I am trying to find an intuitive illustration -- particular of varimaxand quartimax -- to inform the choice of rotation (in the context of a principal components analysis).

I am aware of the formal definition of the two, and these two formidable answers on the topic.

In short, varimax maximizes the variance of loadings on each component (column-wise, so to speak), whereas quartimax minimizes the number of high-loading components on each variable (row-wise, so to speak). In other words, varimax "explains" components with maximally varying variables, whereas quartimax "explains" variables with as few components as possible.

The way I understand rotation procedures, they are a way to make loadings matrices human-readable, much like concepts in general make the world human-readable. (For example, the concept "friendliness" is a way to make the manifold variations of human behaviours intelligible).

Now I suggest that varimax and quartimax (as well as other rotation procedures) can be understood as alternative logics (?) to formulate such concepts.

I'd like to find a good example to illustrate this rotation-as-concept-formulation.

My first attempt at such an illustrative example is this:

  • let us take an example of Q-Mode PCA on foods, where dishes are variables (say, a BLT-sandwhich, a curry, ...), and food characteristics are cases (say, salt content, calories, etc.).
  • let the points in the below loadings plot (not biplot!) be dishes (variables), as they are correlated with other dishes across various food characteristics (as cases).
  • assume further that we're only retaining the first two components from a PCA over various dishes
  • the red axes are the axes as "drawn" by varimax, with variables often loading high on both components (many variable-points are in the corners of the plot)
  • the green axes are the axes as "drawn" by quartimax, with variables often loading high on only one component (many variable-points are on/close to the axes of the plot).

First draft of an illustrative

I have tried to label (interpret) these fictitious axes in a meaningful way; maybe varimax components would be warm-cold, sweet-hearty/savory, whereas quartimax components would be lunch-dinner, domestic-foreign.

Under varimax-rotation, dishes would then be conceptualized in what might (preliminarily) be called clear-cut, abstract terms: any given dish will likely vary strongly along these dimensions (it's rare to have a dish that you're eating somewhat warm, and that is somewhere in-between sweet and savory). On the other hand, it's not really clear what a savory dish would be, that is not also either hot or cold (there are few dishes on the axes).

Conversely, under quartimax-rotation, dishes would be conceptualized in a more mushy, but authentic way: any given dish may be somewhere along the domestic-foreign continuum (say, a pizza hawaii in Italy), but the extremes of the axes do make sense: there is, conceivable, such a thing as a foreign dish (say, sushi), that is neither lunch nor dinner. The axes, are, in other words, occupied, including at the extremes.

Both conceptualizations of dishes, naturally, make sense, but in a different way: quartimax emphasizes categories that can actually be observed in pure form, whereas varimax emphasizes categories that are maximally distinct.

Now, my questions:

  1. Did I get varimax, quartimax approximately right? Does this make sense?
  2. Can you help me come up with a better example?
  3. In particular, can you help me come up with an example in R-mode PCA (not Q-mode)? (I'm used to Q, and couldn't help wrap my head around in R-mode, but that should make this more generally meaningful).
  4. Bonus: How would alternative rotation methods figure in this example: equamax and parsimax (let's stick to orthogonal procedures for now ...).
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    $\begingroup$ At the moment I've not time to do nice explanations here. But as I've seen in your profile that you're in Bremen I've just uploaded an old text of mine which might give at least some introductory, but pretty systematic insight into the problem of rotation. See go.helms-net.de/stat/fa/IR_MethodenManual.doc It is really amateurish and intended as a handbook for my self-programmed factor analytic tool "inside-[r]". The part of quartimax and varimax could be much improved, but embedded in the whole text it might so far at least give an improved idea/intuition. I can come back to this later $\endgroup$ Dec 24 '15 at 8:41
  • $\begingroup$ thanks for pointing me to that excellent, rigorous and highly technical manual of yours @GottfriedHelms! I hadn't fully comprehended the geometric perspective on rotation yet. I am, however, still unsure what especially varimax vs quartimax mean in conceptual terms (see above), and would appreciate any thoughts you might have. $\endgroup$
    – maxheld
    Jan 5 '16 at 16:51
  • $\begingroup$ also @GottfriedHelms, I have a small question (somewhat of an aside): I hadn't heard of centroid rotation before, but just of the centroid factor extraction procedure, which seems to have similar features (indeterminacy, reflections, etc.). Am I correct to assume that centroid rotation is the "default" (unrotated) result from centroid extraction, just as the principal components "rotation" is the "default" (unrotated) result from PCA extraction? $\endgroup$
    – maxheld
    Jan 5 '16 at 16:52
  • $\begingroup$ Well I've taken that "centroid"-thing from two old books (Ueberla[1968],Mulaik[1972]) and it may be they wrote only of "centroid extraction" and not of "~rotation", perhaps my wording is a bit inaccurate here. But it can be implemented by a rotation (and I did it in the program) and so I might have over-generalized the book-texts (not the method itself). Hmm. About the "conceptual difference" - roughly said: varimax maximizes variance (around a floating mean in the loadingssquares of each factor over the iterations) while quartimax works without relating to such a factor-specific mean. $\endgroup$ Jan 5 '16 at 20:31

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