Count Data Models Research Articles

Estimation of Effect Heterogeneity in Rare Events Meta-Analysis

The paper outlines several approaches for dealing with meta-analyses of count outcome data. These counts are the accumulation of occurred events, and these events might be rare, so a special feature of the meta-analysis is dealing with low counts including zero-count studies. Emphasis is put on approaches which are state of the art for count data modelling including mixed log-linear (Poisson) and mixed logistic (binomial) regression as well as nonparametric mixture models for count data of Poisson and binomial type. A simulation study investigates the performance and capability of discrete mixture models in estimating effect heterogeneity. The approaches are exemplified on a meta-analytic case study investigating the acceptance of bibliotherapy.

Flexible models for non-equidispersed count data: comparative performance of parametric models to deal with underdispersion

Flexible models for non-equidispersed count data: comparative performance of parametric models to deal with underdispersion

Bias in instrumental-variable estimators of fixed-effect models for count data

This note looks at the properties of instrumental-variable estimators of models for non-negative outcomes in the presence of individual effects. We show that fixed-effect versions of the estimators of Mullahy (1997) and Windmeijer and Santos Silva (1997) are inconsistent under conventional asymptotics, in general, and that inference based on them in long panels requires bias correction. Such corrections are derived and their effectiveness is investigated in numerical experiments. Consistent estimation in short panels is nonetheless possible in the setting underlying Mullahy’s (1997) approach using a differencing strategy along the lines of Wooldridge (1997) and Windmeijer (2000).

Zero-inflated Modified Borel-Tanner Regression Model for Count Data

By starting from the one-parameter Modified Borel-Tanner distribution proposed recently in the statistic literature, we introduce the zero-inflated Modified Borel-Tanner distribution. Additionally, on the basis of the proposed zero-inflated distribution, a novel zero-inflated regression model is proposed, which is quite simple and may be an interesting alternative to usual zero-inflated regression models for count data. The parameters of the proposed model are estimated by Maximum Likelihood Estimation technique. To check the potentiality of the zero inflated Modified Borel-Tanner regression, an application to the count of infected blood cells is taken. The results suggest that the new zero inflated Modified Borel-Tanner regression is more appropriate to model these count data than other familiar zero-inflated (or not) regression models commonly used in practice.

Comparison of Different Count Models for Investigation of Some Environmental Factors Affecting Stillbirth in Holsteins

Background: The objective of this study is comparing different count data models for stillbirth data. In modeling this type of data, Poisson regression or alternative models can be preferred. Methods: The poisson, negative binomial, zero-inflated poisson, zero-inflated negative binomial, poisson-logit hurdle and negative binomial-logit hurdle regressions were compared and used to examine the effects of the gender, parity and herd-year-season independent variables on stillbirth. Furthermore, the Log-Likelihood statistics, Akaike Information Criteria, Bayesian Information Criteria and rootogram graphs were used as comparison criteria for performance of the models. According to these criteria, Negative Binomial-Logit Hurdle Regression model was chosen as the best model. Result: The parameter estimates obtained by Negative Binomial-Logit Hurdle Regression model in relation to the effects of the gender, parity and herd-year-season independent variables on stillbirth were found to be significant (p less than 0.01). It was found that while stillbirth incidence was higher in males than females, it was found to decrease as the parity increased. As a result, the Negative Binomial Logit Hurdle model was found the best model for stillbirth count data with overdispersion.

