Mantel test with skew-symmetric matrices?

Hello All,

I'm interested in using the mantel()and mantel.partial() functions in the
'vegan' R package to examine fish morphology in several lakes with respect to
distance between lakes, lake elevation (above sea level) and various habitat
measures. For example, several authors have postulated that jaw length varies
with habitat conditions, but jaw length, lake distance, elevation and the
habitat measures co-vary. Applying a partial Mantel to control for elevation
or lake distance seems like a good way of attacking the problem.

Typically, Mantel tests are performed using symmetrical distance matrices
based upon absolute distances. For example, a symmetrical matrix of distances
between 3 lakes might look like this:

0 7 3
7 0 4
3 4 0

This makes sense with distance data (miles, km, etc), but information is lost
on the direction of effect between lakes with data such as jaw length or
elevation. Instead, a so-called skew-symmetric matrix can be generated, which
would instead look something like this:

 0  7  3
-7  0 -4
-3  4  0

I have two questions:

1) Is it wrong to use skew-symmetric matrices - i.e. should I just forget
about the skew data and use absolute values to make all my matrices symmetric?
The original Mantel paper (1967, Cancer Research, 2: 209-220) does talk about
skew-symmetric matrices, but the published applications I've come across only
seem to use symmetric matrices.

 2) If it is ok to use skew-symmetric matrices, do the mantel() and
mantel.partial() functions in 'vegan' (or related functions in other packages,
such as 'ecodist') handle them correctly? I've found that it is possible to
process skew data and generate results using these functions, but I'm
uncertain from the documentation whether the results are meaningful (i.e. is
the coding designed to handle such cases appropriately?)

With thanks, Steve

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Stephen A. Arnott, PhD
Associate Marine Scientist
Marine Resources Research Institute
South Carolina Department of Natural Resources
P.O. Box 12559
Charleston
SC 29422-2559

Phone: (1)-843-953-9794
Fax: (1)-843-953-9820
E-mail: arnotts at dnr.sc.gov<mailto:arnotts at dnr.sc.gov>
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