Biology Reference
In-Depth Information
“size” being separated from “shape” is not really “size”. Also, some workers argue that no
such separation is biologically reasonable; see, for example, the discussion of studies of
heterochrony based on growth models by Klingenberg (1998) . However, even if we accept
that size and shape are intimately linked by biological processes, we still want to know
more about their relationship than the mere fact of its existence.
Extracting information about the relationship between size and shape from a set of
measurements can be especially difficult when the organisms span a broad size range. For
example, when some organisms in a population are 20 mm long and others are 250 mm,
all measurements will differ in length. Even if shape is not much influenced by a tenfold
change in size, all measurements will still be correlated with size. Quantifying that fact
is merely restating the obvious. Consequently, we should expect size to be the dominant
source of variance in traditional morphometric data because these measurements are mea-
surements of size. We have redundantly measured the same factor and it will therefore be
the dominant factor in the data. But we should be concerned about the possibility that the
variance in shape is not so much explained by the variance in size, as it is simply over-
whelmed by it. For instance, in analyses of ontogenetic series of two species of piranha
(one being the running example throughout this chapter), we find that 99.4% of the vari-
ance within each species is explained by PC1. That suggests that there is nothing else to
explain because it is hard to imagine that the remaining 0.6% is anything but noise. And
yet, we do not actually know what proportion of shape variation is actually explained by
size. What that quantity, 99.4%, tells us is the proportion of variation in measurements of
size that is explained by size.
Finally, one serious limitation of traditional morphometrics is that the measurements
convey no information about their geometric structure. If we strip off the line segments
connecting the landmarks in Figure 1.3 and just look at the position of the landmarks
on the page ( Figure 1.4 ), we can see that some are close to each other (e.g. 12 and 13) and
others are far apart (e.g. 1 and 7); some are ventral (9 and 10) and some are dorsal (3 and 5),
others are anterior (e.g. 1 and 2) and still others are posterior (6
8). That information
about position, which is so important to morphologists, is contained in the coordinates of
the landmarks but not in the list of distances among them
not even in the more com-
prehensive list of 120 measurements. But the x - y -coordinates of the 16 landmarks contain
all that positional information in addition to all the information contained in the 120 dis-
tances between all pairs of landmarks. Those distances can be reconstructed from the
coordinates if the units of the coordinate system are known. More importantly, simple
algebraic manipulations allow us to partition the information into size and shape and to
strip off irrelevant information like the position of the organism in the photograph and its
orientation in the photographic plane. After we have removed that irrelevant information,
and separated shape from size, we have fewer than 32 shape variables but we have all
the information about the geometric structure of our landmarks that was captured when
we digitized the specimens. We also have all the information present in the full list of
120 measurements without its redundancy. Consequently, we do not need to cull the data
in advance of the analysis, and we do not thereby lose any information we might have in
the data. In addition, partitioning morphological variation into size and shape means that
variance in size does not overwhelm variance in shape even when variance in size is
relatively large. In the two species mentioned above (in which PC1 accounts for 99.4% of
Search WWH ::




Custom Search