COVID-19 infections in day care centres in Germany: social and organisational determinants of infections in children and staff in the second and third wave of the pandemic

BackgroundDuring the SARS-CoV-2 pandemic, German early childhood education and care (ECEC) centres organised children’s attendance in different ways, they reduced opening hours, provided emergency support for a few children, or closed completely. Further, protection and hygiene measures like fixed children-staff groups, ventilation and surface disinfection were introduced in ECEC centres. To inform or modify public health measures in ECEC, we investigate the occurrence of SARS-CoV-2 infections among children and staff in ECEC centres in light of social determinants (i.e. the socioeconomic status of the children) and recommended structural and hygiene measures. We focus on the question if the relevant factors differ between the 2nd (when no variant of concern (VOC) circulated) and the 3rd wave (when VOC B.1.1.7 (Alpha) predominated).MethodsBased on panel data from a weekly online survey of ECEC centre managers (calendar week 36/2020 to 22/2021, ongoing) including approx. 8500 centres, we estimate the number of SARS-CoV-2 infections in children and staff using random-effect-within-between (REWB) panel models for count data in the 2nd and 3rd wave.ResultsECEC centres with a high proportion of children with low socioeconomic status (SES) have a higher risk of infections in staff and children. Strict contact restrictions between groups like fixed group assignments for children and fixed staff assignments to groups prevent infections. Both effects tend to be stronger in the 3rd wave.ConclusionECEC centres with a large proportion of children with a low SES background and lack of using fixed child/staff cohorts experience higher COVID-19 rates. Over the long run, centres should be supported in maintaining recommended measures. Preventive measures such as the vaccination of staff should be prioritised in centres with large proportions of low SES children.

A Noncentral Lindley Construction Illustrated in an INAR(1) Environment

This paper proposes a previously unconsidered generalization of the Lindley distribution by allowing for a measure of noncentrality. Essential structural characteristics are investigated and derived in explicit and tractable forms, and the estimability of the model is illustrated via the fit of this developed model to real data. Subsequently, this model is used as a candidate for the parameter of a Poisson model, which allows for departure from the usual equidispersion restriction that the Poisson offers when modelling count data. This Poisson-noncentral Lindley is also systematically investigated and characteristics are derived. The value of this count model is illustrated and implemented as the count error distribution in an integer autoregressive environment, and juxtaposed against other popular models. The effect of the systematically-induced noncentrality parameter is illustrated and paves the way for future flexible modelling not only as a standalone contender in continuous Lindley-type scenarios but also in discrete and discrete time series scenarios when the often-encountered equidispersed assumption is not adhered to in practical data environments.

A Flexible Mixed Model for Clustered Count Data

Clustered count data are commonly modeled using Poisson regression with random effects to account for the correlation induced by clustering. The Poisson mixed model allows for overdispersion via the nature of the within-cluster correlation, however, departures from equi-dispersion may also exist due to the underlying count process mechanism. We study the cross-sectional COM-Poisson regression model—a generalized regression model for count data in light of data dispersion—together with random effects for analysis of clustered count data. We demonstrate model flexibility of the COM-Poisson random intercept model, including choice of the random effect distribution, via simulated and real data examples. We find that COM-Poisson mixed models provide comparable model fit to well-known mixed models for associated special cases of clustered discrete data, and result in improved model fit for data with intermediate levels of over- or underdispersion in the count mechanism. Accordingly, the proposed models are useful for capturing dispersion not consistent with commonly used statistical models, and also serve as a practical diagnostic tool.

Quasi-Negative Binomial: Properties, Parametric Estimation, Regression Model and Application to RNA-SEQ Data

Background: The Poisson and the Negative Binomial distributions are commonly used to model count data. The Poisson is characterized by the equality of mean and variance whereas the Negative Binomial has a variance larger than the mean and therefore both models are appropriate to model over-dispersed count data. Objectives: A new two-parameter probability distribution called the Quasi-Negative Binomial Distribution (QNBD) is being studied in this paper, generalizing the well-known negative binomial distribution. This model turns out to be quite flexible for analyzing count data. Our main objectives are to estimate the parameters of the proposed distribution and to discuss its applicability to genetics data. As an application, we demonstrate that the QNBD regression representation is utilized to model genomics data sets. Results: The new distribution is shown to provide a good fit with respect to the “Akaike Information Criterion”, AIC, considered a measure of model goodness of fit. The proposed distribution may serve as a viable alternative to other distributions available in the literature for modeling count data exhibiting overdispersion, arising in various fields of scientific investigation such as genomics and biomedicine.

