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So I ran a factor analysis (principal components method) on a dataset, first using the correlation matrix and then using the covariance matrix, both with varimax rotations. The results of both factor analyses indicate nearly identical first and second factors, and somewhat different third and fourth factors.

I am curious if this could be interpreted as factors 3 & 4 not being useful? Or that only the first two factors are appropriate for analysis? Or is this just showing a pattern in the correlation/covariance structures of my data (i.e. correlated correlations/covariances?)

Gosset
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    There is no need to do FA both on corr and cov matrices, because these are two different analyses (irrespective to whether they give similar results). You should decide apriori which matrix to analyse. – ttnphns May 07 '14 at 14:26
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    possible duplicate of [Is common factor analysis ever performed using the covariance matrix?](http://stats.stackexchange.com/questions/95996/is-common-factor-analysis-ever-performed-using-the-covariance-matrix) – ttnphns May 07 '14 at 14:27

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No. Factors are useful if they are useful. That depends on what you want to use them for.

Factor analysis (as opposed to principal components analysis) is designed to find latent variables - things that are hard or impossible to measure directly. Do your 4 factors correspond to latent variables that make sense? Are they useful? This is more important then whether the eigenvalue is > 1 or the scree plot has a bend or whatever.

Principal components analysis on the other hand, is a data reduction method. It is used if you have a large number of variables and you want to summarize them with a few linear combinations while accounting for as much of the variance as you can.

Peter Flom
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  • the first two factors seem to make sense and are useful; factors 3 and 4 are not very useful. – Gosset May 07 '14 at 18:11