Rotations

The different methods of factor analysis first extract a set of factors from a data set. These factors are almost always orthogonal and are ordered according to the proportion of the variance of the original data that these factors explain. In general, only a (small) subset of factors is kept for further consideration, and the remaining factors are considered either irrelevant or nonexistent (i.e., they are assumed to reflect measurement error or noise).

In order to make the interpretation of the factors that are considered relevant, the first selection step is generally followed by a rotation of the factors that were retained. Two main types of rotation are used: orthogonal, when the new axes are also orthogonal to each other, and oblique, when the new axes are not required to be orthogonal to each other. Because the rotations are always performed in a subspace (the so-called factor space), the new axes will always explainless variance than the original factors (which are computed to be optimal), but obviously, the part of variance explained by the total subspace after rotation is the same as it was before rotation (only the partition of the variance has changed). Because the rotated axes are not defined according to a statistical criterion, their raison d’être is to facilitate the interpretation.

In this entry, the rotation procedures are illustrated using the loadings of variables analyzed with principal components analysis (the so-called R-mode), but the methods described here are valid also for other types of analysis and when analyzing the subjects’ scores (the so-called Q-mode).

Before proceeding further, it is important to stress that because the rotations always take place in a subspace (i.e., the space of the retained factors), the choice of this subspace strongly influences the result of the rotation. Therefore, in the practice of rotation in factor analysis, the strong recommendation is to try several sizes for the subspace of the retained factors in order to assess the robustness of the interpretation of the rotation.

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Notations

To make the discussion more concrete, rotations within the principal components analysis (PCA) framework are described. PCA starts with a data matrix denoted Y with I rows and J columns, where each row represents a unit (in general subjects) described by J measurements that are almost always expressed as Z-scores. The data matrix is then decomposed into scores (for the subjects) or components and loadings for the variables (the loadings are, in general, correlations between the original variables and the components extracted by the analysis). Formally, PCA is equivalent to the singular value decomposition of the data matrix as

with the constraints that QTQ = PTP = I (where I is the identity matrix), and is a diagonal matrix with the so-called singular values on the diagonal. The orthogonal matrix Q is the matrix of loadings (or projections of the original variables onto the components), where one row stands for one of the original variables and one column for one of the new factors. In general, the subjects’ score matrix is obtained as F = P (some authors use P as the scores).

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