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It might be interesting to try out the degrees of freedom formula from Perry and Pillai (2015).
To do this, I'll need to modify pca_naive to divide the residual sums of squares by df, where df is defined using their equations (5.2) and (3.9). I'll also need to return these df's. Right now, vicar has no functionality to allow fa_func to return df's.
I could do a post-hoc multiplication of Sigma by nrow(Y) - r then apply the formulas from Perry and Pillai (2015), but this would be ugly code.