PARAFAC: An “Explanatory” Factor Analysis Procedure

Simple structure and other common principles of factor rotation do not in general provide strong grounds for attributing explanatory significance to the factors which they select. In contrast, it is shown that an extension of Cattell's principle of rotation to proportional profiles (PP) offers a basis for determining explanatory factors for three‐way or higher‐order multimode data. Conceptual models are developed for two basic patterns of multimode data variation, system‐ and object‐variation, and PP analysis is found to apply in the system‐variation case. The generalized PP model provides the basis for a program called PARAFAC (for parallel factors). This program is demonstrated to give unique “correct” solutions with oblique nonsimple structure and even nonlinear factor structures. A series of tests, conducted with synthetic data of known factor composition, provides data on the minimal necessary conditions of uniqueness and the properties of the analysis procedures when these minimal conditions are not fulfilled. PARAFAC has been applied to several different sets of real data with meaningful results (as reported elsewhere in this meeting).