  We develop a model-independent test of General Relativity that allows for the
constraint of the gravitational wave (GW) polarization content with GW
detections of binary compact object inspirals. We first consider three modified
gravity theories (Brans-Dicke theory, Rosen's theory and Lightman-Lee theory)
and calculate the response function of ground-based detectors to gravitational
waves in the inspiral phase. This allows us to see how additional polarizations
predicted in these theories modify the General Relativistic prediction of the
response function. We then consider general power-law modifications to the
Hamiltonian and radiation-reaction force and study how these modify the
time-domain and Fourier response function when all polarizations are present.
From these general arguments and specific modified gravity examples, we infer
an improved parameterized post-Einsteinian template family with complete
polarization content. This family enhances General Relativity templates through
the inclusion of new theory parameters, reducing to the former when these
parameters acquire certain values, and recovering modified gravity predictions
for other values, including all polarizations. We conclude by discussing
detection strategies to constrain these new, polarization theory parameters by
constructing certain null channels through the combination of output from
multiple detectors.
