  The recent announcement of strong evidence for a stochastic
gravitational-wave background (SGWB) by various pulsar timing array
collaborations has highlighted this signal as a promising candidate for future
observations. Despite its non-detection by ground-based detectors such as
Advanced LIGO and Advanced Virgo, Callister \textit{et
al.}~\cite{tom_nongr_method} developed a Bayesian formalism to search for an
isotropic SGWB with non-tensorial polarizations, imposing constraints on signal
amplitude in those components that violate general relativity using LIGO's
data. Since our ultimate aim is to estimate the spatial distribution of
gravitational-wave sources, we have extended this existing method to allow for
anisotropic components in signal models. We then examined the potential
benefits from including these additional components. Using injection campaigns,
we found that introducing anisotropic components into a signal model led to
more significant identification of the signal itself and violations of general
relativity. Moreover, the results of our Bayesian parameter estimation
suggested that anisotropic components aid in breaking down degeneracies between
different polarization components, allowing us to infer model parameters more
precisely than through an isotropic analysis. In contrast, constraints on
signal amplitude remained comparable in the absence of such a signal. Although
these results might depend on the assumed source distribution on the sky, such
as the Galactic plane, the formalism presented in this work has laid a
foundation for establishing a generalized Bayesian analysis for an SGWB,
including its anisotropies and non-tensorial polarizations.
