What does it mean if there is a correlation between underlying factors in factor analysis?

Question

I'm running an EFA using orthogonal/varimax rotation, and assigning variables to a factors based on maximum load (so only each variable only gets one factor). I then want to validate the model using SEM... since the rotation I used to determine the variable<->factor loads was orthogonal, is it "wrong" to let the factors in my model have a covariance with one another? (eg, using RAM: Factor1<->Factor2,theta,NA)

I ask, as I get a much better model fit if I allow for this to occur.

More explicitly, what does it actual mean for underlying factors to have a correlation between them?

Thanks!

Answer 1

OK, in my experience cases like this mean that you should have allowed the factors to correlate in the first place. You should probably rerun a factor analysis using either oblimin or promax rotation and test the fit of your uncorrelated model against your correlated model.

Please do note that SEM loses its utility as a method for testing theories once you start changing the model based on fit indices.

Answer 2

1. The first thing to do is see if the structure changes when you use an oblique rotation that allows the factors to correlated.
2. If you truly want to validate your structure then use a confirmatory factor analysis (CFA, which is the measurement part of SEM) as you are already doing but with a different sample.
3. If you just have one sample and you are interested in whether the factors correlate, look into building a second order CFA that posits a higher order factor explaining the common variance among 2 or more of the lower order factors.