Orthogonal Rotation

Orthogonal Rotation in Factor Analysis of Health Measures

Orthogonal rotation turns the axes of an extracted factor solution through an angle while keeping them at right angles, so that each questionnaire item loads mainly on one factor and only weakly on the others. Because the axes stay perpendicular, the rotated factors are still uncorrelated, and each rotated loading can be read directly as the correlation between an item and a factor. Health economists meet it through patient-reported outcome measures, where the structure found by exploratory factor analysis shapes how items are grouped and scored. The best known case is the SF-36 and SF-12 physical and mental summary scores, whose standard version 1 scoring rests on an orthogonal factor solution. That choice has a measurable cost when physical and mental health are correlated. Orthogonal rotation is one of the two families of factor rotation, the other allowing the factors to correlate.

Why an extracted factor solution is rotated

A factor model describes the correlations among items through a few common factors, each item having a factor loading on each factor, and the extracted solution is not unique. The Stata rotation manual explains that orthogonal transformations of the common factors and their loadings leave the reconstructed correlation matrix unaffected and keep the common factors uncorrelated, so the analyst may pick the orientation easiest to interpret.

Thurstone's informal criteria for a simple structure were later turned into formal rotation criteria. Kaiser's 1958 paper on the varimax criterion set out the scientific case for analytic criteria over subjective graphical rotation, and proposed factorial invariance as the ultimate criterion of a rotation, ahead of simple structure. Costello and Osborne stress the limit of the exercise, since rotation simplifies and clarifies the data structure but cannot improve basic aspects of the analysis such as the amount of variance extracted.

The orthogonality constraint in matrix form

An orthogonal rotation multiplies the unrotated loading matrix by a square matrix whose columns are unit length and mutually perpendicular. The Stata documentation describes such transformations as true reorientations in space and reflections, which leave both the lengths of the item vectors and the angles between them unchanged.

$$\boldsymbol\Lambda^{*} = \boldsymbol\Lambda\mathbf{T}, \qquad \mathbf{T}^{\mathsf T}\mathbf{T} = \mathbf{T}\mathbf{T}^{\mathsf T} = \mathbf{I}$$

where $\boldsymbol\Lambda$ is the $p \times m$ matrix of unrotated loadings for $p$ items and $m$ factors, $\mathbf{T}$ is the $m \times m$ orthogonal rotation matrix, $\boldsymbol\Lambda^{*}$ is the rotated loading matrix, and $\mathbf{I}$ is the $m \times m$ identity matrix.

The common part of the fitted correlation matrix is the same before and after rotation, and so is each item's communality, the diagonal element of that product.

$$\boldsymbol\Lambda^{*}\boldsymbol\Lambda^{*\mathsf T} = \boldsymbol\Lambda\mathbf{T}\mathbf{T}^{\mathsf T}\boldsymbol\Lambda^{\mathsf T} = \boldsymbol\Lambda\boldsymbol\Lambda^{\mathsf T}, \qquad h_i^{2} = \sum_{j=1}^{m} \lambda_{ij}^{*2}$$

where $h_i^{2}$ is the communality of item $i$, the share of its variance explained by the common factors, and $\lambda_{ij}^{*}$ is the rotated loading of item $i$ on factor $j$. The Stata manual adds that the uniquenesses, the variances of the item-specific factors, are not affected because only coordinates in the common factor space change. SAS notes that for uncorrelated factors the variance accounted for by a factor is usually the sum of squares of its column of loadings, and these column sums move with the rotation while their total stays fixed.

Orthomax criteria and normalisation choices

The rotation angle is not chosen by eye. Software searches over all orthogonal matrices for the one that maximises a simplicity criterion, and the common orthogonal criteria belong to the orthomax class that the SAS documentation describes, following Harman.

$$Q_{\gamma}(\boldsymbol\Lambda^{*}) = \sum_{j=1}^{m} \left[ \sum_{i=1}^{p} \lambda_{ij}^{*4} - \frac{\gamma}{p} \left( \sum_{i=1}^{p} \lambda_{ij}^{*2} \right)^{2} \right]$$

where $\gamma$ is the orthomax weight and the other symbols are as above. The SAS procedure maps named methods to weights as shown below.

