In factor analysis, the initial unrotated solution — whether from principal axis factoring, maximum likelihood, or another extraction method — is mathematically indeterminate: any rotation of the factor axes produces an equally valid solution with the same reproduced correlations. Factor rotation exploits this indeterminacy by seeking the transformation that produces the simplest, most interpretable pattern of factor loadings. The goal is to approximate Thurstone's concept of "simple structure," in which each variable loads substantially on one factor and near-zero on the others.
Orthogonal Rotations
where λ̃_ij = λ_ij / h_i (loading normalized by communality)
p = number of variables, j indexes factors
Orthogonal rotations maintain uncorrelated factors. Varimax (Kaiser, 1958) is the most popular: it maximizes the variance of the squared loadings within each factor, pushing loadings toward 0 or ±1. This produces factors on which a few variables load highly and the rest load near zero. Quartimax maximizes the variance of squared loadings for each variable across factors, simplifying variables rather than factors. It tends to produce a dominant general factor. Equamax balances the varimax and quartimax criteria.
Oblique Rotations
Orthogonal rotation forces factors to be uncorrelated — an assumption that is often unrealistic in psychology, where constructs like anxiety and depression, or verbal and quantitative ability, are typically correlated. Oblique rotations allow factors to correlate, yielding both a pattern matrix (regression-like weights of variables on factors) and a structure matrix (correlations between variables and factors), plus a factor correlation matrix.
where P = pattern matrix, Φ = factor correlation matrix
When Φ = I (orthogonal): S = P (pattern equals structure)
Promax begins with a varimax solution and raises loadings to a power (kappa, typically 2–4) to sharpen the pattern, then permits factors to correlate. Direct oblimin minimizes the cross-products of loadings with a parameter δ controlling the degree of obliquity (δ = 0 is the most common default). The choice between orthogonal and oblique rotation should be guided by theory: if constructs are expected to be correlated, oblique rotation is more appropriate. When factor correlations from an oblique rotation are near zero, orthogonal and oblique solutions will be very similar.
Interpreting Rotated Solutions
After rotation, factors are interpreted by examining the pattern of high and low loadings. A loading of |0.30| to |0.40| is often considered the minimum for salience, though this threshold is arbitrary. Variables with high loadings on a single factor define that factor; cross-loading variables (loading substantially on multiple factors) may indicate poor item quality or the need for a different factor model. The interpretability of the rotated solution is the ultimate criterion — a statistically optimal rotation that yields uninterpretable factors is of little use.
Target rotation is a confirmatory approach in which loadings are rotated toward a specified target matrix, useful when theory predicts a particular loading pattern. Procrustes rotation minimizes the discrepancy between the rotated solution and a target, enabling comparisons across samples or studies. These targeted methods bridge the gap between exploratory and confirmatory factor analysis, allowing researchers to test theoretical expectations within an EFA framework while retaining the flexibility to detect unexpected loadings.