# An Overview of Modern Applications of Negative Binomial Modelling in Ecology and Biodiversity

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## Abstract

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## 1. Introduction

`R`[28]. We conclude with a discussion of future directions for analysing ecological and biological data with NB models. Ultimately, we hope that by adopting an expansive approach to this overview, readers can appreciate the growing ease yet broad scope with which NB modelling can be employed, and will subsequently choose to use the NB model as the starting point of their own analyses of count data.

## 2. Traditional Negative Binomial Modelling

#### 2.1. The Negative Binomial Distribution

#### 2.2. Traditional Uses of Negative Binomial Models

## 3. Negative Binomial Modelling in the 21st Century

#### 3.1. Negative Binomial as a Poisson Mixture Model and Beyond

`HierarchicalGOF`package in

`R`, and a Poisson-gamma mixture model fitted using the

`bsamGP`package. Both methods used Bayesian MCMC sampling for estimation with three chains with 10,000 MCMC iterations following a burn-in of 2000; see Conn et al. [11] for more details on priors and model fitting. We also fitted a Poisson (log-linear regression) GLM and a NB GLM, both were fitted using the

`mvabund`package and a Poisson–Tweedie mixture model fitted using the

`ptmixed`package. These methods used maximum likelihood estimation. For all models, we included two environmental covariates to model the (conditional) mean of the response: minimum temperature (${X}_{i,1}$) and stem density of adult trees (${X}_{i,2}$), which are known to correlate with abundance [4]. Thus, we write

#### 3.2. Occurrence/Presence-Absence Data

#### 3.3. Zero-Truncated and Zero-Inflated Data

`R`.

#### 3.4. Species Richness and Biodiversity Estimation

#### 3.5. Occupancy-Detection and Distance Sampling Methods

`HierarchicalGOF`package, where we set ${\alpha}_{\lambda}={\beta}_{\lambda}=0.001$, ${\alpha}_{q}={\beta}_{q}=1$ with 50,000 MCMC iterations following burn-in of 5000. The sampled posterior distribution for predicted bat abundance is given in Figure 3. Based on this, the total number of bats across all sites is predicted to be approximately 610, or between 580 and 630 based on the 2.5th and 97.5th percentiles of the distribution, respectively (dotted blue lines in Figure 3). Compared with the observed number of 450, this suggests there were approximately 160 unaccounted for bats throughout the sampling period.

#### 3.6. Joint Species Distribution and Compositional Data Models

`boral`

`R`-package [91], which utilises Bayesian MCMC modelling; for our analysis, we used the default package settings (i.e., all priors and MCMC parameters) and set the number of latent variables to $d=2$. In Web Figure S2, we give a caterpillar plot of regression coefficients with credible intervals, which were somewhat similar to the estimates given by the fitted mixture models in Table 1. Furthermore, to visualise the residual correlation between species, in Figure 4 we give the following: (a) a model-based residual ordination biplot; (b) a plot of the between-species correlation arising from shared environmental responses; and (c) a plot of correlations between species due to residual correlations. From the biplot (Figure 4a), there were no obvious patterns of site and species clustering, with the exception of Site 11, which was characterised by Myca and Myyu. From the correlation plots, we observed strong positive correlations due to environmental response (large blue circles in Figure 4b)—for example, species Tabr and Pahe—while the residual correlation was primarily dominated by strong, negative correlations (large red circles in Figure 4c)—for example, species Myyu with Lano, Pahe, and Epfu.

## 4. Model Fitting and Software

`R`we refer to the

`glm.nb`function in the

`MASS`package for fitting parametric NB regression models, the

`gam`function in the

`mgcv`package for fitting NB GAMs, and the

`manyglm`and

`traitglm`functions in the

`mvabund`package for fitting stacked NB species distribution models and NB fourth-corner models. Variations of maximum likelihood estimation also exist, including using the conditional likelihood [99], or the bias-corrected likelihood [100], although these are less popular. We also refer to Böhning [31] among others, who examined testing for overdispersion in Poisson and binomial regression models. Turning to NB (spatial) GLMMs, likelihood-based estimation of such mixed-effects models are noticeably more complicated, as the unobserved random effects need to be integrated. While methods for this are available (see Lindgren and Rue [101], along with Table 3 for some example packages in

`R`), an arguably more attractive approach in ecology and biodiversity analyses has been to adopt Bayesian estimation methods, particularly that of MCMC sampling (e.g., [102]). They are widely used in many ecological applications, and a number of different statistical software

`R`-packages have been developed to fit these models.

