We present a comparative study of a set of star-formation rate tracers based on mid-infrared emission in the 12.81$mu$m [Ne II] line, the 15.56$mu$m [Ne III] line, and emission features from polycyclic aromatic hydrocarbons (PAHs) between 5.2 and 14.7$mu$m. We calibrate our tracers with the thermal component of the radio continuum emission at 33 GHz from 33 extranuclear star-forming regions observed in nearby galaxies. Correlations between mid-IR emission features and thermal 33 GHz star-formation rates (SFR) show significant metallicity-dependent scatter and offsets. We find similar metallicity-dependent trends in commonly used SFR tracers such as H$alpha$ and 24$mu$m. As seen in previous studies, PAH emission alone is a poor SFR tracer due to a strong metallicity dependence: lower metallicity regions show decreased PAH emission relative to their SFR compared to higher metallicity regions. We construct combinations of PAH bands, neon emission lines, and their respective ratios to minimize metallicity trends. The calibrations that most accurately trace SFR with minimal metallicity dependence involve the sum of the integrated intensities of the 12.81$mu$m [Ne II] line and the 15.56$mu$m [Ne III] line combined with any major PAH feature normalized by dust continuum emission. This mid-IR calibration is a useful tool for measuring SFR as it is minimally sensitive to variations in metallicity and it is composed of bright, ubiquitous emission features. The Mid-Infrared Instrument (MIRI) on the James Webb Space Telescope will detect these features from galaxies as far as redshift z$sim$1. We also investigate the behavior of the PAH band ratios and find that subtracting the local background surrounding a star-forming region decreases the ratio of PAH 11.3$mu$m to 7.7$mu$m emission. This implies PAHs are more ionized in star-forming regions relative to their surroundings.
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