  We consider the analysis of count data in which the observed frequency of
zero counts is unusually large, typically with respect to the Poisson
distribution. We focus on two alternative modelling approaches: Over-Dispersion
(OD) models, and Zero-Inflation (ZI) models, both of which can be seen as
generalisations of the Poisson distribution; we refer to these as Implicit and
Explicit ZI models, respectively. Although sometimes seen as competing
approaches, they can be complementary; OD is a consequence of ZI modelling, and
ZI is a by-product of OD modelling. The central objective in such analyses is
often concerned with inference on the effect of covariates on the mean, in
light of the apparent excess of zeros in the counts. Typically the modelling of
the excess zeros per se is a secondary objective and there are choices to be
made between, and within, the OD and ZI approaches. The contribution of this
paper is primarily conceptual. We contrast, descriptively, the impact on zeros
of the two approaches. We further offer a novel descriptive characterisation of
alternative ZI models, including the classic hurdle and mixture models, by
providing a unifying theoretical framework for their comparison. This in turn
leads to a novel and technically simpler ZI model. We develop the underlying
theory for univariate counts and touch on its implication for multivariate
count data.