Phase Type Zero Truncated Poisson Lindley Distributions and their application in modeling Secondary Cancer Cases

Insurance of chronic illness is slowly gaining ground in Kenya which has lead to insurance firms introducing insurance products of chronic illness among them being cancer insurance policies. However, unlike other chronic illnesses, cancer can move from the organ of origin to another which will consequently lead to increased cost of treatment. This can not be modeled using ordinary distributions hence it has become an area of interest for many researchers. Zero-truncated phase type distributions are used to solve this drawback of ordinary distributions as it can in-cooperate these transitions while modeling claim count data. They further improve modeling of claim count data as they only consider positive values of claim count excluding zeros. This is the nature of real claim count data as zero claim frequency can not attract any claim severity amount. In this paper aggregate claim losses of secondary cancers in Kenya are estimated using Zero-truncated Poisson Lindley distributions. Zero-truncated one parameter as well as Zero-truncated two parameter Poisson Lindley distributions are derived. Their compound probability generating functions are also constructed. The transitions states of secondary cancer states are estimated using continuous Chapman Kolmogorov equation and used as the matrix parameters for the claim count distributions. Pareto, Generalized Pareto, Weibull, OPPL and TPPL distributions are the distributions considered in this research in modeling claim numbers. This study concludes that aggregate losses of secondary cancer cases using Kenyan data are best modeled by PH-ZTOPPL Generalized Pareto model for PH-ZTOPPL distribution models while for PH-ZTTPPL distribution models the best model was PH-ZTTPPL-Generalized Pareto model. The two best models were compared and PH-ZTTPPL-Generalized Pareto model was proven to be the best model. Comparing this model with PH-TPPL Generalized Pareto model from earlier research PH-TPPL Generalized Pareto model proved to be a better model implying that zero claim count data should be considered in estimation of aggregate losses

Sparse Count Data Clustering Using an Exponential Approximation to Generalized Dirichlet Multinomial Distributions.

Clustering frequency vectors is a challenging task on large data sets considering its high dimensionality and sparsity nature. Generalized Dirichlet multinomial (GDM) distribution is a competitive generative model for count data in terms of accuracy, yet its parameters estimation process is slow. The exponential-family approximation of the multivariate Polya distribution has shown to be efficient to train and cluster data directly, without dimensionality reduction. In this article, we derive an exponential-family approximation to the GDM distributions, and we call it (EGDM). A mixture model is developed based on the new member of the exponential-family of distributions, and its parameters are learned through the deterministic annealing expectation-maximization (DAEM) approach as a new clustering algorithm for count data. Moreover, we propose to estimate the optimal number of EGDM mixture components based on the minimum message length (MML) criterion. We have conducted a set of empirical experiments, concerning text, image, and video clustering, to evaluate the proposed approach performance. Results show that the new model attains a superior performance, and it is considerably faster than the corresponding method for GDM distributions.

A new regression model for Poisson Lindley distribution with application

The main goal of this study is to propose a new regression model using re-parameterization of the Poisson-Lindley distribution for seven parameters. The utility of real-world data is used to assess the accuracy of estimating algorithms. The suggested model is compared to well-known regression models for count data modelling, such as Poisson, on a real data set to demonstrate its utility. While fitting two real data sets, the (GPL7) linear model will be compared to the Poisson for seven parameters and the (GPL4) linear model will be compared to the Poisson for four parameters. The GPL7 linear model was found to be capable of fitting over-dispersed count data and to have the maximum log-likelihood, according to the results.

