Overdispersed Count Data Research Articles

Cell phone use while driving laws and motor vehicle driver fatalities: differences in population subgroups and location

PurposeResearch suggests that cell phone use while driving laws are associated with lower driver fatalities. This study seeks to determine whether this relationship is modified by driver age (16–24, 25–39, 40–59, ≥60), sex (male, female), race/ethnicity (white non-Hispanic, white Hispanic, black non-Hispanic, other), or rurality (rural, urban). MethodsFatality Analysis Reporting System data were merged with state legislation (2000–2014). The exposure was the type of legislation in effect. The outcome was non–alcohol-related driver fatalities by state-quarter-year. Incident rate ratios were estimated using generalized Poisson mixed regression for overdispersed count data with robust standard errors. ResultsAmongst 190,544 drivers, compared to periods without bans, universal hand-held calling bans were associated with 10% (adjusted incident rate ratio = 0.90, 95% confidence interval 0.84, 0.96) lower non–alcohol-related driver fatalities overall and up to 13% lower fatalities across all age groups and sexes but not for race/ethnicity or rurality. When comparing state-quarter-years with bans to those without, universal texting bans were not associated with lower fatalities overall or for any demographic group. ConclusionsThe relationships between cell phone laws and non–alcohol-related driver fatalities are modified by driver demographics, particularly for universal hand-held bans. Universal hand-held calling bans may benefit more types of drivers compared to texting bans.

Racial Difference in Cerebral Microbleed Burden Among a Patient Population in the Mid-South United States

BackgroundAlthough intracerebral hemorrhage (ICH) is more common among African-Americans, data on the burden of cerebral microbleeds (CMBs) among different races is limited. The purpose of this study is to compare the number, associated factors, and topography of CMBs between African-American and Caucasian populations in the Mid-South United States. MethodsUsing natural language processing, we extracted all brain MRIs performed during a 2-year period (2012-2013) when the report indicated the presence of CMB. All the extracted MRI studies were evaluated for number and location of CMBs, prior stroke, and deep white matter lesion. Negative binomial regression was used to model for the overdispersed count data. ResultsA total 167 patients (mean age was 63 ± 15 years, 49% men, 77% African-American, median CMB count: 8) with 1 or more CMBs on their brain MRI were included in this study. There was no significant difference between the 2 groups in terms of CMB locations (P = .086), but there was a significant difference between African-American and Caucasian patients in terms of number of CMBs (16.5 ± 18 versus 6.5 ± 5.5, P < .001). The prevalence of multiple CMBs (CMBs ≥ 5) was similar among African-Americans and Caucasians (72% versus 55%, P = .062). After adjusting for potential confounders, the African-American race was not independently associated with a higher CMB burden (P = .073). ConclusionAfrican-American race was not independently associated with a higher rate of CMB burden when compared to Caucasians after adjusting for potential confounders. We also did not observe a significant racial difference regarding the location and prevalence of multiple CMBs (CMBs ≥ 5).

Alcohol retail sales licenses and sexually transmitted infections in Texas counties, 2008-2015

ABSTRACTBackground: Research suggests that reduced retail alcohol outlet density may be associated with lower prevalence of HIV and other sexually transmitted infections (STIs). On-premise sale of alcohol for immediate consumption is theorized as increasing social interactions that can lead to sexual encounters. Objective: We examined associations between on- and off-premise retail alcohol sales licenses and number of newly diagnosed HIV and STI cases in Texas counties. Methods: Retail alcohol sales license data were obtained from the Texas Alcoholic Beverage Commission. HIV and bacterial STI data were obtained from the Texas Department of State Health Services. Associations between retail alcohol sales licenses and STIs were estimated using spatial linear models and Poisson mixed effects models for over-dispersed count data. Results: Adjusting for county-specific confounders, there was no evidence of residual spatial correlation. In Poisson models, each additional on-premise (e.g., bar and restaurant) alcohol license per 10,000 population in a county was associated with a 1.5% increase (95% CI: 0.4%, 2.6%) in the rate of HIV and a 2.4% increase (95% CI: 1.9%, 3.0%) in the rate of bacterial STIs, adjusting for potential confounders. In contrast, number of off-premise licenses (e.g., take-out stores) was inversely associated with the incidence of STI and HIV, although the association with HIV was not statistically significant. Conclusions: This study adds to the limited literature on the association between retail alcohol availability and STIs. Additional research is needed on the role of alcohol availability (and policies affecting availability) in the spread of HIV and other STIs.

