Structural Equation Models of Factorial Invariance in Parallel Proportional Profiles and Oblique Confactor Problems.

Some problems of multiple group factor rotation based on Cattell's "parallel proportional profiles" and "confactor rotation" are described (see Cattell, 1944, 1966, 1972). Some relations between these classic ideas and contemporary practices in structural equation modeling (e.g., LISREL) are explored. We show how the Confactor approach: (a) is related to Meredith's (1964a) selection model, (b) can be a parsimonious model for multiple group factor analyses, and (c) how this model can be fitted using standard structural equation modeling techniques. We discuss several alternative structural modeling solutions, including (d) selection of a good reference variable solution, (e) rotation of the invariant orthogonal structure by standard rotation routines, and (f) higher-order, latent paths, and latent means structural model restrictions. Mathematical and statistical properties of these models are examined using Meredith's (1964b) four group problem fitted by Jdreskog and Sdrbom's (1979, 1985) LlSREL algorithm. The benefits and limitations of this structural modeling approach to oblique Confactor resolution are examined and opportunities for future research are discussed.