Rotation represents a primary distinction between exploratory and confirmatory factor analysis. Attempts to define simple structure for factor loadings and to define simpler structure will be identified. The latter will include visual versus contextual judgements and the use of rotation criteria.
In the context of orthogonal rotation quadratic and hyperplane distance criteria will be considered in some detail. In the quadratic case recent perfect simple structure theorems will be identified. There is a renewed interest in the ancient almost abandoned hyperplane distance methods of rotation. Recently derived basic properties of these will be identified including theorems on the recovery of perfect simple structure and Thurstone simple structure.
In the context of oblique rotation, the case for direct methods will be made. For quadratic criteria, results on degeneracy and the recovery of perfect simple structure will be given. Component loss criteria are the natural formulation of hyperplane distance criteria when using direct methods. Basic properties similar to those for the orthogonal case will be given. Local minimizer theorems relating partially specified target and Simplimax methods to component loss methods will be identified.
A number of orthogonal and oblique rotation algorithms including general closed form pairwise algorithms for quartic criteria and general pairwise and gradient projection algorithms for arbitrary criteria will be discussed.
Standard errors for rotated loadings using linear approximation methods will be reviewed. These will include information theory standard errors for the normal case and e-jackknife standard errors for the nonparametric case.