# Nice model theoretic properties of a theory adding a new predicate?

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 $L_P=L\cup \{P\}$ (a new predicate). Under which conditions of $T$ we have that the new theory $T_P$ 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 $\exists^\infty$ then $T_P$ 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?