  We analyze the rest-optical emission-line ratios of z~1.5 galaxies drawn from
the MOSDEF survey. Based on composite spectra we investigate the
mass-metallicity relation (MZR) at z~1.5 and measure its evolution to z=0. When
using gas-phase metallicities based on the N2 line ratio, we find that the
difference between the MZR at z~1.5 and z=0 depends on stellar mass, with a
difference of $\Delta\rm log(\rm O/H)\sim0.25$ dex at $M_* <10^{9.75}M_{\odot}$
down to $\Delta\rm log(\rm O/H)\sim0.05$ for galaxies with
$M_*>10^{10.5}M_{\odot}$. In contrast, the O3N2-based MZR shows a constant
offset of $\Delta\rm log(\rm O/H)\sim0.30$ for all masses, consistent with
previous MOSDEF results based on independent metallicity indicators, and
suggesting that O3N2 provides a more robust metallicity calibration for our
z~1.5 sample. We investigated the secondary dependence of the MZR on SFR by
measuring correlated scatter about the mean $M_*$-sSFR and $M_*-\log(\rm O3N2)$
relations. We find an anti-correlation between $\log(\rm O/H)$ and sSFR
offsets, indicating the presence of a $M_*$-SFR-Z relation, though with limited
significance. Additionally, we find that our z~1.5 stacks lie along the z=0
metallicity sequence at fixed $\mu=\log(M_*/M_{\odot})-0.6\times\log(\rm
SFR/M_{\odot}yr^{-1})$, suggesting that the z~1.5 stacks can be described by
the z=0 fundamental metallicity relation (FMR). However, using different
calibrations can shift the metallicities enough to be inconsistent with the
local FMR, indicating that the choice of calibration is essential for
understanding metallicity evolution with redshift. Finally, understanding how
[NII]/H$\alpha$ scales with galaxy properties is crucial for describing the
effects of blended [NII] and H$\alpha$ on redshift and H$\alpha$ flux
measurements in future large surveys utilizing low-resolution spectra such as
with Euclid and the Roman Space Telescope.
