Overdispersion and Poisson Regression

“More extensive discussions of heterogeneity can be found in the influential work Nagin and Land (1993) and in several subsequent papers and books (e.g., D’Unger et al. 1998; Nagin 2005 ).”

“Berk and MacDonald (2010) test the effects of extra police deployment in downtown Los Angeles”

“Negative binomial models do not solve model specification concerns (Berk and MacDonald, 2008).”

“We used Poisson random-effects regression models to estimate the impact of violation compliance and renovation permit on crime outcomes (Y), while controlling for demographic variables (negative binomial models were first attempted but would not converge) [28].”

“According to Berk and MacDonald (2008), overdispersion in a count distribution is quite complex and that omitted variable bias is most likely the major driving influence that leads to this type of empirical distribution.”

“Simulation techniques are the most appropriate methods to accomplish this objective [19] because simulation allows one to vary the model conditions and to know what the values of the true parameters are.”

“In cases of overdispersion, negative-bionimal regression analysis produces more robust results compared to poisson regression (for a discussion of issues related to overdispersion, see Berk & MacDonald, 2008).”

“For criminologists, one wants to know what predicts the number of crime counts assuming the probability of outcomes occurs via a Poisson distribution (Berk and MacDonald 2008).”

“Recall that the Poisson probability function is with the mathematical form [28]:”

“For expositions of the Poisson model family tailored to criminology, see Osgood (2000) and Berk and MacDonald (2008).”

“…included, plus the fact that the Poisson and negative binomial results were vastly different, we opted to use negative binomial regression12 for all models (but see Berk & MacDonald, 2008, for caveats regarding substituting the negative binomial for the Poisson in the presence of overdispersion).”

“The association between exposure variables and number of cigarettes smoked was evaluated using a negative binomial regression model because of the dispersion of the number of cigarettes variable [39].”

“The quasi-Poisson model offers an ad hoc fi x for small amounts of overdispersion (Berk and MacDonald 2008).”

“We use a negative binomial rather than poisson regression model, as the events (denial of service attacks) are not independent, rather there is a simple trend to the data [50][4][11].”

“If the result is greater than one, the data fail to fit the Poisson distribution assumption due to over-dispersion (Berk and MacDonald, 2008).”

“In this model, yi conditional on the covariates has a Poisson distribution with mean li (Berk & MacDonald, 2008: p. 277; Rabe-Hesketh & Skrondal, 2012a: 376).”

“Negative binomial regression is often preferred when count data are overdispersed and the datagenerating mechanisms conform to a negative binomial distribution (Berk & MacDonald, 2008; Gardner, Mulvey, & Shaw, 1995; Piza, 2012).”

“Echoing Berk et al [24], “one risks an arbitrary correction leading to arbitrary results.”

“…of the linkage variables, the 3FE gravity model is expected to be superior in accounting for the heteroskedasticity (overdispersion) present in the data (Zietz, 2001) and to be helpful in removing a potentially important source of omitted-variable bias (Congdon, 1993; Berk and MacDonald, 2008).”

“If the conditional distribution of the outcome variable is over-dispersed, the confidence intervals for NBRM are likely to be narrower than those for PRM (Berk and MacDonald 2008).”

“…a Poisson distribution (or, in the case of overdispersion when the data contains a large number of zeros, a negative binomial (NB) distribution (Berk & MacDonald, 2008; Payne et al., 2017) or a zero‐inflated Poisson (ZIP) distribution (Lewsey & Thomson, 2004) or a zero‐inflated negative…”

“**p< 0.01. alled robust) standard errors instead to see whether the gammaistribution-assumed random intercepts in the negative binomial odels drove our reported results (Berk & MacDonald, 2008; Gould, 011; Rabe-Hesketh & Skrondal, 2012).”

“Given this measure is a count variable, with a skewed distribution and over-dispersion, a Poisson model is employed (Berk & MacDonald, 2008; MacDonald & Lattimore, 2010).”

“Accident data observations on road sections are often characterised by very large deviations, which are referred to as over-dispersion (Berk and MacDonald, 2008; Cox, 1983; Dean and Lawless, 1989; Gschloessl and Czado, 2006; Hauer, 2001; Karlis and Meligkotsidou, 2005).”

“If the count data outcomes are designed as we recommend in this review (see ‘Design of the count data outcome’) [12], then model checking will often show that the underlying assumptions of these generalised linear models are not fulfilled (Table 1) [13].”

“A final issue is that groups or clusters of potential help desk callers may have purchased extended warranty and maintenance contracts for the computer or electronics equipment that they have, which would cause dependence within counts [22, 168].”

“Moreover, to ensure that results aren’t sensitive to the assumptions of the Negative binomial model, with reference on Wooldridge (2010) and Berk and MacDonald (2008), we also present the results of corresponding Poisson regression models with robust standard errors in the appendix.”

“Negative binomial regression models were used to assess the association between ISD and DPs, NPs, and NFTs, which displayed significantly skewed distributions with high standard deviations relative to the mean [31].”

“First, tests for overdispersion do not distinguish between overdispersion and misspecification (see Berk and MacDonald (2008); Blackburn (2015).”

