We present a study of Spitzer/IRAC and X-ray active galactic nuclei (AGNs) selection techniques in order to quantify the overlap, uniqueness, contamination, and completeness of each. We investigate how the overlap and possible contamination of the samples depends on the IR and X-ray depths. We use Spitzer/IRAC imaging, Chandra and XMM X-ray imaging, and PRism MUlti-object Survey (PRIMUS) spectroscopic redshifts to construct galaxy and AGN samples at 0.2<z<1.2 over 8 deg^2. We construct samples over a wide range of IRAC flux limits (SWIRE to GOODS depth) and X-ray flux limits (10 ks to 2 Ms). We compare IR-AGN samples defined using the IRAC color selection of Stern et al. and Donley et al. with X-ray detected AGN samples. For roughly similar depth IR and X-ray surveys, we find that ~75% of IR-AGN are identified as X-ray AGN. This fraction increases to ~90% when comparing against the deepest X-ray data, indicating that only ~10% of IR-selected AGN may be heavily obscured. The IR-AGN selection proposed by Stern et al. suffers from contamination by star-forming galaxies at various redshifts when using deeper IR data, though the selection technique works well for shallow IR data. While similar overall, the IR-AGN samples preferentially contain more luminous AGN, while the X-ray AGN samples preferentially contain lower specific accretion rate AGN, where the host galaxy light dominates at IR wavelengths. The host galaxy populations of the IR and X-ray AGN samples have similar restframe colors and stellar masses; both selections identify AGN in blue, star-forming and red, quiescent galaxies.
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