Three-mode data structures, methods, models, and algorithms. A short survey [abstract]

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Three mode data structures, methods, models, and algorithms. A short survey.

Pieter M. Kroonenberg C Dep. of Education, Univ. of Leiden).

The main purpose of the lecture was to provide a general overview of the main issues in three-mode data analysis. To this end a wide-ranging set of topics was discussed as can be gathered from the title of the lecture.

'/V;;vc modi' data stmcttovf.. In many cases it is useful to follow Carroll & Arabie C1980, Annual Review of Psychology) in making a distinction between the "ways" and the "mode" of a data box. llu' former give the dimensions of the data box, i.e. rows, columns, and tubes, while the latter indicate the research entities that make up the data, i.e. variables, subject, conditions, occasions etc. Using this distinction, many kinds of data are shown to fit data boxes, such as profile data, longitudinal data, (.dis') similarity data, sets of covariance (correla-tion) matrices. Less typical examples are three-way factorial designs and three-three-way contingency tables.

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kind of principal component analysis.

Three-mode models. After a discussion of the singular value decomposition for two-mode data, several three-mode models were presented as gener-alizations of this singular value decomposition, in particular the Tucker 2 model, PARAFAC/CAN-DECOMP, IDIOSCAL, INDSCAL, and three-mode scaling. The interrelationship between these models was pointed out, and finally it was shown that the Tucker models can be seen as higher-order principal component models.

Three-mode algorithms. For the special case of the Tucker 2 model the parallelisms were pointed out between Tucker's original methods (1966, Psychometrika). and recently developed alternating least squares algorithms. In particular, the parallel structure of Kroonenberg and De Leeuw's ( 1980, Psychometrika) algorithm TUCKALS 2 with that of Tucker's Method I, and the similar correspon-dence between Murakami's (1983, Behaviormetri-ka) algorithm and that of Tucker's Method III.

Application of TUCKALS 2. To give an indica-tion of the kind of analysis one might carry out with three-mode models, a data set was reanalysed which contains the information of the physical growth of Japanese girls between the ages 6 and 14. The main purpose of this analysis was to investigate the differences that exist between the rates of growth of these girls.