Kevin,
>
> I'll answer only some of your technical questions. I don't want to
> implicate
> that you should use PCNM, but I only say how to use them if you use them.
> For conceptual issues, you ma also check B. Gilbert & J. R. Bennett Journal
> of Applied Ecology, Volume 47, Issue 5, pages 1071­1082, October 2010.
>
> On 5/11/10 22:59 PM, "Kevin McCluney" <Kevin.McCluney@colostate.edu>
> wrote:
>
> > I am trying to use pcnm in vegan to correct for spatial autocorrelation
> in
> > analyses of influences of environmental factors on multivariate community
> > composition, as well as univariate analyses of diversity and abundance.
>  I
> > have several questions.
> >
> > 1.  PCNM requires a threshold and I am aware that the value that keeps
> all
> > sites connected is most commonly used.  My data has a group of 4 sites
> > spaced <70 m apart that are over 3 km from another group of 30 sites
> spaced
> > less than 105 m apart, and one more site which is 700 m from that group.
>  It
> > seems more reasonable to me to use a threshold of 105 m than one of >3km,
> > especially since my study focuses on ground arthropods.  This would
> create 3
> > groupings of sites.  If I use the >3km threshold I get only 8 pcnm axes,
> > whereas if I use 105 m I get 23 pcnm axes.  What are the dangers of using
> > 105 m for the threshold?
> >
> The choice of threshold is arbitrary and it will influence the results. The
> standard (which also is the default in vegan) is indeed to use longest
> possible threshold to keep the data connected. I cannot see any dangers in
> using any other thresholds. It is not more dangerous to use threshold of
> 700m than to use a threshold of 3000m. You could quite as well as what are
> the danger of using the default of >3km.
>
> The number of PCNM vectors has no relevance for the choice.
>
> If you start from the Euclidean distances of spatial locations on a plane
> (like Earth is for many practical purposes), you would get back two
> principal coordinates. We put there an arbitrary threshold in PCNM and
> these
> non-Euclidify the matrix. Therefore you get more than two PCNM vectors and
> sevaral negative eigenvalues for locations on a plane. Having a low number
> of PCNM vectors and not too many negative eigenvalues is a sign of more
> Euclidean space. You cannot have completely Euclidean (=2 dims) space for
> PCNM since then you just fall back to a simple linear trend surface. With
> PCNM you have trickier surfaces.
>
> > 2.  I know that the standard procedure for pcnm involves removing or
> > ignoring axes with negative eigenvalues.  When I use the pcnm function in
> > vegan, I get more eigenvalues than I have axes.  If I assume that the
> first
> > values correspond to the first axes, then all the axes in my analyses
> have
> > positive eigenvalues, but then the "extra" values are all negative.  What
> > are these "extra" eigenvalues?
> It is a correct assumption that the eigenvalues and corresponding
> eigenvectors are ordered similarly. All PCNMs that you get are for those
> positive eigenvalues, and the first eigenvalue is for the first axis etc.
> You will normally get negative eigenvalues. It is not only a common
> procedure to ignore axes with negative eigenvalues, but it is about the
> only
> practical choice (and the only choice you have in vegan).
> >
> > 3. The next step in pcnm involves selecting pcnm axes with significant
> > effects on responses.  I know that it has been suggested to use forward
> > selection routines in cca to select these axes.  I'm also aware of some
> of
> > the limitations of this technique and the suggestion that forward
> selection
> > should have additional criteria. Namely, that forward selection should
> only
> > be used if the full model with all terms is significant and should also
> > compare the adjusted R2 with each term added to that of the original full
> > model.  If I perform an analysis with all non-negative pcnm terms and the
> > model is not significant, does this mean there is no spatial
> autocorrelation
> > and no selection procedure is needed?  If it is significant, then I need
> > adjusted R2 values to perform the forward selection, but I don't see how
> to
> > get these using cca in vegan.
> >
> There is no way of doing this in vegan. You can do it for rda(), and the
> development version of vegan in repository http://r-forge.r-project.org/has
> an automatic function ordiR2step for rda() or capscale() to do that forward
> selection (it is not yet in the release version, because I have expected
> Guillaume Blanchet's and Pierre Legendre's blessing to the function before
> release). The problem is that vegan does not have adjusted R2 for CCA. I
> have seen a Pedro Peres-Neto's paper on calculating adjusted R2 fro CCA,
> but
> the calculation is pretty tricky and slow, and we haven't implemented that
> in vegan. I guess you refer to Blanchet et al., Ecology 89, 2623-2632
> (2008)
> when you write about recommended procedure: that paper only considered RDA.
> With CCA you must trust your own judgment when you decide how to select
> your
> PCNMs.
>
> I hope you remembered to supply weights to pcnm() function in vegan when
> you
> calculated your PCNM vectors.
>
> Non-significant PCNMs say nothing about spatial autocorrelation. They may
> say something about spatial structure that can be expressed with your set
> of
> spatial basis vectors.
>
> > Additionally, for my multivariate analyses, I am using adonis to perform
> > statistical tests, not cca.  Therefore, should I also be using adonis to
> > look for significant pcnm axes?  I haven't seen this done, but do not see
> > potential drawbacks.  Are there any?
> >
> I wouldn't mix adonis() and pcnm(). The PCNMs were designed to be used with
> RDA or other ordination techniques. adonis() works with dissimilarities. In
> PCNM you start with distances, then you change these to basis vectors, and
> for adonis() you would change those back to distances. I think this is not
> wise. Why not use the PCNM distances directly? One reason is that you
> cannot
> get them directly in vegan or other functions I know without editing vegan
> functions to return them or calculating them manually. That would be rather
> easy since you can just copy the commands of pcnm() functions (it seems
> that
> one line is enough).
>
> I think CCA and adonis is not a good coupling: CCA is weighted method using
> Chi-square metric and adonis is unweighted using Euclidean metric. Coupling
> RDA and adonis or capscale() and adonis (for non-Euclidean case) is more
> natural.
>
> Cheers, Jari Oksanen
>
>


-- 
Kevin E. McCluney, PhD
Post-doctoral Researcher
Affiliate Scientist, Natural Resource Ecology Laboratory, Colorado State
University
Contractor, US Geological Survey, Fort Collins Science Center
Lab phone: 970-491-2414
Cell phone: 480-241-9784
Office: Anatomy/Zoology Building E330
Mail: Dept of Biology
1878 Campus Delivery
Colorado State University
Fort Collins, CO  80523-1878

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