Dealing with under‐ and over‐dispersed count data in life history, spatial, and community ecology

Abstract

Count data arise frequently in ecological analyses, but regularly violate the equi‐dispersion constraint imposed by the most popular distribution for analyzing these data, the Poisson distribution. Several approaches for addressing over‐dispersion have been developed (e.g., negative binomial distribution), but methods for including both under‐dispersion and over‐dispersion have been largely overlooked. We provide three specific examples drawn from life‐history theory, spatial ecology, and community ecology, and illustrate the use of the Conway‐Maxwell‐Poisson (CMP) distribution as compared to other common models for count data. We find that where equi‐dispersion is violated, the CMP distribution performs significantly better than the Poisson distribution, as assessed by information criteria that account for the CMP's additional distribution parameter. The Conway‐Maxwell‐Poisson distribution has seen rapid development in other fields such as risk analysis and linguistics, but is relatively unknown in the ecological literature. In addition to providing a more flexible exponential distribution for count data that is easily integrated into generalized linear models, the CMP allows ecologists to focus on the magnitude of under‐ or over‐dispersion as opposed to the simple rejection of the equi‐dispersion null hypothesis. By demonstrating its suitability in a variety of common ecological applications, we hope to encourage its wider adoption as a flexible alternative to the Poisson.