A discrete Ramos-Louzada distribution for asymmetric and over-dispersed data with leptokurtic-shaped: Properties and various estimation techniques with inference

<abstract> <p>In this paper, a flexible probability mass function is proposed for modeling count data, especially, asymmetric, and over-dispersed observations. Some of its distributional properties are investigated. It is found that all its statistical properties can be expressed in explicit forms which makes the proposed model useful in time series and regression analysis. Different estimation approaches including maximum likelihood, moments, least squares, Andersonӳ-Darling, Cramer von-Mises, and maximum product of spacing estimator, are derived to get the best estimator for the real data. The estimation performance of these estimation techniques is assessed via a comprehensive simulation study. The flexibility of the new discrete distribution is assessed using four distinctive real data sets ԣoronavirus-flood peaks-forest fire-Leukemia? Finally, the new probabilistic model can serve as an alternative distribution to other competitive distributions available in the literature for modeling count data.</p> </abstract>

Performance of the Ridge and Liu Estimators in the zero‐inflated Bell Regression Model

The Poisson regression model is popularly used to model count data. However, the model suffers drawbacks when there is overdispersion—when the mean of the Poisson distribution is not the same as the variance. In this situation, the Bell regression model fits well to the data. Also, there is a high tendency of excess zeros in the count data. In this case, the zero‐inflated Bell regression model is an alternative to the Bell regression model. The parameters of the zero‐inflated Bell regression model are mostly estimated using the method of maximum likelihood. Linear dependency is a threat in a real‐life application when modeling the relationship between the response variable and two or more explanatory variables in a generalized linear model such as the zero‐inflated Bell regression model. It reduced the efficiency of the maximum likelihood estimator. Therefore, we developed the ridge and Liu estimators for the zero‐inflated Bell regression model to deal with this issue. The simulation and application results support the dominance of the proposed methods over the conventional maximum likelihood estimator.

Poisson size-biased Lindley distribution and its applications

In this paper, a new model for count data is introduced by compounding the Poisson distribution with size-biased three-parameter Lindley distribution. Statistical properties, such as reliability, hazard rate, reverse hazard rate, Mills ratio, moments, shewness, kurtosis, moment genrating function, probability generating function and order statistics, have been discussed. Moreover, the collective risk model is discussed by considering the proposed distrubution as the primary distribution and the expoential and Erlang distributions as the secondary ones. Parameter estimation is done using maximum likelihood estimation (MLE). Finally a real dataset is discussed to demonstrate the suitability and applicability of the proposed distribution in modeling count dataset.

On the mean-parameterized Bell–Touchard regression model for count data

The two-parameter Bell–Touchard distribution corresponds to a quite flexible yet tractable parametric family of discrete distributions, which arises prominently as an empirical model for count data. In this paper, we introduce a novel parametric regression model for count response variables on the basis of the mean-parameterized Bell–Touchard distribution, which allows interpretation of the regression coefficients in terms of the expectation of the count response variable, like the generalized linear model framework. The unknown parameters of the mean-parameterized Bell–Touchard regression model are estimated using the traditional maximum likelihood estimation procedure. We also consider the deviance residuals to assess departures from model assumptions. Empirical applications are provided to illustrate the mean-parameterized Bell–Touchard regression model in practice, and comparisons with some popular existing count regression models are made.

Bayesian Inference for Spatial Count Data that May be Over-Dispersed or Under-Dispersed with Application to the 2016 US Presidential Election

We propose a method of spatial prediction using count data that can be reasonably modeled assuming the Conway-Maxwell Poisson distribution (COM-Poisson). The COM-Poisson model is a two parameter generalization of the Poisson distribution that allows for the flexibility needed to model count data that are either over or under-dispersed. The computationally limiting factor of the COM-Poisson distribution is that the likelihood function contains multiple intractable normalizing constants and is not always feasible when using Markov Chain Monte Carlo (MCMC) techniques. Thus, we develop a prior distribution of the parameters associated with the COM-Poisson that avoids the intractable normalizing constant. Also, allowing for spatial random effects induces additional variability that makes it unclear if a spatially correlated Conway-Maxwell Poisson random variable is over or under-dispersed. We propose a computationally efficient hierarchical Bayesian model that addresses these issues. In particular, in our model, the parameters associated with the COM-Poisson do not include spatial random effects (leading to additional variability that changes the dispersion properties of the data), and are then spatially smoothed in subsequent levels of the Bayesian hierarchical model. Furthermore, the spatially smoothed parameters have a simple regression interpretation that facilitates computation. We demonstrate the applicability of our approach using simulated examples, and a motivating application using 2016 US presidential election voting data in the state of Florida obtained from the Florida Division of Elections.

