Publication date: 08/13/2020

Rotation Methods

Rotations are used to change the reference axes of the factors to make the factors more interpretable. Rotations are applied to the factors extracted from the data. Rotation methods are based on various complexity or simplicity functions. For more information about rotations see the FACTOR Procedure chapter in SAS Institute Inc. (2018c), Browne (2001), or Frank and Todeschini (1994).

After the initial extraction, the factors are uncorrelated with each other. If the factors are rotated by an orthogonal transformation, the rotated factors are also uncorrelated. If the factors are rotated by an oblique transformation, the rotated factors become correlated. Oblique rotations often produce more interpretable factors than do orthogonal rotations. However, a consequence of correlated factors is that there is no single unambiguous measure of the importance of a factor in explaining a variable.

Orthogonal Rotation Methods

Varimax

Maximizes the sum of the variances of the squared loadings of a factor on all variables. This common method results in each variable having either a small or large loading on each factor. (Orthomax with γ = 1.)

Biquartimax

An equally weighted solution of the Varimax and Quartimax rotations. (Orthomax with γ = 0.5.)

Equamax

A weighted solution between the Varimax rotation and the Quartimax rotation. (Orthomax with γ = N/2, where N = number of factors.)

Factorparsimax

A solution that aims to minimize the complexity of factors. This method might result in cross-loadings as variable complexity is not considered in the algorithm. (Orthomax with γ = N, where N = number of factors.)

Orthomax

A general weighted rotation method where the weight is denoted by γ. Many specific orthogonal rotation methods are Orthomax rotations with a specific γ.

Parsimax

Balances the variable and the factor complexity. (Orthomax with γ = (I(N-1))/(I+N-2), where I = the number of items and N = number of factors.)

Quartimax

Minimizes the number of factors needed to explain each variable. (Orthomax with γ = 1.)

Oblique Rotation Methods

Biquartimin

A rotation to minimize the ratio of the covariances (Oblimin with τ = 0.5.).

Covarimin

Oblique Varimax rotation. (Oblimin with τ = 1.).

Obbiquartimax

Oblique Biquartimax rotation.

Obequamax

Oblique Equamax rotation.

Obfactorparsimax

Oblique factor Parsimax rotation.

Oblimin

A general weighted oblique rotation method where the weight is denoted by τ. Many specific oblique rotation methods are Oblimin rotations with a specific τ.

Obparsimax

Oblique Parsimax rotation

Obquartimax

Oblique Quartimax rotation, equivalent to the Quartimin method.

Obvarimax

Oblique Varimax rotation.

Quartimin

Oblique Quartimin rotation, equivalent to oblique Quartimax (Oblimin with τ = 0.)

Promax

A two step rotation in which Varimax is performed first and then the Procrustes rotation is used to attain simple structure. This is a computationally efficient method that is an alternative to Oblimin.

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