Stata describes varimax rotation as maximising the variance of the squared loadings within each factor and calls it the most popular rotation, and Costello and Osborne call it by far the most common choice. The Stata manual also warns that varimax is not appropriate if a general factor contributing to all variables is expected, suggesting quartimax instead. Health status instruments in which every item partly reflects overall health may be such a case.

The raw criterion weights items by their communalities. Kaiser suggested that every row should carry the same weight when the optimal rotation is computed, which is done by rescaling each item's loadings so that their squares sum to 1, a step known as Kaiser normalisation. Stata does not normalise by default, while SAS PROC FACTOR defaults to NORM=KAISER, so packages can give slightly different rotated loadings.

Worked example with an illustrative four-item health questionnaire

All numbers in this example are illustrative and are not estimates from any real instrument. A two-factor solution has been extracted for an invented questionnaire with two physical items (mobility and usual activities) and two emotional items (anxiety and low mood). All items load positively on the first unrotated factor and the second splits them by sign, a pattern hard to name.

1. Set the rotation matrix. A rotation through an angle $\theta$ with $\cos\theta = 0.8$ and $\sin\theta = 0.6$, about 36.87 degrees, uses the matrix with first row (0.8, 0.6) and second row (minus 0.6, 0.8). Since 0.64 plus 0.36 is 1 and 0.8 × 0.6 minus 0.6 × 0.8 is 0, $\mathbf{T}^{\mathsf T}\mathbf{T} = \mathbf{I}$.

2. Rotate each row. For mobility, the new first loading is 0.70 × 0.8 plus (minus 0.40) × (minus 0.6), which is 0.56 plus 0.24, giving 0.80, and the new second loading is 0.70 × 0.6 plus (minus 0.40) × 0.8, which is 0.42 minus 0.32, giving 0.10. For anxiety, the first loading is 0.56 × 0.8 minus 0.58 × 0.6, which is 0.448 minus 0.348, giving 0.10, and the second is 0.56 × 0.6 plus 0.58 × 0.8, which is 0.336 plus 0.464, giving 0.80.

3. Check what is preserved. The communality of mobility is 0.49 plus 0.16, which is 0.65, before rotation and 0.64 plus 0.01, also 0.65, after it, and usual activities gives 0.4624 plus 0.0676, which is 0.53, before and 0.49 plus 0.04, also 0.53, after. The common part of the fitted correlation between mobility and anxiety is 0.392 minus 0.232, giving 0.16, before rotation and 0.08 plus 0.08, also 0.16, after it.

4. Follow the variance between factors. Before rotation the column sums of squared loadings are 1.6024 and 0.7576, a total of 2.36. After rotation each column gives 0.64 plus 0.49 plus 0.01 plus 0.04, which is 1.18, and the total is again 2.36, so variance has moved between factors without any being added.

5. Confirm the criterion improved. The raw varimax criterion in the Stata form is the orthomax function with weight 1, divided by the number of items. For each rotated column the sum of fourth powers is 0.4096 plus 0.2401 plus 0.0001 plus 0.0016, which is 0.6514, and the column term is 0.6514 minus 0.3481, giving 0.3033, where 0.3481 is 1.18 squared divided by 4. The criterion is 0.6066 divided by 4, which is 0.15165, against 0.0152 for the unrotated loadings. A search over all angles finds the maximum at 36.87 degrees, so this is the varimax solution.

6. Interpret. After rotation the first factor is plainly physical and the second emotional, and the loadings of 0.80 and 0.70 are the correlations of each item with its factor. The rotation has also assumed by construction that physical and emotional health are uncorrelated. If they are correlated, the solution cannot show it, and scores built from it inherit the assumption.

What uncorrelated factors gain and lose

The gain is simpler output. Stata's postestimation manual notes that the matrix of correlations between items and common factors coincides with the loading matrix after an orthogonal rotation such as varimax, but not after an oblique rotation, where correlated factors partly explain the same variance and factor variances add up to more than in the orthogonal case.

