Biology Reference
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example,
van Snik et al., 1997
). The converse allometric pattern is seen later, in juvenile
growth, as exemplified by the coefficients of S. gouldingi. These patterns are hardly
surprising, which is reassuring if our aim is to make sense of ontogenetic allometry in
functional and ecological terms. A striking example of a study of allometric scaling is one
that tests the hypothesis that scaling maintains functional equivalence of the mandibular
symphysis (where the right and left sides come together) and resistance to “wish-boning”
(lateral transverse bending). Previous study of adults showed that positive evolutionary
allometry of symphyseal measures (particularly its width) maintains similar load resis-
tance capabilities as increasing symphyseal curvature, and consequent stress concentra-
tions increase with size (
Hylander, 1985
). As a result, adults of species that differ in
size maintain a similar capability to resist loads. Studying the ontogeny of symphyseal
curvature and width, as well as relative stress, in two species of macaques, Vinyard and
Ravosa (
Vinyard and Ravosa, 1998
) found no difference between the two ontogenies in
relative stresses, and also that stress does not change significantly throughout ontogeny in
either species. Thus, ontogenetic allometry maintains the functional equivalence in stress
and strain levels during postnatal growth.
Because theories about developmental controls over the spatiotemporal organization of
relative growth, as well as theories about the functional significance of scaling relationship,
are most easily expressed in terms of traditional morphometric measurements, studies of
allometry using traditional morphometric measurements will remain an important part of
evolutionary developmental biology.
Revisiting Geometric Morphometric Analyses of Allometry
In Chapter 8, we introduced multivariate regression, the method we use to analyze the
relationship between shape and size. To review that, the model for allometry is:
f
Y
1
;
Y
2
;
Y
3
;
...
Y
P
g
5
f
m
1
;
m
2
;
m
3
;
...
m
P
g
X
1
f
b
1
;
b
2
;
b
3
;
...
b
P
g
1
fε
1
; ε
2
; ε
3
;
...
ε
g
(11.4)
P
where {Y
1
, Y
2
, Y
3
...
Y
P
} is the vector of shape variables, X is centroid size and {m
1
, m
2
, m
3
,
...
ε
P
} are vectors of slope coefficients, intercepts and
residuals, respectively. We would not use principal components analysis for geometric
analyses of allometry because PC1 of geometric shape data need not be aligned with the
ontogenetic trajectory whenever there are factors other than age in the data (e.g. sexual
dimorphism). Of course it could be aligned with the ontogenetic trajectory, but there is no
reason to use PCA when multivariate regression is guaranteed to give the optimal descrip-
tion of the dependence of shape on size. Also, given that we have a size metric (centroid
size), there is no need to estimate “size” by a linear combination of the measured (shape)
variables.
As is always the case, it does not matter which shape variables we use because we
obtain the same trajectory regardless of whether we use coordinates obtained by a general-
ized Procrustes analysis, by partial warps plus uniform components or the full set of PC
scores. But even though the complete description of the ontogenetic change does not
depend on the choice of variables, the coefficients obviously do. Thus, in striking contrast
m
P
}, {b
1
, b
2
, b
3
,
b
P
} and {
ε
1
,
ε
2
,
ε
3
,
...
...
