# Some Comments about the p-Generalized Negative Binomial (NBp) Model

## Abstract

**:**

## 1. Introduction

^{2}property, most often such that μ < σ

^{2}), further implying that their best distributional description is a Poisson variable with a gamma-distributed mean (i.e., a gamma–Poisson mixture), which translates nicely into a negative binomial (NB) RV. However, a standard NB does not always successfully account for this mean–variance discrepancy in such circumstances. Cameron and Trivedi [1] introduce a generalized version of the NB probability model now known as the NBp model, where p is a positive integer, whose goal is to better capture under/over-dispersion; this variate differs from the negative binomial process (NBP) model (e.g., [2]). The literature contains little discussion about this former generalized model, with the exception of Cameron and Trivedi’s initial treatment of it and papers by Greene [3] and by Di et al. [4]. Of note is that an R package (version ≥ 3.00) exists for implementing this model [4]. The purpose of this paper is to address this gap in the literature by (1) clarifying/correcting several misleading technical details in part of Greene’s paper; (2) presenting new results; (3) illuminating selected small and large sample properties on NB models; and (4) presenting new empirical spatial statistics/econometrics instances involving spatial autocorrelation (SA).

^{2–p}and μ

^{p–1}/α, where μ denotes its mean and 1/α denotes its dispersion parameter, yielding a global NBp model with parameters r = αμ

^{2–p}and probability p = 1/(1 + μ

^{p–1}/α) for some non-negative integer p > 0; p = 1, 2 are standard NB specifications, with the former being the Poisson RV and the latter being the popular NB2 variate.

## 2. Materials and Methods: Clarifying Greene’s Presentation

^{2–p}) rather than 1/(1 + μ

^{p–2}/α)—yield, for independent and identically distributed (iid) NBps, the following probability mass function:

^{2}] = μ(1 + μ

^{p–1}/α).

^{−μ}μ

^{y}/(y!) with μ ~ Γ(αμ

^{2–p}, μ

^{p–1}/α), nor the outcome from Greene’s replacing α with αμ

^{2–p}in this mixture, which renders E(Y) = μ and E[(Y − μ)

^{2}] = μ(1 + μ

^{p–3}/α).

## 3. Results: Articulating NBp Variates

#### 3.1. The NBp Moment-Generating Function and Asymptotic Normality

#### 3.1.1. The NBp Moment-Generating Function (mgf)

^{p–1}/α).

#### 3.1.2. Asymptotic Normality and an NBp RV

^{2}is a function of (y

_{i}− $\stackrel{\_}{y}$); Figure 1a]. Figure 1 portrays scatterplots of relationships between sample s

^{2}and s

^{2}from simulation experiment results for 10,000 replications of samples of size n = 100 (a well-known appropriate size to ensure that the classical Central Limit Theorem is at work). For this experiment, the linear correlation between $\stackrel{\_}{y}$ and s

^{2}for a standard normal RV (i.e., iid) is –0.014 (Figure 1a), whereas that for an NBp RV with p = 2, μ = 1000, and α = 1000 (i.e., iid) is 0.027 (Figure 1b). An extension of this conventional, well-known theorem is as follows:

**Theorem**

**1.**

^{2}for a mixture of independent and non-identically distributed (i~id) normal RVs is zero.

**Proof**

**of Theorem 1.**

^{2}can be any positive real number. Because these two sets of values can be paired in any of the infinite number of possible ways that exist, each value of μ is paired with relatively large, intermediate, and small values of σ

^{2}, and each value of σ

^{2}is paired with relatively large, intermediate, and small values of μ. Consequently, the linear correlation between these two variates is zero. □

**Theorem**

**2.**

^{2}for a mixture of i~id NBp RVs is $\frac{\sqrt{3(2\mathrm{p}+1)}}{p+2}>0$, p ≥ 1.

**Proof**

**of Theorem 2.**

- This new theorem indicates that if p = 1 (i.e., a Poisson RV), then ρ = 1, and for the standard NB2 case of p = 2, ρ ≈ 0.968. The parameter p needs to be 20 before this linear correlation is approximately 0.5. The dispersion parameter α does not affect this linear correlation. This new theorem indicates that for the aforementioned NBp, the sample mean and variance, $\stackrel{\_}{y}$ and s
^{2}, are correlated (Figure 1d). In other words, the linear correlation between $\stackrel{\_}{y}$ and s^{2}for a mixture of normal distributions is zero, whereas for a mixture of NBp distributions, it is asymptotically zero as p → ∞. Although this linear correlation remains zero when normal distributions are pooled, it can be as large as 1 for pooled NBp distributions.

