  We introduce a Pfaffian formula that extends Schur's $Q$-functions
$Q_\lambda$ to be indexed by compositions $\lambda$ 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
$Q_\lambda$ 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.
