Analysis of the Discriminant Function in Educational and Psychological Research

Abstract

AN EXTREMELY useful statistical tool that has been largely ignored by educational and psychologi cal researchers is the analysis of the discriminant function. In the December, 1963, issue of the Re view of Education Research, which is devoted en tirely to statistical methodology, discriminant anal ysis is not even mentioned, although there is a chap ter on regression and correlation, to which the dis criminant function is closely alliedc Of books on statistics familiar to educators and psychologists, only that by McNemar (4) and another by Guilford (2) devote some space to the topic. Further evi dence of the neglect of this technique is furnished by the fact that Tatsuoka and Tiedeman, in their chap ter on statistics in the Handbook of Research on Teaching (5) mentioned no more than five studies that used discriminant analysis since 1938. By con trast, the same authors said that factor analysis, which is another multivariate technique, was exten sively used in educational and psychological re search. This phenomenon is puzzling in view of the fact that discriminant analysis is both useful and com paratively easy to comprehend as a statistical con cept. While the usual theoretical discussion of the rationale of the technique is couched in terms of ma trix algebra, an understanding of it is possible to anyone with a background of high school algebra? It is the belief of the author that as researchers be come more familiar with the technique they will make greater use of it than they have hitherto. While discriminant analysis can be used for more than two groups, this paper will deal with only the two group situation. Even with the two group situa tion the potentials of the technique are practically unlimited. In the study of high school dropouts, for instance, discriminant analysis can be used to dif ferentiate successful high school students and poten tial dropouts. Over achievers and underachievers can be studied in a similar fashion. School counse lors, seeking information as to which of two similar occupations or professions a counselee belongs i n, may find himself in a better position to counsel if he knows the discriminant function that best differenti ates the two occupations. In brief, the discriminant function is useful in any situation where subjects can be dichotomized in any meaningful way, such as suc cess or failure, extraversion or introversion, med icine or dentistry, etc. This paper will not go into a detailed discussion of the theoretical basis of discriminant analysis, as excellent articles are available for those with a knowledge of matrix algebra as, for instance, one by Travers (6). Rather, this paper will attempt to de scribe what the discriminant function does and illus trate its use by a concrete example that c an be fol lowed by anyone familiar with high school algebra. The computations, which were made on a Monroe calculator, were checked against a computer pro gram1 which made use of matrices and inverse ma trices in its calculations. The results were identi cal within rounding errors. In a real sense, discriminant analysis is the mul tiple regression technique with a different purpose. Whereas in multiple regression the purpose is to seek a composite score (predicted score) that is a linear function of several independent variables such that it has maximum predictive power for agiven group, in discriminant analysis the purpose is to seek a composite score (the discriminant function) that is a linear function of several independent vari ables such that it can best differentiate two groups. In either case the problem is to find the proper co efficients (weights) for the independent variables such that the composite score is functionally op timal. Algebraically, if we let the composite score be Z, and Xi, X2, X3 . . . Xfc be the independent vari ables, then Z is a linear function o?XlyX2f. . . Xfc, where k is the number of independent variables.