  It has been proposed that the mass-metallicity relation of galaxies exhibits
a secondary dependence on star formation rate (SFR), and that the resulting
M-Z-SFR relation may be redshift-invariant, i.e., "fundamental." However,
conflicting results on the character of the SFR dependence, and whether it
exists, have been reported. To gain insight into the origins of the conflicting
results, we (a) devise a non-parametric, astrophysically motivated analysis
framework based on the offset from the star-forming ("main") sequence at a
given stellar mass (relative specific SFR), (b) apply this methodology and
perform a comprehensive re-analysis of the local M-Z-SFR relation, based on
SDSS, GALEX, and WISE data, and (c) study the impact of sample selection, and
of using different metallicity and SFR indicators. We show that metallicity is
anti-correlated with specific SFR regardless of the indicators used. We do not
find that the relation is spurious due to correlations arising from biased
metallicity measurements, or fiber aperture effects. We emphasize that the
dependence is weak/absent for massive galaxies ($\log M_*>10.5$), and that the
overall scatter in the M-Z-SFR relation does not greatly decrease from the M-Z
relation. We find that the dependence is stronger for the highest SSFR galaxies
above the star-forming sequence. This two-mode behavior can be described with a
broken linear fit in 12+log(O/H) vs. log (SFR$/M_*$), at a given $M_*$.
Previous parameterizations used for comparative analysis with higher redshift
samples that do not account for the more detailed behavior of the local M-Z-SFR
relation may incorrectly lead to the conclusion that those samples follow a
different relationship.
