  In this paper, we focus on testing gravity theories in the radiative regime
using pulsar timing array observations. After reviewing current techniques to
measure the dispersion and alternative polarization of gravitational waves, we
extend the framework to the most general situations, where the combinations of
a massive graviton and alternative polarization modes are considered. The atlas
of the Hellings-Downs functions is completed by the new calculations for these
dispersive alternative polarization modes. We find that each mode and
corresponding graviton mass introduce characteristic features in the
Hellings-Downs function. Thus, in principal, we can not only detect each
polarization mode, measure the corresponding graviton mass, but also
discriminate the different scenarios. In this way, we can test gravity theories
in the radiative regime in a generalized fashion, and such method is a direct
experiment, where one can address the gauge symmetry of the gravity theories in
their linearised limits. Although current pulsar timing still lacks enough
stable pulsars and sensitivity for such practices, we expect that future
telescopes with larger collecting area could make such experiments be feasible.
