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:
- 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. - 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. - 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?
Best Answer
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:
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.