Rotation to simple loadings using component loss functions: The orthogonal case

Component loss functions (CLFs) are used to generalize the quartimax criterion for orthogonal rotation in factor analysis. These replace the fourth powers of the factor loadings by an arbitrary function of the second powers. Criteria of this form were introduced by a number of authors, primarily Katz and Rohlf (1974) and Rozeboom (1991), but there has been essentially no follow-up to this work. A method so simple, natural, and general deserves to be investigated more completely. A number of theoretical results are derived including the fact that any method using a concave CLF will recover perfect simple structure whenever it exists, and there are methods that will recover Thurstone simple structure whenever it exists. Specific CLFs are identified and it is shown how to compare these using standardized plots. Numerical examples are used to illustrate and compare CLF and other methods. Sorted absolute loading plots are introduced to aid in comparing results and setting parameters for methods that require them.The author is very indebted to a reviewer for pointing him to the generalized hyperplane count literature and to all the reviewers for valuable comments and suggestions.     

Rotation to Simple Loadings Using Component Loss Functions: The Oblique Case

Component loss functions (CLFs) similar to those used in orthogonal rotation are introduced to define criteria for oblique rotation in factor analysis. It is shown how the shape of the CLF affects the performance of the criterion it defines. For example, it is shown that monotone concave CLFs give criteria that are minimized by loadings with perfect simple structure when such loadings exist. Moreover, if the CLFs are strictly concave, minimizing must produce perfect simple structure whenever it exists. Examples show that methods defined by concave CLFs perform well much more generally. While it appears important to use a concave CLF, the specific CLF used is less important. For example, the very simple linear CLF gives a rotation method that can easily outperform the most popular oblique rotation methods promax and quartimin and is competitive with the more complex simplimax and geomin methods. The author would like to thank the editor and three reviewers for helpful suggestions and for identifying numerous errors.     

A simple general method for oblique rotation

A simple and very general algorithm for oblique rotation is identified. While motivated by the rotation problem in factor analysis, it may be used to minimize almost any function of a not necessarily square matrix whose columns are restricted to have unit length. The algorithm has two steps. The first is to compute the gradient of the rotation criterion and the second is to project this onto a manifold of matrices with unit length columns. For this reason it is called a gradient projection algorithm. Because the projection step is very simple, implementation of the algorithm involves little more than computing the gradient of the rotation criterion which for many applications is very simple. It is proven that the algorithm is strictly monotone, that is as long as it is not already at a stationary point, each step will decrease the value of the criterion. Examples from a variety of areas are used to demonstrate the algorithm, including oblimin rotation, target rotation, simplimax rotation, and rotation to similarity and simplicity simultaneously. While it may be, the algorithm is not intended for use as a standard algorithm for well established problems, but rather as a tool for investigating new methods where its generality and simplicity may save an investigator substantial effort.The author would like to thank the review team for their insights and recommendations.