**Principal components + orthoblique rotation (all three cases)**

Syntax file: orthoblique.sps (sps / 12,99K / 03.09.2021.)

How to run it: runorthoblique.sps (sps, 125,00B, 03.09.2021.)

Harris and Kaiser (1964) have developed a model that defines all possible analytic transformation procedures. Their orthoblique equations describe the computation of both orthogonal (Case I) and oblique (Case II & III) transformation solutions using only orthonormal transformation matrices.

The independent cluster solution (Case II) is most appropriate when each variable loading is on no more than one factor, while "the pattern proportional" is most appropriate when some of the variables are factorially complex (some variables loading on more than one factor, Case III). However, Case II is the most used technique.

There are several procedures for the oblique parsimonious rotation of the principal components by orthonormal transformations, but the orthoblique is probably the best. Orthoblique usually produces the best approximation of the simple structure both in the pattern and structure matrix. The obtained latent dimensions have non-null correlations in space of entities, but are orthogonal in space of variables (columns are orthogonal). Algebraic and conceptual simplicity makes orthoblique easy to interpret and fast analytic factor rotation, recomended as optimal rotation in exploratory factor analyses.

Harris, C. W., & Kaiser, H. F. (1964). Oblique factor analytic solutions by orthogonal transformations. *Psychometrika*, *29*(4), 347-362.