Biology Reference
In-Depth Information
over all individuals in the population. The difference between those two averages is the
variation explained “Side”. Measurement error is quantified by the variation over repli-
cates for each individual and side. This error is a combination of the photographic error
plus digitizing error. The statistical significance of “Side”, the fixed factor, is tested against
the interaction term and the statistical significance of FA is tested against measurement
error. The degrees of freedom for each term are the univariate degrees of freedom (e.g. 1
for side) multiplied by the dimensionality of the data.
The analysis of object asymmetry is more complex because we have only one configura-
tion per replicate. As a result, we cannot superimpose the right side onto a reflected left
side. Instead, we superimpose each configuration onto a reflected copy of itself. We thus
start by making that reflected copy, which involves first reversing the sign of the
x
-coordi-
nate of each landmark and then renumbering those landmarks to correspond with the
number it had in the original copy. For example, if landmark 2 on the left side is bilater-
ally homologous with landmark 4 on the right side, we would relabel the reflected land-
mark 4 as landmark 2 and similarly relabel landmark 2 as landmark 4. Then the original
and reflected configurations are superimposed (using the standard least squares
Procrustes superimposition). Following superimposition, each individual is symmetric and
the midline landmarks line up as do the midpoints between the paired landmarks.
Because the right and left sides of the paired landmarks are now redundant, the whole
configuration is described by the landmarks on just one side.
The variation of the paired landmarks is unconstrained
they can vary in all direc-
tions. But the variation of the midline landmarks is constrained
they can only vary in
one direction, which is along the midline. Any variation away from the midline would
mean that the midline is not the midline. Thus, when calculating the dimensionality of
shape (and the degrees of freedom for each effect) the constraint on the variation of mid-
line landmarks must be taken into account. For the symmetric component of variation,
there are 2
K
2 dimensions, where
K
is the number of paired landmarks and
P
is the
number of midline landmarks (this notation differs from that of
Klingenberg et al. (2002)
because they use
L
for the number of midline landmarks but we have used
L
for the num-
ber of semilandmarks). So, if there are 20 paired landmarks and 5 unpaired landmarks,
there are 40
P
1
2
2 dimensions. For the symmetric component of a two-dimensional con-
figuration, only one dimension is used up by translation and none are used up by rotation
(hence
5
1
2
4).
In the analysis of object symmetry, the “Sides” component of the variation is calculated
from the difference between the original and reflected configurations. In the asymmetric
component of variation, just like in the symmetric component, the asymmetry of the
paired landmarks can be in any direction. This means that for
K
pairs of two-dimensional
landmarks there are 2
K
dimensions of variation. But asymmetry of the midline landmarks
is possible only in the direction
perpendicular
to the midline (or median plane)
2 rather than
2
2
if the vari-
ation was along the midline it would not be asymmetric. One consequence of this
restricted variation is the reduced dimensionality of the asymmetric component; the mid-
line landmarks add
P
dimensions, but their sum must be zero, eliminating a degree of
freedom (from both two- and three-dimensional data). The rotation step of the Procrustes
superimposition removes one additional degree of freedom from two-dimensional data
and two from three-dimensional data but none are used up by scaling or orientation of
