We introduce a Pfaffian formula that extends Schur's $Q$-functions $Q_λ$ to be indexed by compositions $λ$ with negative parts. This formula makes the Pfaffian construction more consistent with other constructions, such as the Young tableau and Vertex Operator constructions. With this construction, we develop a proof technique involving decomposing $Q_λ$ into sums indexed by partitions with removed parts. Consequently, we are able to prove several identities of Schur's $Q$-functions using only simple algebraic methods.