The frequency and lifetime of the phonon and the electron correlation effect in monovalent metal are formulated simultaneously using the two-time Green's-function method, and the screening of the electron-phonon interaction is studied. It is shown that when only the direct Coulomb force is taken into account, the contribution of the electron-phonon interaction to the dispersion relation and half-width formula is proportional to the first and the second powers of the screening factor ∊-1, respectively. The half-width formula becomes complicated when the exchange Coulomb force is taken into account, and the effective electron-phonon interaction is not a well-defined concept in this case. The electron-correlation effect is taken into account by increasing the order at which the hierarchy equation is decoupled to one higher than the Hartree-Fock approximation. The correlation effect including the nonadiabatic effects is obtained and is compared with the results of Bailyn, Hone, and Vosco. The numerical calculation of the dispersion and lifetime of the phonon for sodium metal is carried out and is compared with the results of Toya and Dayal-Srivastava. The Kohn anomaly in the phonon width appears in the transverse branches T1 and T2 in the  direction.