Rotational Strategies in Factor Analysis

Various rotational strategies have been proposed. The goal of all of these strategies is to obtain a clear pattern of loadings, that is, factors that are somehow clearly marked by high loadings for some variables and low loadings for others. This general pattern is also sometimes referred to as simple structure (a more formalized definition can be found in most standard statistical textbooks).

Typical rotational strategies are varimax, biquartimax, quartimax, and equamax. Some authors (e.g., Cattell & Khanna; Harman, 1976; Jennrich & Sampson, 1966; Clarkson & Jennrich, 1988) have discussed in some detail the concept of oblique (non-orthogonal) factors in order to achieve more interpretable simple structure. Specifically, computer (algorithmic) strategies have been developed to rotate factors so as to best represent "clusters" of variables without the constraint of orthogonality of factors. However, the oblique factors produced by such rotations are often not easily interpreted. Click the Hierarchical analysis of oblique factors button on the Loadings tab of the Factor Analysis Results dialog box instead in order to identify (correlated, oblique) clusters of variables (see also Hierarchical factor analysis).

Note that you can use the Structural Equation Modeling (SEPATH) module to test the adequacy (goodness of fit) of specific orthogonal or oblique factor solutions.

Varimax raw. This selection from the Factor rotation drop-down list performs a varimax rotation of the factor loadings. This rotation is aimed at maximizing the variances of the squared raw factor loadings across variables for each factor; this is equivalent to maximizing the variances in the columns of the matrix of the squared raw factor loadings.

Varimax normalized. This selection from the Factor rotation drop-down list performs a varimax rotation of the normalized factor loadings (raw factor loadings divided by the square roots of the respective communalities). This rotation is aimed at maximizing the variances of the squared normalized factor loadings across variables for each factor; this is equivalent to maximizing the variances in the columns of the matrix of the squared normalized factor loadings. This is the method that is most commonly used and referred to as varimax rotation.

Biquartimax raw. This selection from the Factor rotation drop-down list performs a biquartimax rotation of the raw factor loadings. This rotation can be considered to be an "even mixture" of the varimax and quartimax rotation. Specifically, it is aimed at simultaneously maximizing the sum of variances of the squared raw factor loadings across factors and maximizing the sum of variances of the squared raw factor loadings across variables; this is equivalent to simultaneously maximizing the variances in the rows and columns of the matrix of the squared raw factor loadings.

Biquartimax normalized. This selection is equivalent to the biquartimax raw rotation, except that it is performed on normalized (standardized) factor loadings.

Quartimax raw. This selection from the Factor rotation drop-down list performs a quartimax rotation of the (raw) factor loadings. This rotation is aimed at maximizing the variances of (the squared raw) factor loadings across factors for each variable; this is equivalent to maximizing the variances in the rows of the matrix of the squared raw factor loadings.

Quartimax normalized. This selection from the Factor rotation drop-down list performs a quartimax rotation of the normalized factor loadings, that is, the raw factor loadings divided by the square roots of the respective communalities. This rotation is aimed at maximizing the variances of the squared normalized factor loadings across factors for each variable; this is equivalent to maximizing the variances in the rows of the matrix of the squared normalized factor loadings. This is the method that is commonly referred to as quartimax rotation.

Equamax raw. This selection from the Factor rotation drop-down list performs an equamax rotation of the raw factor loadings. This rotation can be considered to be a "weighted mixture" of the varimax and quartimax rotation. Specifically, it is aimed at simultaneously maximizing the sum of variances of the squared raw factor loadings across factors and maximizing the sum of variances of the squared raw factor loadings across variables; this is equivalent to simultaneously maximizing the variances in the rows and columns of the matrix of the squared raw factor loadings. However, unlike the biquartimax rotation, the relative weight assigned to the varimax criterion in the rotation is equal to the number of factors divided by 2.

Equamax normalized. This selection from the Factor rotation drop-down list performs an equamax rotation, as described for Equamax raw; however, this rotation will be performed on the normalized factor loadings.