  Observations of long gamma-ray bursts (LGRBs) offer a unique opportunity for
probing the cosmic star formation history, although whether or not LGRB rates
are biased tracers of star formation rate history is highly debated. Based on
an extensive sample of LGRBs compiled by Robertson & Ellis (2012), we analyze
various models of star formation rate and the possible effect of the evolution
of cosmic metallicity under the assumption that LGRBs tend to occur in
low-metallicity galaxies. The models of star formation rate tested in this work
include empirical fits from observational data as well as a self-consistent
model calculated in the framework of the hierarchical structure formation.
Comparing with the observational data, we find a relatively higher metallicity
cut of $Z\gtrsim0.6Z_{\odot}$ for the empirical fits and no metallicity cut for
the self-consistent model. These results imply that there is no strong
metallicity preference for the host galaxy of LGRBs, in contrast to previous
work which suggest a cut of $Z\sim0.1-0.3Z_{\odot}$, and that the inferred
dependencies of LGRBs on their host galaxy properties are strongly related to
the specific models of star formation rate. Furthermore, a significant fraction
of LGRBs occur in small dark matter halos down to
$3\times10^{8}\,\mathrm{M_{\odot}}$ can provide an alternative explanation for
the discrepancy between the star formation rate history and LGRB rate history.
