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What does all this mean? I'm a factor analysis 'noob' and although I've read a book, it didn't tell me everything apparently.

Since the chi square statistic is so high and the p-value so low, it would seem that the data is close to being coplanar (2 dimensions) within the 6-dimensional space. Yet that only accounts for 89.4% of the variance (am I interpreting this right?)

Also, I thought factors were orthogonal to each other, so how can both factors have positive loadings for every variable?

And what do the uniquenesses mean?

> factanal(charges[3:8],2)

Call:
factanal(x = charges[3:8], factors = 2)

Uniquenesses:
      APT    CHELPG   Natural       AIM Hirshfeld       VDD 
    0.217     0.250     0.082     0.052     0.005     0.033 

Loadings:
          Factor1 Factor2
APT       0.609   0.642  
CHELPG    0.657   0.564  
Natural   0.571   0.769  
AIM       0.382   0.896  
Hirshfeld 0.910   0.408  
VDD       0.844   0.504  

               Factor1 Factor2
SS loadings      2.817   2.544
Proportion Var   0.470   0.424
Cumulative Var   0.470   0.894

Test of the hypothesis that 2 factors are sufficient.
The chi square statistic is 77.1 on 4 degrees of freedom.
The p-value is 7.15e-16 
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  • $\begingroup$ @SeanMurphy: Thanks! Now, I understand that factor analysis "factors" (approximately) the 160x6 data matrix into a 160x2 scores matrix and a 2x6 loadings matrix. The output gives me the loadings matrix, but I'm also interested in the scores matrix. How do I get that? Preferably in a form that I can export and/or plot. $\endgroup$ Commented Oct 26, 2014 at 15:46

1 Answer 1

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The chi-square statistic and p-value in factanal are testing the hypothesis that the model fits the data perfectly. When the p value is low, as it is here, we can reject this hypothesis - so in this case, the 2-factor model does not fit the data perfectly (this is opposite how it seems you were interpreting the output).

It's worth noting that 89.4% of the variance explained by two factors is very high, so I'm not sure why the 'only'.

The factors themselves are uncorrelated (orthogonal) but that doesn't mean individual measures cannot correlate with both factors. Think about the directions North and East on a compass - they're uncorrelated, but North-East would 'load' onto both of them positively.

Uniquenesses are the variance in each item that is not explained by the two factors.

This link might be useful to your interpretation.

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