On Zero-Modified Poisson-Sujatha Distribution to Model Overdispersed Count Data

In this paper we propose the zero-modified Poisson-Sujatha distribution as an alternative to model overdispersed count data exhibiting inflation or deflation of zeros. It will be shown that the zero modification can be incorporated by using the zero-truncated Poisson-Sujatha distribution. A simple reparametrization of the probability function will allow us to represent the zero-modified Poisson-Sujatha distribution as a hurdle model. This trick leads to the fact that proposed model can be fitted without any previously information about the zero modification present in a given dataset. The maximum likelihood theory will be used for parameter estimation and asymptotic inference concerns. A simulation study will be conducted in order to evaluate some frequentist properties of the developed methodology. The usefulness of the proposed model will be illustrated using real datasets of the biological sciences field and comparing it with other models available in the literature.

Pǿlya–Aeppli regression model for overdispersed count data

The log-linear Poisson model, characterized by linear variance function and a logarithmic relation between means and covariates, embedded in the exponential family regression framework provided by generalized linear models (GLM) is still the standard approach for analyzing count data responses with regression models. In practice, however, count data are often overdispersed and, thus, not conducive to Poisson regression. Therefore, the main goal of this article is to introduce a log-linear model based on the P[Formula: see text]lya–Aeppli (PA) distribution, which is an extension of the Poisson distribution by including a dispersion parameter ρ, to address the problem of overdispersion. Maximum likelihood (ML) estimation procedure is discussed as well as a test for determining the need for a PA regression over a standard Poisson regression. In addition, a simple EM-type algorithm for iteratively computing ML estimates is presented. In order to study departures from the error assumption as well as the presence of outliers, we perform residual analysis based on the standardized Pearson residuals. Furthermore, for different parameter settings and sample sizes, various simulations are performed. Finally, we also illustrated the new method on three real datasets, two of them are from biological researches and the other is from a violence study.

Developing a Random Parameters Negative Binomial-Lindley Model to analyze highly over-dispersed crash count data

The existence of preponderant zero crash sites and/or sites with large crash counts can present challenges during the statistical analysis of crash count data. Additionally, unobserved heterogeneity in crash data due to the absence of important variables could negatively impact the estimated model parameters. The traditional negative binomial (NB) model with fixed parameters might not adequately handle highly over-dispersed data or unobserved heterogeneity. Many research efforts that have involved the negative binomial–Lindley (NB-L) model or the random parameters negative binomial (RPNB) model, for example, have attempted to improve the inference of estimated coefficients by explicitly accounting for extra variation in crash data. The NB-L is a mixed modeling approach which provides flexibility to account for additional dispersion in data. The RP modeling approach accommodates the effect of unobserved variables by allowing the model parameters to vary from one observation to another. The following study proposes a combination of these models – the random parameters NB-L (RPNB-L) generalized linear model (GLM) – to account for underlying heterogeneity and address excess over-dispersion. The results show that the RPNB-L model not only provides a superior goodness-of-fit (GOF) with the sample data, but also offers a better understanding about the effects of potential contributing factors. The paper uses the Bayesian framework to provide a strategy for eliminating the potential for poor mixing in the Markov Chain Monte Carlo (MCMC) chains during the estimation of the RPNB-L model.

Models for analyzing zero-inflated and overdispersed count data: an application to cigarette and marijuana use.

This paper describes different methods for analyzing counts and illustrates their use on cigarette and marijuana smoking data. The Poisson, zero-inflated Poisson (ZIP), hurdle Poisson (HUP), negative binomial (NB), zero-inflated negative binomial (ZINB) and hurdle negative binomial (HUNB) regression models are considered. The different approaches are evaluated in terms of the ability to take into account zero-inflation (extra zeroes) and overdispersion (variance larger than expected) in count outcomes, with emphasis placed on model fit, interpretation, and choosing an appropriate model given the nature of the data. The illustrative data example focuses on cigarette and marijuana smoking reports from a study on smoking habits among youth e-cigarette users with gender, age, and e-cigarette use included as predictors. Of the 69 subjects available for analysis, 36% and 64% reported smoking no cigarettes and no marijuana, respectively, suggesting both outcomes might be zero-inflated. Both outcomes were also overdispersed with large positive skew. The ZINB and HUNB models fit the cigarette counts best. According to goodness-of-fit statistics, the NB, HUNB, and ZINB models fit the marijuana data well, but the ZINB provided better interpretation. In the absence of zero-inflation, the NB model fits smoking data well, which is typically overdispersed. In the presence of zero-inflation, the ZINB or HUNB model is recommended to account for additional heterogeneity. In addition to model fit and interpretability, choosing between a zero-inflated or hurdle model should ultimately depend on the assumptions regarding the zeros, study design, and the research question being asked. Count outcomes are frequent in tobacco research and often have many zeros and exhibit large variance and skew. Analyzing such data based on methods requiring a normally distributed outcome are inappropriate and will likely produce spurious results. This study compares and contrasts appropriate methods for analyzing count data, specifically those with an over-abundance of zeros, and illustrates their use on cigarette and marijuana smoking data. Recommendations are provided.

