A Synthesis of Multivariate Techniques to Distinguish Patterns of Growth in Grasshoppers

Abstract

The discriminant function which best separates the means of two groups of well-defined and objectively distinguishable organisms is, in effect, a vector which expresses the contrast between the patterns of growth of the organisms: when the two groups are polymorphic forms of a species, the generalised distance associated with the discriminant function is a measure of the polymorphism. This generalised distance represents the efficacy of the discriminant function in separating the two groups in a space of as many dimensions as there are characters measured (Rao [1952]). Furthermore, when the characters which enter into the discriminant function are the dimensions of parts of the body or of its appendages, the vector then represents a change of shape, adequately represented in proportion to the extent to which the chosen suite of characters covers the main parts of the organism. A description, with worked examples, of the way in which such suites of characters may be combined into a single measure of differences of shape is given by Rao [1952]. When more than two groups are compared in this way, the discriminant functions which link them to one another may differ in direction as well as in length, and the generalised distance between the positions of the means of each group can be used as a measure of the degree of likeness of one shape to another, whereas the direction of the discriminant functions reveals the qualitative distinction between dissimilar changes of shape. In this way each vector may readily be given its appropriate biological identification, although it may not be so easy to determine the number of different kinds of polymorphism