\sum_{A} \beta_{A}(f) \Omega_{\rm GW}^A(f) = \sum_A \beta_A(f) \Omega_{\rm ref}^A \left(\frac{f}{f_{\rm ref}}\right)^{\alpha_A},
\label{eq:nongrY}$$ where $\beta_A = \gamma_A(f)/\gamma_T(f)$, and $A$ labels the polarization, $A=\{T,V,S\}$. The functions $\gamma_T(f)$, $\gamma_V(f)$, and $\gamma_S(f)$ are the overlap reduction functions for tensor, vector, and scalar polarizations [@TestingGR_stoch]. Because these overlap reduction functions are distinct, the spectral shape of $\hat C(f)$ enables us to infer the polarization content of the . While we use the notation $\Omega^A_{\rm GW}(f)$ in analogy with the general relativity (GR) case, in a general modification of gravity, the quantities $\Omega^T_\mathrm{GW}(f)$, $\Omega^V_\mathrm{GW}(f)$, and $\Omega^S_\mathrm{GW}(f)$ are best understood as a measurement of the two-point correlation statistics of different components of the  rather than energy densities [@Isi_Stein_NonGR_Energy].

Following Refs. [@TestingGR_stoch; @stoch_nongr_O1], we compute two Bayesian odds: odds $\mathcal{O}^\textsc{s}_\textsc{n}$ for the presence of a stochastic signal of any polarization(s) versus Gaussian noise, and odds $\mathcal{O}^\textsc{ngr}_\textsc{gr}$ between a hypothesis allowing for vector and scalar modes and a hypothesis restricting to standard tensor polarizations. Using the combined O1 and O2 measurements, we find $\log\mathcal{O}^\textsc{s}_\textsc{n} = \SNRLOGBFlogUniform$ and $\log\mathcal{O}^\textsc{ngr}_\textsc{gr} = \NGRGRLOGBFlogUniform$, consistent with Gaussian noise. Given the non-detection of any generic stochastic background, we use Eq.  to place improved upper limits on the tensor, vector, and scalar background amplitudes, after 