Seattle, February 3, 2021
Dear MorphMetters,
You probably know already that over the last several years I
have been rethinking many of the fundamental tools of
geometric morphometrics associated with Procrustes shape coordinates
and their principal components. My published critiques typically begin
with a detailed inspection of the explicit mathematics embodied in
these tools in order to arrive at a clearer understanding
of when their multivariate analysis and the ensuing thin-plate spline
depictions are likely to lead to valid biological explanations vs. when
this happy outcome is UNlikely. One main message of these papers is that
when there are more than just a few Procrustes shape coordinates,
the basic toolkit of GMM is dangerously risky: wielded in
a rote or unsophisticated way, it leads predictably to fallacious inferences
in a range of typical biological applications. Sometimes the critique
simply requires us to abandon a previously popular tool (e.g.,
between-groups principal components analysis, 2019). In other settings, as in
my 2015 method for studying integration, it is possible to
suggest a replacement that circumvents the original problems. Here
is another of those suggested replacements, this one designed
to substitute for current Procrustes-style analyses of
allometric growth in landmark data. An open-access paper
just published in Benedikt Hallgrimsson's journal Evolutionary
Biology demonstrates this replacement, unusual perhaps in that
its basic idea is more than a century old -- even older than
D'Arcy Thompson, in fact. Anybody interested may download the paper from
https://link.springer.com/article/10.1007/s11692-020-09530-w
I attach its Abstract.
Here's hoping you enjoy this challenge. Yours, Fred Bookstein
=========
{Abstract.} The geometric morphometric (GMM)
construction of Procrustes shape coordinates from a data set of
homologous landmark configurations puts exact algebraic constraints
on position, orientation, and geometric scale. While position as digitized
is not ordinarily a biologically meaningful quantity, and orientation is
relevant mainly when some organismal function
interacts with a Cartesian positional gradient such as
horizontality, size per se is a crucially important biometric concept,
especially in contexts like growth, biomechanics, or bioenergetics.
"Normalizing" or "standardizing" size (usually by dividing
the square root of the summed squared distances from the centroid out
of all the Cartesian coordinates specimen by specimen),
while associated with the elegant symmetries of the
Mardia-Dryden distribution in shape space, nevertheless
can substantially impeach the validity of any organismal inferences
that ensue. This paper adapts two variants of standard morphometric
least-squares, principal components and uniform strains, to circumvent
size standardization while still accommodating
an analytic toolkit for studies of differential growth
that supports landmark-by-landmark graphics and thin-plate splines.
Standardization of position and orientation but not size yields the
coordinates Franz Boas first discussed in 1905. In studies of growth, a first
principal component of these coordinates often appears to involve
most landmarks shifting almost directly away from their centroid,
hence the proposed model's name, "centric allometry." There is also a
joint standardization of shear and dilation resulting in a variant of
standard GMM's "nonaffine shape coordinates" where scale information is
subsumed in the affine term. Studies of growth allometry should
go better in the Boas system than in the Procrustes shape space that
is the current conventional workbench for GMM analyses. I demonstrate two
examples of this revised approach (one developmental, one phylogenetic)
that retrieve all the findings of a conventional shape-space-based
approach while focusing much more closely on the phenomenon
of allometric growth per se. A three-part Appendix provides an
overview of the algebra, highlighting both similarities to
the Procrustes approach and contrasts with it.
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