Should zeros count? Modeling the galaxy-globular cluster scaling relation with(out) zero-inflated count models

The scaling relation between the size of a galaxy's globular cluster (GC) population ($N_{GC}$) and the galaxy's stellar mass ($M_*$) is usually described with a continuous, linear model, but in reality it is a count relationship that should be modeled as such. For massive galaxies, a negative binomial (NB) model has been shown to describe the data well, but it is unclear how the scaling relation behaves at low galaxy masses where a substantial portion of galaxies have $N_{GC}=0$. In this work, we test the utility of Poisson and NB models for describing the low-mass end of the $N_{GC}-M_*$ scaling relation. We introduce the use of zero-inflated versions of these models, which allow for larger zero populations (e.g. galaxies without GCs) than would otherwise be predicted. We evaluate our models with a variety of predictive model comparison methods, including predictive intervals, leave-one-out cross-validation criterion, and posterior predictive comparisons. We find that the NB model is consistent with our data, but the naive Poisson is not. Moreover, we find that zero inflation of the models is not necessary to describe the population of low-mass galaxies that lack GCs, suggesting that a single formation and evolutionary process acts over all galaxy masses. Under the NB model, there does not appear to be anything unique about the lack of GCs in many low-mass galaxies; they are simply the low-mass extension of the larger $N_{GC}-M_*$ scaling relation.