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size of each configuration and the coordinates of the landmarks from which differences in
position, scale and rotational effects have all been removed. These new configurations,
shown in
Figure 1.12A
, represent the shapes of all the specimens. To answer the first ques-
tion about the existence of an effect, we regress shape on centroid size using a multivariate
regression in which “shape” is the dependent variable and “centroid size” (or its logarithm)
is the independent variable. For this example, we can conclusively reject the null hypothesis
of no effect; we obtain an
F
-ratio of 94.02 with 28 and 1008 degrees of freedom;
P
10
2
5
.
We can also determine that 72.3% of the shape variation is explained by size. To answer
the second question, we depict the changes either by relative landmark displacement
(
Figure 1.12B
), a deformed grid (
Figure 1.12C
)orboth(
Figure 1.12D
).
Replacing distances with coordinates also does not require us to abandon familiar
ordination methods, such as principal components analysis and canonical variates analy-
sis. These methods are often used to explore patterns in the data in the hope that their
results will suggest the factors responsible for variation among individuals or differences
among groups. At the very least, these analyses can extract the dimensions along which
individuals vary most and groups differ most. The results include scatter-plots of speci-
mens that depict patterns of variation or differences. The interpretation of these scatter-
plots is by the accompanying graphics of the dimensions along which specimens most
vary (
Figure 1.13
) or groups most differ (
Figure 1.14
).
The important distinction between analyses of geometric shape data and conventional mor-
phometric data is that analyses of landmark configurations are necessarily multivariate. By
definition, shape is a feature of the whole configuration of landmarks. Even the simplest shape,
a triangle, cannot be analyzed univariately. Shape data are multidimensional in that each indi-
vidual datum, i.e. each configuration, is described by multiple coordinates. Because we have
defined shape in terms of a whole configuration of landmarks, our analyses must be of that
1
,
3
FIGURE 1.13
Principal components analysis of piranha body shape.
