Personality and Individual Differences
Twelve markers in a plane. Turn the axes and every loading changes while not one point moves.
Simulated — fictional adjective markers with illustrative coordinates
By the end you should be able to say exactly what rotation changes and what it leaves untouched, and defend simple structure without claiming it finds the true solution.
About 20 minutes. Nothing you do here is saved or sent anywhere.
A researcher extracts two factors, then rotates them.
Fictional adjective markers with illustrative coordinates. The positions came out of the extraction and never change; only the axes move.
Labels are an interpretive act, not a result. These are the highest-loading markers, nothing more.
| Marker | F1 | F2 | Communality | Salient on |
|---|
Rotate until you have found the simplest structure you can, then answer.
Every loading changes; the configuration of points does not. The rotated model reproduces the correlations exactly as well as the unrotated one, each variable's communality is untouched, and the total variance explained is identical. Rotation is a change of coordinate system, and coordinate systems are not discoveries.
Two opposite overreactions are common. The first is that rotation manufactures results — it cannot, because fit is invariant. The second is that the rotated solution is the true structure — it is not, because the mathematics does not privilege it. Simple structure is a criterion people adopted, for good reasons: it yields solutions that are interpretable and comparable between studies. Those are real virtues, and they are not the same as truth.
Forcing the axes to 90° asserts that the constructs are uncorrelated. For most psychological constructs that is implausible, and the third marker set shows what it costs. When the clusters genuinely sit closer to 36° apart than to 90°, no orthogonal rotation puts an axis through both, and the solution looks worse than it is. Allowing the axes to move fixes the picture and produces correlated factors — which is a claim about the world, to be defended rather than avoided.
In the second marker set, one pair of markers sits between the clusters and no rotation makes it simple. That is what a genuine cross-loading is: not a nuisance to be rotated away but a fact about a variable that belongs partly to both factors. Deleting such items to improve simple structure is common and quietly changes what the factors mean.