We revisit the correlation between the mid-infrared (6 $mu$m) and hard X-ray (2--10 keV) luminosities of active galactic nuclei (AGNs) to understand the physics behind it. We construct an X-ray flux-limited sample of 571 type 1 AGNs with $f_{0.5-2.0 ,{rm keV}} > 2.4 times 10^{-12}$ erg cm$^{-2}$ s$^{-1}$, drawn from the ROSAT Bright Survey catalog. Cross-matching the sample with infrared data taken from Wide-field Infrared Survey Explorer, we investigate the relation between the rest-frame 6 $mu$m luminosity ($L_{rm 6}$) and the rest-frame 2--10 keV luminosity ($L_{rm X}$), where $L_{rm 6}$ is corrected for the contamination of host galaxies by using the spectral energy distribution fitting technique. We confirm that $L_{rm 6}$ and $L_{rm X}$ are correlated over four orders of magnitude, in the range of $L_{rm X} = 10^{42-46}$ erg s$^{-1}$. We investigate what kinds of physical parameters regulate this correlation. We find that $L_{rm X}$/$L_{rm 6}$ clearly depends on the Eddington ratio ($lambda_{rm Edd}$) as $log lambda_{rm Edd} = -(0.56 pm 0.10) log , (L_{rm X}/L_{rm 6}) - (1.07 pm 0.05)$, even taking into account quasars that are undetected by ROSAT as well as those detected by XMM-Newton in the literature. We also add hyper-luminous quasars with $L_{rm 6}$ $>$ 10$^{46}$ erg s$^{-1}$ in the literature and perform a correlation analysis. The resultant correlation coefficient is $-0.41 pm 0.07$, indicating a moderately tight correlation between $L_{rm X}$/$L_{rm 6}$ and $lambda_{rm Edd}$. This means that AGNs with high Eddington ratios tend to have lower X-ray luminosities with respect to the mid-infrared luminosities. This dependence can be interpreted as a change in the structure of the accretion flow.
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