Seattle, April 2, 2022
Dear MorphMetters,
A few days ago I submitted a new manuscript, "Dimensions
of Morphological Integration," to Benedikt Hallgrimsson's
journal, Evolutionary Biology, where many of us have been
publishing morphometric discussions lately. This time,
the website included an offer I hadn't seen before: would you
like us to automatically submit your manuscript to Research
Square, a free preprint posting service? I was intrigued enough
to agree, and indeed my complete manuscript was posted there,
every word and every figure, yesterday evening. You can get a copy by
pointing your browser to
https://doi.org/10.21203/rs.3.rs-1498707/v1
and clicking on the Download PDF button.
The piece is currently under review, and any constructive comments
you care to send me will be taken into account if a revision is requested.
To help you decide if you want to bother with this, here's the abstract.
The associated list of keywords also might intrigue you:
"integration, geometric morphometrics, Boas coordinates,
allometry, calvaria, thin-plate spline, eigenvalues and
eigenvectors, covariance matrix, concentration matrix, leghorn
chickens, Vilmann data set, polar coordinates, spiderweb diagram."
Thanks for thinking about integration, Fred Bookstein
======
{\bf Abstract.}
Over several generations of evolutionary and developmental biologists,
ever since Olson and Miller's pioneering work of the 1950's,
the concept of ``morphological integration''
as applied to Gaussian representations $N(\mu,\Sigma)$
of morphometric data has been a focus equally
of methodological innovation and methodological perplexity.
Reanalysis of a century-old example from Sewall Wright shows how some
fallacies of distance analysis by correlations can be
avoided by careful matching of the distance rosters involved
to a different multivariate approach, factor analysis.
I reinterpret his example by restoring the information (means and
variances) ignored by the correlation matrix, while confirming
what Wright called ``special size factors'' by a different
technique, inspection of the concentration matrix $\Sigma^{-1}.$
In geometric morphometrics (GMM), data accrue instead as
Cartesian coordinates of labelled points; nevertheless,
just as in the Wright example, statistical manipulations do better
when they reconsider the normalizations that went into the
generation of those coordinates. Here information about both
$\mu$ and $\Sigma,$ the means and the variances/covariances,
can be preserved via the Boas coordinates
(Procrustes shape coordinates without the size adjustment) that
protect the role of size per se as an essential explanatory factor
while permitting the analyst to acknowledge the realities of
animal anatomy and its trajectories over time or size
in the course of an analysis. A descriptive quantity for
this purpose is suggested, the correlation of
vectorized $\mu$ against the first eigenvector of $\Sigma$ for the
Boas coordinates. The paper reanalyzes two GMM data sets from this
point of view. In one, the classic Vilmann rodent neurocranial growth data,
a description of integration can be aligned with the purposes of
evolutionary and developmental biology by a graphical exegesis based
mainly in the loadings of the first Boas principal component.
There results a multiplicity of morphometric patterns, some homogeneous and
others characterized by gradients. In the other, a Vienna data set
comprising human midsagittal skull sections mostly sampled along curves,
a further integrated feature emerges, thickening of
the calvaria, that requires a reparametrization
and a modified thin plate spline graphic distinct from the
digitized configurations per se. This new GMM protocol fulfills
the original thrust of Olson and Miller's (1951)
``$\rho$F-groups,'' the alignment of statistical and
biological explanatory guidance, while respecting the
enormously greater range of morphological descriptors afforded
by well-designed landmark/semilandmark configurations.
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