Modelle der dreimodalen Faktorenanalyse: formale Eigenschaften, theoretische Zusammenhänge und ihre Implikationen für das Konzept individueller Differenzen

in Psychologische Beiträge (1991), 133

The present paper is concerned with methods of three-mode factor analysis to obtain a dimensional representation of three-way data. Classifying the methods by the number of derived spaces and their ... [more ▼]

The present paper is concerned with methods of three-mode factor analysis to obtain a dimensional representation of three-way data. Classifying the methods by the number of derived spaces and their interrelations yields two distinct classes of models: CANDECOMP (Carroll & Chang, 1970), PARAFAC (Harshman, 1976) and SUMMAX (Orlik, 1980) rest on a basic trilinear decornposition of the data defining a separate space for each mode, whereas Tucker's (1964a thrce-mode factor analysis and SUMMA X in its extended form use a quadeilinear model specifying an additional core matrix. Associated with the current classification are different properties of the two model classes which refer to the number of substantial dimensions, their interpretation and the orientation of dimensions which is subject to rotations within the quadrilinear dass and uniquely determined by trilinear methods. Considering the different characteristics of the methods, formal relations between the classes have been found under very restrictive conditions only. However, there exist some general connections between trilinear and quadrilinear models. CANDECOMP and PARAFAC derive from the trilinear SUMMAX model by rescaling and permutation of axes, and the methodological link between the Tucker model and SUMMAX is given by orthogonal rotations of the SUMMAX configuration. These relationships are shown in an empirical example and their implications for the distinct concepts of individual differences within the two classes of methods are discussed. [less ▲]