September 30, 2005
This is a reply to Luke Finley's question of earlier today
about using nonmetric techniques with shape coordinates.
I'll refer to the questioner in the second person, as "you".
Your "rationale for using MDS" is actually the list of
reasons for using PCA, which in this specific context does
PRECISELY what you are asking MDS to do!
Remember that the Procrustes shape coordinates arise
all at once as an essentially unique
representation of one single distance function,
namely, Procrustes distance. The shape coords should not be thought of
as separate variables, but only as one joint set. (The same
is true of the partial warp scores, which are just a rotation
of the natural basis [assuming you've preserved the zeroth
PW, which is to say, the uniform component in the right metric].)
It makes no sense to do a "nonparametric" analysis of coordinates
that have the metric built right in down to the level of
the fundamental definition the way these do.
In particular, there is just no point to a technique like MDS in
this connection, as there is no choice about the projection
metric you MUST use: it MUST BE the principal coordinates of the
shape coordinate data (the best representation of the original
distance metric that's already been enforced), and for the first pair
this is identical with the ordinary RW1/RW2 plot, by
definition. I'm glad
you observed that in your example, because you need to be
using the RW's, not any other output from MDS: this plane is,
by definition, the unique best representation of the original
Procrustes distances -- you just needn't bother with MDS.
If your results aren't IDENTICAL between MDS and PCA of the PW
scores, it can only be that you have done something wrong with
one computation or the other.
Likewise there is just
no point to using "similarity in the RW1/RW2 plane" or anywhere
else. The Procrustes technique has already locked you into
a measure of shape distance, and you gain nothing whatever
by using only the part in the first two dimensions -- that's
just a limitation of your retina, having nothing to do with
krill OR statistics.
So your rationale for using MDS instead of PCA is
actually the argument for PCA, or, rather, the argument for PCOORD
(which is identical with PCA but is reported using different
words), and you have to
use PCOORD because you already assumed that Procrustes distance
was worth paying attention to when you used the Procrustes shape
representation in the first place, and it doesn't matter
if you're using the PW scores vs. the shape coordinates because
they have exactly the same metric.
But the analysis of shape against geography doesn't go very
well in terms of these distances. In other words, your
last concern, "whether samples that are far apart are more different
than samples close together," isn't actually a very good way
to ask a question, as it presumes two distance measures when
you don't actually have to presume even one.
Scientifically, it's better to do a Partial
Least Squares analysis of shape coordinates against geographical
coordinates (or ecological, or whatever). For the latitude/longitude
example, see Frost et al., Cranial allometry, phylogeography,
and systematics of large-bodied
papionins (Primates/Cercopithecinae) inferred
from geometric morphometric analysis of landmark data.
{\sl Anatomical Record} 275A:1048--1072 (with cover), 2003,
where we do it for baboon skulls over a map of Africa.
What you get are prediction gradients that have (finally)
lifted the requirement of symmetry that underlies the
Procrustes method, and you get to learn what is predicted
by latitude and by longitude separately. Only if these
are the same does it make any sense to proceed in terms of
distance apart, which was the question as you posted it
(so, to repeat, I think that's not a good way to phrase hte
question -- unless you know for certain that you're looking
at a shape diffusion process rather than a selective
process! -- such knowledge is rarely granted to ordinary
mortals, even though Sewall Wright published some astonishingly
good examples in the '40's).
Thus I see no role for either ANOSIM or MDS in
a good competent analysis of the data you've described for us.
Every biometric data set has problems; yours is about average
in this regard.
Fred Bookstein
[EMAIL PROTECTED]
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