`R`-packages for fitting various NB models, covering both traditional uses and modern applications. In particular, while some of the modern usages of NB modelling have inspired relatively user-friendly software packages (e.g., the

`unmarked`package for fitting occupancy-detection and N-mixture models discussed in Section 3.5, and the

`boral`and

`gllvm`packages for fitting NB JSDMs discussed in Section 3.6), many of the techniques described in Section 3 either do not have associated

`R`-packages or in-built functions, or are more likely require bespoke

`R`-code implementations (e.g., through the use of generic MCMC samples like

`JAGS`https://mcmc-jags.sourceforge.io/, accessed on 17 April 2022, or automatic differentiation tools like Template Model Builder (TMB, https://cran.r-project.org/web/packages/TMB/index.html, accessed on 17 April 2022)). We also refer to Hilbe [33], Zuur et al. [52], and Kéry and Royle [74], who give excellent examples of fitting a variety of different NB models to various applications with all code provided (the latter two references focus on numerous ecological examples). Finally, we note that other software such as SAS, MATLAB, and SPSS also have NB implementations. However, given the popularity of

`R`in ecology and biology, we do not cover their utilities here.

## 5. Discussion

`R`in Section 4. Whilst we primarily focused on modelling $\mu $ with simple structures in our examples, it is also possible to fit so-called double GLMs, where the (log of the) overdispersion parameter is also regressed against covariates (e.g., $log\left({\kappa}_{i}\right)={\alpha}_{0}+{X}_{i}^{\top}\alpha $, where ${\kappa}_{i}$ now also depends on the observation unit through the covariates, and ${\alpha}_{0}$ and $\alpha $ denote the associated regression coefficients). Indeed, we hinted at this in Section 3.1 with Poisson mixture models where both parameters of the mixture distribution are regressed against covariates. A more substantial example of such an application is seen in Bonat et al. [103], who simultaneously modelled the mean and covariance structure using count data collected on prey animals in Pico Basilè, Bioko Island, Equatorial Guinea, via generalised estimating equations; see also the discussion below.

`R`-packages have also been developed to fit NB regularisation models (e.g.,

`glmnet`and

`rpql`[106]). Moreover, variable selection in more complex settings such as NB JSDMs and spatio-temporal/occupancy-detection count models continues to remain an active area of research, and an increasingly large array of statistical (e.g., [107,108]) and computational techniques (e.g., [44,60,80,101]) are starting to become available for tackling these cutting-edge model estimation and inferential challenges.

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

`R`-code supporting reported results have been included in the submission.

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Flowchart of selected examples for existing, modern, and extensions of negative binomial (NB) models for count data to address a variety of ecological applications.

**Figure 2.**The sample variance of observed counts against the sample mean for seven bat species (spp.) using the acoustic count data. The mean-variance relationship is clearly non-linear (in fact, close to quadratic), indicating evidence of overdispersion. Seven bat species were recorded in oak woodlands of California labelled by the following abbreviation: Tadarida brasiliensis (Tabr), Eptesicus fuscus (Epfu), Lasionycteris noctivagans (Lano), Lasiurus cinereus (Laci), Parastrellus hesperus (Pahe), Myotis yumanensis (Myyu), and Myotis californicus (Myca).

**Figure 3.**Sampled posterior distribution for predicted bat abundance when fitting a Poisson-gamma N-mixture model with no covariates. In this example, we combined all species counts and account for imperfect detection as well as overdispersion. Based on the median (blue dashed line), the predicted bat abundance is just below 610. The dotted blue lines represent the 2.5th and 97.5th percentiles of the distribution, respectively.

**Figure 4.**(

**a**) A residual ordination biplot based on latent variable posterior medians; (

**b**) a plot of the between-species correlation arising from shared environmental responses; and (

**c**) a plot pf correlations between species due to residual correlations when fitting an NB joint species distribution to the bat acoustic data. The 20 site numbers are labeled in black, and the seven bat species are shown in red and labelled by the following abbreviated names: Tadarida brasiliensis (Tabr), Eptesicus fuscus (Epfu), Lasionycteris noctivagans (Lano), Lasiurus cinereus (Laci), Parastrellus hesperus (Pahe), Myotis yumanensis (Myyu), and Myotis californicus (Myca).