Power and sample size calculation incorporating misspecifications of the variance function in comparative clinical trials with over-dispersed count data.

Over-dispersed count data are frequently observed in clinical trials where the primary endpoint is occurrence of clinical events. Sample sizes of comparative clinical trials with these data are typically calculated under negative binomial models or quasi-Poisson models with specified variance functions, or under the assumption that the specified "working" variance functions are correctly specified. In this article, we propose a sample size formula anticipating misspecifications of the working variance function. We derived a method based on the asymptotic distribution of a Wald test statistic with a sandwich-type robust variance estimator under quasi-Poisson models. Under misspecifications of the working variance function, the asymptotic variance of the estimator of the treatment effect is expressed as a form involving both true and working variance functions. Our sample size formula includes several existing formulas as special cases when the working variance function is correctly specified as the true variance function. We also consider a sensitivity analysis for possible misspecifications of the "true" variance function when estimating sample sizes using our formula. A simulation study demonstrated the adequacy of our formulas in finite sample size settings. An application to a clinical trial to evaluate the treatment effect on prevention of COPD exacerbation is provided.

Game Changers: Detecting Shifts in Overdispersed Count Data

In this paper, I introduce a Bayesian model for detecting changepoints in a time series of overdispersed counts, such as contributions to candidates over the course of a campaign or counts of terrorist violence. To avoid having to specify the number of changepoint ex ante, this model incorporates a hierarchical Dirichlet process prior to estimate the number of changepoints as well as their location. This allows researchers to discover salient structural breaks and perform inference on the number of such breaks in a given time series. I demonstrate the usefulness of the model with applications to campaign contributions in the 2012 U.S. Republican presidential primary and incidences of global terrorism from 1970 to 2015.

On the zero-modified Poisson–Shanker regression model and its application to fetal deaths notification data

In this paper, we propose the zero-modified Poisson–Shanker regression model as an alternative to model overdispersed count data exhibiting inflation or deflation of zeros in the presence of covariates. The zero modification has been incorporated using the zero-truncated Poisson–Shanker distribution. The zero-modified Poisson–Shanker distribution has been written as a hurdle model using a simple reparameterization of the probability function which leads to the fact that the proposed model can be fitted without any previous information about the zero modification present in a given dataset. The standard Bayesian procedures have been considered for estimation and inference. A simulation study has been presented to illustrate the performance of the developed methodology. The usefulness of the proposed model has been evaluated using a real dataset on fetal deaths notification data in Bahia State, Brazil. A sensitivity study to detect points which can influence the parameter estimates has been performed using Kullback–Leibler divergence measure. The randomized quantile residuals have been considered for the model validation issue. General comparison of the proposed model with some well-known discrete distributions has been provided.

The Socio-Economic Determinants of Terrorism: A Bayesian Model Averaging Approach

This paper introduces model uncertainty into the empirical study of the determinants of terrorism at country level. This is done by adopting a Bayesian model averaging approach and accounting for the over-dispersed count data nature of terrorist attacks. Both a broad measure of terrorism and incidents per capita have been analyzed. Our results suggest that, among the set of regressors considered, those reflecting labor market conditions and economic prospects tend to receive high posterior inclusion probabilities. These findings are robust to changes in the model specification and sample composition and are not meaningfully affected by the generalized linear regression model applied. Evidence of a geographically heterogeneous relationship between terrorism and its determinants is also provided.

