Biology Reference
In-Depth Information
There is one special case of multiple triangles that is worth singling out: multiple
triangles describing two sides of a bilaterally symmetric organism. If we are interested
specifically in asymmetry, both sides contain relevant information because we are then
interested in the difference between sides. Otherwise, the two sides are redundant and
using both sides in the analysis implicitly treats them as if they were independent and
inflates our degrees of freedom. We can avoid this problem using the coordinates we
have obtained, reflecting one side across the midline and averaging the coordinates of
the two sides. This approach also allows us to use partially fragmentary specimens with
landmarks present on only one side or the other because, for these specimens, the “average”
for a landmark is obtained from the one side on which it was preserved.
Choosing the Baseline
When we calculated shape coordinates, we chose one side of the triangle to serve as a
baseline. An obvious question is whether our results might depend on that choice. One
important consideration is that variation will be transferred from the baseline landmarks
to the others. There are few, if any, truly invariant landmarks and when two are fixed,
their variance must be put
somewhere
. Some landmarks are difficult to digitize and these
should not serve as an endpoint of the baseline because their noise will be transferred to
all the other landmarks. What makes the transfer of variance really worrisome is that it is
not necessarily unbiased, rather it is related to the distances of the free landmarks to the
baseline (
Dryden and Mardia, 1998
). A baseline that runs through the centroid of the form
is preferable to one that is far from most other landmarks, as it minimizes this effect.
Additionally, when choosing the endpoints of the baseline, we do not want points that are
too close to each other because any highly localized variation in shape may be common to
both those points and the variance local to those baseline landmarks will be transferred to
all the other landmarks. As
Bookstein (1991)
has argued, the scatters for different sets of
shape coordinates of the same triangle to different baselines differ mainly by translation,
rotation and rescaling but the inhomogeneity of variance and the correlations among land-
marks induced by fixing two points could be problematic for statistical analysis. Another
consideration, but one that is primarily a matter of interpretability, is the orientation of the
baseline. If the baseline rotates relative to a body axis it does not compromise the statistical
analyses, but it can make interpretations based on graphics difficult
it might seem that
all the landmarks are moving away from the baseline in the posterodorsal direction,
for example, when the baseline rotates in the anteroventral direction. Ideally, therefore,
we want endpoints of the baseline to be along the longest diameter of the form that passes
through the centroid of the form, so long as those points are not especially unreliable and
the longest diameter does not rotate.
To see the consequences of fixing various endpoints, we can consider the ontogenetic
series of the piranha
Serrasalmus gouldingi
. We have used landmarks 1 and 7 as the
endpoints of the baseline; both the scatter of coordinates and the depiction of ontogenetic
change resulting from that choice can be seen in
Figure 3.6A,B
. We could have used two
dorsal landmarks (3 and 5), producing a strikingly different scatter-plot that implies
considerably more variation (
Figure 3.6C
), as well as a strikingly different picture of
