Shigeo Kashiwagi

A MOMENT APPROACH TO THE ORTHOGONAL FACTOR ROTATION

The possibility of describing the orthogonal factor rotation in terms of statistical moment based on squared factor loadings is discussed with the purpose of attaining simple structure type of answers and level contributions of rotated factors.In the first place, a short historical review of the analytical rotation approach based on squared factor loadings is presented and the characteristics of some methods of current use including the normal varimax, parsimax, and factor parsimony criteria are investigated. And it is shown mathematically that both the parsimax -which is advocated to be most recommendable when the definite number of factors is defined before rotation- and the factor parsimony criteria may have some inherent difficulties because of which they produce answers with high complexities. The empirical examples are presented in Table 1.Two kinds of generalized moments based on squared factor loadings are introduced. They are nΣj=1mΣi=1(aji2-1/mmΣi=1aji2)K (1) and mΣi=1nΣj=1(aji2-1/nnΣj=1aji2)K (2) where i (=1, 2, ……, m), j (=1, 2, ……, n), aji, and K refer to factor, test, rotated factor loadings, and positive integer except unity respectively, and more attention is paid on the formula (2) which is herein called the generalized moment with respect to factor column and its mathematical aspects are discussed.In the formula (2) with the values of 2 and 3 for K, the rotational angles in the single plane procedure can be obtained easily. The former case includes both the quartimax and the varimax criteria and the latter both the communality weighted quartimax and the skewmax ones which are herein developed.The normal skewmax criterion is shown to be most recommendable from the practical point of view and the possibility of improving the normal varimax solution is suggested with a numerical example of Table 2, in that the former criterion attains both simple structure type of answers and level contributions of factors more satisfactorily than the latter does. The formula for rotation angle is to be θ=1/4arctan2nΣj=1(B+C)(B-C)D/nΣj=1(B+C)((B-C)-D2) (3) where B=aj12-1/nnΣj=1aj12, C=aj22-1/nnΣj=1aj22, and D=2 (aj1aj2-1/nnΣj=1aj1aj2)Related miscellaneous topics are discussed and some needs for further studies are suggested from another line of approach.