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[ODP] The international general socioeconomic factor: Factor analyzing international

This version has fixed intercorrelation results. They are hardly any different, e.g. mean intercor with factor scores in SPI was given as 0.9923575 before, now it's corrected to 0.990829. Similarly for the mean intercor loading, and mean factor congruence coefficient.

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.pdf   international_socioeconomic_general.pdf (Size: 376.64 KB / Downloads: 446)
Can you add a table to the paper with the factor scores for the countries?
Yes. The question is more which of the factors specifically you want. They correlate very highly for sure, but do you want PCA, ML, minres, GLS, WLS, PA, oblimin third order and which dataset?
Both datasets. You choose the scores obtained with the factor analytic approach you think produced the better results.
Concerning rotation in PC/FA, I have always considered Promax to be the best, see here, although Oblimin should produce similar results.

Also, if you want a test (requested by Dalliard) for knowing which number of factors to retain in EFA, you should better avoid eigenvalue>1 and scree plot. You should better use Parallel analysis. It's a Monte Carlo simulation technique which evaluates what minimum eigenvalues are needed to reject the null hypothesis by subjecting a data set of random numbers to analysis (repeated many times, e.g., 1000 replications) when the data is adjusted for sample size and the number of variables.

See below for why you should prefer this one :

Determining the Number of Factors to Retain in EFA: an easy-to-use computer program for carrying out Parallel Analysis

In R, it's nFactors package you need.

It should also give you a plot that helps you to determine the n factors to be retained.

I used four different methods to determine number of factors to retain. They were all always in agreement except for the author's custom method which always gave the answer 1. Read more here: http://www.er.uqam.ca/nobel/r17165/RECHE...Scree.html

This is the function from the nFactors package. :)

I will try with promax too.
So, i will wait for your final version before approving. I don't have lot of things to say, that's bad. It was only about methods and stats. Not related with the subject of the article. But I don't disagree with you here. In case like that, I generally remain silent.

I almost forget. If you add new analyses on R, I recommend you to add these new lines of coding in your appendix .doc files.
New lines of code? But I have >500 lines of R code for this project, and two scripts for Python (200 lines approx.)

For the final version, I will of course post a final version of all the supplementary material including any new code.
Oddly, promax does not produce similar results.

r = 0.63 for SPI and 0.92 for DR, using ML as the extraction method. Very strange.

And there seems not to be much agreement regarding which method to use.


Quote:There is no widely preferred method of oblique rotation; all tend to produce similar results (Fabrigar et al., 1999), and it is fine to use the default delta (0) or kappa (4) values in the software packages. Manipulating delta or kappa changes the amount the rotation procedure “allows” the factors to correlate, and this appears to introduce unnecessary complexity for interpretation of results. In fact, in our research we could not even find any explanation of when, why, or to what one should change the kappa or delta settings.

The cited paper is: http://search.proquest.com.ez.statsbibli...ite=summon

Anyone have any ideas how to interpret this? Clearly the promax is at odds with all the other results. I will ask A. Beaujean. I have asked on /r/statistics too. http://www.reddit.com/r/statistics/comme...s_oblimin/
So, it seems that promax was mostly used because it gives similar results to oblimin but requires less computing. That was relevant back when computers were weak or rotation was done in hand, but not very relevant any more.

To further explore the idea of oblique rotations in higher levels, I did the following:

Used schmid() in every combination of factor method and rotation. Correlated the loadings (factor congruence didn't work here) with the loadings from the first unrotated factor using the same extraction method. Did for both datasets. Scores were not available, so it wasn't possible to correlate them.

> y.sl.df
       oblimin simplimax promax
minres    1.00      1.00   1.00
pa        0.99      0.90   0.98
ml        0.98      0.85   0.97
> z.sl.df
       oblimin simplimax promax
minres    0.97      0.97   0.99
pa        0.97      1.00   0.99
ml        0.98      0.99   0.99

Results were remarkable similar.

Given uncertainty about how schmid works precisely, I wanted to compare it with a manual extraction of the 3rd order general factor. Concretely, I extracted the first unrotated factor. Then I extracted the first 8/9 factors. Then I used those to extract 3 factors. Then I used those to extract 1 general factor. Loadings were not available because the last factor is extracted from 2nd order factors, not the manifest variables. Factor scores were available so I used them.

> y.oblique.df
       promax oblimin simplimax bentlerQ geominQ biquartimin
minres   0.80    0.87      0.04     0.86    0.63        0.05
wls      0.55    0.87     -0.01    -0.86    0.80        0.10
gls      0.61    0.85     -0.01    -0.86    0.78        0.14
pa       0.64    0.92      0.01    -0.85    0.66       -0.04
ml       0.63    0.97      0.11     0.99    0.83        0.34
minchi   0.83    0.97     -0.12     0.84    0.99        0.09
> z.oblique.df
       promax oblimin simplimax bentlerQ geominQ biquartimin
minres   0.55    0.97     -0.05     0.91    0.34        0.00
wls      0.90    0.86      0.00     0.92    0.16       -0.04
gls      0.90    0.88      0.00     0.93    0.03       -0.01
pa       0.87    0.68     -0.07     0.50    0.86        0.03
ml       0.59    0.98      0.12     0.95   -0.27        0.00
minchi   0.94    0.97      0.01     0.92    0.76       -0.11

What to make of this? Average by rotation method:

> round(apply(y.oblique.df,2,mean),2)
     promax     oblimin   simplimax    bentlerQ     geominQ biquartimin
       0.68        0.91        0.00        0.02        0.78        0.11

> round(apply(z.oblique.df,2,mean),2)
     promax     oblimin   simplimax    bentlerQ     geominQ biquartimin
       0.79        0.89        0.00        0.86        0.31       -0.02

Average by factor method:
> round(apply(y.oblique.df,1,mean),2)
minres    wls    gls     pa     ml minchi
  0.54   0.24   0.25   0.22   0.64   0.60
> round(apply(z.oblique.df,1,mean),2)
minres    wls    gls     pa     ml minchi
  0.45   0.47   0.46   0.48   0.39   0.58

We see that oblimin consistently gives the highest correlations with promax somewhat behind. The strongly divergent results for bentlerQ is due to the factor being reversed half of the time in SPI. As for factor method, we see varied results, perhaps chance flukes.

Thoughts about what to make of this?

The minchi method is apparently another method of extraction that I either missed when I looked the first time, or was recently added. I have rerun the earlier analyses with it and they are more of the same.
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