Common Mistakes in Rotation, Scores, and Fit Assessment

Why These Errors Matter

A rotated factor solution can be easy to read but easy to overinterpret. A simple-looking loading matrix does not, by itself, establish good model fit; an estimated factor score is not the person’s known latent value; and two sets of loadings can look different merely because their factor axes are ordered or signed differently. These distinctions matter because rotation changes the coordinates used to describe common variation, while fit concerns how well the model reproduces the observed covariance structure.

Consider the common factor model for a \(p\)-dimensional observation \(X\) and \(q\) factors:

Here \(\Lambda\) is the \(p\times q\) pattern loading matrix, \(\Phi\) is the factor covariance matrix, and \(\Psi\) is the uniqueness covariance matrix, commonly taken to be diagonal. Assume \(F\) and \(\varepsilon\) are uncorrelated and have finite second moments. Then the model-implied covariance matrix is \(\Sigma=\Lambda\Phi\Lambda^\mathsf{T}+\Psi\). In an orthogonal solution, \(\Phi=I\); in an oblique solution, factors may be correlated.

Oblique Loadings: Do Not Square the Pattern Row and Stop

In an oblique solution, the pattern matrix describes the conditional contribution of each factor to a variable, given the other factors. The structure matrix describes the variable–factor covariances; with standardized variables and factors, its entries are correlations and it is \(S=\Lambda\Phi\). The two matrices are not interchangeable when factors correlate. In particular, summing squared entries of a pattern row omits cross-products involving the factor correlations.

Why Communality Survives a Change of Factor Coordinates

A useful check is that the communality does not depend on the particular admissible factor coordinates used to represent the common covariance. Let \(T\) be any nonsingular \(q\times q\) matrix, and define \(F^*=T^{-1}F\). The corresponding loadings and factor covariance are \(\Lambda^*=\Lambda T\) and \(\Phi^*=T^{-1}\Phi T^{-\mathsf{T}}\).

Proof. Using \(\Lambda^*=\Lambda T\) and \(\Phi^*=T^{-1}\Phi T^{-\mathsf{T}}\), multiply the transformed quantities:

The diagonal entries of these equal matrices are the communalities, so they are equal as well. The uniqueness matrix \(\Psi\) is unchanged in this coordinate transformation, and therefore the full model covariance is unchanged. \(\square\)

This result also explains why squaring an oblique pattern row is not a coordinate-invariant way to calculate communality: it discards \(\Phi\), which must change along with the loadings when the factor coordinates change.

Compare Factors Only After Addressing Alignment

Factor labels have no inherent order, and the sign of a factor can be reversed together with the signs of its loadings. Consequently, raw column-by-column comparisons may confuse an equivalent relabeling with a substantive difference. One standard summary for comparing two loading columns \(a\) and \(b\) is Tucker’s congruence coefficient:

It is a normalized inner product, so reversing one column’s sign reverses the coefficient’s sign. A negative coefficient can therefore signal a sign mismatch rather than a different factor. Permutations and sign changes should be considered before interpreting column matches. For oblique solutions, matching may require more than these simple adjustments: distinct oblique coordinate systems can mix factors. Any alignment should respect the model and substantive constraints, not just make columns look similar.

If two solutions are not related by a simple permutation or sign change, a low congruence may reflect a different orientation, sampling variation, or genuinely different structure. It is not, by itself, a verdict about fit. In particular, aligning factors is an interpretive comparison; it does not make two fitted covariance models equivalent.

Factor Scores Are Estimates, Not Observed Latent Values

A factor score estimate summarizes information about an unobserved factor given observed variables and a chosen scoring method. It is not a direct measurement of the factor. Score indeterminacy and conditional uncertainty mean that multiple plausible latent values can be compatible with the same observed data. A point estimate should therefore be reported as an estimate, and its uncertainty should not be suppressed when it affects interpretation.

Scores also depend on the factor coordinates. If a model is transformed consistently, the latent coordinates change with it; estimates should be interpreted in the coordinates of the reported solution. Do not compare score values across solutions until factor alignment, scaling, and the scoring method have been checked.

A Rotation Criterion Is Not a Fit Test

A rotation criterion evaluates an orientation of the loading matrix, often rewarding a feature such as simple structure. Fit assessment instead asks how adequately the model reproduces the data, for example through the discrepancy between the observed covariance matrix and \(\Sigma=\Lambda\Phi\Lambda^\mathsf{T}+\Psi\), or through an inferential criterion appropriate to the fitted models. These are different questions. A low rotation-criterion value is not a test statistic for covariance fit and does not establish that the model is adequate.

Proof. Let \(Q\) be orthogonal, so \(Q Q^\mathsf{T}=I\), and set \(\Lambda^*=\Lambda Q\). For an orthogonal factor model, \(\Phi=I\); the rotated common covariance is

Keeping \(\Psi\) fixed therefore gives the same full covariance matrix before and after rotation. Any fit measure depending only on the fitted covariance and the data has the same value for these equivalent representations. A rotation criterion may still change because it evaluates the loading orientation, not covariance reproduction. \(\square\)

The corresponding oblique coordinate change also preserves the common covariance when \(\Phi\) is transformed consistently, as shown above. By contrast, comparing models with different restrictions, factor counts, or fitted covariance matrices is a model-comparison problem. A rotation criterion cannot substitute for a fit diagnostic or an appropriate comparison procedure.

Common Mistakes and Corrections

• Wrong: “The oblique communality is the sum of squared pattern loadings.” Correction: Use \(\lambda_i^\mathsf{T}\Phi\lambda_i\), including the factor-correlation cross-products.
• Wrong: “The first factor in each solution must represent the same construct.” Correction: Check permutations and signs, then assess whether a more general alignment is warranted.
• Wrong: “A person’s estimated score is their known latent value.” Correction: Call it an estimate and acknowledge the uncertainty and scoring assumptions.
• Wrong: “The rotation with the best simple-structure criterion fits the data best.” Correction: Assess covariance fit separately; a criterion for orientation is not a fit test.

The practical lesson is to keep four objects distinct: the loading coordinates, the common covariance they represent, the estimated scores in those coordinates, and the diagnostics used to judge fit. Confusing them can turn an equivalent representation into an apparent disagreement, or a convenient interpretation into an unjustified model claim.