Dear Nicol�s
Sender: [EMAIL PROTECTED]
Precedence: bulk
The points you raise are important ones.
>On the other hand, if one subtracts the means from a matrix of
>log-transformed variables (i.g. if one subtracts the isometric
>component), could be named the residual matrix a form matrix because it
>have both size and shape information (allometry)?
Yes, the centered log-distance variables contain information on size
and shape and therefore are about form. However, the "form matrix" in
Euclidean Distance Matrix Analysis (EDMA) is a special format of this
sort of data: the matrix of the distances between all possible pairs
of landmarks in a configuration. That means, such an entire matrix is
used to characterize the form of a single specimen (normally with
some redundant information).
Because the term "form matrix" is now usually used in this EDMA
context, it is better not to use that term for a data matrix
consisting of centered log-distance variables for multiple specimens.
Your term residual matrix would be fine if you want a specific name.
>In multivariate analyses as CPCA or mgPCA, size effects could be
>sequestered in one variable (first PC) rendering matrices free of
>allometric effects as can be verified indirectly by linear
>regression. In this case, how I could
>name those matrices?
What's left after a "size correction" by eliminating the CPC1 or
mgPC1 is what one might call "size-free variation" or "size-invariant
variation" in the log-distance data. It is important to realize that
this is not the same as (geometric) shape, except for the special
case of isometry. Therefore, one should resist to call the second and
subsequent CPC or mgPC "shape", as this has been done traditionally
(producing lots of confusion).
>It is not easy for me to understand the concept of shape free of
>allometry. Is it defined only for need (or convenience) in the
>multivariate statistics, or actually can it be found in biological
>structures?
The geometric concept of shape is just another way to partition
variation, which is very intuitive because many of us think in terms
of images -- shape is a very intuitive idea. As a result, geometric
shape analyses have many options for graphical presentations of the
results. But this benefit comes at the cost of the more ambiguous
biological meaning of shape.
In contrast, the allometric approach partitions variation into
components parallel and perpendicular to some axis of variation
(growth trajectory, static allometry, etc.). So the biological
meaning is immediately clear, but it is a bit more difficult to
visualize the findings, and we should resist to use the term shape
(with its geometric baggage) in this context.
Best wishes,
Chris
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Christian Peter Klingenberg
School of Biological Sciences
University of Manchester
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