Biology Reference
In-Depth Information
FIGURE 12.7 The two
forms of symmetry for bilater-
ally symmetric forms. (A)
Matching symmetry; (B) object
symmetry
structures, one on the right side and the other on the left. In this kind of asymmetry,
typified by the two halves of the mammalian mandible, all landmarks are present on
both sides ( Figure 12.7A ). To analyze matching symmetry, we can compare the right to the
left side, using one configuration of landmarks for each side. The alternative is “object
symmetry”, typified by the mammalian cranium, a single structure with an axis of symme-
try along its midline ( Figure 12.7B ). Instead of analyzing two configurations, we analyze
a single one, and there are midline landmarks in addition to the bilaterally paired
landmarks. The ANOVA design and the analysis of the paired landmarks are similar for
both kinds of symmetry. But the analyses differ according to the type of symmetry
because matching symmetry is analyzed using two configurations per individual whereas
object symmetry is analyzed using just one. One important consequence of having all the
bilateral landmarks plus the midline landmarks within a single configuration is that
we can analyze the relationship between the two halves when we have object asymmetry
but not if we have matching asymmetry ( Klingenberg et al., 2002 ).
The analysis of matching asymmetry is more straightforward so we begin with it. We
will assume that every individual is photographed twice on each side. The first step in the
analysis is to reflect all the configurations from one side, e.g. the left side, onto the other
so that we can compare the two sides in the same orientation. If the photographs were not
reflected before they were digitized, the reflection is done by changing the sign of the x -
coordinate for every landmark on one side. Following that reflection, the configurations
for both sides (for all replicates for all individuals) are superimposed using a standard
least squares Procrustes superimposition. Then the right and left sides of all replicates for
each individual are averaged, giving the estimate of the mean shape for that individual.
Variation among the mean shapes of the individuals is the variation explained by the
main factor “Individual”. To calculate the variation explained by “Side”, we compute the
difference between the two sides from the difference in average shapes of each side. That
is, for each individual we calculate its average right-side shape (over the replicate photo-
graphs) and its average left-side shape, and we then calculate the average for each side
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