This issue arose using R's psych package, specifically the fa function, but the problem is a theoretical one:
When generating the factors for a dataset, the reported "cumulative variance", i.e. the proportion of variance explained by the factors, appears to change based on the rotation method chosen. Specifically, varimax rotation appears to capture more of the overall variance than oblimin rotation.
Surely this is impossible? My understanding is that, when modelling p observed variables as manifestations of q latent variables (where q < p), a q-dimensional subspace is created within p-dimensional space. Rotation changes the basis used to describe this subspace, but doesn't change the subspace itself. In other words, rotation should not affect the proportion of variance explained, as the p-dimensional data is still being projected onto the same q-dimensional subspace.
Is my understanding incorrect? Or is the issue with my use of the psych package (perhaps, when returning the loadings generated by the fa function, the "Cumulative Var" row doesn't mean what I think it means?)
Thanks in advance for any help!
a q-dimensional subspace is created _within_ p-dimensional spaceThis is true for PCA and not for FA. See https://stats.stackexchange.com/a/288646/3277, https://stats.stackexchange.com/a/95106/3277. – ttnphns Feb 16 '23 at 21:05