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

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!

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Not a "proper" answer, but if two variables are correlated, but you put them into different factors, then it does make intuitive sense that the two factors should also be correlated. – probabilityislogic Feb 25 '11 at 8:48
How do you measure the increase in goodness-of-fit? – chl Feb 25 '11 at 9:15
Hey chl - looking at RMSEA, SRMS, and BIC. – Scott Feb 25 '11 at 18:46

## 2 Answers

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.

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Thanks! So if I want to validate a orthogonal-rotated factor-loading with CFA (as I have done), it makes sense to validate that with a CFA model where the factors are uncorrelated. Since it's not a great fit, you are saying I should re-run my EFA using an oblique rotation, and then build a CFA model where the factors may correlate. – Scott Feb 25 '11 at 18:48
yup, thats it. If you're going to be doing EFA and CFA on one dataset, you might want to look at splitting the data, running EFA on one half and CFA on the other, to increase replicability of the factor structure. – richiemorrisroe Feb 25 '11 at 22:04
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.
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Thanks M- Adams. In response to 1: I've noticed that the structures are nearly consistent. For 2) Thanks! I am validating against a sample (split my big set in half) 3) Are you suggesting a model where they may be a factor driving other factors? Thanks! – Scott Feb 25 '11 at 18:52