The loss is realism. Costello and Osborne call the conventional advice to prefer orthogonal rotation for interpretability a flawed argument, because factors in the social sciences are generally expected to correlate, and orthogonal rotation then loses valuable information. When the factors really are uncorrelated, the two approaches give nearly identical results. As an oblique rotation can reproduce an orthogonal solution but not the reverse, they recommend oblique rotation.

SF-36 and SF-12 summary scores

Farivar, Cunningham and Hays report that the standard scoring algorithm for the SF-36 and SF-12 version 1 summary measures is based on a factor analytic technique that forces the scores to be orthogonal, fixing the correlation between the physical and mental health constructs at zero. Ware and colleagues argued that the orthogonal method is mathematically simpler, yet Farivar and colleagues cite studies placing the correlation between the physical and mental factors between 0.32 and 0.66.

The constraint leaves a trace in the scoring weights. The physical component summary gives negative weights to the social functioning, role-emotional and emotional well-being scales, and the mental component summary gives negative weights to the four physical scales, so better mental health scale scores push the physical summary down. In one study they cite, patients starting antidepressant treatment improved by 0.28 to 0.49 standard deviation units on the physical health scales over three months, yet the orthogonal physical summary barely moved, at 51 at baseline and 50 afterwards. In their hypothetical profile with physical scales 1 standard deviation and mental scales 0.3 standard deviations above the mean, the orthogonal mental summary was 49.6, exactly at the mean, despite the better mental scale scores.

Farivar and colleagues warn that when summary scores are reported alone a study may fail to detect an effect of an intervention, and for economic evaluation this means analysts using SF-12 or SF-36 summaries as outcomes, or as inputs to a mapping function, need to know which scoring version was used and should look at the scale scores too.

Boundaries with neighbouring methods

Orthogonal rotation follows extraction in exploratory factor analysis, and its result is one piece of evidence on structural validity, not proof of it. In confirmatory factor analysis the analyst specifies in advance which loadings are free and whether factors correlate, so there is no rotation step to choose.

Rotated principal components need caution, since the Stata manual notes that some of their optimality properties are not preserved under rotation. Loadings from either method describe measurement structure only and are not preference weights for QALYs.

Common misreadings

Most errors with orthogonal rotation come from treating a property imposed by the method as a property of the data. Others come from leaving out the analytic choices that a reader needs to repeat the result.

1. Treating uncorrelated factors as a finding. Zero factor correlation is imposed by the method, so it says nothing about whether the constructs are related in patients.
2. Thinking rotation improves fit. Communalities, uniquenesses and the fitted correlations are unchanged, and only the description of the same solution changes.
3. Reading factor order or sign as meaningful. Stata notes that rotation criteria are invariant to column order and sign changes, so both are conventions.
4. Assuming varimax suits every instrument. Where a general health factor is expected to run through all items, the Stata manual judges varimax inappropriate and points to quartimax.
5. Leaving out the options. Reports should name the criterion, normalisation and software, since defaults differ.

Sources

• Costello AB, Osborne JW. Best practices in exploratory factor analysis: four recommendations for getting the most from your analysis. Practical Assessment, Research and Evaluation. 2005;10(7). https://doi.org/10.7275/jyj1-4868
• 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
• Kaiser HF. The varimax criterion for analytic rotation in factor analysis. Psychometrika. 1958;23(3):187-200. https://doi.org/10.1007/BF02289233
• SAS Institute. SAS/STAT 9.2 User's Guide, Second Edition. The FACTOR Procedure. SAS Institute; 2009. https://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/statug_factor_sect003.htm
• StataCorp. Stata Multivariate Statistics Reference Manual. rotate: Orthogonal and oblique rotations after factor and pca. StataCorp; 2025. https://www.stata.com/manuals/mvrotate.pdf
• StataCorp. Stata Multivariate Statistics Reference Manual. rotatemat: Orthogonal and oblique rotations of a Stata matrix. StataCorp; 2025. https://www.stata.com/manuals/mvrotatemat.pdf
• StataCorp. Stata Multivariate Statistics Reference Manual. factor postestimation: Postestimation tools for factor and factormat. StataCorp; 2025. https://www.stata.com/manuals/mvfactorpostestimation.pdf