I would like to know what nice model theoretic properties (for example simplicity, NIP, stability, etc) can be preserved when we add a new predicate to the language.

Explicitly, if T is an L-theory and LP=L∪{P} (a new predicate). Under which conditions of T we have that the new theory TP obtained by a suitable interpretation of the predicate P becomes simple? or NIP? or stable?

I know, for example, that if T is simple and eliminates the quantifier ∃∞ then TP is simple (Chatzidakis, Pillay). But, what other theorems like this are known? Are there easy examples witnessing the failure of this ``preserving nice properties'' phenomena?

Thank you in advance for the possible answers...