Type Ia supernovae (SNe Ia) are standardizable candles, but for over a decade, there has been a debate on how to properly account for their correlations with host galaxy properties. Using the Bayesian hierarchical model UNITY, we simultaneously fit for the SN Ia light curve and host galaxy standardization parameters on a set of 103 Sloan Digital Sky Survey II SNe Ia. We investigate the influences of host stellar mass, along with both localized ($r<3$ kpc) and host-integrated average stellar ages, derived from stellar population synthesis modeling. We find that the standardization for the light-curve shape ($alpha$) is correlated with host galaxy standardization terms ($gamma_i$) requiring simultaneous fitting. In addition, we find that these correlations themselves are dependent on host galaxy stellar mass that includes a shift in the color term ($beta$) of $0.8 mathrm{mag}$, only significant at $1.2sigma$ due to the small sample. We find a linear host mass standardization term at the $3.7sigma$ level, that by itself does not significantly improve the precision of an individual SN Ia distance. However, a standardization that uses both stellar mass and average local stellar age is found to be significant at $>3sigma$ in the two-dimensional posterior space. In addition, the unexplained scatter of SNe Ia absolute magnitude post standardization, is reduced from $0.122^{+0.019}_{-0.018}$ to $0.109pm0.017$ mag, or $sim10%$. We do not see similar improvements when using global ages. This combination is consistent with either metallicity or line-of-sight dust affecting the observed luminosity of SNe Ia.
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