Biology Reference
In-Depth Information
FIGURE 8.4
The ontogenetic allometry of Serrasalmus gouldingi, depicted as a deformation.
In our second example, the null hypothesis is less obviously dubious because all indivi-
duals are adults and they are nearly the same size; centroid size in this sample ranges
from 69.54 to 76.24, with a coefficient of variation of just 2.1%. We first test the null
hypothesis that there is no relationship between size and shape for the 15 landmarks,
which makes it possible to use the multivariate tests (e.g. Wilks'
Λ
) as well as Goodall's F.
We then test the null hypothesis of no relationship between size and shape using the full
data set of 15 landmarks and 85 semilandmarks, which cannot be tested multivariately
because the number of coordinates (200) vastly exceeds our sample size (104). For the data
restricted to the 15 landmarks, we obtain a value for Wilks'
of 0.5567, which yields an
approximate F-ratio of 2.38, with 26 and 77 degrees of freedom (for numerator and denom-
inator, respectively). The probability of obtaining an F-ratio this large, with those degrees
of freedom, is 0.002. Using Goodall's F-test, the percent unexplained by the regression is
97.61, and the F-ratio is 2.49, with 26 and 2652 degrees of freedom; the probability of
obtaining an F-ratio this large, with these degrees of freedom, is less than 0.005. Because
the assumptions are so dubious, we use permutations to determine the significance of F.
In this case, 0.2% of the permuted values exceed the observed one, so we again conclude
that shape is allometric rather than isometric. The effect, as shown in
Figure 8.5A
, is subtle
so we exaggerate it 10-fold to make it visible. Given that these 15 landmarks provide so lit-
tle information about mandibular shape, especially the complex curvature of the jaw, we
redo this analysis including the 85 semilandmarks. Using Goodall's F-test, we determine
that size does not explain 96.67% of the variation in shape and thus that it does explain
3.33% of the variation. Goodall's F for this case is 3.51, with 196 and 19992 degrees of free-
dom; the probability of obtaining an F-ratio this large, with these degrees of freedom, is
less than 0.005. Using permutations instead to determine the significance of F, we find that
0.1% of the values for F obtained by permutation equal or exceed the observed one, thus
we again conclude that size has a statistically significant, if relatively small, impact on
Λ