^{2}= μ(1 + μ

^{p–1}/α), with suitable restrictions on both p and μ (e.g., p must be even and the μ domain interval [–1, 0] must be removed from the range of σ

^{2}). Linear correlation occurs for this particular normal distribution case by, for example, imposing a constraint such as μ > 0, which then also allows p to be odd and linear correlation results to be described by Theorem 2.

#### 3.2. MLE and Method of Moments Estimation (MME) of the Dispersion Parameter α

#### 3.2.1. Estimation of Parameter Exponent p

_{3}, to the set of estimation equations. Accordingly,

^{p–1}/α) is similar to quasi-likelihood estimation with regard to parameter p. Denote $\frac{1}{\widehat{\alpha}}$ with ${\widehat{\alpha}}^{-1}$; if ${\widehat{\alpha}}^{-1}$ seems too large, then a k

^{th}multiple of $\widehat{\mu}$ (i.e., ${\widehat{\mu}}^{\mathrm{k}}$), where k is a positive integer, can be factored from it, effectively setting p = 2 + k. This factorization is nonconstant across the values of μ

_{i}, with subscript i denoting an individual observation, when the mean varies from observation to observation (this is the situation Greene [1] presents).

#### 3.2.2. A Simple n = 3 Numerical Example

_{i}= {2, 5, 8}. Then

#### 3.2.3. Moran Eigenvector Spatial Filtering: A Brief Overview

**C**, that often is binary (0–1) and that ties n geographic objects together in space (indicating which are pairwise directly correlated by an entry of one), and then adds these vectors as control variables to a regression model specification; SWMs are analogous to undirected planar graph adjacency matrices. These control variables identify and isolate stochastic spatial dependencies among a set of georeferenced observations, filtering these dependencies out of a model’s residuals and adding them to the model’s mean response, thus allowing regression model building to proceed with georeferenced observations that mimic being independent.

**1**

^{T}

**C1**)

**Y**

^{T}(

**I**−

**11**

^{T}/n)

**C**(

**I**−

**11**

^{T}/n)

**Y**/

**Y**

^{T}(

**I**−

**11**

^{T}/n)

**Y**,

**Y**is an n-by-1 vector of attribute values;

**I**is an n-by-n identity matrix;

**1**is an n-by-1 vector of ones; superscript T denotes matrix transpose and, here,

**C**is a binary 0–1 SWM for which 1 denotes that the row and column areal units (i.e., locations, e.g., shapefile polygons) share a non-zero length common boundary, and 0 denotes that they do not (i.e., the rook adjacency definition, phraseology based upon chess game pieces and their moves). The eigenfunctions (i.e., the paired eigenvalues and eigenvectors) are extracted from the following matrix, which is doubly centered because matrix

**C**is pre- and post-multiplied by the projection matrix (

**I**−

**11**

^{T}/n), appearing in the numerator of expression (4) as follows:

**I**−

**11**

^{T}/n)

**C**(

**I**−

**11**

^{T}/n).

**1**

^{T}

**C1**), an eigenvalue of matrix (5) is converted to the MC measuring the SA in its associated eigenvector map pattern of real number elements [24,25]. The sign of an eigenvalue indicates the nature of SA represented by its corresponding eigenvector, whereas its magnitude indicates the degree of SA. The extreme eigenvalues of matrix expression (4) determine the limits of MC, which are not necessarily ±1 [26].

#### 3.2.4. An Empirical 2010 Puerto Rico Population Density Toy Illustration

_{max}) being 1.094, and GR = 0.474]. This scenario acknowledges a well-recognized source of over-dispersion, namely correlation among observations (i.e., dependent data, autocorrelation), which is frequent in spatial statistics; for example, SA accounts for roughly half of any detected over-dispersion (e.g., extra-Poisson variation).

^{5.6361}≈ 280.4 people per square mile, with a dispersion parameter MLE of ${\widehat{\alpha}}^{-1}$ ≈ 0.0010 (i.e., nearly a Poisson RV). The eigenvector spatial filter (ESF) specification, a linear combination of a subset of the eigenvectors extracted from the adjusted SWM, describes a spatially varying mean population density by region [i.e., 283.6 (SJ), 278.6 (A), 295.7 (M), 271.6 (P), and 272.4 (C)] about a global mean of 280.3 and reduces the over-dispersion parameter to 0.0006 (even more closely resembling a Poisson RV).

