Oblimin and Promax Rotation Criteria

Why These Oblique Criteria Are Useful

A rotated loading matrix is meant to make a factor solution easier to describe: ideally, each measured variable has a few substantial loadings and the rest are small. When factors may be correlated, an oblique rotation can seek such a pattern without requiring the rotated factor axes to remain perpendicular. Oblimin specifies a family of criteria for pursuing simple structure; promax constructs a target from an initial orthogonal solution and finds an oblique transformation that approximates it.

These methods change the representation of a fitted factor model, not the observed variables or the number of factors. The resulting loadings must be considered together with the factor-covariance matrix: an oblique loading pattern, by itself, does not state how strongly the factors correlate.

Definition: Let $\Lambda=(\lambda_{ij})$ be a $p\times q$ loading matrix, with $p$ measured variables and $q$ factors. One commonly used generalized oblimin criterion is $$ Q_\gamma(\Lambda)= \sum_{i=1}^{p}\left(\sum_{j=1}^{q}\lambda_{ij}^{2}\right)^2 -\frac{\gamma}{p}\sum_{j=1}^{q}\left(\sum_{i=1}^{p}\lambda_{ij}^{2}\right)^2, $$ where $\gamma$ is a chosen tuning parameter. An oblimin rotation seeks a transformation that reduces the selected criterion, subject to the scaling and identification conventions of the rotation procedure. Different values of $\gamma$ trade off the row and column contributions differently.

The criterion is a function of the loading matrix, so its numerical value depends on the convention used to scale the factors and loadings. It is not a goodness-of-fit statistic for the original factor model. In particular, a lower criterion value does not mean that the rotated representation fits the observed covariance matrix better; rotation is intended to change interpretability while preserving the fitted covariance representation.

Calculating an Oblimin Criterion

The row contribution in $Q_\gamma$ uses each variable’s squared loadings summed across factors, then squares that sum. The column contribution first sums squared loadings down each factor column, then squares those column totals. Keeping these operations in the stated order is essential.

Worked Example: Evaluating an Oblimin Objective

Consider the loading matrix $$ \Lambda= \begin{pmatrix} .7&.2\\ .2&.7\\ .7&.2\\ .2&.7 \end{pmatrix}, $$ with $p=4$. Evaluate $Q_{.5}(\Lambda)$.

For each row, the sum of squared loadings is \[ .7^2+.2^2=.49+.04=.53. \] There are four such rows, so the first term is \[ 4(.53^2)=4(.2809)=1.1236. \]

For either column, the sum of squared loadings is \[ 2(.7^2)+2(.2^2)=2(.49)+2(.04)=1.06. \] Thus the sum of the squared column totals is \[ 2(1.06^2)=2(1.1236)=2.2472. \] The penalty term is $(.5/4)(2.2472)=.2809$, giving \[ Q_{.5}(\Lambda)=1.1236-.2809=.8427. \]

This calculation illustrates a frequent arithmetic trap: the column total is $1.06$, not $1.8$. The total is formed from the squared loadings, not from the loadings themselves.

In practice, an oblimin algorithm searches over admissible oblique transformations, and the final solution depends on the starting point, normalization choices, and selected $\gamma$. The objective helps define the search; it does not imply that every run will find the same numerical solution. The next example illustrates how the criterion can be evaluated for a second loading pattern without claiming that either matrix is an optimum for a particular data set.

Worked Example: Comparing Two Loading Patterns by the Criterion

Compare the criterion for the matrix above with the more sharply separated pattern $$ \widetilde{\Lambda}= \begin{pmatrix} .8&.1\\ .1&.8\\ .8&.1\\ .1&.8 \end{pmatrix} $$ at $\gamma=.5$.

