  Three examples of free field constructions for the vertex operators of the
elliptic quantum group ${\cal A}_{q,p}(\hat{sl}_2)$ are obtained. Two of these
(for $p^{1/2}=\pm q^{3/2},p^{1/2}=-q^2$) are based on representation theories
of the deformed Virasoro algebra, which correspond to the level 4 and level 2
$Z$-algebra of Lepowsky and Wilson. The third one ($p^{1/2}=q^{3}$) is
constructed over a tensor product of a bosonic and a fermionic Fock spaces. The
algebraic structure at $p^{1/2}=q^{3}$, however, is not related to the deformed
Virasoro algebra. Using these free field constructions, an integral formula for
the correlation functions of Baxter's eight-vertex model is obtained. This
formula shows different structure compared with the one obtained by Lashkevich
and Pugai.
