  We revisit the question of the contributions of two-photon exchange with
$\Delta(1232)$ excitation to the electron-proton scattering in a hadronic
model. Three improvements over the previous calculations are made, namely,
correct vertex function for $\gamma N\rightarrow\Delta$, realistic $\gamma
N\Delta$ form factors, and coupling constants. The discrepancy between the
values of $R\equiv \mu_p G_E/G_M$ extracted from Rosenbluth technique and
polarization transfer method can be reasonably accounted for if the data of
Andivahis {\it et al.} \cite{Andivahis94} are analyzed. However, substantial
discrepancy remains if the data of Qattan {\it et al.} \cite{Qattan05} are
used. For the ratio $R^\pm$ between $e^\pm p$ scatterings, our predictions
appear to be in satisfactory agreement with the preliminary data from VEPP-3.
The agreement between our model predictions and the recent measurements on
single spin asymmetry, transverse and longitudinal recoil proton polarizations
ranges from good to poor.
