Biology Reference
In-Depth Information
independent variable, as discussed above in the context of a simple bivariate regression.
Under the null hypothesis that the dependent variable is uncorrelated with the indepen-
dent variable, the significance of the observed F-ratio (the mean summed squared
Procrustes distance explained by the model relative to the mean square residual expressed
in the same units) is tested by comparing that observed value to the distribution of F-ratios
produced via random permutation (discussed in greater detail, below). If the observed F-
ratio is extreme relative to that obtained from the permuted values (at some desired alpha
level), then the null hypothesis (i.e. that X is not a linear predictor of Y) may be rejected at
that alpha level.
Examples: Testing the Null Hypothesis that
Y
We test the null hypothesis that X is not a linear predictor of Y, using centroid size as
our X variable, for a sample comprising an ontogenetic series of Serrasalmus gouldingi,a
data set measured at 16 landmarks (the case shown above [see
Figures 8.1, 8.2
]whenwe
checked the assumption of linearity). The null hypothesis is thus that shape is unrelated to
size over ontogeny, i.e. that growth is isometric. If we can reject that null hypothesis, we
can say that shape is a function of size. We also test this same null hypothesis for a second
case, a sample of adult alpine chipmunks, T. alpinus, captured between 1911 and 1919
from an elevational transect through the Sierra Nevada mountains near Yosemite. This
second example differs from the first in that the sample comprises solely adults and also
because the data comprise 85 semilandmarks as well as 15 landmarks.
In the analysis of S. gouldingi, a regression of the full set of partial warps on the natural
log of centroid size yields a value of Wilk's
X
is Not a Linear Predictor of
Λ
of 0.006785 corresponding to an F-statistic of
10
2
7
). Thus, it is highly improbable
that the null hypothesis is true; we would therefore reject it in favor of the alternative,
which is that shape is allometric, meaning that it changes as a function of size. However,
the sample size is fairly small (N
47.05 with 28 and 9 degrees of freedom (P
5
4.46
3
38) and, as noted above, there is good reason to worry
5
about the reliability of the Wilk's
statistic when the sample size is small. We could
instead use Goodall's F-test, which determines the proportion of the shape variation that
is not predicted by size, summing the squared Procrustes distances between the observed
and expected shape for each individual, given its size. From that sum, we conclude that
27.66% of the shape variance is not explained by the regression. Thus, 100%
Λ
27.66%
5
72.34% of the shape variance is explained by size. We obtain an F-ratio of 94.199, with 28
and 1008 degrees of freedom (for numerator and denominator, respectively). From the tab-
ulated values of F, we would conclude that the probability of obtaining such an extreme
value when the null hypothesis is true is less than 0.00001. Considering the dubious
assumptions of this test, we can use permutations to determine the significance of the
F-ratio and, not surprisingly, only 0.1% of the F-ratios obtained by permutations are as
large as the one obtained from the data. Thus, we can reject the null hypothesis that shape
is unrelated to size over ontogeny of this species. The magnitude of this effect can be
appreciated visually by the depiction of the regression as a deformation (
Figure 8.4
). That
magnitude can be quantified by the Procrustes distance between the shapes at the lowest
and highest values of the independent variable, which is 0.21.
2
