Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-model factor analysis

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 multi-mode data. Conceptual models are developed for two basic patterns of multi-mode data variation, systemand object-variation, and PP analysis is found to apply in the system-variation case. Although PP was originally formulated as a principle of rotation to be used with classic two-way factor analysis, it is shown to embody a latent three-mode factor model, which is here made explicit and generalized frown two to N "parallel occasions". As originally formulated, PP rotation was restricted to orthogonal factors. The generalized PP model is demonstrated to give unique "correct" solutions with oblique, non-simple structure, and even non-linear factor structures. A series of tests, conducted with synthetic data of known factor composition, demonstrate the capabilities of linear and non-linear versions of the model, provide data on the minimal necessary conditions of uniqueness, and reveal the properties of the analysis procedures when these minimal conditions are not fulfilled. In addition, a mathematical proof is presented for the uniqueness of the solution given certain conditions on the data. Three-mode PP factor analysis is applied to a three-way set of real data consisting of the fundamental and first three formant frequencies of 11 persons saying 8 vowels. A unique solution is extracted, consisting of three factors which are highly meaningful and consistent with prior knowledge and theory concerning vowel quality. The relationships between the three-mode PP model and Tucker's multi-modal model, McDonald's non-linear model and Carroll and Chang's multi-dimensional scaling model are explored.