Biology Reference

In-Depth Information

1. Calculate sample estimates of
ˆ
M
and

*
K
for the
entire object
,

including all bilateral landmarks and midline points.

2. Divide the mean form into
half-forms
. The right half-form

consists of all right-side bilateral landmarks and all (option-

al) midline landmarks. Similarly, the left half-form consists of

all left-side bilateral landmarks and, again, all (optional) mid-

line landmarks.

3. Within each half-form, calculate all distances between bilat-

eral landmarks. Also, if midline landmarks are used, calculate

the distances between each bilateral landmark and each of

the midline landmarks. We do not calculate the distances

between midline landmarks, as they have no bearing on the

analysis. Place the right-side distances into a vector called
R

and the left-side distances into a vector called
L
.

4. Define an
asymmetry vector
(
A
) by elementwise division of the

half-form vectors:
A
=
R/L
. Under the null hypothesis of bilat-

eral symmetry, we expect all of the elements of
A
to be 1.

Elements other than 1 indicate directional asymmetry. If an

element of
A
is greater than 1, it indicates a distance where

the right side is larger. Conversely, an element less than 1

indicates a distance where the left side is larger.

5. Compute marginal confidence intervals for the elements of
A
,

using either nonparametric bootstrapping (Lele & Richts-

meier, 1995) or parametric bootstrapping (Lele & Cole, 1996).

[These methods were introduced in
Chapter 4
.] If the confi-

dence interval for an element of
A
contains 1, the null

hypothesis of symmetry for the corresponding distance cannot

be rejected. However, if the confidence interval excludes 1, the

null hypothesis is rejected, and there is evidence of significant

directional asymmetry.

To illustrate with real data, let us consider how EDMA can be used

to study asymmetry in a clinical context. Our sample consists of eight

children affected with unicoronal craniosynostosis, where the coronal

suture has prematurely fused on one side of the neurocranium (
Figure

vault, the cranial base, and the face. All of the children have left-side

fusion, so we can treat this antisymmetry problem as a directional

asymmetry problem. To make our descriptions clearer, we hereafter

refer to the sides of the skull as either “fused” or “unfused” (as opposed

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