  For investigating the relationship between the star formation rate and gas
surface density, we develop a Bayesian linear regression method that rigorously
treats measurement uncertainties and accounts for hierarchical data structure.
The hierarchical method simultaneously estimates the intercept, slope, and
scatter about the regression line of each individual subject and the
population. Using synthetic datasets, we demonstrate that the method recovers
the underlying parameters of both the individuals and the population,
especially when compared to commonly employed ordinary least squares
techniques, such as the bisector fit. We apply the hierarchical method to
estimate the Kennicutt-Schmidt (KS) parameters of a sample of spiral galaxies
compiled by Bigiel et al. (2008). We find significant variation in the KS
parameters, indicating that no single relationship holds for all galaxies. This
suggests that the relationship between molecular gas and star formation differs
between galaxies, possibly due to the influence of other properties within a
given galaxy, such as metallicity, molecular gas fraction, or stellar mass. In
four galaxies the slope estimates are sub-linear (at the 95% level). We
estimate the mean index of the KS relationship for the population to be 0.84,
with 95% range [0.63, 1.0]. For the galaxies with sub-linear KS relationships,
a possible interpretation is that CO emission is tracing some molecular gas
that is not directly associated with star formation. Equivalently, a sub-linear
KS relationship may be indicative of an increasing gas depletion time at higher
surface densities, as traced by CO emission. The Bayesian method can account
for all sources of uncertainties, including variations in the conversion of
observed intensities to star formation rates and gas surface densities, and is
therefore well suited for a thorough statistical analysis of the KS law.