Quantile regression for overdispersed count data: a hierarchical method

Generalized Poisson regression is commonly applied to overdispersed count data, and focused on modelling the conditional mean of the response. However, conditional mean regression models may be sensitive to response outliers and provide no information on other conditional distribution features of the response. We consider instead a hierarchical approach to quantile regression of overdispersed count data. This approach has the benefits of effective outlier detection and robust estimation in the presence of outliers, and in health applications, that quantile estimates can reflect risk factors. The technique is first illustrated with simulated overdispersed counts subject to contamination, such that estimates from conditional mean regression are adversely affected. A real application involves ambulatory care sensitive emergency admissions across 7518 English patient general practitioner (GP) practices. Predictors are GP practice deprivation, patient satisfaction with care and opening hours, and region. Impacts of deprivation are particularly important in policy terms as indicating effectiveness of efforts to reduce inequalities in care sensitive admissions. Hierarchical quantile count regression is used to develop profiles of central and extreme quantiles according to specified predictor combinations.

Purely sequential bounded-risk point estimation of the negative binomial mean under various loss functions: one-sample problem

A negative binomial (NB) distribution is useful to model over-dispersed count data arising from agriculture, health, and pest control. We design purely sequential bounded-risk methodologies to estimate an unknown NB mean $$\mu (>0)$$ under different forms of loss functions including customary and modified Linex loss as well as squared error loss. We handle situations when the thatch parameter $$\tau (>0)$$ may be assumed known or unknown. Our proposed methodologies are shown to satisfy properties including first-order asymptotic efficiency and first-order asymptotic risk efficiency. Summaries are provided from extensive sets of simulations showing encouraging performances of the proposed methodologies for small and moderate sample sizes. We follow with illustrations obtained by implementing estimation strategies using real data from statistical ecology: (1) weed count data of different species from a field in Netherlands and (2) count data of migrating woodlarks at the Hanko bird sanctuary in Finland.

Gas dynamic analogous exposure approach to interaction intensity in multiple-vehicle crash analysis: Case study of crashes involving taxis

Exposure is a frequency measure of being in situations in which crashes could occur. In modeling multiple-vehicle crash frequency, traditional exposure measures, such as vehicle kilometrage and travel time, may not be sufficiently representative because they may include situations in which vehicles rarely meet each other and multiple-vehicle crashes can never happen. The meeting frequency of vehicles should be a better exposure measure in such cases. This study aims to propose a novel Gas Dynamic Analogous Exposure (GDAE) to model multiple-vehicle crash frequency. We analogize the meeting frequency of vehicles with the meeting frequency of gas molecules because both systems consider the numbers of the meetings of discrete entities. A meeting frequency function of vehicles is derived based on the central idea of the classical collision theory in physical chemistry with consideration of constrained vehicular movement by the road alignments. The GDAE is then formulated on the basis of the major factors that contribute to the meeting frequency of vehicles. The proposed GDAE is a more representative proxy exposure measure in modeling of multiple-vehicle crash frequency because it further investigates and provides insight into the physics of the vehicle meeting mechanism. To demonstrate the applicability of the GDAE, zonal crash frequency models are constructed on the basis of multiple-vehicle crashes involving taxis in 398 zones of Hong Kong in 2011. The GDAE outperforms the conventional time exposure in multiple-vehicle crash modeling. To account for any unobservable heterogeneity and to cope with the over-dispersed count data, a random-parameter negative binomial model is established. Explanatory factors that contribute to the zonal multiple-vehicle crash risk involving taxis are identified. The proposed GDAE is a promising exposure measure for modeling multiple-vehicle crash frequency.

Generalized Linear Latent Variable Models for Multivariate Count and Biomass Data in Ecology

In this paper we consider generalized linear latent variable models that can handle overdispersed counts and continuous but non-negative data. Such data are common in ecological studies when modelling multivariate abundances or biomass. By extending the standard generalized linear modelling framework to include latent variables, we can account for any covariation between species not accounted for by the predictors, notably species interactions and correlations driven by missing covariates. We show how estimation and inference for the considered models can be performed efficiently using the Laplace approximation method and use simulations to study the finite-sample properties of the resulting estimates. In the overdispersed count data case, the Laplace-approximated estimates perform similarly to the estimates based on variational approximation method, which is another method that provides a closed form approximation of the likelihood. In the biomass data case, we show that ignoring the correlation between taxa affects the regression estimates unfavourably. To illustrate how our methods can be used in unconstrained ordination and in making inference on environmental variables, we apply them to two ecological datasets: abundances of bacterial species in three arctic locations in Europe and abundances of coral reef species in Indonesia. Supplementary materials accompanying this paper appear on-line.

