7 - Ordination methods

Abstract

This chapter discusses two methods for describing the diversity of shapes in a sample: principal components analysis (PCA) and canonical variates analysis (CVA). The discussion of these methods draws heavily on expositions presented by Morrison (1967), Chatfield and Collins (1980), and Campbell and Atchley (1981). Both methods are used to simplify descriptions rather than to test hypotheses. PCA is a tool for simplifying descriptions of variation 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 specimens along the new variables, the methods are called “ordination methods.” The most important difference between PCA and CVA is that PCA constructs variables that can be used to examine variation among individuals within a sample, whereas CVA constructs variables to describe the relative positions of groups (or subsets of individuals) in the sample. PCA and CVA both serve a similar purpose, and the mathematical transformations performed in the two analyses are similar. The chapter describes PCA first because it is somewhat simpler, and because it provides a foundation for understanding the transformations performed in CVA. It begins the description of PCA with some simple graphical examples, and then presents a more formal exposition of the mathematical mechanics of PCA. This is followed by a presentation of an analysis of a real biological data set.