We consider three new classes of exponential dispersion models of discrete probability distributions which are defined by specifying their variance functions in their mean value parameterization. In a previous paper (Bar-Lev and Ridder, 2020a), we have developed the framework of these classes and proved that they have some desirable properties. Each of these classes was shown to be overdispersed and zero inflated in ascending order, making them as competitive statistical models for those in use in statistical modeling. In this paper we elaborate on the computational aspects of their probability mass functions. Furthermore, we apply these classes for fitting real data sets having overdispersed and zero-inflated statistics. Classic models based on Poisson or negative binomial distributions show poor fits, and therefore many alternatives have already proposed in recent years. We execute an extensive comparison with these other proposals, from which we may conclude that our framework is a flexible tool that gives excellent results in all cases. Moreover, in most cases our model gives the best fit.
Marginalized models are in great demand by most researchers in the life sciences particularly in clinical trials, epidemiology, health-economics, surveys and many others since they allow generalization of inference to the entire population under study. For count data, standard procedures such as the Poisson regression and negative binomial model provide population average inference for model parameters. However, occurrence of excess zero counts and lack of independence in empirical data have necessitated their extension to accommodate these phenomena. These extensions, though useful, complicat…
Generalized Poisson Matrix Factorization for Overdispersed Count Data
Graduate School of Science and Engineering, Chuo University（中央大学理工学研究科）；School of Pharmaceutical Sciences, Wakayama Medical University（和歌山医科大学药学部）；Faculty of Science and Engineering, Chuo University（中央大学理工学部）
Dynamic treatment regimens (DTRs), also known as treatment algorithms or adaptive interventions, play an increasingly important role in many health domains. DTRs are motivated to address the unique and changing needs of individuals by delivering the type of treatment needed, when needed, while minimizing unnecessary treatment. Practically, a DTR is a sequence of decision rules that specify, for each of several points in time, how available information about the individual's status and progress should be used in practice to decide which treatment (e.g., type or intensity) to deliver. The sequen…
This paper studies the non-asymptotic merits of the double $\ell_1$-regularized for heterogeneous overdispersed count data via negative binomial regressions. Under the restricted eigenvalue conditions, we prove the oracle inequalities for Lasso estimators of two partial regression coefficients for the first time, using concentration inequalities of empirical processes. Furthermore, derived from the oracle inequalities, the consistency and convergence rate for the estimators are the theoretical guarantees for further statistical inference. Finally, both simulations and a real data analysis demo…
Count data are ubiquitous in ecology and the Poisson generalized linear model (GLM) is commonly used to model the association between counts and explanatory variables of interest. When fitting this model to the data, one typically proceeds by first confirming that the data is not overdispersed and that there is no excess of zeros. If the data appear to be overdispersed or if there is any zero-inflation, key assumptions of the Poison GLM may be violated and researchers will then typically consider alternatives to the Poison GLM. An important question is whether the potential model selection bia…
We introduce a new class of Poisson-exponential-Tweedie (PET) mixture in the framework of generalized linear models for ultra-overdispersed count data. The mean-variance relationship is of the form $m+m^{2}+ϕm^{p}$, where $ϕ$ and $p$ are the dispersion and Tweedie power parameters, respectively. The proposed model is equivalent to the exponential-Poisson-Tweedie models arising from geometric sums of Poisson-Tweedie random variables. In this respect, the PET models encompass the geometric versions of Hermite, Neyman Type A, Pólya-Aeppli, negative binomial and Poisson inverse Gaussian models. Th…
Kolmogorov-Type Maximal Inequalities for Independent and Dependent Negative Binomial Random Variables: Sharp Bounds, Sub-Exponential Refinements, and Applications to Overdispersed Count Data
National Technical University of Athens（雅典国立技术大学）；Archimedes/Athena Research Center（阿尔基米德/雅典研究中心）；Canisius University（卡尼休斯大学）
