  With the strong evidence for a gravitational wave (GW) background in the
nanohertz frequency band from pulsar timing arrays, the detection of continuous
GWs from individual supermassive black hole binaries is already at the dawn.
Utilizing continuous GWs to test theories of gravity, especially to test the
polarizations of GWs is becoming more and more realistic. In this theoretical
study, assuming a detection of signals from individual supermassive binary
black holes, we use the null stream to estimate the capability of identifying
the nontensorial polarizations of GWs. We consider cases for the nontensorial
polarizations where the dipole radiation and quadrupole radiation dominate
separately. With a frequentist method, we estimate the threshold of the
nontensor-to-tensor relative amplitude above which extra polarizations can be
detected. We also conduct Bayesian analysis to do parameter estimation with the
null stream data. Our treatment provides a data-analysis methodology using the
null stream to probe the nontensorial GW polarizations with pulsar timing
arrays.
