Biology Reference
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all the coordinates obtained by a GLS superimposition and sum those variances over all
landmarks (this is exactly the same as calculating the trace of the variance covariance
matrix, and can be done in any spreadsheet). This method, while quick and intuitive, will
not provide confidence intervals. It can also be risky if it leads to thinking of variances as
being at landmarks (recall that changes in relative landmark positions are distributed
across landmarks, a topic discussed in context of superimposition methods). Just as change
is not located at a landmark, neither is variance.
ANALYZING THE STRUCTURE OF DISPARITY
To this point we have talked solely about the magnitudes of disparity and variance; but
in many studies we want to know if shapes are randomly distributed throughout the mor-
phospace. A closely related question (do samples occupy the same subspace?) will be
addressed in Chapter 11. Here, we focus on two questions concerning the homogeneity of
morphospace occupation. The first asks how widely shapes are dispersed, i.e. are they as
close together as we would expect if they were randomly distributed. This question is
answered using nearest-neighbor analysis. The second question asks whether there are
clusters and gaps indicating hierarchical structure (which could be phylogenetic or
ecological or both). This question is answered by combining cluster analysis to infer the
hierarchical structure with the cophenetic correlation test to determine whether the
inferred clustering accurately reflects the morphological distances between samples.
Nearest-Neighbor Analysis
Nearest-neighbor analysis, as the term implies, examines the smallest distances between
shapes. From those distances, we can ask whether shapes are more (or less) similar than
expected by chance. If they are closer than expected by chance, we would reject the null
hypothesis in favor of one of clustering; conversely, if they are further apart than expected
by chance, we would reject the null model in favor of a hypothesis of “over-dispersion”
(or “repulsion”). Because the null model is the distribution expected by chance, it is impor-
tant to consider what the reasonable null model might be. One reasonable null model is
that the probability of being at any location in the morphospace is equal (uniform) over
the entire space, and is independent of the shape of any other species. Another reasonable
null model is that shapes follow a normal (Gaussian) distribution. The uniform model
is a reasonable null for comparisons among species, whereas the Gaussian model is
more reasonable when analyzing distributions of individuals around the mean of a
homogeneous sample. Having two null models allows us to guard against accepting a
hypothesis of a particular random distribution.
Nearest-neighbor analysis is another method pioneered by Foote (1990) , so we begin by
reviewing his approach, and then we extend it to geometric shape data. The first step in a
nearest-neighbor analysis is to compute the nearest-neighbor distance D i for each of the N
species (or other groups) in the study. For the sake of brevity, we will refer to “species” as
the units of analysis, but the analysis follows the same protocol even when the units are
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