^{5.6361}≈ 280.4. However, the dispersion parameter MME of ${\widehat{\alpha}}^{-1}$ is –0.0001, which would result in its being rounded to 0. Finally, the spatially varying mean population density by region is as follows: 280.1 (SJ), 280.6 (A), 280.8 (M), 280.4 (P), and 280.0 (C). These outcomes seem inferior to their preceding MLE counterparts.

## 4. Discussion: Two MESF Empirical Examples

#### 4.1. An Empirical 2010 Puerto Rico Urban Population Density Case Study

^{7.0549}≈ 1158.5, with a dispersion parameter of 0.6707 (i.e., sizeable extra-Poisson variation). Historically, the population has been concentrated on the coastal lowlands of the island, the preferred locations of the initial Spanish settlers because of convenience and accessibility, furnishing a rationale for including mean elevation as a covariate in an NB2 regression. Primarily because of the way urban areas expand (e.g., the establishment of infrastructure at the rural–urban fringe), this variable should contain PSA. In this case, executing NB2 stepwise regression estimation results in the selection of five PSA and no NSA eigenvectors from the candidate set extracted from the adjusted SWM for which MC

_{j}/MC

_{max}≥ 0.25, with subscript j denoting the jth eigenvalue. Table 2 summarizes the MLE results. It reveals that SA accounts for roughly 60% of the over-dispersion in these data, and the single covariate coupled with the constructed ESF accounts for more than three-fourths of the spatial variation in urban population density across the main island of Puerto Rico. With a final ${\widehat{\alpha}}^{-1}$ < 0.2, a dramatic shrinkage from nearly 0.7, p = 2 seems adequate for the NB model specification.

#### 4.2. An Empirical 2010 Puerto Rico Rural Population Density Illustration

^{4.1298}≈ 62.2, with a dispersion parameter of 1.8760. Rural and urban land are mutually exclusive, furnishing a rationale for including the density of rural land as a covariate. Because rural land tends to concentrate outside of urban areas, suggesting it contains PSA, and primarily because the expansion of urban land means the contraction of rural land, particularly along the rural–urban fringe, with this spatial competition suggesting it contains NSA, this variable should contain a PSA-NSA mixture. In this case, the stepwise NB2 estimation results in the selection of five eigenvectors, four with PSA and one with NSA (i.e., a mixture). Table 2 summarizes the MLE results. It reveals that SA accounts for roughly 26% of the over-dispersion in these data, and the single covariate coupled with the constructed ESF accounts for more than half of the spatial variation in rural population density across the main island of Puerto Rico. With a final ${\widehat{\alpha}}^{-1}$ > 1, p = 2 may seem inadequate for the NB model specification. Setting p = 3—with estimation executed via SAS PROC NLMIXED (see Appendix B), in which the NB3 probability density log-likelihood function was written expressing the modified NB as a general distribution [see https://stats.oarc.ucla.edu/sas/faq/how-can-i-compute-negative-binomial-models-with-random-intercepts-and-slopes-using-nlmixed/ (last accessed on 20 March 2024)], alters ${\widehat{\alpha}}^{-1}$, which remains greater than one, while increasing the pseudo-R

^{2}values when an ESF is present; these latter values decrease when p = 4, implying that p = 3 is the adequate value.

#### 4.3. An Empirical 2010 Puerto Rico Rural Population Den

## 5. Concluding Comments and Implications

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Estimation of iid NBp Parameters

## Appendix B. SAS PROC NLMIXED Code

**STEP3**) contains the needed adjusted SWM eigenvectors, the response (i.e., population counts) and offset (i.e., log-area) variables, and the covariate (i.e., the logarithm of rural land area percentage; to avoid undefined terms for zero rural land, zero was replaced with one, which is trivial because the smallest non-zero measure is 9703):

