What is the contribution of base variables to principal components (PCA) after rotation?

I've run a PCA with PCAmixdata using R. My dataset consists of about 1'600 individuals and 600 variables. After trying several options, I've chosen to retain the 6 first PC (elbow plot, eigenvalue, etc.).

I am particularly inteterested in the contribution of some of my base variables to the PC. For this, I can check ...$quanti$contrib from PCAmixdata. Now in order to interpret my PC, I've used a varimax rotation (also using PCAmixdata). Unfortunately, there is no ...quanti\$contrib for a rotated object. Is there a way to get this information? Is there even a reason to do so?

My understanding was that a rotation was not really "changing" the analysis, just making it easier for interpretation (and redistributing the proportion of total variance explained). I've been wondering if the contribution of the base variables would be the same before and after rotation. To convince myself, I've tried to plot the correlation between the coordinates of each base variables before and after rotation.

As we can see, there is a strong correlation between base variables coordinates for PC1 and PC1.rot, PC2 and PC2.rot, PC3 and PC3.rot, and PC4 and PC4.rot. This would make me think that it could be "okay" to say that contribution of the base variables to the PC is somewhat similar before and after rotation. But clearly this does not seem to be the case for PC5 and PC6... Why? I don't get how a rotation could be a way to ease interpretation if the coordinates of the variables are very different before and after rotation? Thanks!

1 Answer

The idea of principal component analysis is to transform your original variables into new uncorrelated or "perpendicular" ones. So if your original variables were already pretty much uncorrelated, it is to be expected that the rotated variables will ressemble them (and thus correlate with them).

What's really surprising me is to see correlation among the rotated variables themselves. This should definitely not be the case.