Note 10.6. The loadings are therefore determined only up to an orthogonal rotation: the data cannot distinguish \(L\) from \(LT\), whatever \(T\) may be. This is an embarrassment and an opportunity in equal measure. It is an embarrassment because no rotation is more correct than another, so any claim that the factors have been identified must be read carefully. It is an opportunity because one may as well choose the rotation that is easiest to interpret — which is what varimax does, seeking loadings near \(0\) or \(\pm 1\) so that each variable loads heavily on one factor and negligibly on the rest.

Interpretation of rotated factors is the part of multivariate analysis where judgement most easily outruns evidence. The rotation was chosen for interpretability; that the result is interpretable is therefore not evidence that it is true.