“The number of weekly dengue cases was assumed to follow an over-dispersed Poisson distribution [34].”

“We use this approach because estimates from a Poisson regression with robust standard errors are more robust than negative binomial regression (Berk & MacDonald, 2008; Wooldridge, 2010).”

“…didactic articles (Coxe, West, & Aiken, 2009), tutorials in substantive areas (Atkins, Baldwin, Zheng, Gallop, & Neighbors, 2013; Atkins & Gallop, 2007; Berk & McDonald, 2008; Xie, Tao, McHugo, & Drake, 2013) and entire textbooks (Cameron & Trivedi, 2013; Hilbe, 2011, 2014) devoted to the issue.”

“In such cases, it can be advisable to use statistical procedures that specifically allow for modeling of data with skewed distributions (e.g., Berk & MacDonald, 2008; Coxe, West, & Aiken, 2009; Hilbe, 2008).”

“Overdispersion arises because observations are correlated (Berk and MacDonald 2007): in the case of marine mammals, the same process (for example epizootics (Carpenter 2013; Di Guardo 2012)) affecting a demographic unit may induce a multiple strandings event, thus violating the implicit assumption…”

“As such, we used robust standard errors to adjust for overdispersion in our Poisson regressions (Berk & MacDonald, 2008) and report the results of negative binomial regressions in appendix B. ratings, .806 for frisk lawfulness ratings, and .730 for search lawfulness ratings.”

“In certain circumstances, it is found that the empirical variance is greater than the mean, known as overdispersion [17, 18].”

“…total number of crashes (Ma and Kockelman, 2006; Park and Lord, 2007; Ma et al., 2008; Yannis et al., 2008; Aguero-Valverde and Jovanis, 2009; El-Basyouny and Sayed, 2009; Ye et al., 2009), which would lead to excess variation around fitted values that cannot be captured (Berk and MacDonald, 2007).”

“To accommodate this overdispersion of the data, we ran a negative binomial regression model (Berk & MacDonald, 2008; Cameron & Trivedi, 1990; Cox, 1983) using the R package msme (v0.”

“Overdispersion refers to excess variation when the systematic structure of the model is correct (Berk 2007).”

“2018; Jaya and Folmer 2020, 2021a, b): 5 A negative binomial distribution is appropriate when there is large overdispersion of zeros (Berk and MacDonald 2008; Payne et al.”

“We later discuss how departures from this Poisson assumption, in particular how crime tends to be over-dispersed compared to the Poisson distribution (Berk and MacDonald 2008), will subsequently affect the proposed test statistic.”

“The standard errors for the models we present were adjusted for model deviance, and thus take into account the under- or overdispersion in any given model [48,49].”

“…overdispersion is found, it can have a number of causes: outliers, model misspecification (e.g., omission of interaction terms or cur-vilinearity in the model), heteroskedasticity, outliers, and unmeasured sources of heterogeneity (Cameron and Trivedi 1998; Hilbe 2007; Berk and MacDonald 2008).”

“Since there was evidence of over-dispersion in our count data (too many zeros), we opted to use a negative binomial distribution (Berk & MacDonald, 2007).”

“The remaining repeats after fitting the Poisson distribution, which are called overdispersion (Berk & MacDonald, 2008; Osgood, 2000; Paternoster & Brame, 1997), can be attributed to nonrandom factors; however, the repeat victimization fitting the Poisson distribution should still be considered as…”

“See Berk & Macdonald (2008) for a more general discussion of why Poisson models can be appropriate in the presence of overdispersion. is focused on level differences in arrests between races over the life course, not on whether time-varying covariates explain variation in the age–crime curve (e.g.,…”

“Overdispersion occurs when the observed variance is higher than the variance of a theoretical model [39].”

“The negative binomial is a frequently used alternative count model that will be more efficient if the underlying data generating process is negative binomial rather than Poisson, though it will not solve potential bias or inconsistency in our estimators (Berk and MacDonald 2008).”

“So far, PR has been successfully applied to many real-world scenarios, such as crowd counting [9]–[11], age estimation [12], ecology related organisms [13], criminology applications [14], survival analysis [15], [16], etc.”

“It is more likely the Poisson distribution assumption is not reasonable, and so the extra variance (likely from shootings being overdispersed, Berk and MacDonald, 2008) is what accounts for different in the overall rates over time.”

“We use a negative binomial regression model to account for the overdispersion of the count dependent variable (Berk & MacDonald, 2008).”

“JRHS 2017; 17(3): e00392 therefore it is more suitable for health system to enhance index of constancy as much as possible.”

“Then, we analyzed each vegetation type (secondary vegetation and agricultural fields) with a Poisson regression (Berk and MacDonald 2008) to determine which spatial attributes of the vegetation cover (within our 100-m radius) explained the number of carcasses found.”

“We checked the model assumptions and found that overdispersion [104] was not an issue (dispersion parameter 0.”

“According to Berk and MacDonald (2008), when overdispersion exists the reliance on the negative binomial distribution to “correct” for the mean-variance inequality would not address the specification errors due to the incomplete (i.e., misspecified) models.”

“18 The adjusted incident rate ratios (IRRs) were reported for each regression.”