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Three-Mode Orthomax Rotation

Published online by Cambridge University Press:  01 January 2025

Henk A. L. Kiers
Henk A. L. Kiers*
Affiliation:
University of Groningen
*
Requests for reprints should be sent to Henk A. L. Kiers, Department of Psychology, Grote Kruisstraat 2/l, 9712 TS Groningen, THE NETHERLANDS.
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Abstract

Factor analysis and principal components analysis (PCA) are often followed by an orthomax rotation to rotate a loading matrix to simple structure. The simple structure is usually defined in terms of the simplicity of the columns of the loading matrix. In Three-mode PCA, rotational freedom of the so called core (a three-way array relating components for the three different modes) can be used similarly to find a simple structure of the core. Simple structure of the core can be defined with respect to all three modes simultaneously, possibly with different emphases on the different modes. The present paper provides a fully flexible approach for orthomax rotation of the core to simple structure with respect to three modes simultaneously. Computationally, this approach relies on repeated (two-way) orthomax applied to supermatrices containing the frontal, lateral or horizontal slabs, respectively. The procedure is illustrated by means of a number of exemplary analyses. As a by-product, application of the Three-mode Orthomax procedures to two-way arrays is shown to reveal interesting relations with and interpretations of existing two-way simple structure rotation techniques.

Keywords

varimaxquartimaxthree-mode principal components analysissimple structure
Type
Original Paper
Information
Psychometrika , Volume 62 , Issue 4 , December 1997 , pp. 579 - 598
Copyright
Copyright © 1997 The Psychometric Society

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Footnotes

This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the author. The author is obliged to Jos ten Berge and two anonymous reviewers for useful comments on an earlier version of this paper.

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