Interpretation of Factor scores in STATA

[Josefine Reimer Lynggaard]

Hi,
I am running a factor analysis using ten variables. I selected two eigenvalues as these fell above the threshold of 1 as set out in the Kaiser rule. Since I am assuming correlation between my variables, I am using oblique rotation. (See the 1st image with the factor analysis "Factor Analysis_STATA").
I am able to interpret the factor loadings from the three factors I have (See the 2nd picture "Rotated Factor Loadings_STATA").
However how do you interpret the factor scores obtained? I am not sure which table is actually the factor scores from the output obtained in STATA. Would anybody be able to advice me on possible interpretation of these numbers?
Thanks!

[Nick Cox]

None of those tables are the factor scores. There are as many scores as observations used, as the scores are just the values of the factors, which are just (potentially) new variables in your dataset. You need to use predict to get the factor scores.

By the way, you say you retained two factors, but your output says three.

Correlation between variables is not so much an assumption, as something you need for this kind of analysis to help at all. Correlation between factors is what you're alluding to.

Wherever factors can be interpreted in terms of just a few of the original variables, it's often a better research choice to work with those variables, not factors that are just somewhat arbitrary mushed-together composites of the originals. There's a lot of propaganda that says different, so that is just one point of view.

On a very rough guess, your first factor looks like some measure of size (of economy?) and your second factor is picking up something oblique to that. Unless your supervisor/boss/teacher is insisting on factor analysis, it's not compulsory.

Factor analysis is often like using a ladder to climb to a higher floor only to realise that you could have taken the stairs any way.

[daniel klein]

I selected two eigenvalues as these fell above the threshold of 1 as set out in the Kaiser rule.

I believe that rule only applies to principal component factor analysis, where you assume no unique factors.

Best
Daniel

[Nick Cox]

I think all these rules are at best rules of thumb (meaning, rough indicators based mostly on experience) and rules in the sense of regulative principles only if you want them to be.

[daniel klein]

I agree.

[Josefine Reimer Lynggaard]

Dear Nick,
Thanks for the input above. I noticed that I had indeed retained three factors rather than two for the ensuing calculations. From your comments, you are suggesting that factor analysis would not provide any additional interesting information on the variables? My understanding was that factor analysis could provide further information the correlations between variables and whether this correlation was consistent. Would it perhaps be more appropriate to use fixed-effect and/or random-effect regressions instead?

I used the command predict f1 f2 f3 and I obtained a set of scores. Just to confirm is this the output you would use for analysing factor scores instead? Would a higher score then suggest that the factor explains more variance within the individual variable?

Yes the first factor is a measure of economic-related variables measuring expenditures, revenues and GDP of provincial authorities. The 2nd factor contains variables measuring provincial authorities' investments.

Thanks Daniel for the advice!

[Nick Cox]

Factor analysis resembles religion insofar as people arrange themselves from believers to sceptics (or beyond). If I use anything in that territory I tend to use PCA, but people again differ. Fans of PCA tend to regard FA as a debased relative and fans of FA tend to regard PCA as a limiting (and limited) special case of FA.

I can't tell you what's best for your research goals, not least because I don't know what they are.

I don't believe there is any sense in which factor analysis provides further information on correlations, as it is based on those correlations. It's true that people have often lost, or never learned, the art of looking at correlations and thinking about correlation structure based on substantive thinking about variables and the strength of correlations (and what graphs show you!), so sometimes factor analysis output helps.

I have to suggest that you need to study basic texts rather more, as you seemed confused on fundamentals:

Would a higher score then suggest that the factor explains more variance within the individual variable?

This doesn't make sense to me unless you're confusing scores and loadings. Factor scores are values on new variables and will be high or low according to what data imply. A high (or low) factor score doesn't say anything about variance explained any more than someone being tall or short (having particular values of height) says anything whatsoever about the correlation between height and something else, say weight.

Most positively, factor analysis and PCA seem to work best when you have

1. a cluster of variables all at least loosely of the same kind (so an arbitrary rag-bag is suspect before you even start)

2. relatively simple structure capable of summarizing in linear terms or of being understood within a low-dimensional space.

In your case, as you know that (1) size of economy (2) certain kinds of investments are among the key drivers, I would choose variables representing those best.