I have a linear model that requires a large number of interactions (there are as many interactions as there are IVs) and I want to reduce collinearity using PCA, then regressing the DV on the principal components (PCR).
lm(Y ~ NUM1 + NUM2 + NUM3 + NUM4 + NUM2*NUM3 + NUM2*NUM4 + FACT1*NUM1*NUM2)
FACT1 is a 10 levels categorical variable. All other IVs are numerical.
Identifying principal components on the full model specifications, including interactions, doesn't feel like the right approach since interactions inevitably introduce correlation among the model variables, that is between the basic predictors and the interactions themselves.
How should I conduct PCR in this case?