**Table 1.**Parameter estimates with either 95% confidence or credible intervals when fitting a Poisson GLM, a Poisson-log-normal mixture model, a Poisson-gamma mixture model, and a Poisson–Tweedie mixture model to each species (spp.) in the bat acoustic count data. Seven bat species were recorded in oak woodlands of California, labelled by the following abbreviations: Tadarida brasiliensis (Tabr), Eptesicus fuscus (Epfu), Lasionycteris noctivagans (Lano), Lasiurus cinereus (Laci), Parastrellus hesperus (Pahe), Myotis yumanensis (Myyu), and Myotis californicus (Myca).

Spp. | Model | ${\widehat{\mathit{\beta}}}_{0}^{\left(\mathit{k}\right)}$ | ${\widehat{\mathit{\beta}}}_{1}^{\left(\mathit{k}\right)}$ | ${\widehat{\mathit{\beta}}}_{2}^{\left(\mathit{k}\right)}$ |
---|---|---|---|---|

Laci | Poisson | −0.38 (−0.69, −0.06) | −0.01 (−0.30, 0.29) | −0.47 (−0.83, −0.11) |

Poisson-log-normal mixt. | −0.42 (−0.76, −0.13) | 0.01 (−0.28, 0.31) | −0.45 (−0.77, −0.12) | |

Poisson-gamma mixt. | −0.39 (−0.73, −0.04) | 0.07 (−0.29, 0.43) | −0.46 (−0.87, −0.06) | |

Poisson–Tweedie mixt. | −0.37 (−0.43, −0.31) | −0.00 (−0.06, 0.06 | −0.44 (−0.52, −0.37) | |

Lano | Poisson | −0.88 (−1.30, −0.46) | 0.63 (0.19, 1.08) | −0.02 (−0.35, 0.32) |

Poisson-log-normal mixt. | −1.09 (−2.07, −0.54) | 0.71 (0.21, 1.34) | −0.04 (−0.51, 0.34) | |

Poisson-gamma mixt. | −0.97 (−1.50, −0.44) | 0.77 (0.08, 1.46) | 0.00 (−0.42, 0.42) | |

Poisson–Tweedie mixt. | −0.89 (−0.98, −0.80) | 0.64 (0.55, 0.74) | −0.08 (−0.16, −0.00) | |

Myyu | Poisson | −0.36 (−0.72, −0.01) | 0.81 (0.55, 1.07) | −1.39 (−1.72, −1.05) |

Poisson-log-normal mixt. | −0.86 (−1.63, −0.31) | 0.46 (−0.06, 0.98) | −0.99 (−1.65, −0.43) | |

Poisson-gamma mixt. | −0.03 (−0.47, 0.40) | 0.36 (−0.14, 0.86) | −1.02 (−1.58, −0.46) | |

Poisson–Tweedie mixt. | 0.13 (0.07, 0.19) | 0.53 (0.45, 0.61) | −0.87 (−0.97, −0.78) | |

Tabr | Poisson | 2.08 (1.98, 2.17) | 0.34 (0.25, 0.44) | −0.53 (−0.63, −0.42) |

Poisson-log-normal mixt. | 0.87 (0.39, 1.29) | 0.82 (0.37, 1.33) | −0.48 (−0.93, −0.05) | |

Poisson-gamma mixt. | 1.98 (1.64, 2.33) | 0.74 (0.15, 1.33) | −0.75 (−1.17, −0.33) | |

Poisson–Tweedie mixt. | 2.09 (2.06, 2.12) | 0.56 (0.53, 0.59) | −0.22 (−0.25, −0.19) | |

Epfu | Poisson | 1.39 (1.26, 1.52) | 0.30 (0.18, 0.42) | 0.62 (0.54, 0.70) |

Poisson-log-normal mixt. | 0.07 (−0.59, 0.61) | 0.61 (0.06, 1.25) | 0.57 (0.06, 1.10) | |

Poisson-gamma mixt. | 1.50 (1.02, 1.97) | 0.18 (−0.58, 0.93) | 0.59 (0.06, 1.12) | |

Poisson–Tweedie mixt. | 1.58 (1.53, 1.62) | 0.36 (0.32, 0.40) | 0.31 (0.2, 0.34) | |

Myca | Poisson | 2.16 (2.07, 2.24) | 0.31 (0.21, 0.40) | 0.14 (0.06, 0.21) |

Poisson-log-normal mixt. | 0.64 (0.11, 1.10) | 0.32 (−0.15, 0.78) | 0.09 (−0.35, 0.55) | |

