Oblique Rotation

Oblique Rotation in Health Outcome Measurement

In an exploratory factor analysis of a health questionnaire, an oblique rotation repositions the factor axes so that each item has a clear home, while the factors themselves remain free to correlate. Physical and emotional functioning can then be modelled as separate domains that still move together. The rotated solution carries a factor correlation matrix, and its loadings divide into a pattern matrix of regression-like weights and a structure matrix of item and factor correlations. The choice matters in practice, since orthogonal scoring of the SF-36 summary measures gave negative weights to some scales. Oblique results inform instrument appraisal and domain scoring but never yield the preference weights needed for QALYs.

Why correlated factors suit health domains

Factor rotation is possible because an extracted factor solution is not unique, so the axes can be turned to make the loadings easier to read. SAS notes that after extraction in an exploratory factor analysis the factors are uncorrelated, that an orthogonal rotation keeps them so, and that oblique rotations often produce more useful patterns.

Stata's manual sees little reason in many substantive theories to impose uncorrelated factors. Its own health example, a two-factor analysis of 19 survey items from 9,999 respondents, uses an oblique oblimin rotation because the literature suggests physical and mental health are related, and the rotated factors correlate at 0.36. Studies cited by Farivar and colleagues put the correlation between physical and mental health factors at 0.32 to 0.66, so treating these latent variables as independent misdescribes them.

The oblique transformation and the factor correlation matrix

The common factor model reproduces the item correlations from loadings, factor correlations and uniquenesses. Stata gives the fitted matrix in this form.

$$\boldsymbol\Sigma = \boldsymbol\Lambda\boldsymbol\Phi\boldsymbol\Lambda^{\mathsf T} + \boldsymbol\Psi$$

where $\boldsymbol\Sigma$ is the model-implied correlation matrix of the standardised items, $\boldsymbol\Lambda$ is the loading matrix (items in rows, factors in columns), $\boldsymbol\Phi$ is the correlation matrix of the common factors and $\boldsymbol\Psi$ is the diagonal matrix of unique variances.

Before rotation $\boldsymbol\Phi$ is the identity matrix. An oblique rotation changes the loadings and introduces factor correlations together, as Stata's factor postestimation entry sets out.

$$\boldsymbol\Lambda_{r} = \boldsymbol\Lambda_{u}\left(\mathbf{T}^{\mathsf T}\right)^{-1}, \qquad \boldsymbol\Phi = \mathbf{T}^{\mathsf T}\mathbf{T}, \qquad \operatorname{diag}\left(\mathbf{T}^{\mathsf T}\mathbf{T}\right) = \mathbf{1}$$

where $\boldsymbol\Lambda_{u}$ is the unrotated loading matrix, $\boldsymbol\Lambda_{r}$ is the rotated pattern matrix, $\mathbf{T}$ is the factor rotation matrix and the last condition gives each column of $\mathbf{T}$ unit length. If $\mathbf{T}$ is orthonormal, the inverse of its transpose is $\mathbf{T}$ itself and $\boldsymbol\Phi$ stays the identity, which is the orthogonal case. With two factors, the unit-length condition makes the factor correlation the cosine of the angle between the rotated axes, so a correlation of 0.50 means axes 60 degrees apart.

Pattern, structure and reference structure matrices

An oblique solution has several loading matrices, and they answer different questions. SAS warns that with correlated factors there is no single unambiguous measure of a factor's importance for a variable, so the pattern alone is not enough and the structure and reference structure must also be examined.

SAS describes the oblique pattern as standardised regression coefficients for predicting the items from the factors, so each loading shows an item's association with one factor holding the others fixed. The structure matrix holds plain item and factor correlations, and Stata computes it as the pattern times the factor correlation matrix.

$$\mathbf{S} = \boldsymbol\Lambda_{r}\boldsymbol\Phi, \qquad h_i^2 = \sum_{j}\sum_{k}\lambda_{ij}\lambda_{ik}\phi_{jk} = \sum_{j}\lambda_{ij}s_{ij}$$

where $\mathbf{S}$ is the structure matrix with elements $s_{ij}$, $\lambda_{ij}$ is the pattern loading of item $i$ on factor $j$, $\phi_{jk}$ is the correlation between factors $j$ and $k$, and $h_i^2$ is the communality of item $i$.

