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the shapes of male and female jaws. For the data restricted to the 15 landmarks, we obtain
a value of Wilks'
Λ
of 0.6964, with 26 and 77 degrees of freedom (for numerator and
denominator, respectively, yielding an approximate F-ratio of 1.29. The probability of
obtaining an F-ratio this large, with those degrees of freedom, when the null hypothesis is
true, is 0.1946. We thus would not reject the null hypothesis that mandibular shape is not
sexually dimorphism. Using Goodall's F-test, the percent of the variance unexplained by
the regression is 96.18, so 3.82% of the variance is explained by the regression. The F-ratio
is 4.06, with 26 and 2652 degrees of freedom, and the probability of obtaining an F-ratio
this large, with these degrees of freedom, is less than 0.005. Only 0.1% of permutations
yield an F-ratio this large or larger, so we can reject the null hypothesis that males and
females do not differ in shape. The effect, shown in
Figure 8.6A
, is exaggerated fivefold
because it is subtle, being most pronounced on the coronoid process, depth of the ramus
and (perhaps) curvature of the incisor. Given that these 15 landmarks provide so little infor-
mation 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 sex
does not explain 95.59% of the variation in shape and thus that it does explain 4.41% of the
variation. Goodall's F for this case is 4.71, with 196 and 19992 degrees of freedom; the proba-
bility 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 con-
clude that sex has a statistically significant, if relatively small, impact on shape. Depicting
FIGURE 8.6
Sexual dimorphism of T. alpinus (A)
based on 15 landmarks, exaggerated 10-fold; (B)
based on 15 landmarks plus 85 semilandmarks,
exaggerated 10-fold.