Poisson-gamma mixt. | 2.16 (1.70, 2.62) | 0.52 (0.03, 1.02) | 0.25 (−0.24, 0.74) | |

Poisson–Tweedie mixt. | 2.20 (2.17, 2.24) | 0.09 (0.05, 0.13) | 0.11 (0.07, 0.15) | |

Pahe | Poisson | 0.75 (0.55, 0.94) | 0.34 (0.18, 0.51) | −0.83 (−1.04, −0.62) |

Poisson-log-normal mixt. | −1.39 (−2.58, −0.57) | 1.16 (0.40, 2.12) | −1.32 (−2.34, −0.51) | |

Poisson-gamma mixt. | 0.70 (0.14, 1.26) | 0.71 (−0.07, 1.49) | −1.04 (−1.75, −0.29) | |

Poisson–Tweedie mixt. | 0.71 (0.64, 0.79) | 0.65 (0.57, 0.74) | −0.66 (−0.76, −0.57) |

**Table 2.**Parameter estimates with either 95% confidence or credible intervals when fitting a negative binomial (NB GLM), zero-inflated Poisson (ZI-Poisson), and zero-inflated NB (ZI-NB) model to each species (spp.) in the bat acoustic count data. Seven bat species were recorded in oak woodlands of California, labelled by the following abbreviations: Tadarida brasiliensis (Tabr), Eptesicus fuscus (Epfu), Lasionycteris noctivagans (Lano), Lasiurus cinereus (Laci), Parastrellus hesperus (Pahe), Myotis yumanensis (Myyu), and Myotis californicus (Myca).

Spp. | Model | ${\widehat{\mathit{\beta}}}_{0}^{\left(\mathit{k}\right)}$ | ${\widehat{\mathit{\beta}}}_{1}^{\left(\mathit{k}\right)}$ | ${\widehat{\mathit{\beta}}}_{2}^{\left(\mathit{k}\right)}$ |
---|---|---|---|---|

Laci | NB GLM | −0.38 (−0.74, −0.01) | 0.03 (−0.33, 0.38) | −0.47 (−0.88, −0.05) |

ZI-Poisson | 0.08 (−0.40, 0.57) | −0.05 (−0.41, 0.31) | −0.34 (−0.85, 0.17) | |

ZI-NB | 0.08 (−0.40, 0.57) | −0.05 (−0.41, 0.31) | −0.34 (−0.85, 0.17) | |

Lano | NB GLM | −0.91 (−1.41, −0.41) | 0.72 (0.16, 1.27) | 0.01 (−0.42, 0.43) |

ZI-Poisson | −0.62 (−1.05, −0.19) | 0.50 (0.02, 0.97) | 0.61 (0.08, 1.14) | |

ZI-NB | −0.64 (−1.14, −0.14) | 0.57 (−0.03, 1.17) | 0.66 (0.01, 1.31) | |

Myyu | NB GLM | −0.02 (−0.50, 0.46) | 0.34 (−0.14, 0.81) | −0.96 (−1.53, −0.39) |