Multivariate Bayesian spatio-temporal P-spline models to analyze crimes against women.

Univariate spatio-temporal models for areal count data have received great attention in recent years for estimating risks. However, models for studying multivariate responses are less commonly used mainly due to the computational burden. In this article, multivariate spatio-temporal P-spline models are proposed to study different forms of violence against women. Modeling distinct crimes jointly improves the precision of estimates over univariate models and allows to compute correlations among them. The correlation between the spatial and the temporal patterns may suggest connections among the different crimes that will certainly benefit a thorough comprehension of this problem that affects millions of women around the world. The models are fitted using integrated nested Laplace approximations and are used to analyze four distinct crimes against women at district level in the Indian state of Maharashtra during the period 2001-2013.

Exact fisher information of generalized Dirichlet multinomial distribution for count data modeling

Despite the multiple benefits of the Fisher information matrix, it is generally disregarded and substituted by the identity matrix or an approximation format. However, when dealing with complicated real-world applications, ignoring the correlation between data features may compromise the modeling capability. To address this problem we present the exact calculation of the Fisher information matrix (EFIM) for the generalized Dirichlet multinomial (GDM) mixture that has proven its efficiency when modeling count data. We present a parametrization of GDM mixture model that allows the determination of the Fisher matrix’s elements by means of the Beta-binomial probability function. We also propose a novel count data modeling approach with the benefit of EFIM. In particular, we tackle the problem of mixture model estimation and selection using the Fisher scoring algorithm and minimum message length within the Deterministic Annealing Expectation-Maximization learning framework. Experiments on detecting depression in tweets, dialogue-based emotion recognition, and image-based sentiment analysis confirm the capability of the proposed approach and the merits of using the EFIM as compared with existing state-of-the-art methods and techniques that ignore the full determination of the Fisher information matrix’s elements.

TascCODA: Bayesian Tree-Aggregated Analysis of Compositional Amplicon and Single-Cell Data.

Accurate generative statistical modeling of count data is of critical relevance for the analysis of biological datasets from high-throughput sequencing technologies. Important instances include the modeling of microbiome compositions from amplicon sequencing surveys and the analysis of cell type compositions derived from single-cell RNA sequencing. Microbial and cell type abundance data share remarkably similar statistical features, including their inherent compositionality and a natural hierarchical ordering of the individual components from taxonomic or cell lineage tree information, respectively. To this end, we introduce a Bayesian model for tree-aggregated amplicon and single-cell compositional data analysis (tascCODA) that seamlessly integrates hierarchical information and experimental covariate data into the generative modeling of compositional count data. By combining latent parameters based on the tree structure with spike-and-slab Lasso penalization, tascCODA can determine covariate effects across different levels of the population hierarchy in a data-driven parsimonious way. In the context of differential abundance testing, we validate tascCODA’s excellent performance on a comprehensive set of synthetic benchmark scenarios. Our analyses on human single-cell RNA-seq data from ulcerative colitis patients and amplicon data from patients with irritable bowel syndrome, respectively, identified aggregated cell type and taxon compositional changes that were more predictive and parsimonious than those proposed by other schemes. We posit that tascCODA 1 constitutes a valuable addition to the growing statistical toolbox for generative modeling and analysis of compositional changes in microbial or cell population data.