**PROC NLMIXED DATA = STEP3;**

**PARMS B0 = 0 B1 = 0 B2 = 0 B3 = 0 B4 = 0 B5 = 0 B6 = 0 A = 1.4;**

**P = 3;**

***XB = B0 + OFFSET;**

***XB = B0 + B1*PCTRL + OFFSET;**

**XB = B0 + B1*PCTRL + B2*COL10 + B3*COL12 + B4*COL15 + B5*COL20**

**+B6*COL44 + OFFSET;**

**MU = EXP(XB);**

**M = 1/A;**

**LL = LGAMMA(Y + M) − LGAMMA(Y + 1) - LGAMMA(M)**

**+Y*LOG(A*MU**(P - 1)) - (Y + M)*LOG(1 + A*MU**(P - 1));**

**MODEL Y ~ GENERAL(LL);**

**PREDICT MU OUT = MU;**

**RUN;**

**XB**differentiates between a constant mean, a bivariate regression with the rural area percentage covariate, and an MESF regression that includes this preceding covariate.**P = 3**allows the estimation of NB3; setting**P**to 2 estimates NB2, and setting it to 1 estimates NB1 (i.e., a Poisson). Finally, the temporary SAS file**WORK.MU**stores predicted counts for post-processing. The NB1 and NB2 options support comparative output checks with other software modules.

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**Figure 1.**Selected simulated (10,000 replications, n = 100) relationships between $\stackrel{\_}{y}$ and s

^{2}. Left (

**a**): iid normal RVs. Left middle (

**b**): iid NB2 RVs. Right middle (

**c**): i~id normal RVs. Right (

**d**): i~id NBp RVs.

**Figure 2.**Monte Carlo simulation experiment boxplots (as usual, asterisks denote outliers). Left (

**a**): NB2[5, 5(1 + 5/0.0476)]. Right (

**b**): NB2[5, 5(1 + 5/0.1600)].

**Figure 3.**The 1899 Puerto Rico agricultural administrative regions (AAR; denoted by red borders). Black lines denote municipio boundaries. Tertile choropleth map: black, gray, and white, respectively, denote relatively high, intermediate, and relatively low population density. Left (

**a**): the municipality-AAR choropleth map. Right (

**b**): the AAR spatial weights matrix and its relevant eigenvector.

**Figure 4.**Municipio geographic resolution 2010 population density (i.e., people per square mile) across the main island of Puerto Rico; green, yellow, and red, respectively, denote relatively low, intermediate, and high density values. Left (

**a**): urban population (extra-Poisson variation is 41,183 >> 1). Right (

**b**): rural population (extra-Poisson variation is 1427 >> 1).

**Table 1.**Summary Monte Carlo simulation experiment results for NB2[5, 5(1 + 5/0.0476)] and NB2[5, 5(1 + 5/0.1600)].

Parameter | Estimate | Standard Deviation | Range | % 0 s | % < 0 |
---|---|---|---|---|---|

using MLEs as the population parameters | |||||

μ | 4.984 | 1.410 | 0.7–11.7 | ||

1/α_{MLE} | 0.073 | 0.231 | 0.0–7.2 | 71.1 | |

1/α_{MME} | 0.113 | 0.213 | 0.0–2.7 | 5.4 | 52.2 |

using MMEs as the population parameters | |||||

μ | 5.003 | 1.714 | 0.3–14.3 | ||

1/α_{MLE} | 0.137 | 0.315 | 0.0–7.7 | 58.4 | |

1/α_{MME} | 0.190 | 0.295 | 0.0–2.8 | 1.3 | 40.6 |

Model Specification | NB2 Urban Population Density | Rural Population Density | ||||
---|---|---|---|---|---|---|

NB2 | NB3 | |||||

${\widehat{\mathit{\alpha}}}^{-1}$ | Pseudo–R^{2} | ${\widehat{\mathit{\alpha}}}^{-1}$ | Pseudo–R^{2} | ${\widehat{\mathit{\alpha}}}^{-1}$ | Pseudo–R^{2} | |

Intercept only | 0.6707 | 0 | 1.8760 | 0 | 1.8285 | 0 |

Single covariate | 0.4848 | 0.18 | 1.3964 | 0.48 | 1.1847 | 0.48 |

Covariate + ESF | 0.2021 | 0.79 | 1.0399 | 0.52 | 1.0065 | 0.65 |

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**MDPI and ACS Style**

Griffith, D.A.
Some Comments about the p-Generalized Negative Binomial (NBp) Model. *AppliedMath* **2024**, *4*, 731-742.
https://doi.org/10.3390/appliedmath4020039

**AMA Style**

Griffith DA.
Some Comments about the p-Generalized Negative Binomial (NBp) Model. *AppliedMath*. 2024; 4(2):731-742.
https://doi.org/10.3390/appliedmath4020039

**Chicago/Turabian Style**

Griffith, Daniel A.
2024. "Some Comments about the p-Generalized Negative Binomial (NBp) Model" *AppliedMath* 4, no. 2: 731-742.
https://doi.org/10.3390/appliedmath4020039