A novel method for quantifying overdispersion in count data and its application to farmland birds

Abstract

The statistical modelling of count data permeates the discipline of ecology. Such data often exhibit overdispersion compared with a standard Poisson distribution, so that the variance of the counts is greater than that of the mean. Whereas modelling to reveal the effects of explanatory variables on the mean is commonplace, overdispersion is generally regarded as a nuisance parameter to be accounted for and subsequently ignored. Instead, we propose a method that models the overdispersion as a biologically interesting property of a data set and show how novel inference is provided as a result. We adapted the double hierarchical generalized linear model approach to create an easily extendible model structure that quantifies the influence of explanatory variables on the overdispersion of count data, and apply it to farmland birds. These data were from a study within Irish agricultural ecosystems, in which total bird species abundance and the abundance of farmland indicator species were compared on dairy and non‐dairy farms in the winter and breeding seasons. In general, overdispersion in bird counts was greater on dairy farms than on non‐dairy farms, and for total bird numbers, overdispersion was greatest on dairy farms in winter. Our code is fitted using the Bayesian package Rstan, and we make all code and data available in a GitHub repository. Within a Bayesian framework, this approach facilitates a meaningful quantification of the effects of categorical explanatory variables on any response variable with a tendency to overdispersion that has a meaningful biological or ecological explanation.

Modeling spatiotemporal abundance of mobile wildlife in highly variable environments using boosted GAMLSS hurdle models

“…Variation in abundance is often a key component of a species’ response to habitat quality and is crucial for accurately predicting species distributions (Howard, Stephens, Pearce‐Higgins, Gregory, & Willis, ; Johnston et al, ). The statistically challenging features of count data—particularly overdispersion, in which the variance of the data exceeds the mean—may in fact represent important biological responses to environmental features and conditions (McMahon, Purvis, Sheridan, Siriwardena, & Parnell, ; Richards, ). Modeling count data also often requires accounting for zero‐inflation (Martin et al, ), non‐linear responses to covariates (Cunningham & Lindenmayer, ), and spatial and temporal autocorrelation (Hoeting, ), which require a highly flexible modeling approach with few assumptions about either underlying distribution or response functions.…”

“…GAMs do not assume linear effects on the response but flexibly adapt to the observed data, which makes them especially applicable to systems in which the form of the relationship between species occupancy, abundance, and underlying environmental conditions is often non‐linear and unknown (Guisan & Zimmermann, ). Moreover, unlike other generalized modeling approaches, GAMLSS allow both the mean and dispersion of the response to be modeled as a function of environmental covariates (Rigby & Stasinopoulos, ), which incorporates additional information about count data not reflected by mean values alone (McMahon et al, ). Despite these promising features, although GAM has recently gained popularity as a predictive distribution modeling approach (Miller, Burt, Rexstad, & Thomas, ), GAMLSS have yet to be widely adopted for modeling the spatial distribution of species based on biophysical features.…”

“…Variation in abundance is often a key component of a species' response to habitat quality and is crucial for accurately predicting species distributions (Howard, Stephens, Pearce-Higgins, Gregory, & Willis, 2014;Johnston et al, 2015). The statistically challenging features of count dataparticularly overdispersion, in which the variance of the data exceeds the mean-may in fact represent important biological responses to environmental features and conditions (McMahon, Purvis, Sheridan, Siriwardena, & Parnell, 2017;Richards, 2008).…”

“…Moreover, unlike other generalized modeling approaches, GAMLSS allow both the mean and dispersion of the response to be modeled as a function of environmental covariates (Rigby & Stasinopoulos, 2005), which incorporates additional information about count data not reflected by mean values alone (McMahon et al, 2017). Despite these promising features, although GAM has recently gained popularity as a predictive distribution modeling approach (Miller, Burt, Rexstad, & Thomas, 2013), GAMLSS have yet to be widely adopted for modeling the spatial distribution of species based on biophysical features.…”

Modelling excess zeros in count data: A new perspective on modelling approaches

“…Regression analysis of count data arises in many fields, including agriculture (Blasco-Moreno et al, 2019), ecology (Mcmahon et al, 2017), climatology (Salter-Townshend and Haslett, 2012), finance (Benson and Friel, 2017), and pharma (Min and Agresti, 2005). The simplest modelling framework for such analysis is generalised linear modelling using the Poisson and Negative Binomial (NB) families.…”