The objective of simple structure rotation is to facilitate the interpretation and use of factor analytic results. However, a given factor analysis may fall short of this goal for many reasons other than ineffective rotation, It is therefore desirable to have a means for obtaining and/or recognizing a reliably good approximate rotation to simple structure in a given situation, when this rotation is uniquely determined.
One basis is provided by identifying simple structure with the maximized kurtosis of the distribution of factor loadings, when each variable is tallied in both its normal and reflected form to eliminate the artifact of scoring. For orthogonal rotation this only requires maximization of the sum of fourth powers of all factor loadings, Q, where .
Differentiation shows that
Figure 1.
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must necessarily hold for each pair of factors, j and k. It is further shown that these functions vary sinusoidally with rotation in any plane, with period of ¼.
These facts are used to develop two convenient computational routines – for hand computing and for high-speed electronic computing – and to demonstrate the convergence of these routines to a stable position.
The method is illustrated with two real and two hypothetical examples, and it is shown that the real data tend to yield a simple structure better than that obtainable when the same vectors are placed into random configurations. Certain difficulties of the method are discussed.