SIGNIFICANCE TESTS FOR DISCRIMINANT FUNCTIONS AND LINEAR FUNCTIONAL RELATIONSHIPS

In previous papers (Bartlett, 1951; Williams, 1952 a) certain exact tests for the adequacy of a hypothetical discriminant function were derived. Later papers (Williams, 1952b, 1952c, 1953) showed how these tests couild be applied in a number of situations of practical usefulness. The first object of the present paper is to extend the work in the above-mentioned papers and to show how the results obtained may be interpreted in terms of multiple linear regression. The calculations may indeed be carried out in the manner of a covariance analysis. The second object is to develop, along the same lines, exact tests for an assumed linear relationship among v ariables; this is a problem which has been discussed in various contexts by Koopnians (1937), Tintner (1945, 1946, 1950), Geary (1948, 1949), Bartlett (1948), Anderson (1951) and others. Since the question of determining underlying relationships has been given considerable attention in the literature from a number of different points of viewv the opportunity is taken also to discuss and to atternpt to unify the different approaches made-the use of information provided by instrurmental variates, by grouping of the data, and by higher moments. The reason for discussing discriminant functions and functional relationships in the same paper is because the two problems are really different aspects of the same problem. This has been well demonstrated by Geary (1948). If a single discriminant function is assumed adequate to describe differences among a number of p-variate populations, this assumption is equivalent to assuming that there exist p 1 linear relations among the means for the p var iates; the means then lie on a line. In general, postulating that the differences among the populations are described by r discriminant functions is equivalent to postulating p r linear relationships (provided always that the number of populations considered is not less than p). The quantity r, the number of dimensions in which the population means lie. may be called the rank of the populations, and p r the degeneracy. Thus the test for a single linear relationship is equivalent to the test for the adequacy of p 1 discriminant functions. In deriving significance tests for either a discriminant function or a linear relationship, the same principle is applied, though the function being tested has a different role in the two cases, and thus enters differently into the tests. In the simple bivariate case, the test for a linear relationship is exactly the same as the test for the discriminant function which is orthogonal to it. The problems of this paper have been framed above in terms of an analysis of variance model, for testing the significance of differences between populations. This has been done in order to link them with those discuissed in the earlier work (Williams, 1952 a, b). A more general specification is in terms of a regression model, wherein the interrelationships between a set of p variates and another set of q variates are investigated. In such a model a discriminant function is better described as a canonical variate. Throughout the remainder of