Biology Reference
In-Depth Information
FIGURE 1.3
All 120 measurements between endpoints defined by the 16 landmarks of
Figure 1.2
.
coefficients, principal component loadings) for each trend or difference. Analyzing
all 120 requires specialized methods beyond the scope of this topic (see
Lele and
Richtsmeier, 1991, 2001; Richtsmeier and Lele, 1993
). We might be tempted to cull the
120 measurements, retaining only those that seem most likely to be informative but, until
we have done the analysis, we cannot know which can be safely culled. Clearly, we need
another way to get the same shape information as the 120 measurements, but without
the excessive redundancy.
Another problem common to the truss and more traditional schemes is that all the
measurements are of size
each measurement is the magnitude of some dimension, such
as length, width, area, all of which are measures of size. That does not mean that the data
include no information about shape
they do. But that information is contained in the
ratios among the lengths, and it can be surprisingly difficult to extract it because that
requires separating information about shape from that about size. Some studies have ana-
lyzed ratios directly but ratios pose serious statistical problems (debated by
Atchley et al.,
1976; Corruccini, 1977; Albrecht, 1978; Atchley and Anderson, 1978; Dodson, 1978; Hills,
1978
). The more usual approach is to construct shape variables from linear combinations
of length measurements, such as Principal Component (PC) loadings. Here, one compo-
nent, usually the first (PC1), is interpreted as a measure of size and all the others are inter-
preted as measures of shape. However, PC1 includes information about both shape and
size, as do all the other PCs. The raw measurements include information about both shape
and size and so do their linear combinations.
Not only are the methods of separating size from shape problematic; the whole
idea
of “size and shape” has been one of the most controversial subjects in traditional morpho-
metrics. One reason for this controversy is the multiplicity of definitions of size (and there-
fore also of shape), several of which are articulated by
Bookstein (1989)
. Virtually any
approach to effecting this separation can be disputed on the grounds that the notion of
