Three-mode factor analysis.

Abstract

Abstract : Three-mode factor analysis is an extension of classical factor analysis for treatment of threeway classification data. It performs a simultaneous factor analysis of all three ways. There are several logical-intuitive interpretations of three-mode factor analysis. The most obvious is in terms of idealized factors. It can be illustrated with sematic differential data. Given a three-way classification set of data like subjects by scales by concepts, arranged in the form of a box, it is possible to reduce this box to a box of idealized subjects by idealized scales by idealized concepts. Each idealized subject is a matrix of scores of the idealized concepts on the idealized scales. Each idealized scale is a matrix of score of the idealized concepts by the idealized subjects and so on. These idealized matrices are sections of the reduced box called the ''inner core matrix.'' Corresponding to the idealized entities we obtain factor-weight-matrices. The factor weights for subjects give a column of weights for each subject, and the score matrix of each subject (scales by concepts) is obtained as a weighted summation of the idealizedsubject matrices; similarly, for concepts and scales. The original scores are reproduced by a triple operation applying consecutively the three sets of weights. Two empirical applications of three-mode factor analysis are reported. (Author)