Rotation Methods
FactorRotations.jl implements multiple orthogonal and oblique rotation methods.
Let us consider the p-by-k factor loadings matrix L for p variables and k factors. Most of the rotation methods aim to find the full-rank k-by-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.
| 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Infomax | Browne (2001) | based on the unpublished manuscript McKeon (1968) |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 U to be an arbitrary full-rank k-by-k matrix.
| 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Infomax | Browne (2001) | based on the unpublished manuscript McKeon (1968) |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.