  In this article we consider the distribution arising when two zero-inflated
Poisson count processes are constrained by their sum total, resulting in a
novel zero & $N$-inflated binomial distribution. This result motivates a
general class of model for applications in which a sum-constrained count
response is subject to multiple sources of heterogeneity, principally an excess
of zeroes and $N$'s in the underlying count generating process. Two examples
from the ecological regression literature are used to illustrate the wide
applicability of the proposed model, and serve to detail its substantial
superiority in modelling performance as compared to competing models. We also
present an extension to the modelling framework for more complex cases,
considering a gender study dataset which is overdispersed relative to the new
likelihood, and conclude the article with the description of a general
framework for a zero & $N$-inflated multinomial distribution.
