I have discovered that factor rotation has had detrimental effects in a lot of studies applying factor analysis in cross-cultural studies. I have made a meta-analysis of cultural differences between countries. Most of the published studies of cultural differences use rotated factor analysis. I discovered that the unrotated results of the different analyses were very similar, while the published rotated results all looked different. The authors of each study invented new confusing names for the factors they found, believing that they had found something new and useful. Using unrotated factors instead, I found that most of the cultural variables had high correlations with one or two general factors, while the remaining factors seemed unimportant and non-reproducible. The common practice of factor rotation has obscured the fact that many researchers have made very similar findings with strong loadings on the same two general factors. My discovery immediately caused attention among researchers in the field of cross-cultural studies.

My study is published in the journal Cross-Cultural Research: https://journals.sagepub.com/doi/abs/10.1177/1069397120956948 (open access preprint: https://www.researchgate.net/publication/343670790_A_test_of_the_reproducibility_of_the_clustering_of_cultural_variables )

I find it hard to believe that factor rotation has been a standard recommendation for many years without mentioning that it may obscure the existence of strong general factors in the case where many variables are correlated with each other. The only mentioning of this phenomenon that I can find is in this post Is there a reason to leave an exploratory factor analysis solution unrotated?

Have I missed something in the literature that is discussing this? Should I publicize my findings somewhere so that researchers in other fields than cross-cultural studies can learn from it?

  • $\begingroup$ I'm leaving an instant answer below, but I haven't read your article. $\endgroup$
    – ttnphns
    Sep 20, 2020 at 18:15
  • $\begingroup$ authors invented [through rotations they did] new confusing names for the factors they found, believing that they had found something new and useful I would say this is a matter of creativity or of vanity more than a statistical or mathematical matter. $\endgroup$
    – ttnphns
    Sep 20, 2020 at 22:43
  • $\begingroup$ I found that most of the cultural variables had high correlations with one or two general factors, while the remaining factors seemed unimportant. (i) If you are extracting (modeling) m factors, you already pose all the m ones "important" (existing); otherwise why not create less factors? (ii) In factor extraction methods such as PAF (based on PCA method), the 1st factor maximizes its loadinds as its goal, so it is natural that most variables get big loadings (correlations) with it. $\endgroup$
    – ttnphns
    Sep 20, 2020 at 23:01
  • $\begingroup$ I discovered that the unrotated results of the different analyses were very similar If unrotated loadings of two analyses (with equal num. of factors) are "very similar" why would the rotated ones (by the same rotation method, of course) differ any much? $\endgroup$
    – ttnphns
    Sep 21, 2020 at 2:22
  • $\begingroup$ why would the rotated ones differ any much? Because they are different studies done by different researchers with different data and different goals. They are unaware that many of their variables are correlated with the same 'general factor', hence they do not look for it. $\endgroup$
    – A Fog
    Sep 22, 2020 at 4:46

1 Answer 1


Do we need factor rotation? Of all the factors? Do the strongest unrotated factor reveal the "general factor"?

Books do not urge, "rotate, don't leave your factors unrotated". Rather, they say that rotation can benefit in interpretation.

Factor rotations are done for the sake of more easy and "better" interpretation of the meaning of factors (the latent features). One is absulutely free in how to rotate their factors if to rotate at all. Rotation does not change juxtaposition of the variable vectors in the space of loadings, but only coordinates thereof.

In particular, you may rotate not all the factors (i.e. not the entire post-extraction loading matrix columns), but only selected factors (selected loading matrix columns). The factors that don't participate in the rotation retain their pre-rotation loadings, as well as their factor or component scores (at least as computed by the regression method). The initial orthogonality of the factors abstained from rotation with the factors undergone rotation is preserved. (In fact, imagine a 3D loading plot with factors - the axes - F1, F2, F3. You can rotate the F2-F3 plane orthogonal to F1 around the F1 axis. You can even bring close or move apart the F2 and F3 axes, making the rotation of F2-F3 subsystem an oblique one, - but F1 remains fixed and orthogonal to them both, and loadings for F1 won't change.)

