A continuous-time approach to the oblique Procrustes problem

In the paper proposed we will make use of the gradient flow approach to consider a generalization of the well-known oblique Procrustes rotation problem, involving oblique simple structure rotation of both the core and component matrices resulting from three-mode factor analysis. The standard oblique Procrustes rotations to specified factor-structure and factor-pattern follow as special cases. The approach adopted leads to globally convergent algorithm and includes solving of initial value problem for certain matrix ordinary differential equation. Necessary conditions are established for the solution of the problem. The same approach is extended easily to the weighted oblique Procrustes rotation. Finally, some simulated numerical results are given and commented

Rotation to a partially specified target matrix in exploratory factor analysis in practice

The purpose of the current study was to explore the influence of the number of targets specified on the quality of exploratory factor analysis solutions with a complex underlying structure and incomplete substantive measurement theory. Previous research in this area was extended by (a) exploring this phenomenon in situations in which both the common factor model and the targeted pattern matrix contained specification errors and (b) comparing the performance of target rotation to an easier to use default rotation criterion (i.e., geomin) under conditions commonly observed in practice. A Monte Carlo study manipulated target error, number of targets, model error, overdetermination, communality, and sample size. Outcomes included bias (i.e., accuracy) and variability (i.e., stability) with regard to the rotated pattern matrix. The effects of target error were negligible for both accuracy and stability, while small effects were observed for the number of targets for both outcomes. Further, target rotation outperformed geomin rotation with regard to accuracy but generally performed worse than geomin rotation with regard to stability. These findings underscore the potential importance (or caution in the case of stability) of using extant, even if incomplete and somewhat inaccurate, substantive measurement theory to inform the rotation criterion in a non-mechanical way