Dear David,

Great question!  I disagree with the statement that the samples' variance
in shape space is not biologically real or, perhaps more accurately, is
less real than the variance in any other space.  As far as I see it, the
basic strategy in any biostatistical study, be it GM or otherwise, is that
a researcher represents a real biological population as an abstract
statistical population.  They then use this abstract statistical population
as a proxy for the real one so that inferences in the statistical space
have implications for the real world.

For example, a PCA finds a direction in the statistical space in which the
statistical population tends to be spread out.  This is interesting to the
researcher because this direction has a correspondence to certain real
world variables.  As a result, the PCA tells the researcher in what ways
the real population tends to vary.  The key point, though, is that the
researcher transitioned from the statistical space to the real world.

Moving from shape space to the real world is no different in principle.  We
have a real population of specimens, whose shape are of interest to us, and
we represent them using vectors of shape variables.  The vectors are
abstractions; it is not as if we can hold a vector in our hands.  However,
this is irrelevant because they are just proxies, no less real than any
other quantitative representation.  What matters is if we can tie them back
to the real world.  This is why morphometricians implement a visualization
step.  In a PCA, the PCs describe how our proxies vary, and we visualize in
order to see how this variation appears in the real world.  It is
infeasible to visualize every point along this axis, so we instead
visualize a handful.  Since the core goal in PCA (at least in this context)
is to describe variance, we generally describe the locations where a
visualization occurs in units of standard deviations from the mean.  We
could use absolute distances along an axis, but this is probably more
arbitrary than standard deviation units.  The standard deviations come from
the data's distribution, whereas the absolute distance is really only
well-defined in the mathematical space.

To summarize: i) Nearly all quantitative analyses involve an abstraction to
a mathematical space.  ii) The description of points in a mathematical
space is useful to the researcher because the researcher is able to
translate the abstract mathematical space into a real world
interpretation.  iii) In GM, the shape variables are traditionally
translated into the real world via visualization.  Ergo, morphometricians
often interpret PCA results via visualizations along individual PCs.  To
aid in interpretation, this tends to occur in standard deviation units
because the standard deviation is more easily tied to the real world
relative to arbitrary selecting a unit of distance.

Perhaps some of these points are up for debate, but remember that
statistics is largely the study of VARIATION.  If the variation in shape
space did not have any biological significance, almost no analysis after
alignment would be possible.

Hope somewhere in this long commentary, you found something helpful,


On Tue, May 9, 2017 at 4:56 PM, dsbriss_dmd <> wrote:

> Good afternoon all, I have a question about interpretation of PCs.  I have
> come across several articles in orthodontic literature having to do with
> morphometric analysis of sagittal cephalograms that discuss warping a
> Procrustes analysis along a principal component axis.  Essentially the
> authors discuss finding whatever principal components represent shape
> variance, then determining the standard deviation(s) of those PC's, and
> applying the standard deviations to the Procrustes shape to warp the
> average shape plus or minus.  So if you have an average normodivergent
> Procrustes shape, one warp perhaps in the negative direction might give you
> a brachycephalic shape, while the opposite would give you a dolichocephalic
> shape.  But I don't know where this idea comes from.  I have been involved
> with 8 or 9 morphometrics projects over the last few years and I have never
> been able to figure this out or the rationale for performing such an
> application with the PC results.
> As an example of what I am talking about here is a passage from the
> Journal of Clinical & Diagnostic research, doi:  10.7860/JCDR/
> 2015/8971.5458 <>
> "Here, the first 2 PCs are shown & the Average shape (middle) was warped
> by applying each PC by amount equal to 3 standard deviations in negative
> (left) and positive (right) direction {[Table/Fig-10
> <>]: PC1
> with standard deviation, [Table/Fig-11
> <>] PC 2
> with standard deviation}."
> I did not include the graphs from the article but if it would help to
> answer this question I can supply them.
> What I do not quite understand is what exactly is the purpose of applying
> standard deviation(s) to the PCA and then warping the Procrustes average
> shape to these standard deviations?  Maybe my understanding of PCA is
> limited, but I was under the impression that in GPA the principal
> components are only statistical variance, and don't represent something
> biologically real.  So to see how an individual varies from the shape
> average you have to go back and look at whatever landmark(s) represent that
> specific individual and compare that shape to the Procrustes average.
> Maybe this is not correct?
> Thanks in advance, I appreciate any help you can give me.
> David
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