The Overlooked Potential of Generalized Linear Models in Astronomy-III:
  Bayesian Negative Binomial Regression and Globular Cluster Populations
  In this paper, the third in a series illustrating the power of generalized
linear models (GLMs) for the astronomical community, we elucidate the potential
of the class of GLMs which handles count data. The size of a galaxy's globular
cluster population $N_{\rm GC}$ is a prolonged puzzle in the astronomical
literature. It falls in the category of count data analysis, yet it is usually
modelled as if it were a continuous response variable. We have developed a
Bayesian negative binomial regression model to study the connection between
$N_{\rm GC}$ and the following galaxy properties: central black hole mass,
dynamical bulge mass, bulge velocity dispersion, and absolute visual magnitude.
The methodology introduced herein naturally accounts for heteroscedasticity,
intrinsic scatter, errors in measurements in both axes (either discrete or
continuous), and allows modelling the population of globular clusters on their
natural scale as a non-negative integer variable. Prediction intervals of 99%
around the trend for expected $N_{\rm GC}$comfortably envelope the data,
notably including the Milky Way, which has hitherto been considered a
problematic outlier. Finally, we demonstrate how random intercept models can
incorporate information of each particular galaxy morphological type. Bayesian
variable selection methodology allows for automatically identifying galaxy
types with different productions of GCs, suggesting that on average S0 galaxies
have a GC population 35% smaller than other types with similar brightness.
