  We re-analyze the data of the elastic electron proton scattering to look for
model-independent evidence of the two-photon-exchange (TPE) effect. In contrast
to previous analyses, TPE effect is parametrized in forms which are free of
kinematical-singularity, in addition to being consistent with the constraint
derived from crossing symmetry and the charge conjugation. Moreover, we fix the
value of $R=G_E/G_M$ as determined from the data of the polarization transfer
experiment. We find that, at high $Q^2 \geq 2 GeV^2$ values, the contribution
of the TPE effect to the slope of $\sigma_R$ vs. $\epsilon$ is large and
comparable with that arising from $G_{E}$. It also behaves quasi-linearly in
the region of current data, namely, in the range of $0.2 < \epsilon < 0.95$.
Hence the fact that the current elastic $ep$ cross section data shows little
nonlinearity with respect to $\epsilon$ can not be used to exclude the presence
of the TPE effect. More precise data at extreme angles will be crucial for a
model-independent extraction of the TPE effect.
