Circumplex Models: Can You Use Factor Analysis?

I'm doing a study for my thesis involving using SEM to predict therapy outcome (Outcome Questionnaire -45) using the Inventory of Interpersonal Problems (IIP). The IIP is a circumplex model and my advisor insists that "using factor analysis on circumplex models is stupid." As CFA is part of my SEM, he has a problem with my using the IIP. I'm trying to find research discussing the concepts of a circumplex model and measurement/analytic issues, but I can't find anything about the general concept of a circumplex, only individual circumplex models.

Does anybody know of any resources that could help me understand this? Or does anybody have any knowledge of circumplex models that might be able to shed some light on what my advisor may be talking about?


Doesn't actually exist
Well, your advisor is kind of right and kind of not, but I can see where he’s coming from (he’s probably a little old school).

There are many “idealized” measurement models of scales out there that were conceived in the dawn of Psychometrics. The most popular one by far is Thurstone’s Simple Structure. People like scales/tests/measures that have Simple Structure because it implies that each factor has high loadings on only the set of items that define it and 0 loadings everywhere else (aka cross-loadings are 0 in the population). Simple structure is desirable from the perspective that you can claim only ONE latent variable is being measured at a time by the items intended to measure it.

A different model (which is what your advisor is alluding to) is what used to be called the Radex model or the Circumplex and that was proposed by Guttman. Circumplex models posit a 2-dimensional axes and assume people exist somewhere along the two dimensions. If you were to imagine a circle, circumplex models classify people by seeing where they land along the circle. Or other people imagine it as the 4-quadrant model.

Now, how does this all tie in with measurement? Well, if you are intending to do a “traditional” CFA (i.e., factors load on their respective items, all cross-loadings fixed at 0) to a Circumplex scale, then your advisor is right. Simple structure implies items would cluster together in their respective axes and nowhere in between. Circumplex structure claims the opposite: the items DO NOT cluster in the axes. They should distribute somewhat evenly along a circle defined through the axes that are now your 2 latent variables. Finding evidence for your CFA would actually contradict the model implied by the scale, if it is a true circumplex.

The reason of why I say your prof is a little old school is because we *can* use Factor Analysis to test for circumplex structure. But it would not be through the easy-cheezy, “traditional” CFA approach. You need to implement older methods (like the squared loading index) to find evidence for it.

The classic reference that introduces the Circumplex model to the psychometric literature is here:

I’m sure if you go down the rabbit hole of citations (i.e., who’s cited it for what) you can get a better sense of how people use Circumplex models.