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-
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
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