The communality follows from the diagonal of the fitted matrix, and rotation leaves it unchanged, as the identical uniqueness columns before and after rotation in Stata's output show. The reference structure holds semipartial correlations between items and factors once the other factors' effects are removed from each factor, and it equals the pattern when factors are uncorrelated. Jennrich and Sampson's direct oblimin simplifies the primary loadings, where older analytic methods simplified the reference structure.

Oblimin, promax and the constants behind them

Oblimin and promax each have a constant that trades simpler loadings against higher factor correlations. The direct oblimin family is indexed by a constant called gamma in Stata and tau in SAS, and a value of zero gives quartimin, Stata's default. Stata reports Jennrich's recommendation of zero or below, because for positive values an optimal oblique rotation may not exist and the iteration can drift to a degenerate solution.

Hendrickson and White proposed promax in 1964 as a quick method that reaches orthogonal simple structure first and then transforms that solution to an oblique one. In Stata the steps are a varimax rotation, raising each varimax loading to a power while keeping its sign, and an approximate oblique Procrustes rotation of the original loadings towards that target. The default power is 3, Stata recommends a value above 1 and no higher than 4, and larger powers push loadings towards zero and one at the cost of more factor correlation. SAS also offers Harris-Kaiser and Procrustes rotations, and Stata offers rotation towards a target matrix, a more confirmatory use.

Kaiser normalisation, which rescales each row of the loading matrix to unit sum of squares so that every item carries equal weight, can change the rotated solution. Iterative criteria such as oblimin can also stop at a local optimum, so Stata provides random restarts.

Worked example with a five-item physical and mental health scale

All numbers in this example are illustrative. A team developing a short patient-reported outcome measure retains two factors from five items and applies direct oblimin with gamma at zero. The rotated factor correlation is $\phi = 0.50$.

1. Structure coefficients. Each is the item's own pattern loading plus the factor correlation times its loading on the other factor. For item 3, $s_{31} = 0.50 + 0.50 \times 0.28 = 0.64$ and $s_{32} = 0.28 + 0.50 \times 0.50 = 0.53$. For item 1, $s_{12} = 0.00 + 0.50 \times 0.80 = 0.40$, so an item with no direct mental loading still correlates 0.40 with the mental factor.

2. Communalities. With two factors, $h_i^2 = \lambda_{i1}^2 + \lambda_{i2}^2 + 2\phi\lambda_{i1}\lambda_{i2}$. For item 3, $h_3^2 = 0.25 + 0.0784 + 2 \times 0.50 \times 0.50 \times 0.28 = 0.4684$, and the pattern and structure products agree, $h_3^2 = 0.50 \times 0.64 + 0.28 \times 0.53 = 0.4684$. Had the factors been uncorrelated, item 3's communality would be $h_3^2 = 0.25 + 0.0784 = 0.3284$, so the overlap between the domains adds 0.14.

3. Total common variance. The communalities sum to $H = 0.64 + 0.5488 + 0.4684 + 0.5932 + 0.6412 = 2.8916$, so the two factors explain $H / 5 = 0.578$, about 58%, of item variance. Summing each factor's variance ignoring the other factor, the column sums of squared structure coefficients (1.8941 and 1.8338), would give 3.7279 and overstate this, because correlated factors partly explain the same variance.

4. Cross-loadings depend on the matrix read. With 0.30 as the salience cut-off, no item has a pattern loading of 0.30 or more on both factors, yet all five have structure coefficients of 0.30 or more on both.

5. Interpretation. Read through the pattern, items 1 and 2 define physical function, items 4 and 5 define mental wellbeing and item 3 is mainly physical with a modest mental component. The 0.50 correlation is part of the finding and would be reported, since a scoring rule or a later confirmatory factor analysis must allow for it.

Oblique scoring of the SF-36 summary scores

The SF-36 shows rotation choice changing applied results. Farivar, Cunningham and Hays explain that the version 1 Physical and Mental Component Summary scores (PCS and MCS) rest on a factor technique that forces the two to be orthogonal. The PCS therefore gives negative weights to the social functioning, role-emotional and emotional wellbeing scales, so better mental health scales pull the PCS down, and the MCS behaves the same way in reverse.

