We measure the cosmological bias factor of damped Ly α absorbers (DLAs) from their crosscorrelation with the Ly α forest absorption, as a function of the DLA metal strength, defined from an average of equivalent widths of the strongest detectable low-ionization metal lines. A clear increase of the bias factor with metal strength is detected, as expected from a relation of metallicity and velocity dispersion with host halo mass. The relation is stronger after the metal strength is corrected for the H I column density, to make it more related to metallicity instead of metal column density. After correcting for the effects of measurement errors of the metal strength parameter, we find that the bias factor of DLAs with the weakest metal lines is close to unity, consistent with an origin in dwarf galaxies with host halo masses ∼1010M , whereas the most metal rich DLAs have a bias factor as large as bDLA ∼ 3, indicative of massive galaxies or galaxy groups in host haloes with masses ∼1012M . Our result confirms the physical origin of the relation of bias factors measured from cross-correlation studies to the host haloes of the absorbers.