Varimax Rotation

Varimax Rotation in Health Questionnaire Factor Analysis

Varimax rotation is the step in an exploratory factor analysis that turns the extracted factors until each one has a few large loadings and many near zero, while the factors stay uncorrelated. Henry Kaiser developed it in his 1956 doctoral thesis and published it in 1958 as an analytic replacement for rotating factor axes by eye, and it is the default rotation in packages such as Stata and R. Health economists meet it in the papers that develop and test patient-reported outcome measures, where varimax-rotated loadings are often the evidence for how items group into domains. Its criterion is a sum of variances, and the angle that maximises it can be found in closed form for each pair of factors. Varimax belongs to the orthogonal rotation family, whose shared properties, such as unchanged communalities, apply to it in full.

Kaiser's criterion and the simple structure it rewards

An extracted factor solution can be rotated without changing how well it fits, so a rule is needed to pick one orientation. Kaiser's 1958 paper in Psychometrika argued for analytic criteria over subjective graphical rotation, reviewed the earlier quartimax criterion and set out the varimax criterion in detail. The paper's abstract states that the normal varimax solution probably coincides closely with the principle of simple structure, but proposes factorial invariance as the ultimate test of a rotation. A note to the paper records that the name varimax was suggested by John Caffrey, who also wrote the original code for the criterion on an IBM 602A.

The Stata reference manual describes varimax as maximising the variance of the squared loadings within factors, the columns of the loading matrix. A column has a large variance of squared loadings when it holds some large and some small values, so the rotation pushes each factor towards a clear split between items that define it and items that do not. Quartimax, by contrast, maximises the variance within rows, so that each item loads on as few factors as possible. The manual adds that the two can give rather similar results, though they need not.

The criterion written as column variances

The IBM SPSS Statistics algorithms document gives the criterion that its software tracks across iterations. Written for $p$ items and $m$ factors, that quantity is the sum over factors of the variance of the squared loadings in each column.

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

where $V$ is the varimax criterion, $p$ is the number of items, $m$ is the number of factors, and $\lambda_{ij}$ is the loading of item $i$ on factor $j$, which SPSS takes after Kaiser normalisation. The first term inside the brackets is the mean of the fourth powers in a column and the second is the square of the mean squared loading, so their difference is the variance of the squared loadings.

The rotation leaves each item's communality, its sum of squared loadings across factors, unchanged. Variance therefore moves between factors and within columns, and none is added. A larger $V$ means sharper columns, and the rotation stops when $V$ no longer increases by more than a small tolerance, set at 0.00001 in the SPSS algorithm.

Rotating one pair of factors at a time

Stata states that its default varimax is computed by the classic algorithm of cycling through rotations of two factors at a time, and SPSS describes orthogonal rotations as done cyclically on pairs of factors until a convergence criterion is met. For each pair the best angle has a closed form, which the SPSS algorithms document gives following Harman.

$$\tan 4\phi = \frac{X}{Y}, \qquad X = D - \frac{2AB}{p}, \qquad Y = C - \frac{A^{2} - B^{2}}{p}$$

where $\phi$ is the rotation angle for factors $j$ and $k$, taken so that $\sin 4\phi$ has the sign of $X$ and $\cos 4\phi$ the sign of $Y$ (the other root of the equation gives the minimum), and $A$, $B$, $C$ and $D$ are sums over the $p$ items defined below.

$$u_i = \lambda_{ij}^{2} - \lambda_{ik}^{2}, \qquad v_i = 2\lambda_{ij}\lambda_{ik}, \qquad A = \sum_{i=1}^{p} u_i, \qquad B = \sum_{i=1}^{p} v_i, \qquad C = \sum_{i=1}^{p} \left( u_i^{2} - v_i^{2} \right), \qquad D = \sum_{i=1}^{p} 2u_i v_i$$

where $u_i$ and $v_i$ are working quantities for item $i$ and $\lambda_{ij}$ and $\lambda_{ik}$ are its current loadings on the two factors. The SPSS document gives the same structure for quartimax and equamax with different weights on $A$ and $B$. The pair is then rotated through $\phi$.

$$\lambda_{ij}^{\text{new}} = \lambda_{ij}\cos\phi + \lambda_{ik}\sin\phi, \qquad \lambda_{ik}^{\text{new}} = -\lambda_{ij}\sin\phi + \lambda_{ik}\cos\phi$$

where $\lambda_{ij}^{\text{new}}$ and $\lambda_{ik}^{\text{new}}$ are the loadings after this step. With more than two factors every pair is rotated in turn and the cycle repeats until the criterion settles.

