Factor analysis: What to do when determinant is almost zero and when KMO for a variable is low? I'm conducting a factor analysis on 40 interval-level variables, and have two immediate concerns:


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*The determinant is 6.608E-006, which is much lower than the cut-off of 0.00001. I went back and screened the correlation matrix to find significant, too-high correlations between variables; there's nothing even approaching 0.8. What now? Do I proceed anyway, just noting the determinant value, or is there something else I should be doing to adjust the data to exclude multicollinearity?

*KMO is a respectable .846, but on examining the anti-correlation matrix, I found five variables whose individual KMO values were less than 0.5. Now what? Exclude? Ignore, as overall KMO is high enough already?
 A: Singularity problems:
Bivariate correlations are not the only way to diagnose singularity in your dataset. You may want to see what happens when you predict each of your variables from the set of other variables. A common mistake in psychology datasets is to include scale scores as well as items where the scale scores are a function of the items. 
Item removal:
KMO relates to properties of the overall correlation matrix. You could for example add a random variable unrelated to any of the other variables and still get a decent overall KMO. 
In general, there are many reasons to justify removal of a variable from a factor analysis. This is a bit of an art. 
If you are truly doing factor analysis and you have a variable that is unrelated to the other variables then this may be a reason for item removal. Alternatively, it may flag that you should have had more variables measuring the construct that the variable relates to.
If you are doing PCA, then you may just be concerned with data reduction, and as such an independent item may not be a problem.
You should make an assessment of the degree of independence of the variable from the others. Think conceptually about whether you think this independent variance is substantively interesting or whether it is because the variable failed to measure anything interesting.
In general, you want to integrate content knowledge and the statistics to make a reasoned judgement.