The main object of this article is to present an extension of the zero-inflated Poisson-Lindley distribution, called of zero-modified Poisson-Lindley. The additional parameter $π$ of the zero-modified Poisson-Lindley has a natural interpretation in terms of either zero-deflated/inflated proportion. Inference is dealt with by using the likelihood approach. In particular the maximum likelihood estimators of the distribution's parameter are compared in small and large samples. We also consider an alternative bias-correction mechanism based on Efron's bootstrap resampling. The model is applied to …
Real count data time series often show the phenomenon of the underdispersion and overdispersion. In this paper, we develop two extensions of the first-order integer-valued autoregressive process with Poisson innovations, based on binomial thinning, for modeling integer-valued time series with equidispersion, underdispersion and overdispersion. The main properties of the models are derived. The methods of conditional maximum likelihood, Yule-Walker and conditional least squares are used for estimating the parameters, and their asymptotic properties are established. We also use a test based on o…
The Dirichlet-multinomial (DM) distribution plays a fundamental role in modern statistical methodology development and application. Recently, the DM distribution and its variants have been used extensively to model multivariate count data generated by high-throughput sequencing technology in omics research due to its ability to accommodate the compositional structure of the data as well as overdispersion. A major limitation of the DM distribution is that it is unable to handle excess zeros typically found in practice which may bias inference. To fill this gap, we propose a novel Bayesian zero-…
Spearman's rho for zero-inflated count data: formulation and attainable bounds
Eindhoven University of Technology（埃因霍温理工大学）；University of Windsor（温莎大学）；University of Quebec in Trois-Rivères（特鲁瓦河魁北克大学）；Université Libre de Bruxelles（布鲁塞尔自由大学）
We propose a new class of discrete generalized linear models based on the class of Poisson-Tweedie factorial dispersion models with variance of the form $μ+ ϕμ^p$, where $μ$ is the mean, $ϕ$ and $p$ are the dispersion and Tweedie power parameters, respectively. The models are fitted by using an estimating function approach obtained by combining the quasi-score and Pearson estimating functions for estimation of the regression and dispersion parameters, respectively. This provides a flexible and efficient regression methodology for a comprehensive family of count models including Hermite, Neyman…
Mediation analysis seeks to understand the mechanism by which a treatment affects an outcome. Count or zero-inflated count outcome are common in many studies in which mediation analysis is of interest. For example, in dental studies, outcomes such as decayed, missing and filled teeth are typically zero inflated. Existing mediation analysis approaches for count data assume sequential ignorability of the mediator. This is often not plausible because the mediator is not randomized so that there are unmeasured confounders associated with the mediator and the outcome. In this paper, we develop caus…
We present a new modelling approach for longitudinal count data that is motivated by the increasing availability of longitudinal RNA-sequencing experiments. The distribution of RNA-seq counts typically exhibits overdispersion, zero-inflation and heavy tails; moreover, in longitudinal designs repeated measurements from the same subject are typically (positively) correlated. We propose a generalized linear mixed model based on the Poisson-Tweedie distribution that can flexibly handle each of the aforementioned features of longitudinal overdispersed counts. We develop a computational approach to …
The problem of estimating the structure of a graph from observed data is of growing interest in the context of high-throughput genomic data, and single-cell RNA sequencing in particular. These, however, are challenging applications, since the data consist of high-dimensional counts with high variance and over-abundance of zeros. Here, we present a general framework for learning the structure of a graph from single-cell RNA-seq data, based on the zero-inflated negative binomial distribution. We demonstrate with simulations that our approach is able to retrieve the structure of a graph in a vari…
Cluster randomized trails (CRT) have been widely employed in medical and public health research. Many clinical count outcomes, such as the number of falls in nursing homes, exhibit excessive zero values. In the presence of zero inflation, traditional power analysis methods for count data based on Poisson or negative binomial distribution may be inadequate. In this study, we present a sample size method for CRTs with zero-inflated count outcomes. It is developed based on GEE regression directly modeling the marginal mean of a ZIP outcome, which avoids the challenge of testing two intervention e…
