A family that has children who are 18 years or older.
A family with children who have become independent.
A family with children grown-up and living by themselves separately.
家、というのはこの状況では家庭なので「family」を使います。
18歳以上の子供がいる家 = A family that has children who are 18 years or older.
子供が独立した家 = A family with children who have become independent.
（おまけ）上記の独立した家をもう少し詳しく説明して
子供が大きくなって自分で住んでいる = A family with children grown-up and living by themselves separately.
日本 