For $\widetilde{\Lambda}$, every row has squared-loading sum $.8^2+.1^2=.65$. The first term is therefore \[ 4(.65^2)=4(.4225)=1.69. \] Each column has squared-loading total $2(.64)+2(.01)=1.30$, so the second sum is $2(1.30^2)=3.38$. Its penalty is $(.5/4)(3.38)=.4225$, and \[ Q_{.5}(\widetilde{\Lambda})=1.69-.4225=1.2675. \]

Under this stated objective and these scaling conventions, the first matrix has the smaller criterion: $.8427<1.2675$. That comparison is only about the objective values of the two displayed matrices. It does not establish that either matrix is a valid rotation of the other or that one is a better-fitting factor model.

Promax: A Powered Target and an Oblique Transformation

Promax begins with a loading matrix $V$ obtained from an orthogonal rotation, commonly varimax. It then raises each loading to a power in magnitude while retaining its sign, creating a target with small entries closer to zero and large entries emphasized. A least-squares transformation maps $V$ as closely as possible to this target. This direct construction is what distinguishes promax from optimizing an oblimin criterion.

Definition: Given an initial loading matrix $V$, a positive integer power $k$ produces the powered target $T$ with entries \[ t_{ij}=\operatorname{sign}(v_{ij})|v_{ij}|^k. \] In the unnormalized least-squares version of promax, choose a matrix $A$ to minimize $\lVert VA-T\rVert_F^2$, where $\lVert\cdot\rVert_F$ is the Frobenius norm. If $V'V$ is invertible, the solution is $A=(V'V)^{-1}V'T$, and the promax loading matrix is $VA$. Implementations may apply additional column rescaling after this step.

The least-squares formula follows by differentiating the sum of squared residuals, or equivalently by solving its normal equations. The next example displays every matrix product so that the transformation and its consequences can be checked directly.

Worked Example: Constructing a Promax Solution

Take $$ V= \begin{pmatrix} .8&.1\\ .7&.2\\ .2&.7\\ .1&.8 \end{pmatrix} \quad\text{and choose } k=2. $$ All entries are positive, so squaring them gives $$ T= \begin{pmatrix} .64&.01\\ .49&.04\\ .04&.49\\ .01&.64 \end{pmatrix}. $$

The cross-products are $$ V'V= \begin{pmatrix} 1.18&.44\\ .44&1.18 \end{pmatrix}, \qquad V'T= \begin{pmatrix} .864&.198\\ .198&.864 \end{pmatrix}. $$ For instance, the first entry of $V'T$ is $.8(.64)+.7(.49)+.2(.04)+.1(.01)=.864$. Since $\det(V'V)=1.18^2-.44^2=1.1988>0$, the matrix is invertible, and $$ A=(V'V)^{-1}V'T = \begin{pmatrix} 7/9&-11/90\\ -11/90&7/9 \end{pmatrix} \approx \begin{pmatrix} .7778&-.1222\\ -.1222&.7778 \end{pmatrix}. $$

Multiplying $V$ by $A$ gives $$ VA\approx \begin{pmatrix} .61&-.02\\ .52&.07\\ .07&.52\\ -.02&.61 \end{pmatrix}. $$ The two main loading clusters are now more distinct than in $V$, although the least-squares approximation does not reproduce every target entry exactly.

If the original factors have covariance $I$, the transformed loading matrix $VA$ represents the same common part of the covariance matrix when the transformed factor vector is $A^{-1}f$. Its covariance matrix is then $\Phi=A^{-1}A^{-T}$. For this $A$, the two eigenvalues of $A$ are $59/90$ and $9/10$; hence the eigenvalues of $\Phi$ are $(90/59)^2$ and $(10/9)^2$. The diagonal entries of $\Phi$ are equal, and its off-diagonal entries are positive. Standardizing those entries gives a factor correlation of approximately $0.307$.

Why the Factors Correlate

The origin of the factor correlations is the nonorthogonal transformation. If a model has loading matrix $\Lambda$ and factor covariance $\Phi$, its common covariance component is $\Lambda\Phi\Lambda'$. Under an invertible transformation $A$, set $\Lambda^*=\Lambda A$ and $\Phi^*=A^{-1}\Phi A^{-T}$. The model’s common covariance component is unchanged:

\Lambda^*\Phi^*(\Lambda^*)' =\Lambda A(A^{-1}\Phi A^{-T})A'\Lambda' =\Lambda\Phi\Lambda'.