They cite a study of 482 patients starting antidepressants in which physical scales improved by 0.28 to 0.49 standard deviation units over three months while the orthogonal PCS barely moved, at 51 and then 50. In their own sample of 6,931 patients, refitting the eight scales with a two-factor promax solution gave a factor correlation of 0.62 and far fewer negative scoring coefficients. In their hypothetical case, with physical scales 1 standard deviation and mental scales 0.3 standard deviations above the mean, the orthogonal scores were PCS 62.2 and MCS 49.6, leaving the MCS at the mean, while the oblique scores were PCS 60.0 and MCS 54.6. The authors recommend using the correlated scores along with the uncorrelated ones, so any trial or cost-effectiveness study using SF-36 summary scores needs to say which version it used.

Reading oblique results for instrument appraisal and economic evaluation

The COSMIN guideline for systematic reviews of patient-reported outcome measures, version 2.0, rates structural validity from an exploratory analysis as sufficient when each item loads at least 0.30 on its factor, at most 10% of items load at least 0.30 on several factors, and explained variance reaches 50% with a structure in line with theory, or scree plot or Kaiser criterion (eigenvalues above 1) results agree with theory. Because these criteria speak of loadings, an appraiser of an oblique solution needs to know whether the table is the pattern or the structure, and the worked example passes the cross-loading test on one and fails it on the other.

Factor and domain scores describe health status and carry no preference weights. For SF-36 data the preference-based option is the SF-6D, which Brazier, Roberts and Deverill built for economic evaluation by revising the SF-36 into a six-dimension classification valued by standard gamble in a UK general population sample. An oblique model can support the dimensional structure of a health-related quality of life instrument, but a health state utility value still needs a valuation study.

Where oblique rotation ends and neighbouring methods begin

Orthogonal rotation such as varimax is the restricted case with every factor correlation fixed at zero, so an oblique solution with near-zero correlations gives much the same loadings. The Stata manual notes that most researchers would not accept highly correlated common factors. Confirmatory factor analysis estimates factor correlations directly under a pre-specified pattern and needs no rotation, and Stata advises caution with rotated principal components, because some of their optimality properties are not preserved under rotation.

Misreadings and limits of oblique solutions

Most errors come from reading one matrix as the whole result. The pattern describes unique association and the structure describes overall correlation.

1. Treating structure coefficients as loadings. They include the factor correlation, so items look cross-loaded when the pattern is clean.
2. Adding up factor variances. Stata notes that oblique factors partly explain the same variance, so it shows no cumulative proportion.
3. Leaving out the factor correlations. Without them a report cannot be checked, and the correlation is often the substantive finding.
4. Omitting the criterion and its constant. Promax power, oblimin gamma and Kaiser normalisation all move the solution.
5. Mixing SF-36 summary score versions. Orthogonal and oblique PCS and MCS scores are not interchangeable across studies.
6. Expecting rotation to fix weak items or a wrong factor count. Rotation only reorients a solution with fixed communalities.

Sources

• Brazier J, Roberts J, Deverill M. The estimation of a preference-based measure of health from the SF-36. Journal of Health Economics. 2002;21(2):271-292. https://doi.org/10.1016/S0167-6296(01)00130-8
• Farivar SS, Cunningham WE, Hays RD. Correlated physical and mental health summary scores for the SF-36 and SF-12 Health Survey, V.1. Health and Quality of Life Outcomes. 2007;5:54. https://doi.org/10.1186/1477-7525-5-54
• Hendrickson AE, White PO. Promax: a quick method for rotation to oblique simple structure. British Journal of Statistical Psychology. 1964;17(1):65-70. https://doi.org/10.1111/j.2044-8317.1964.tb00244.x
• Jennrich RI, Sampson PF. Rotation for simple loadings. Psychometrika. 1966;31(3):313-323. https://doi.org/10.1007/BF02289465
• Mokkink LB, Elsman EBM, Terwee CB. COSMIN guideline for systematic reviews of patient-reported outcome measures version 2.0. Quality of Life Research. 2024;33(11):2929-2939. https://doi.org/10.1007/s11136-024-03761-6
• SAS Institute. The FACTOR procedure, outline of use and displayed output. SAS/STAT 9.2 User's Guide, 2nd ed. 2009. https://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/statug_factor_sect003.htm and https://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/statug_factor_sect024.htm
• StataCorp. rotate; rotatemat; factor postestimation. Stata Multivariate Statistics Reference Manual, release 19. 2025. https://www.stata.com/manuals/mvrotate.pdf, https://www.stata.com/manuals/mvrotatemat.pdf and https://www.stata.com/manuals/mvfactorpostestimation.pdf