The Comparability of the Simple Discriminant Function and Multiple Regression Techniques

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FIRST SUGGESTED by R. A. Fisher (1) and introduced to psychologists by Travers (4) the simple linear discrimin?t function has also been considered at length by Garrett (2) and briefly by Wherry (5). Although Garrett has shown numerically the identity, or proportionality, of the weights obtained by the discriminant func tion to those realized by a multiple regression approach in which each of the independent vari ables is related to the dichotomous crite r i o n of classification through use of the point bi serial coefficient, an analytic solution of the equivalence of the two techniques was not forth coming. Through use of either the point biser ial coefficient or biserial coefficient of correl ation relative to the existence of either a genu ine dichotomy or an artificial dichotomy in the criterion variable Wherry has shown that the application of familiar multiple regression tech niques will lead to the determination of two sets of proportional weights. Although suggesting in the summary of his article that the weights ob tained from application of multiple regression procedures are identical (or proportional) to those found by the discriminant function a p proach irrespective of the assumption regard ing the nature of the criterion variable, Wherry has not shown clearly the comparability of h i s multiple-regression approach to that of the dis criminant function involving use of analysis of variance. Despite the fact that Wherry's normal equa tions resemble closely those associated with the determination of weights in the discriminant function procedure as described by F i s her (1), Travers (4), Garrett (2), andHoel(3), thesums of squares of scores on a given independent var iable (e. g., a test) and the product moments of scores on the two given independent variables (e. g., two tests) are calculated with respect to two different sets of means. In the instance of the multiple-regression approach the two groups composing the criterion variable are combined, and calculations are effected with respect to the single composite mean for each independent var iable; however, when the discriminant function approach is followed two means (one for each of the two criterion groups) are employed with re spect to the determination of sums of squares and product moments.