# Logistic principal component regression where PCs are correlated with an additional binary predictor

My scenario is this: I collected a bunch of vegetation data (% cover counts in a quadrant at different heights) in patches where birds were seen foraging and also in control patches where no foraging was known to occur. This habitat data all suffers (as expected) from multi-collinearity so I did a principal component analysis (PCA) which yielded 3 new habitat variables: PC1, PC2 and PC3.

I then decided to model patch type (1=forage patch, 0=control patch) via logistic regression:

patch type ~ pc1+pc2+pc3


to see how the probability that a bird used a patch for foraging was influenced by habitat variables. However since I collected this data in two different habitat types (grazed and ungrazed) I included this 2-level categorical habitat type variable as an interaction with each of the PC variables like this:

patch type ~ pc1*habitat type + pc2*habitat type + pc3*habitat type


My problem is that my PC variables themselves are correlated with my habitat type variable. Is there anything I can do to make this work with this approach?

• Unless the correlation of PCs with habitat type is very high, this should not be an issue. – Aksakal Dec 5 '14 at 19:20
• What would qualify as very high? VIF score is 9 and the visualization of the data shows a definite positive trend. – user1658170 Dec 5 '14 at 19:25
• Then start with building separate models for grazed and ungrazed. – Aksakal Dec 5 '14 at 19:52
• Can't you include habitat type in your PCA? It seems a bit arbitrary to use principal component regression, but to include only a subset of predictors in the PCA, leaving other predictors (habitat type in your case) "raw". – amoeba Dec 5 '14 at 23:42
• Habitat type is binary "grazed or ungrazed". I am not sure that it would be ideal to include this in PCA. I wouldn't know how to interpret it – user1658170 Dec 8 '14 at 13:56