  We use direct method oxygen abundances in combination with strong optical
emission lines, stellar masses ($M_{\star}$), and star formation rates (SFRs)
to recalibrate the N2, O3N2, and N2O2 oxygen abundance diagnostics. We stack
spectra of $\sim$200,000 star-forming galaxies from the Sloan Digital Sky
Survey in bins of $M_{\star}$ and SFR offset from the star forming main
sequence to measure the weak emission lines needed to apply the direct method.
All three new calibrations are reliable to within $\pm 0.10$ dex from
$\log(M_{\star}/M_{\odot}) \sim 7.5 - 10.5$ and up to at least $200~M_{\odot}$
yr$^{-1}$ in SFR. The N2O2 diagnostic is the least subject to systematic
biases. We apply the diagnostics to galaxies in the local universe and
investigate the $M_{\star}$-$Z$-${\rm SFR}$ relation. The N2 and O3N2
diagnostics suggest the SFR dependence of the $M_{\star}$-$Z$-${\rm SFR}$
relation varies with both $M_{\star}$ and $\Delta \log(SSFR)$, whereas the N2O2
diagnostic suggests a nearly constant dependence on SFR. We apply our
calibrations to a sample of high redshift galaxies from the literature, and
find them to be metal poor relative to local galaxies with similar $M_{\star}$
and SFR. The calibrations do reproduce direct method abundances of the local
analogs. We conclude that the $M_{\star}$-$Z$-${\rm SFR}$ relation evolves with
redshift.
