Permutimin: Factor Rotation to Simple Structure with Permutation of Variables

Abstract

Factor rotation is usually performed for a p-variables $\times r$-factors loading matrix so that the resulting rotated matrix has a simple structure. This simple structure was originally defined by Thurstone (1947) by specifying how zero elements are arranged in the loading matrix. In this article, we propose a new rotation technique, which is directly based on Thurstone’s definition. It can give a p-variables $\times r$-factors target matrix of zero and nonzero elements, which stands for the properties to be possessed by the rotated loading matrix. However, it is unknown how the rows of the target matrix are associated with those of the loading matrix. In the proposed procedure, a loading matrix is rotated simultaneously with a permutation of the rows of the target matrix, so that the rotated loading matrix is optimally matched to the permuted target matrix in a least squares sense. Its novel feature is the use of permutation, thus we call the technique Permutimin. Its algorithm is presented, with Thurstone’s definition of simple structure modified so as to specify the target matrix uniquely. Permutimin is illustrated with real data examples. Finally, we discuss the relationships between Permutimin and Procrustes rotation.

Keywords:

• Exploratory factor analysis
• factor rotation
• simple structure
• permuting variables
• Procrustes rotation
• Thurstone’s definition