Raw and normal varimax across software

Raw varimax works on the loadings as they are, so items with high communalities pull harder on the angle. The Stata manual explains that Kaiser suggested every row should carry the same weight, and the R documentation describes the step as rescaling each row to unit length before rotation and scaling it back afterwards. This is Kaiser normalisation.

Defaults differ between packages, so the same data can give slightly different loadings. Stata's rotate command performs raw varimax unless the normalize option is given. R's varimax function uses Kaiser normalisation by default, and R's factanal function uses varimax as its default rotation. SAS PROC FACTOR applies no rotation unless ROTATE= is given, defaults to NORM=KAISER, and its orthomax weight defaults to 1, which gives varimax. A methods section should therefore say whether Kaiser normalisation was applied.

Worked example rotating an invented four-item health scale

All numbers in this example are illustrative and do not come from any real instrument. A two-factor solution has been extracted for a four-item scale with two physical items, mobility and usual activities, and two emotional items, anxiety and low mood. The unrotated loadings are (0.80, 0.10) for mobility, (0.70, 0.20) for usual activities, (0.40, minus 0.60) for anxiety and (0.30, minus 0.50) for low mood, with communalities of 0.65, 0.53, 0.52 and 0.34.

1. Compute the working quantities. For mobility, u is 0.64 minus 0.01, giving 0.63, and v is 2 × 0.80 × 0.10, giving 0.16. Usual activities gives u of 0.49 minus 0.04, which is 0.45, and v of 0.28. Anxiety gives u of 0.16 minus 0.36, which is minus 0.20, and v of 2 × 0.40 × (minus 0.60), which is minus 0.48. Low mood gives u of minus 0.16 and v of minus 0.30.

2. Form the four sums. A is 0.63 plus 0.45 minus 0.20 minus 0.16, giving 0.72, and B is 0.16 plus 0.28 minus 0.48 minus 0.30, giving minus 0.34. The terms u squared minus v squared are 0.3713, 0.1241, minus 0.1904 and minus 0.0644, so C is 0.2406. The terms 2uv are 0.2016, 0.2520, 0.1920 and 0.0960, so D is 0.7416.

3. Find the angle. X is 0.7416 minus 2 × 0.72 × (minus 0.34) / 4, which is 0.7416 plus 0.1224, giving 0.8640. Y is 0.2406 minus (0.5184 minus 0.1156) / 4, which is 0.2406 minus 0.1007, giving 0.1399. The ratio is 6.1758, so four times the angle is 80.80 degrees and the angle is 20.20 degrees, with cosine 0.9385 and sine 0.3453.

4. Rotate each item. For mobility the new first loading is 0.80 × 0.9385 plus 0.10 × 0.3453, which is 0.7508 plus 0.0345, giving 0.7853, and the second is 0.0938 minus 0.2762, giving minus 0.1824. For anxiety the first is 0.3754 minus 0.2072, giving 0.1682, and the second is minus 0.1381 minus 0.5631, giving minus 0.7012. The second factor now loads negatively on its defining items, so its signs are reversed, as the SPSS algorithm does for factors with negative sums.

5. Check the criterion. Before rotation the squared loadings in the first column are 0.64, 0.49, 0.16 and 0.09, with mean 0.345 and fourth powers summing to 0.6834. The column variance is 0.6834 / 4 minus 0.345 squared, which is 0.17085 minus 0.11903, giving 0.0518. The second column gives 0.04845 minus 0.02723, which is 0.0212, so V is 0.0731. After rotation the columns give 0.16478 minus 0.08762, which is 0.0772, and 0.08764 minus 0.04580, which is 0.0418, so V is 0.1190. V is 0.1057 at 10 degrees, 0.1160 at 25 degrees and 0.1067 at 30 degrees, so 20.20 degrees is the maximum.

