Introduction - Geometric Morphometrics for Biologists - page 13

that are not in shape, we are left with only the differences in shape. We can now use the
coordinates of the final configurations ( Figure 1.10D ) to analyze these shape differences.
When we have sampled the curves between landmarks, we also have to remove any
non-shape variation resulting from our (arbitrary) choice of where to sample those points
along a curve.

Siz Kendall's definition of shape mentions scale as one of the effects to be filtered out.
The implication is that “scale” is the definition of size that is complementary to shape
under some models of error. The two are ideally geometrically independent (i.e. orthogo-
nal). The concept underlying geometric scale is quite simple, and may be intuitively obvi-
ous by visual inspection. As you can see in Figure 1.10A , the landmarks are generally
further apart in one configuration than in the other. That is obviously what we expect
when a configuration is larger, whether because the organism is larger or the photograph
of it is larger. To calculate geometric scale, we compute the distances of all the landmarks to
the center of the form (its “centroid”); Figure 1.11 shows the location of the centroid and the
segments connecting the landmarks to it. Now we can compute geometric scale by calculat-
ing the square of each distance from landmark to the centroid, summing those squares and
taking the square root of their sum. This quantity is called “centroid size”.
Centroid size is the one measure of size that is mathematically independent of shape.
It is, more precisely, orthogonal to shape. That is a matter of definition, not biology. In bio-
logical data, centroid size may often be empirically correlated with shape because larger
organisms often are shaped differently than smaller ones. The fact that we have defined
and measured shape and size separately does not mean that we are assuming them to be
biologically separate. Nor does their separation cause us to lose information about the rela-
tionship between size and shape. We can easily analyze that relationship by conventional
statistical methods.

FIGURE 1.11 A visual repre-

sentation of centroid size as
computed for 16 landmarks on a
piranha. The open circle is the
centroid; the segments connecting
the centroid to the landmarks
represent the distances used to
compute centroid size.