Rotation Methods

FactorRotations.jl implements multiple orthogonal and oblique rotation methods.

Let us consider the p×k factor loadings matrix L for p variables and k factors. Most of the rotation methods aim to find the full-rank k×k rotation matrix U, so that the rotated loadings matrix Λ = L × U optimizes the given criterion function Q(Λ).

Orthogonal methods

Orthogonal criteria restrict the rotation matrix U to be orthogonal ($U⋅Uᵀ = I$).

criterion	reference	note
`Biquartimax`		equivalent to `Oblimin(gamma = 0.5, orthogonal = true)`
`Biquartimin`	Jennrich and Bentler (2011)
`ComponentLoss`	Jennrich (2004), Jennrich (2006)
`CrawfordFerguson`	Crawford and Ferguson (1970)
`Equamax`	Crawford and Ferguson (1970)	equivalent to `Oblimin(gamma = k/2, orthogonal = true)`
`Infomax`	Browne (2001)	based on the unpublished manuscript McKeon (1968)
`KatzRohlf`
`LinearRightConstant`	Jennrich (2004)
`MinimumEntropyRatio`	McCammon (1966)
`MinimumEntropy`	Jennrich (2004)
`Oblimax`
`Oblimin`
`Parsimax`	Crawford and Ferguson (1970)	equivalent to `Oblimin(gamma = p*(k-1)/(p+k-2), orthogonal = true)`
`PatternSimplicity`	Bentler (1977)
`Quartimax`	Neuhaus and Wrigley (1954)	equivalent to `Oblimin(gamma = 0, orthogonal = true)`
`TandemCriteria`	Comrey (1967)
`TandemCriterionII`	Comrey (1967)	second step of `TandemCriteria`
`TandemCriterionI`	Comrey (1967)	first step of `TandemCriteria`
`TargetRotation`
`Varimax`	Kaiser (1958)	equivalent to `Oblimin(gamma = 1, orthogonal = true)`

Oblique methods

Oblique criteria allow the rotation matrix R to be any full-rank k×k matrix. Oblique rotations do not preserve the orthogonality of the factors. To preserve the loadings matrix communalities ($c_i = ∑_{j=1}^k L_{i,j}²$), the oblique rotation matrix R has to satisfy $\mathrm{diag} (R⋅Rᵀ) = I$.

criterium	reference	note
`Absolmin`	Jennrich (2006)
`Biquartimin`	Jennrich and Bentler (2011)
`ComponentLoss`	Jennrich (2004), Jennrich (2006)
`Concave`	Jennrich (2006)
`CrawfordFerguson`	Crawford and Ferguson (1970)
`Geomin`
`Infomax`	Browne (2001)	based on the unpublished manuscript McKeon (1968)
`Oblimax`
`Oblimin`
`PatternSimplicity`	Bentler (1977)
`Simplimax`
`TargetRotation`

References

Bentler, P. (1977). Factor simplicity index and transformations. Psychometrika 42, 277–295.
Browne, M. W. (2001). An overview of analytic rotation in exploratory factor analysis. Multivariate behavioral research 36, 111–150.
Comrey, A. L. (1967). Tandem criteria for analytic rotation in factor analysis. Psychometrika 32, 143–154.
Crawford, C. B. and Ferguson, G. A. (1970). A general rotation criterion and its use in orthogonal rotation. Psychometrika 35, 321–332.
Jennrich, R. I. (2004). Rotation to simple loadings using component loss functions: The orthogonal case. Psychometrika 69, 257–273.
Jennrich, R. I. (2006). Rotation to simple loadings using component loss functions: The oblique case. Psychometrika 71, 173–191.
Jennrich, R. I. and Bentler, P. M. (2011). Exploratory bi-factor analysis. Psychometrika 76, 537–549.
Kaiser, H. F. (1958). The varimax criterion for analytic rotation in factor analysis. Psychometrika 23, 187–200.
McCammon, R. B. (1966). Principal component analysis and its application in large-scale correlation studies. The Journal of Geology 74, 721–733.
Neuhaus, J. O. and Wrigley, C. (1954). The quartimax method: An analytic approach to orthogonal simple structure. British Journal of Statistical Psychology 7, 81–91.