A generalised NGINAR(1) process with inflated‐parameter geometric counting series

SummaryIn this paper we propose a new stationary first‐order non‐negative integer valued autoregressive process with geometric marginals based on a generalised version of the negative binomial thinning operator. In this manner we obtain another process that we refer to as a generalised stationary integer‐valued autoregressive process of the first order with geometric marginals. This new process will enable one to tackle the problem of overdispersion inherent in the analysis of integer‐valued time series data, and contains the new geometric process as a particular case. In addition various properties of the new process, such as conditional distribution, autocorrelation structure and innovation structure, are derived. We discuss conditional maximum likelihood estimation of the model parameters. We evaluate the performance of the conditional maximum likelihood estimators by a Monte Carlo study. The proposed process is fitted to time series of number of weekly sales (economics) and weekly number of syphilis cases (medicine) illustrating its capabilities in challenging cases of highly overdispersed count data.

Conditional vs marginal estimation of the predictive loss of hierarchical models using WAIC and cross-validation

The predictive loss of Bayesian models can be estimated using a sample from the full-data posterior by evaluating the Watanabe-Akaike information criterion (WAIC) or using an importance sampling (ISCVL) approximation to leave-one-out cross-validation loss. With hierarchical models the loss can be specified at different levels of the hierarchy, and in the published literature, it is routine for these estimators to use the conditional likelihood provided by the lowest level of model hierarchy. However, the regularity conditions underlying these estimators may not hold at this level, and the behaviour of conditional-level WAIC as an estimator of conditional-level predictive loss must be determined on a case-by-case basis. Conditional-level ISCVL does not target conditional-level predictive loss and instead is an estimator of marginal-level predictive loss. Using examples for analysis of over-dispersed count data, it is shown that conditional-level WAIC does not provide a reliable estimator of its target loss, and simulations show that it can favour the incorrect model. Moreover, conditional-level ISCVL is numerically unstable compared to marginal-level ISCVL. It is recommended that WAIC and ISCVL be evaluated using the marginalized likelihood where practicable and that the reliability of these estimators always be checked using appropriate diagnostics.

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

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.

A Dirichlet-tree multinomial regression model for associating dietary nutrients with gut microorganisms.

Understanding the factors that alter the composition of the human microbiota may help personalized healthcare strategies and therapeutic drug targets. In many sequencing studies, microbial communities are characterized by a list of taxa, their counts, and their evolutionary relationships represented by a phylogenetic tree. In this article, we consider an extension of the Dirichlet multinomial distribution, called the Dirichlet-tree multinomial distribution, for multivariate, over-dispersed, and tree-structured count data. To address the relationships between these counts and a set of covariates, we propose the Dirichlet-tree multinomial regression model for which we develop a penalized likelihood method for estimating parameters and selecting covariates. For efficient optimization, we adopt the accelerated proximal gradient approach. Simulation studies are presented to demonstrate the good performance of the proposed procedure. An analysis of a data set relating dietary nutrients with bacterial counts is used to show that the incorporation of the tree structure into the model helps increase the prediction power.

Variational Approximations for Generalized Linear Latent Variable Models

ABSTRACTGeneralized linear latent variable models (GLLVMs) are a powerful class of models for understanding the relationships among multiple, correlated responses. Estimation, however, presents a major challenge, as the marginal likelihood does not possess a closed form for nonnormal responses. We propose a variational approximation (VA) method for estimating GLLVMs. For the common cases of binary, ordinal, and overdispersed count data, we derive fully closed-form approximations to the marginal log-likelihood function in each case. Compared to other methods such as the expectation-maximization algorithm, estimation using VA is fast and straightforward to implement. Predictions of the latent variables and associated uncertainty estimates are also obtained as part of the estimation process. Simulations show that VA estimation performs similar to or better than some currently available methods, both at predicting the latent variables and estimating their corresponding coefficients. They also show that VA estimation offers dramatic reductions in computation time particularly if the number of correlated responses is large relative to the number of observational units. We apply the variational approach to two datasets, estimating GLLVMs to understanding the patterns of variation in youth gratitude and for constructing ordination plots in bird abundance data. R code for performing VA estimation of GLLVMs is available online. Supplementary materials for this article are available online.