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Oblique RotationBy J. Edward JacksonKeywords: principalcomponentanalysis,factoranalysis,rotationofaxes,obliquerotation, Obliminrotation,Orthoblique rotation, Promax rotation, Optres rotationAbstract: Any rotation of vectors associated with principal components analysis or factoranalysis designed to produce new set approximating simple structure is said to be obliqueif the resultant set of vectors are not at right angles to each other. Oblique rotations mayobtain a better approximation to simple structure but with the loss of orthogonality. Someof these rotations are obtained by a general procedure called Oblimin rotation but someof the more widely used rotations such as Orthoblique, Promax and Optres do not use thisform but instead are two- and three-stage procedures, usually using orthogonal rotationsas the first stage.Given a matrix V of dimension (p k) consisting of a set of k vectors, usually defining a set of principalcomponents or factors (See Factor Analysis: Exploratory), a new set of transformed variables may beobtained by a rotation of V, namely B=VTheta1. V is often the factor loading matrix or factor matrix from theinitial step in a principal components analysis or a factor analysis. Such a rotation is said to be oblique iftheresultantrotatedaxesinthevectorspacearenotatrightanglestoeachother.Mostrotationproceduresare designed to approximate a simple structure. The matrixTheta1of dimension (kk) defines the angles ofrotationandthematrixBofdimension(pk)definesthevectorsdeterminingthenewvariablesobtainedby rotation.The purpose of using oblique rotation is to obtain a matrix B that exhibits a better pattern of simplestructure than would be obtained by using an orthogonal rotation. This improvement towards a simplestructure comes at the cost of loss of orthogonality. If the main purpose of rotation is to cluster groups ofthe original variables, then this may present no problem. As in the case of orthogonal rotation, there aremany methods of obtaining oblique rotations and, again, there is a general quartic formula, the Obliminrotation, which produces a number of these procedures. Unlike orthogonal rotation, the more popularoblique procedures are not members of this family and are generally two-stage procedures. Among theseprocedures are Orthoblique or HarrisKaiser, Promax, and Optres rotations.Rochester, NY, USAThis article was originally published online in 2005 in Encyclopedia of Biostatistics, cJohn Wiley & Sons, Ltd and republishedin Wiley StatsRef: Statistics Reference Online, 2014.Copyright c2005 John Wiley & Sons, Ltd. All rights reserved.

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