So, if you want to preserve an extracted factor as it is, such as the first factor which you consider to be enough "general" factor, then just don't touch it, and rotate all the other factors towards some "simple structure" facilitating their interpretation. Another possible approach might be first to perform quartimax rotation on all the factors (quartimax may aid for further "generalization" of the most strong, i.e. 1st, extracted factor), and only then to rotate the rest of the factors by varimax or other method.

Does "general factor" exist? This is a philosophical question (and also connected with this one). Many researchers don't admit the factor, and when they do they may differ in a theoretical concept of it. Some proponents of the general factor may demand, for example, first to do PCA to skim the 1st component away from the data (or from correlation matrix), and then turn to do FA of the residual data/correlations. This approach is not unreasonable, because the general factor (embodied in the 1st PC here) is being removed from all the variability before the conception of unique factors (as the variabilities orthogonal to the common factors) is introduced via FA. (Indeed, do variables have to have any uniqueness protected from the general factor of correlatedness? It depends on what you think that factor is.)

Another problematic topic is whether extracted (unrotated) factors uncover "general factor" at all. Methods of factor extraction differ. Some methods may yield the same solution however somewhat rotated differently relative each other. This fact questions it whether unrotated result could be of any value at all and suggests that a rotation is perhaps necessary? Next to remark, PAF method maximizes loadings of the 1st factor, then of the 2nd one, etc. as its primary goal (and so as if fitting ones expectations for the "general" factor), but other methods do not "hunt" this goal. Will they (unrotated) uncover "general factor" and what one? These are not very easy questions.

A bonus question from a comment. "How to get the loading matrix after any method of factor extraction where, in the manner of PAF method, the variance (i.e. sum of squared loadings) of the 1st factor is maximized, the variance of the 2nd factor is next possible maximal, etc?" I mean: variance maximized, not simply that factors are sorted by the amount of their variance. The answer is obvious: just perform PCA of the loading matrix as if it were some "data" and the columns (factors) were the data "variables". But do not center the columns, perform PCA without centering. (That is, apply SVD to raw loadings as they are.) The "PC scores" from this PCA will be the solution you seek.

  • $\begingroup$ Thanks for your clarification. FA is typically applied by researchers whose expertise is elsewhere - psychology, sociology, or whatever. They will just do what the textbook recommends, which is FA with rotation. That's why the many published studies of cultural differences all use rotation. They failed to discover the strong general factor because they didn't know that an unrotated result could even be meaningful. $\endgroup$
    – A Fog
    Sep 22, 2020 at 4:56
  • $\begingroup$ When I say unrotated solution, I am referring to PAF. Is there a name for the rotation method that will produce the same effect from other factor extraction methods? The umxEFA function in R is excellent for dealing with missing values, yet it cannot produce the 'unrotated' solution. Quartimax gives a reasonable approximation, yet the 'unrotated' solution is the one that gives the best reproducibility among studies. Any recommended literature that I can refer to? $\endgroup$
    – A Fog
    Sep 22, 2020 at 5:08
  • $\begingroup$ @AFog, Unrotated result could be meaningful. FA does not give a unique solution, and it is left on the researcher to bring in sense and meaning for the factors. He may rotate or not. Or rotate only a subset of the factors (this was my main point). $\endgroup$
    – ttnphns
    Sep 22, 2020 at 8:33
  • $\begingroup$ I never saw a book of FA which says "rotate, don't leave them unrotated". Rather, they all say "rotation can benefit in interpretation". As for researchers, the majority of them nowadays are not interested or even do not believe in the "general factor". If you insist it is there, persuade them. $\endgroup$
    – ttnphns
    Sep 22, 2020 at 8:43
  • $\begingroup$ Is there a name for the rotation method that will produce the same effect from other factor extraction methods? If you want a solution where the 1st factor is maximal variance (i.e. sum of squared loadings), 2nd is next max., etc., as it is with PAF, only that you used not PAF but another extraction method, - please see my addendum to the answer. $\endgroup$
    – ttnphns
    Sep 22, 2020 at 9:58

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