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07-23-2014 10:43 AM

I recently came across a site (see below) which covered some aspects of doing principal component analysis. It states at one point:

"`Loadings are eigenvectors normalized to respective eigenvalues: A value = V value * sqrt(L value)`

"

Loadings are the covariances between variables and components.

The 'L' in their example represents the eigenvalues and the 'V' represents the eigenvectors.

After some searching I came across others that also mention it (see second site below and search for the phrase "I want especially to stress twice here the terminologic difference between eigenvectors").

Would this imply that when a PCA is being performed using proc princomp, that the eigenvector scores are not loadings and that for loadings you need to multiply the eigenvector scores by the square root of the eigenvalue of the corresponding component?

Any help on this would be much appreciated as the SAS help document for proc princomp does not say much about loadings.

Steps done in factor analysis compared to steps done in PCA - Cross Validated

pca - Making sense of principal component analysis, eigenvectors & eigenvalues - Cross Validated

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07-23-2014 11:43 AM

Eigenvalues and loadings can be scaled in different ways ... different authors will use the terms to mean different things ... it all depends on whether or not you want the scores to have variance of 1, or variance equal to the eigenvalue ... SAS allows you to scale things either way, see the STANDARD option. So instead of worrying about terminology, you can obtain the results that are scaled the way you want ...

Jackson (1991), "A Users Guide to Principal Components", John Wiley and Sons, covers the different possible scalings of the eigenvalues and/or different scalings of the scores. His table 1.2 shows actually three different scalings.

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07-23-2014 11:57 AM

Normalization is done because PROC FACTOR assumes all factors/components to have variance of 1, while PROC PRINCOMP creates components that have variances equal to the eigenvalues.