Understanding factor correlations and factor score correlations in CFA and EFA

I've come across something a little puzzling when comparing factor scores estimated using EFA and CFA models, which I'm hoping someone here can explain to me.

I have survey data featuring three sets of items. Each set of items is a different method of measuring the same psychological construct. I have fit three types of factor analysis model:

1. Three separate EFAs, one for each set of items, extracting one factor. When I extract factor scores, the three factor scores variables are correlated at .28 to .35.
2. One three-factor CFA, with one factor specified for each set of items and factor correlations fixed at zero. When I extract factor scores, they are still correlated at .28 to .35.
3. One three-factor CFA, with one factor specified for each set of items and factor correlations estimated. The factor correlations are estimated to be .39 to .44, and the extracted factor scores are now correlated at .49 to .55.

Ultimately I want to fit something like a second-order CFA, but my concern now is with understanding the relationship between correlations between factors and correlations between factor scores. In particular, in model 2, the 3-factor orthogonal CFA assumes no correlation between factors, yet the factor scores are correlated. How should I understand these two sets of correlations?

• measuring the same psychological construct. Why then didn't you focus on testing that the combined tree sets produce only one major factor? – ttnphns Dec 7 '15 at 12:33
• My comment is about why specifically 3 factor models? When there is 3 sets of items which had been designed to measure the same thing and only one thing then combining the sets is expected to "produce" 1 strong factor plus addidional weaker factors, some of them possibly (but not necessarily) reflecting the diferences between the sets. But the number of weak factors is not known apriori, it could be 1, 2, 3 or more. – ttnphns Dec 7 '15 at 12:43
• Regarding your point 1. Three EFAs with subsequent correlating the three factor scores variables is crude - because factor scores are not the true factor values (which are unknown) but only approximations. – ttnphns Dec 7 '15 at 12:47
• I'm replicating other work that has measured these things as separate scales and argued that they should be separate. Ultimately the 2nd-order CFA is where I'll go. However, conducting this exercise has thrown up the question of the difference between factor correlations and factor score correlations that I'm puzzled over. – nomad545 Dec 7 '15 at 13:58

I actually do not think you have conducted CFAs, as you think you have, for your second and third models. Instead, for a couple of reasons, it reads as though you have just conducted three separate EFAs. For one, you mention the term "orthogonal"--a rotation method type--and factor scores, but rotation and factor scores are only features of EFA, not CFA. And in CFA, if you fit a model specifying three uncorrelated factors, the estimated correlations of those models would in fact be zero, and if the factors were correlated, this specification would worsen the fit of your model.

With that, there is still the question of why your estimated factor correlations and factor score correlations are changing from model to model. You actually have identified the likely cause of these discrepancies yourself:

In particular, in model 2, the 3-factor orthogonal CFA assumes no correlation between factors, yet the factor scores are correlated.

Orthogonal rotation methods assume factors are uncorrelated; orthogonal methods do not make factors uncorrelated (Fabrigar & Wegener, 2011). Thus, when using this rotation method, you could still end up with factors that are correlated when you somehow estimate their correlations (e.g., as you did using factor scores). But if factors are truly correlated, and you assume no correlation, the true shared variance between factors needs to go somewhere, so it ends up getting suppressed back down to the factor loadings (Osborne, 2015). Lay the factor matrix of your orthogonal solution next to the pattern matrix of your oblique solution (i.e., with correlations estimated); I'm willing to bet you will see higher "cross-loadings" with the former than with the latter. Put another way, your orthogonal solution will exhibit worse "simple structure" (Fabrigar & Wegener, 2011).

The end result is that the factor scores from your orthogonal and oblique models are computed using fairly different factor loading estimates, and the orthogonal solution suppresses the correlations between factors. So you shouldn't be surprised that the oblique rotation factor scores show stronger correlations.

The reason your factor score correlations from your oblique solution differ from the estimated factor correlations from the same solution is a bit complicated, but ttnphns comment above is a good summary--the factor scores are only approximations, and therefore their correlations are only approximations, whereas the estimated correlations are based on the unobserved error-free latent variables from the EFA (see DeStefano, Zhu, & Mîndrilă, 2009; Grice, 2001 for more details on the nature of factor scores).

References

DeStefano, C., Zhu, M., & Mîndrilă, D. (2009). Understanding and using factor scores: Considerations for the applied researcher. Practical Assessment Research & Evaluation, 14, 1-11.

Fabrigar, L. F., & Wegener, D. T. (2011). Exploratory factor analysis. New York, NY: Oxford.

Grice, J. W. (2001). Computing and evaluating factor scores. Psychological Methods, 6, 430-450.

Osbourne, J. W. (2015). What is rotating in exploratory factor analysis? Practical Assessment Research & Evaluation, 20, 1-7.

• +1. Just a minor note for clarity, if you don't object. [with orthogonal rotation] you could still end up with factors that are correlated when you somehow estimate their correlations (e.g., as you did using factor scores). Factors are extracted as orthogonal in loadings, and orthogonal rotation applied then to loadings preserves orthogonality. It however does not guarantee uncorrelated factor scores (computed by regressional method then), since scores are not exact values. – ttnphns Dec 7 '15 at 16:17
• Do you know, is it possible to have the Fabrigar & Wegener book as pdf for free somewhere? – ttnphns Dec 7 '15 at 16:20
• The book is not, but their earlier article (much of their conceptual overview remains the same) can be found relatively easily: Fabrigar, L. R., Wegener, D. T., MacCallum, R. C., & Strahan, E. (1999). Evaluating the use of exploratory factor analysis in psychological research. Psychological Methods, 4, 272-299. – jsakaluk Dec 7 '15 at 16:24
• @jsakaluk Thank you for the detailed response. A note of clarification however: I am in fact running CFAs for the latter two models (cfa function from the lavaan library for R). I edited my initial question slightly to hopefully make this clearer to others. It appears that a good part of your response rests on the interpretation of factor rotation, which is an EFA property. And the Grice paper appears to address EFA as well. Would you add any commentary or suggested citations given the use of CFA? – nomad545 Dec 7 '15 at 16:36
• Indeed, the 3rd model fits the best and fits just as well as the 2nd order CFA. Regarding my question, thinking about your response, and looking at the Grice and DiStefano et al papers, I realize now that the difference between estimated factors and true factors is the key to the answer. Thank you! – nomad545 Dec 7 '15 at 17:05