ZI-Poisson | 0.51 (0.08, 0.94) | 0.44 (0.09, 0.78) | −1.05 (−1.46, −0.63) | |

ZI-NB | 0.17 (−0.31, 0.65) | −0.06 (−0.63, 0.51) | −1.35 (−1.99, −0.72) | |

Tabr | NB GLM | 1.96 (1.58, 2.34) | 0.71 (0.32, 1.11) | −0.73 (−1.13, −0.34) |

ZI-Poisson | 2.41 (2.32, 2.51) | −0.02 (−0.13, 0.09) | −0.46 (−0.57, −0.35) | |

ZI-NB | 2.16 (1.75, 2.58) | 0.24 (−0.36, 0.85) | −0.69 (−1.11, −0.28) | |

Epfu | NB GLM | 1.44 (0.99, 1.88) | 0.24 (−0.21, 0.69) | 0.51 (0.07, 0.95) |

ZI-Poisson | 1.98 (1.85, 2.12) | −0.09 (−0.27, 0.10) | 0.56 (0.47, 0.65) | |

ZI-NB | 1.68 (1.22, 2.13) | −0.44 (−1.10, 0.22) | 0.76 (0.34, 1.18) | |

Myca | NB GLM | 2.12 (1.67, 2.56) | 0.51 (0.06, 0.95) | 0.25 (−0.19, 0.69) |

ZI-Poisson | 2.49 (2.40, 2.57) | 0.26 (0.17, 0.34) | 0.07 (0.00, 0.14) | |

ZI-NB | 2.12 (1.67, 2.56) | 0.51 (−0.02, 1.04) | 0.25 (−0.25, 0.74) | |

Pahe | NB GLM | 0.62 (0.03, 1.21) | 0.67 (0.05, 1.29) | −1.08 (−1.75, −0.41) |

ZI-Poisson | 1.84 (1.63, 2.04) | −0.23 (−0.42, −0.03) | −0.55 (−0.79, −0.31) | |

ZI-NB | 1.43 (0.53, 2.32) | −0.27 (−1.09, 0.55) | −0.81 (−1.81, 0.18) |

**Table 3.**A selective list of

`R`-packages for fitting various traditional and modern negative binomial (NB) models described in the main text. We denote as s the number of species/taxa, n the number of sampling units (e.g., sites), and p the number of covariates. Note that generic Bayesian MCMC packages such as the

`rjags`and

`R2jags`

`R`-packages can also be used to fit many of the models listed below, with some additional coding required for specifying the NB likelihood. Indeed, some of the modern NB models discussed in Section 3 either do not have currently associated

`R`-packages, or require bespoke

`R`-code. Furthermore, some of the packages listed can fit multiple types of NB models (e.g., the

`gamlss`package can also fit NB GLMs and GAMs).

Model: | Modelling Usage and Notes: | R-Package(s): | Common Function: |
---|---|---|---|

Generalised linear model (GLM) | Single species ($s=1$) | MASS | glm.nb() |

Generalised additive model (GAM) | Smoothing | mgcv | gam(family = nb()) |

gamlss | gamlss(family = NBI) | ||

Generalised linear mixed model (GLMM) | Random/mixed effects | lme4 | glmer.nb() |

glmmTMB/glmmadmb | glmmTMBfamily = nbinom2() | ||

Generalised additive mixed model | mgcv | gamm(family = nb()) | |

GLM with regularisation penalties | High-dimension $(n>p)$ | glmnet | glmnet(family = negative.binomial) |

GLMM with regularisation penalties | rpql | rpql(family = "nb2") | |

Species distribution model | Stacked SDM ($s>1$) | mvabund | manyglm(family = "negative.binomial") |

Stacked and Reduced-rank SDMs/GAMs | VGAM | vglm(family = negbinomial()) | |

Joint species distribution model | (Residual) correlation across species | boral | boral(family = "negative.binomial") |

gllvm | gllvm(family = "negative.binomial") | ||

Poisson-log-normal mixture model | HierarchicalGOF | pois.overd.no.spat() | |

Poisson-gamma mixture model | bsamGP | gblr(family = "poisson.gamma") | |

Poisson–Tweedie mixture model | ptmixed | ptglm() | |

Zero-inflated GLM | pscl/countreg | zeroinfl(dist = "negbin") | |

VGAM | vglm(family = zinegbin()) | ||

gamlss | family = ZIP()/ZINBI() | ||

Zero-truncated GLM | countreg | zerotrunc(dist = "negbin") | |

VGAM | vglm(family = posnegbinom()) | ||

gamlss.tr | gen.trun(family = "PO") | ||

N-mixture models | Imperfect detection | unmarked | pcount(mixture = "NB") |

Generalised multinomial N-mixture | Three hierarchical levels | unmarked | gmultmix(mixture = "NB") |

Hierarchical distance sampling | unmarked | gdistsamp(mixture = "NB") |

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Stoklosa, J.; Blakey, R.V.; Hui, F.K.C.
An Overview of Modern Applications of Negative Binomial Modelling in Ecology and Biodiversity. *Diversity* **2022**, *14*, 320.
https://doi.org/10.3390/d14050320

**AMA Style**

Stoklosa J, Blakey RV, Hui FKC.
An Overview of Modern Applications of Negative Binomial Modelling in Ecology and Biodiversity. *Diversity*. 2022; 14(5):320.
https://doi.org/10.3390/d14050320

**Chicago/Turabian Style**

Stoklosa, Jakub, Rachel V. Blakey, and Francis K. C. Hui.
2022. "An Overview of Modern Applications of Negative Binomial Modelling in Ecology and Biodiversity" *Diversity* 14, no. 5: 320.
https://doi.org/10.3390/d14050320