  A successful measurement of the Stochastic Gravitational Wave Background
(SGWB) in Pulsar Timing Arrays (PTAs) would open up a new window through which
to test the predictions of General Relativity (GR). We consider how these
measurements might reveal deviations from GR by studying the overlap reduction
function -- the quantity that in GR is approximated by the Hellings-Downs curve
-- in some sample modifications of gravity, focusing on the generic prediction
of a modified dispersion relation for gravitational waves. We find a distinct
signature of such modifications to GR -- a shift in the minimum angle of the
angular distribution -- and demonstrate that this shift is quantitatively
sensitive to any change in the phase velocity. In a given modification of
gravity, this result can be used, in some regions of parameter space, to
distinguish the effect of a modified dispersion relation from that due to the
presence of extra polarization modes.
