A Biometrics Invited Paper. Factor Analysis: An Introduction to Essentials II. The Role of Factor Analysis in Research

Abstract

If, as the extensive findings with simple structure, supported by the few presently available confactor rotations, suggest, oblique factors are the rule in nature, with orthogonality as a special, rather rare, case, it behoves us to outline more fully the implications of the oblique model, for it has certain important new properties. Two main consequences appear, both unknown in the orthogonal case: (1) The correlations of a factor with its variables, on the one hand, and its loadings on them on the other, which fuse into the same values in orthogonal factors, here become recognizably and importantly different; (2) From factoring the correlations among factors one can derive second and higher order factor systems which do not exist in the orthogonal case. The differentiation of a correlation and a loading can be shown geometrically in Figure 3, which also brings out the distinction between reference vectors and factors. The reference vector is the perpendicular to a hyperplane, and it is by examination of the projections formed by correlations with the reference vector that we perform rotations for simple structure. The corresponding factor is the line of intersection of the hyperplanes of all the reference vectors other than the one perpendicular to the hyperplane. Corresponding reference vectors and factors are shown in Figure 3, whence one sees that where the former are positively correlated the latter are inversely correlated. In the orthogonal case, reference vectors and factors are one and the same. As Figure 3 shows, the lines of projection of variables on factor axes, as in all nmathematical oblique representation, must run parallel, in getting projections on one factor, to the other factor axis. Thus, as is easily seen, the correlations of variables with a reference vector are