Biology Reference
In-Depth Information
CHAPTER
6
O
rdination Method
s
In this chapter, we discuss two methods for describing the diversity of shapes in a sam-
ple: principal components analysis (PCA) and canonical variates analysis (CVA). Our dis-
cussion of these methods draws heavily on expositions presented by
Morrison (1990)
,
Chatfield and Collins (1980)
and
Campbell and Atchley (1981)
. Both methods produce
ordinations that simplify descriptions, or provide tools for exploratory data analysis.
These ordinations are descriptions of the data, not tests of hypotheses. However, CVA
may also be used as multigroup discriminant function, in which the rate of correct assign-
ment of specimens to groups based on shape is used to support specific hypotheses related
to the ability to assign individuals to different species (
Nolte and Sheets., 2005; Costa
et al., 2008; Van Bocxlaer and Schultheiß, 2010; Williams et al., 2012
) or as a diagnostic tool
(
Menesatti et al., 2008
,
Yee et al., 2009
). PCA is a tool for simplifying descriptions of varia-
tion among individuals, whereas CVA is used for simplifying descriptions of differences
between groups. Both analyses produce new sets of variables that are linear combinations
of the original variables. They also produce scores for individuals on those variables, and
these can be plotted and used to inspect patterns visually. Because the scores order speci-
mens along the new variables, the methods are called “ordination methods”. It is hoped
that the ordering provides insight into patterns in the data, perhaps revealing patterns
that are convenient for addressing biological questions. The most important difference
between PCA and CVA is that PCA constructs variables that can be used to examine vari-
ation among individuals within a sample, whereas CVA constructs variables to describe
the relative positions of groups (or subsets of individuals) in the sample.
We discuss PCA and CVA in the same chapter because they serve a similar purpose,
and because the mathematical transformations performed in them are similar. We describe
PCA first because it is somewhat simpler, and because it provides a foundation for under-
standing the transformations performed in CVA and in other related methods. We begin
the description of PCA with some simple graphical examples, and then present a more for-
mal exposition of the mathematical mechanics of PCA. This is followed by a presentation
of an analysis of a real biological data set. The description of CVA follows a similar out-
line; the only difference is that we begin with a discussion of groups and grouping
