Biology Reference
In-Depth Information
TABLE 11.6
Two-way Multivariate Analysis of Variance (MANOVA) of Piranha Body Shape Between
Species (“Taxa”) and Developmental Stage (“Stage”)
Effect
df
Pillai's Trace
Approx F
Num df
Den df
P
Taxa
1
0.99751
1642
30
123
,
0.0001
Size
1
1
1230192
30
123
,
0.0001
taxa:level
1
0.9971
1411
30
123
,
0.0001
Residuals
152
Examples: Applying These Methods to Data
We first compare the ontogenetic trajectories of the two piranha species (S. gouldingi
and S. manueli) and then those of the two rodent species, the cotton rat (S. fulviventer) and
house mouse (M. musculus domesticus) using traditional morphometric data. Next we
conduct both comparisons using geometric morphometric data.
The hypothesis of ontogenetic scaling predicts that species differ only in adult body
size, not in either juvenile shape (measured at the intercept, or at a comparable develop-
mental stage), or the direction of ontogenetic shape change. Given this hypothesis, we can
use MANCOVA (see Chapter 9) to test it. But MANCOVA presents a problem because the
substantive biological hypothesis (ontogenetic scaling) is equivalent to the statistical null.
Normally, the null is the hypothesis that we would like to reject, and we use various strat-
egies to ensure that we do not reject it too readily. But the hypothesis of scaling is the one
we wish to accept, so we are put in an odd position. Procedures that prevent rejecting a
false null hypothesis, such as using a very conservative test, and factors that can reduce
our ability to reject it, such as small sample size, can lead us to accept a false hypothesis of
scaling. Thus, the inference of ontogenetic scaling will not be convincing if the test is
conservative or the sample size is small. Presuming that the sample size is large enough to
reject a false null hypothesis, finding that only the covariate is a significant term supports
a hypothesis of ontogenetic scaling. In the analyses of both the two piranhas and the
two rodent species, we use MANOVA rather than MANCOVA because we are comparing
the mean shapes for each species at two developmental stages rather than using a continu-
ous factor (size) as a covariate. We find that both factors, plus the interaction term, are
highly significant statistically in the comparison of the two piranhas (
Table 11.6
).
Nevertheless, the angle between allometric vectors appears to be very small, 6.27
, which
corresponds to a very high correlation of 99.4. Nonetheless, using the bootstrapping proce-
dure described above, the within-species angles are just 1.5
and 2.0
so 6.27
is signifi-
cantly greater than 0.0
(P
0.05). A permutation test similarly determines that the two
species differ significantly (at P
,
0.001). Both factors, plus the interaction term, are also
highly significant statistically in the case of the two rodents (
Table 11.7
) but, again, the
correlation between the vectors is very high: 0.984, corresponding to an angle of just 10.4
.
Again, using the bootstrapping procedure to test the statistical significance of the angle
between vectors, we find that within-species angles are just 4.2
and 3.4
so the observed
angle of 10.4
is significantly greater than expected by chance. Thus, in both cases, the
,
