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I am developing a scale which consists of six scale dimensions, six items per dimension, with hypothesized positive relationships between all factors. All scale dimensions have positive correlations between each other. However, in an EFA of the scale which gives a working six-factor solution (based on Kaiser's K1 rule and scree plot; principal axis factoring, direct oblimin rotation with deltas =0), even though almost all items load well on their "expected" factors, one of the factors correlates negatively to other factors at the factor-level correlations. Let's call this scale dimension "dimension A" below for clarity. So, at the manifest level, "dimension A" correlates positively with all other dimensions, but at latent level it correlates negatively to other factors.

I further tested "dimension A" in separate EFAs with all other scale dimensions, looking for whether the two-factor solutions all worked. Turns out that in one of these EFAs, there are two factors where on the first factor all items from both scale dimensions load positively in the .7 to .8 range, and on the second factor the items from "factor A" load positively and items from the other factor load negatively in the .3 to .5/ -.3 to -.5 range. There is no clear theoretical explanation for this, so I was wondering if someone had an idea what might be happening here. Low sample size (n=100) could affect the results.

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  • $\begingroup$ @MiguelIC Welcome to the site! At the manifest level, I am guessing you scale scored each factor, e.g. (item1 + item2 + ... item6)/6. How large exactly are correlations between A and other factors at the scale-scored manifest level? Also, how large are the actual respective negative correlations? $\endgroup$ – PsychometStats Nov 20 '19 at 22:49
  • $\begingroup$ @PsychometStats Thank you! Yes, you guessed right. At the manifest level, correlations between A and other factors are .5 to .7 so quite high (rather similar as other correlations between factors). Whereas the factor-level negative correlations between A and other factors are -.3 to -.5. $\endgroup$ – MiguelC Nov 21 '19 at 11:37
  • $\begingroup$ thank you for providing more information. Couple more questions. What software are you using? Also, do you observe negative variance for any of the items or factors? Bonus question: are you confident with Confirmatory Factor Analysis? $\endgroup$ – PsychometStats Nov 21 '19 at 16:55
  • $\begingroup$ I am using SPSS (version 25). Unfortunately I am not quite sure how to compute the factor-level variances with SPSS, are these commonly used also in EFA like in CFA? The item communalities are all >.5. I have a little bit of experience with CFA, and I will try to implement that at least later on. Starting with an exploratory approach seemed reasonable to me, since the studied phenomenon is new (although I have an expected structure for the scale in mind, so in that sense a CFA could also work). $\endgroup$ – MiguelC Nov 22 '19 at 11:56
  • $\begingroup$ that's great! So the suggestion for trying CFA would be to see if CFA-based latent correlations also give you a trouble, i.e. A is still negatively correlated with other factors. Admittedly, it is perplexing why this is happening $\endgroup$ – PsychometStats Nov 22 '19 at 16:21
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Factor analysis solutions can be indeterminate with respect to sign, because covariances between items are built out of products of loadings and factor covariances. With a product of multiple parameter estimates, you can flip the signs on many parameter estimates but end up at exactly the same end point.

So a solution with one factor having negative covariances and negative loadings will show the same item covariances as a solution where the factor covariance is positive and all loadings are positive.

It is a matter of starting values--if you start parameter estimates on one side of 0, they will tend to stay on that side, unless the solution really calls for something different. If the software allows, specify a positive starting value for one of the loadings for that factor. That may resolve the situation.

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