Zero & $N$-inflated overdispersed binomial models for sum-constrained Poisson count processes

A frequent challenge encountered with compositional ecological data is how to interpret and model data with a high proportion of zeros and $N$'s. Such data frequently occur in ecological applications where counts of species are collected until a pre-specified total imposed (typically) by sampling cost is reached. In the bivariate count (two-species) setting we focus on in this article, zero-inflation of one species will result in $N$-inflation of the other. This can lead to species absence being attributed to an unsuitable habitat as opposed to missingness by chance. Similarly, an excess of $N$'s will lead to misleading inferences about habitat preference and abundance estimates. Our contribution is to identify that two independent zero-inflated Poisson processes subject to a sum constraint provide a novel biologically-motivated generating mechanism for the occurrence of binomial count data exhibiting zero and $N$-inflation. We identify an extension to the model to capture additional overdispersion within the data resulting in a novel zero and $N$-inflated beta-binomial model. We consider two motivating datasets, one involving a pesticide treatment for an invasive species, and a second involving the abundance of two plant species. We demonstrate that incorporation of covariates in each case enable learning about sources of zero and $N$-inflation as well as abundance. We show that the models result in improved understanding of underlying biological processes as well as improved predictive performance.