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 underdispersion 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.

Drought and presence of ants can influence hemiptera in tropical leaf litter

“…For the GLMM we used a Conway–Maxwell–Poisson distribution, using function glmmTMB in package glmmTMB (Magnusson et al, ) which can model this distribution, for all metrics except relative species richness and abundance, for which we used a beta distribution as they are proportion data. Compared with other distributions used for analyzing count data, Conway–Maxwell–Poisson distribution has increased model performance because it addresses violations of equi‐dispersion flexibly (Lynch, Thorson, & Shelton, ). In all models, we treated year and treatment as fixed effects, and included their interaction.…”

A novel method to estimate prey contributions to predator diets

“…Analysis methods that include likelihood frameworks have been developed for count data (de Valpine and Harmon-Threatt 2013;Fletcher et al 2005;Lynch et al 2014), but this mixture model is explicitly created for proportional data. Formal model selection is standard in other areas such as regression (Burnham and Anderson 1998;Hilborn and Mangel 1997) and is typically done using AIC or likelihood ratio tests.…”

Statistical modeling of patterns in annual reproductive rates

“…These constraints often lead to underdispersion where the variance is smaller than the mean (Kendall and Wittmann , Lynch et al. ). In addition, distributions of reproductive effort are often “zero‐inflated”, meaning that there are more zeros observed than would be expected under a given statistical model.…”

“…Lynch et al. () subsequently demonstrated that the Conway‐Maxwell‐Poisson (CMP) distribution, which has the same flexibility to model under‐ and overdispersion, is also well‐suited to modeling reproductive data. Although regression models using the GP distribution commonly fail to converge when residuals are underdispersed (Famoye et al.…”