Biology Reference
In-Depth Information
To understand why angles between random vectors of traditional data are so far from
90 , recall how allometric coefficients (k) are calculated as well as how the angles between
vectors are calculated. The coefficients are the power to which body size is raised in the
power law ( Equation 11.1 ). As long as structures grow rather than shrink over ontogeny,
k is invariably a positive number. Moreover, the coefficients rarely differ by much the
most extreme values for S. gouldingi are 0.75 (for eye diameter) and 1.23 (for mid-body
depth and posterodorsal head length). That difference may seem very large because one is
highly negatively allometric whereas the other is highly positively allometric, but the
difference is still numerically very small (in this case, it is less than 0.5). The angles are com-
puted by taking the dot product between the two vectors (after they are normalized to unit
length), which means that we multiply k 1 of one species by k 1 of the other, and add that to
the product of k 2 in one species by k 2 of the other, and so forth. We are therefore summing
products of corresponding allometric coefficients. Because all the elements of both vectors
are positive numbers, the sum of their products cannot be zero much less negative. To
produce an angle of 90 , the sum would have to be zero (because the cosine of 90 is zero).
The angle is necessarily smaller than that often very much smaller (as in the two compari-
sons, above). In striking contrast, allometric coefficients obtained from geometric data can
be negative as well as positive, the angle between random vectors is near 90.0 . Thus, results
from studies based on geometric data yield angles that are more easily interpreted in light
of our expectation that the angle between random vectors should be 90 .
Re-evaluating the Traditional Method for Estimating the Variation
Explained by Scaling
Given that ontogenetic scaling clearly does not explain either data set well, we need
to re-evaluate the inference that we drew from the PCA. When analyzing the traditional
morphometric data, we found that PC1 explained an overwhelming proportion of the
variation in piranhas and the rodents alike. Conducting a PCA of the geometric data for
the two piranhas ( Figure 11.15 ), we again find that PC1 accounts for far more variance
than PC2 does: 64.81% versus 9.45%. The eigenvalue of PC1 is 0.0023 and that for PC2 is
0.00034 so PC1 clearly dominates. But the picture no longer suggests either ontogenetic
scaling or parallel trajectories. It even looks as if the ontogenetic trajectories of both species
bend. What we know from computing the two trajectories that each is linear, and we also
know from computing the angle between that the two vectors are at 34.9 to each other.
The picture shows neither the linearity nor the degree of divergence well. The multigroup
PCA ( Figure 11.16 ) for the two rodents suggests that the two ontogenies are parallel except
that the ontogenetic trajectories are oblique to PC1 which, in this data set, is not the size or
age axis. PC1, which accounts for 53.61%, separates the two species at all ages. PC2, which
accounts for 27.88% of the variance, is aligned with the averaged ontogeny. What the
plot suggests is that the difference between the two species is not constant throughout
ontogeny, implying a divergence in growth allometries.
The most notable difference between the PCs derived from traditional and geometric
data is predictable in analyses of geometric data, size no longer dominates the analysis.
Changes in shape related to size are preserved in data, but geometric scale is not and
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