Zero & N-Inflated Overdispersed Binomial Models for Sum-Constrained Poisson Count Processes

Abstract

Ecological monitoring programmes and palaeoecological analyses play a key role in understanding how biodiversity, including plants and wildlife, respond to environmental factors such as climate change or biotic competition. Such data are typically compositional in nature, and a common challenge in the analysis of ecological compositional data is the interpretation and modeling of datasets containing a high proportion of zeros and $N$'s. Such data frequently occur in ecological applications where counts of species are collected until a sum total imposed by sampling cost or sampling effort 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 proposed models provide for an improved understanding of the biological processes underlying ecological patterns and support more accurate biodiversity assessment, environmental monitoring, and palaeoclimate reconstruction.