I would like to contribute another perspective to this discussion
concerning methods for hypothesis testing. Fred, Chris and others have
pointed out the advantages and potential pitfalls of MANOVA like
procedures using shape variables as data, so I will not rehash them. The
alternative Fred advocated is in the spirit of Goodall's F-test, using
Procrustes sum of squares and permutation tests. Both options allow one
to assess the magnitude of shape difference (or change) among samples.
However, for certain hypotheses, it is the direction of shape change
that is of interest. Direction is not captured by Procrustes distance,
so for such cases use of the multivariate shape data in other procedures
is required.
Here is a simple example. Say one has 2 groups (males and
females) collected at 2 time periods.Either MANOVA or the
Procrustes-permutation approach could be used to assess whether the
sexes differ, whether time periods differ, and whether there was an
interaction between sex and time. If an interaction was identified, the
shape change in males through time is different than the shape change in
females though time, implying the direction of shape change is not
concordant. To assess this more fully one
could quantify the direction of shape change by calculating the vector
between population means in the multivariate shape space and comparing
the angle between vectors for males and females. Other common
hypotheses in evolutionary ecology generate similar situations (e.g.,
comparison of species in allopatry versus sympatry).
For those interested, the magnitude vs. direction topic has
come up before in the context of using GM data for studies in
quantitative genetics. Monteiro and colleagues have advocated a
Procrustes-distance based approach to assessing shape heritability
(Evolution, 2002: 56:563; Evolution, 2003: 57:196), while Klingenberg
and colleagues have advocated a multivariate shape data approach
(Evolution, 2001: 55:2342; Evolution, 2003: 57:191). Both can assess the
magnitude of shape heritability, but only the multivariate approach can
assess direction (for full discussion I
refer the readers to the above articles). In our research, we found
this does make a difference. We examined shape heritability in multiple
populations and found similar magnitudes of shape heritability among
populations, but also found that shape is evolving in different
directions in the two populations (Myers et al., submitted). This is
important, as it shows that the 2 populations are evolving along
different trajectories. With the Procrustes-distance approach, one fails
to identify this important
aspect of shape change through evolutionary time. As the GM data
captures this rich information, it seems to me that one should use any
and all approaches to assess shape variation, and shape change, through
a combination of analyses.
Best,
Dean
Dean C. Adams, Ph.D.
Assistant Professor
Department of Ecology, Evolution, and Organismal Biology
Department of Statistics
Iowa State University
Ames, IA 50011
tel: (515) 294-3834
fax: (515) 294-8457
web: http://www.public.iastate.edu/~dcadams
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