This identity explains both why a rotation can preserve the model representation and why an oblique rotation generally requires a nonidentity factor-covariance matrix. The loadings describe how variables relate to the chosen factor coordinates; $\Phi$ describes the covariance among those coordinates. When the factor variances are standardized to one, the off-diagonal entries of $\Phi$ are factor correlations.

Result: If $A$ is invertible and the original factors have covariance $I$, then $\Lambda^*=\Lambda A$ and $f^*=A^{-1}f$ give $\operatorname{Cov}(f^*)=A^{-1}A^{-T}$. Moreover, $\Lambda^*\operatorname{Cov}(f^*)(\Lambda^*)'=\Lambda\Lambda'$.

To verify the result, use the definition of covariance under a linear transformation: \[ \operatorname{Cov}(A^{-1}f)=A^{-1}\operatorname{Cov}(f)(A^{-1})'. \] Since $\operatorname{Cov}(f)=I$ and $(A^{-1})'=A^{-T}$, this is $A^{-1}A^{-T}$. Substituting this covariance and $\Lambda^*=\Lambda A$ into the common covariance gives \[ \Lambda A(A^{-1}A^{-T})A'\Lambda' =\Lambda I I\Lambda' =\Lambda\Lambda', \] because $A^{-T}A'=I$. Thus the transformation changes the factor coordinates but not the represented common covariance. $\square$

Comparing the Two Approaches

Both methods can produce oblique loading patterns, but their routes differ. Oblimin chooses a criterion and searches for a transformed solution that reduces it. Promax first constructs an explicit powered target and then uses least squares to approximate that target. In either case, practical solutions can depend on scaling conventions, normalization, and the details of the numerical algorithm.

Feature	Oblimin	Promax
How the target is specified	Through an oblimin criterion and its tuning parameter	By powering entries of an initial orthogonal loading matrix
Main numerical task	Optimize the criterion over admissible transformations	Fit the powered target by least squares
Factor correlations	Allowed by the oblique transformation	Allowed by the generally nonorthogonal transformation
Interpretation of criterion or target	A criterion value guides the rotation search	The target is an intermediate construction, not a model-fit test

The choice between methods is not a choice between correlated and uncorrelated factors: both allow correlation. It is a choice about how the loading pattern is encouraged. Promax makes its target construction especially visible; oblimin makes the selected criterion and tuning parameter central to the search. After either rotation, the loading and factor-correlation matrices must be read together. Detailed interpretation and comparison of rotated solutions require attention to the orientation and scaling of the factors.

Common Pitfalls

Using unsquared loadings in the oblimin column totals. In the first worked example, each squared-loading column total is $1.06$, not $1.8$. The correct squared-total aggregate is $2(1.06^2)=2.2472$.
Treating the powered promax target as the final loading matrix. The target $T$ is approximated by $VA$; it need not equal the resulting loadings.
Assuming a nonidentity transformation leaves factors uncorrelated. The transformed factor covariance is $A^{-1}A^{-T}$ when the original covariance is $I$. It is generally not $I$.
Reading loadings without the factor covariance matrix. In an oblique solution, loadings alone do not supply the correlations among factors.
Interpreting a lower rotation criterion as better model fit. The criterion defines a rotation preference. It is not a replacement for assessing the factor model’s fit to the data.

Check Your Understanding

Use the definitions and calculations above to answer the following.

For an oblimin calculation, why must each column total be formed from squared loadings before it is squared?
What role does the tuning parameter $\gamma$ play in the stated generalized oblimin objective?
How is the powered target in promax constructed from an initial loading matrix?
What condition on $V'V$ permits the least-squares formula $A=(V'V)^{-1}V'T$?
If the original factor covariance is $I$, what covariance matrix results from the transformation $f^*=A^{-1}f$?
Why does a lower rotation criterion not, by itself, establish better fit of the factor model to the observed data?