# correlations between factors in r- scores or loadings?

I've conducted a factor analysis in r with three factors (function=fa {psych};rotation=promax ; method=GLS).

 fa1 =fa(d,nfactors=3,rotate="promax",oblique.scores=TRUE,method="gls",missing=TRUE,scores=TRUE,impute="mean")


Now I would like to add a correlation matrix between the three factors. Which matrix will be better in defining the correlation between the factors? using the scores correlation?

cor(fa1$scores)  or the loadings correlation? cor(fa1$loadings)

• When you say "correlations between factors" do you mean (1) true correlations between the factors, or (2) observed correlations between the estimated factor scores? These two are not generally the same thing because factor scores are only approximations to the unknown true factor values. Apr 27, 2015 at 12:17
• well, I'm not sure. My gole is to explore the relationship between different carecristics which I found by redusing the data to three latent variables. Apr 27, 2015 at 18:22
• What do you mean by "add a correlation matrix"? Add to what? Apr 27, 2015 at 20:27
• I mean that in addition to the actual factor analysis, I would like to present in my study a correlation matrix that defins the relationship between those three factors. Apr 27, 2015 at 21:12

The psych package's fa function should already provide you with the matrix of correlations between factors; there's no need for you to do this yourself. This information will be located in a section of your output labeled "With factor correlations of", just below the output pertaining the the proportion of variance explained by your factor solution. Though you could examine correlations between factor scores, this would introduce the problem of rotational indeterminacy--correlations between latent factors (what fa provides automatically), as opposed to observed factor scores, do not share this limitation. In practice though, both approaches will likely produce similar estimates of factor correlations. A matrix of the correlations between factor loadings, on the other hand, would not provide you with the kind of information you are seeking about the association between your factors.