Commonly in biomedical research, studies collect data in which an outcome measure contains informative excess zeros; for example when observing the burden of neuritic plaques in brain pathology studies, those who show none contribute to our understanding of neurodegenerative disease. The outcome may be characterized by a mixture distribution with one component being the `structural zero' and the other component being a Poisson distribution. We propose a novel variance components score test of genetic association between a set of genetic markers and a zero-inflated count outcome from a mixture …
Bayesian palaeoclimate reconstruction from zero-inflated count-compositional pollen data: A case study of Lago Grande di Monticchio in southern Italy
Hamilton Institute and Department of Mathematics and Statistics, Maynooth University（梅努斯大学汉密尔顿研究所与数学统计系）；School of Mathematics and Statistics, University College Dublin（都柏林大学学院数学统计学院）；Department of Biosciences, Durham University（杜伦大学生物科学系）
Relational count data are often obtained from sources such as simultaneous purchase in online shops and social networking service information. Bi-clustering such relational count data reveals the latent structure of the relationship between objects such as household items or people. When relational count data observed at multiple time points are available, it is worthwhile incorporating the time structure into the bi-clustering result to understand how objects move between the cluster over time. In this paper, we propose two bi-clustering methods for analyzing time-varying relational count dat…
In microbiome studies, it is of interest to use a sample from a population of microbes, such as the gut microbiota community, to estimate the population proportion of these taxa. However, due to biases introduced in sampling and preprocessing steps, these observed taxa abundances may not reflect true taxa abundance patterns in the ecosystem. Repeated measures, including longitudinal study designs, may be potential solutions to mitigate the discrepancy between observed abundances and true underlying abundances. Yet, widely observed zero-inflation and over-dispersion issues can distort downstrea…
This article introduces a k-Inflated Negative Binomial mixture distribution/regression model as a more flexible alternative to zero-inflated Poisson distribution/regression model. An EM algorithm has been employed to estimate the model's parameters. Then, such new model along with a Pareto mixture model have been employed to design an optimal rate--making system. Namely, this article employs number/size of reported claims of Iranian third party insurance dataset. Then, it employs the k-Inflated Negative Binomial mixture distribution/regression model as well as other well developed counting mod…
Models such as the zero-inflated and zero-altered Poisson and zero-truncated binomial are well-established in modern regression analysis. We propose a super model that jointly and maximally unifies alteration, inflation, truncation and deflation for counts, given a 1- or 2-parameter parent (base) distribution. Seven disjoint sets of special value types are accommodated because all but truncation have parametric and nonparametric variants. Some highlights include: (i) the mixture distribution is exceeding flexible, e.g., up to seven modes; (ii) under-, equi- and over-dispersion can be handled u…
Finite mixture representations of zero-and-$N$-inflated distributions for count-compositional data
Hamilton Institute and Department of Mathematics and Statistics, Maynooth University（梅努斯大学数学与统计系汉密尔顿研究所）；School of Mathematics and Statistics, Insight Centre for Data Analytics, University College Dublin（都柏林大学学院数据洞察中心数学与统计学院）
We consider modeling and prediction of Capelin distribution in the Barents sea based on zero-inflated count observation data that vary continuously over a specified survey region. The model is a mixture of two components; a one-point distribution at the origin and a Poisson distribution with spatio-temporal intensity, where both intensity and mixing proportions are modeled by some auxiliary variables and unobserved spatio-temporal effects. The spatio-temporal effects are modeled by a dynamic linear model combined with the predictive Gaussian process. We develop an efficient posterior computati…
The analysis of count data is commonly done using Poisson models. Negative binomial models are a straightforward and readily motivated generalization for the case of overdispersed data, i.e., when the observed variance is greater than expected under a Poissonian model. Rate and overdispersion parameters then need to be considered jointly, which in general is not trivial. Here we are concerned with evidence synthesis in the case where the reporting of data is rather heterogeneous, i.e., events are reported either in terms of mean event counts, the proportion of event-free patients, or rate esti…
