Biology Reference
In-Depth Information
CANONICALVARIATES ANALYSIS
The purpose of CVA is to simplify the description of differences among
groups
and to
form mathematical functions which may be used to assign specimens to groups, acting as
a multiaxis discriminant function. For example, CVA could be used to describe differences
in mandible shape among queens, soldiers and workers in a colony of ants, and to deter-
mine the rate at which individual ants could be correctly assigned to one of these classes.
It could also be used to describe differences in soldier morphology among colonies, spe-
cies, or more inclusive categories. If individuals in a study can be sorted into mutually
exclusive sets, CVA can be used to describe the differences among those sets. CVA does
not provide a test of differences in the mean forms, that test is best performed using the
General Linear Model framework (see Chapters 8 and 9), which encompasses the familiar
MANOVA methods. CVA does allow estimation of the rate at which specimens may be
effectively sorted or assigned as members of
a priori
groups, which implies both differ-
ences in the mean shape and also some degree of non-overlap in the distributions of traits.
This analysis of assignments or classifications makes CVA a useful complement to tests of
difference in mean form.
There are many similarities between CVA and PCA. Like PCA, CVA constructs a new
coordinate system (the canonical variates, CVs) and determines the scores on those axes
for all individuals in a study. Also, the CVs are linear combinations of the original vari-
ables and are constrained to be mutually orthogonal. However, whereas PCA is used to
describe differences among individuals, CVA is used to describe differences among group
means. In this sense, CVA is analogous to a PCA of the group means. Another difference
between CVA and PCA is that CVA uses the patterns of within-group variation to scale
the axes of the new coordinate system. Because of this rescaling, CVs are not simply rota-
tions of the original coordinate system, and distances in CV space are not equal to dis-
tances in the original coordinate system. (This is where the analogy breaks down.) As a
result of the rescaling, CV1 is the direction in which groups are most effectively discrimi-
nated, which is not necessarily the direction in which the group means are most different.
An important difference between PCA and CVA is that distances computed along the
CVA axes are not equivalent to Procrustes distances and, thus, some care must be taken in
interpreting positions in morphospaces defined by CVA axes. The number of CVA axes
that can appear in an analysis is also limited to the number of distinct groups present
minus one (assuming this is smaller than the degrees of freedom per individual), another
difference relative to PCA, in which the number of meaningful axes is controlled by the
degrees of freedom per measured individual.
Groups and Grouping Variables
A group is a set of individuals that share a particular state of a discontinuous trait.
Examples of groups include sexes, color morphs, species, and supraspecific categories like
guilds. The groups analyzed by CVA must be mutually exclusive, meaning that they can-
not comprise nested or intersecting sets. In other words, the groups differ in the values of
a categorical variable, which is sometimes called a “qualitative trait” or a “grouping
