  Family algebraic structures indexed by a semigroup first appeared in the
algebraic aspects of renormalizations in quantum field theory. The concept of
the Rota-Baxter family and its relation with (tri)dendriform family algebras
have been recently discovered. In this paper, we first consider a notion of
$\mathcal{O}$-operator family as a generalization of the Rota-Baxter family and
define two variations of associative Yang-Baxter family that produce
$\mathcal{O}$-operator families. Given a Hochschild $2$-cocycle on the
underlying algebra, we also define a notion of twisted $\mathcal{O}$-operator
family (in particular twisted Rota-Baxter family). We also introduce and study
NS-family algebras as the underlying structure of twisted
$\mathcal{O}$-operator families. Finally, we define suitable cohomology of
twisted $\mathcal{O}$-operator families and NS-family algebras (in particular
cohomology of Rota-Baxter families and dendriform family algebras) that govern
their deformations.