6. Apply Kaiser normalisation. Dividing each row by the square root of its communality turns low mood into 0.5145 and minus 0.8575, since that square root is 0.5831. Steps 1 to 3 on the normalised rows give an angle of 22.11 degrees, and rescaling gives the last two columns of the table. Low mood, the item with the smallest communality, now weighs as much as mobility, and the angle moves by about 2 degrees.

7. Interpret. Both solutions show a physical and an emotional factor, with mobility keeping a cross-loading of about 0.2 on the emotional one.

Varimax evidence in health measurement and economic evaluation

Costello and Osborne surveyed two years of studies in PsycINFO and found over 1,700 that used some form of exploratory factor analysis. Well over half reported principal components analysis with varimax rotation. They judged this common pairing not optimal and recommended maximum likelihood extraction with oblique rotation, partly because factors in the social sciences are usually expected to correlate.

A study of the EQ-5D-5L in alopecia areata by Lloyd and colleagues shows the method in an HTA setting. Using data from adults in the ALLEGRO-2b/3 trial, with 601 in the factor analysis, the authors analysed 51 variables that included items of the EQ-5D-5L, the SF-36v2 and a condition-specific measure, the AAPPO, and planned to choose between promax and varimax rotation. Oblique rotation was run first, and because the factor correlations were 0.30 or less the analysis was re-run with orthogonal rotation. The nine-factor solution captured 56% of the variance. The AAPPO items loaded on their own factors, which contained no EQ-5D-5L or SF-36v2 items, and the authors read this as variance that the EQ-5D-5L did not assess and a possible limit to its content validity. They argued that if the EQ-5D cannot measure the full impact of alopecia areata, HTA bodies may underestimate the value of a treatment.

For an economic evaluation, varimax loadings are evidence on measurement structure. They can support structural validity arguments and guide which domains a mapping function or a bolt-on item should target. They are not preference weights and do not value health states for QALYs.

Where varimax sits among rotation methods

Varimax is a special case of several families of criteria. Stata notes that varimax is equivalent to the oblimin criterion with weight 1 and to the Crawford and Ferguson criterion with weight 1 divided by the number of items. In the orthomax family set out on the orthogonal rotation page, varimax is the member with weight 1. The Stata manual warns that varimax is not appropriate when a general factor is expected to contribute to all variables, and suggests quartimax in that case.

Varimax is also the usual starting point for promax, an oblique rotation. Stata, SAS and SPSS all build promax from a varimax solution. Varimax and promax are both kinds of factor rotation. In confirmatory factor analysis the loading pattern is fixed in advance, so there is no rotation step.

Common misreadings of varimax output

One mistake confuses the two Kaiser rules. Others leave out settings that another analyst would need to repeat the result.

1. Kaiser criterion and Kaiser normalisation confused. The rule of keeping factors with eigenvalues above 1 decides how many factors to rotate, while Kaiser normalisation weights rows during rotation.
2. Unreported settings. Normalisation defaults differ between Stata, R and SAS, so loadings cannot be reproduced without them.

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
• IBM Corporation. IBM SPSS Statistics Algorithms (edition for IBM SPSS Statistics 25 and later). FACTOR Algorithms. IBM; 2017. https://public.dhe.ibm.com/software/analytics/spss/documentation/statistics/26.0/en/client/Manuals/IBM_SPSS_Statistics_Algorithms.pdf
• Kaiser HF. The varimax criterion for analytic rotation in factor analysis. Psychometrika. 1958;23(3):187-200. https://doi.org/10.1007/BF02289233
• Lloyd A, Aggio D, Dixon C, Law EH, Price T. Psychometric properties of the EQ-5D-5L in patients with alopecia areata. PharmacoEconomics Open. 2024;8(5):715-725. https://doi.org/10.1007/s41669-024-00504-8
• R Core Team. varimax {stats}: Rotation methods for factor analysis, and factanal {stats}: Factor analysis. R documentation, stats package version 4.6.0, accessed 31 October 2026. https://stat.ethz.ch/R-manual/R-devel/library/stats/html/varimax.html
• 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_sect006.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