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I've been reading about performing exploratory factor analysis via principal axis factor extraction (PAF) and principal component analysis (PCA). I'm a bit confused about why the difference between the two methods is important, and the overarching question is why/when to use one method over the other?

Both methods use an eigen-decomposition of a matrix. In the PAF case, the input matrix is a reduced correlation matrix with communalities on the diagonal, whereas in PCA the input is just the correlation matrix

Why is it important that one method uses the reduced correlation matrix with communalities, and the other uses just the correlation matrix? How do the results differ, and how does that then change the interpretation of factor loadings?

Further reading states that PAF is a factor method, whereas PCA is a component method. I'm not sure why this distinction is important

The book states the following: "in component analysis, all of the variance in the variables, rather than just the common variance is analyzed. There is therefore no uniqueness term in the equation for component analysis". How does this relate to the use of communalities?

Simon
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  • PCA is PCA. PAF is PCA-based bona fide factor analysis. [See](https://stats.stackexchange.com/q/102882/3277) of an example of comparison of PCA and PAF on iris dataset. See [here](https://stats.stackexchange.com/q/50745/3277) about some methods of exploratory FA. For fundamental differences between PCA and FA, read great [thread](https://stats.stackexchange.com/q/1576/3277) where I would recommend to you my own answer there as a thorough one, as well as [this thread](https://stats.stackexchange.com/q/94048/3277). – ttnphns Oct 10 '17 at 07:13
  • The difference in short: in PAF, we are interested in communalities and we train them to restore correlations (by means of loadings). In PCA, we aren't interested in communalities (they just come down as a "dross" of the analysis) and loadings are not meant to reproduce correlations, rather, they are after to explain as more variance as possible. – ttnphns Oct 10 '17 at 07:16
  • The thing that confuses me is that regardless of whether you use a regular correlation matrix (PCA) or one with communalities (PAF), the relationship between variables is still given by the off-diagonals (the correlations). And yet the only thing that is changing between the 2 methods is what you put on the diagonal. Given that, why does 1 method aim to maximize variance (PCA) while only the other aims to restore correlations (PAF), if correlations are in both input matrices? – Simon Oct 10 '17 at 23:03
  • In other words, why are communalities important (what are they telling you?), and how/why does using communalities automatically change what the technique is trying to maximize? Does using communalities tell you more about the correlations than just a regular correlation matrix does? – Simon Oct 10 '17 at 23:04
  • Communalities are the only portion of the variances which are suffice to explain the observed amount of correlations (This is the fundamental difference between FA and PCA.) FA says: you can keep only these diminished values on the diagonal and I'll restore the off-diagonal elements by few `m` factors as fully as PCA will be able to do with full variances on the diagonal and by just all the `p` pr. components. – ttnphns Oct 11 '17 at 08:51
  • Having communalities on the diagonal is a more "narrow" or "precise" information than the full correlation matrix. Communality is the portion of the variable variability due to which correlations of the variable with the other variables arise, - given the `m` common factors behind the data. – ttnphns Oct 11 '17 at 08:56

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