Further Applications of EDMA - An Invariant Approach to Statistical Analysis of Shapes - page 263

asymmetry and antisymmetry is a matter of degree, where the distri-
butions of A are unimodal and bimodal, respectively. Antisymmetries
are seen both in clinical samples and natural populations. A well-
known clinical example is hemifacial microsomia, where the bones on
one side of the face are markedly undersized (compared to normal-
sized bones on the opposite side). Another example is posterior
plagiocephaly, where one side of the posterior neurocranium appears
“flattened.” In both of these examples, there is no handedness that
characterizes the patient population as a whole, so that right- and left-
side deformities are expected to occur with equal frequency.
Antisymmetries also occur in nature, where most species of fiddler
crabs in the genus Uca are a familiar example (Jones & George, 1982).
In these taxa, each male fiddler crab has one claw that is dramatical-
ly enlarged; however, some males will have an enlarged right-hand
claw and others will have an enlarged left-hand claw (so there is,
again, no handedness that characterizes the population as a whole).
7.1.1. Studying Asymmetry Using EDMA
We now consider how EDMA can be applied to the analysis of
asymmetry when form is measured using landmark coordinate data. In
using EDMA, we can generalize the univariate examples given in the
previous section to analyses where many interlandmark distances are
considered. As with other EDMA applications, we are not only interest-
ed in determining whether a form difference occurs, but we are also
interested in where it occurs. In other words, our studies of asymmetry
aim to localize the differences in form between the left and right sides.
To quantify asymmetry patterns, we use form matrices, but we
modify the way that they are constructed. In all previous applications,
we described form differences between two observations or two sample
means. However, in studying asymmetry, we want to measure form dif-
ferences within observations (that is, between the right and left sides
of the same observation). The algorithm for studying directional asym-
metry , which is the most straightforward type to study, is presented
below. The basics of this algorithm (where a form difference matrix is
used to compare sides of the same observation) were introduced by
O'Grady and Antonyshyn (1999); we have extended it to include the
computation of confidence intervals.