Bivariate zero-inflated generalized Poisson regression model with flexible covariance

This paper introduces several forms of nested bivariate zero-inflated generalized Poisson (BZIGP) regression model which can be fitted to bivariate and zero-inflated count data. The main advantage of having several forms of BZIGP regression model is that they are nested and allow likelihood ratio test to be performed for choosing the best model. In addition, the BZIGP regression models have flexible forms of marginal mean–variance relationship, can be fitted to bivariate and zero-inflated count data with positive or negative correlations, and allow additional overdispersion of the two response variables. The BZIGP regression models are fitted to the Australian Health Survey data.     

Score test for testing zero-inflated Poisson regression against zero-inflated generalized Poisson alternatives

In several cases, count data often have excessive number of zero outcomes. This zero-inflated phenomenon is a specific cause of overdispersion, and zero-inflated Poisson regression model (ZIP) has been proposed for accommodating zero-inflated data. However, if the data continue to suggest additional overdispersion, zero-inflated negative binomial (ZINB) and zero-inflated generalized Poisson (ZIGP) regression models have been considered as alternatives. This study proposes the score test for testing ZIP regression model against ZIGP alternatives and proves that it is equal to the score test for testing ZIP regression model against ZINB alternatives. The advantage of using the score test over other alternative tests such as likelihood ratio and Wald is that the score test can be used to determine whether a more complex model is appropriate without fitting the more complex model. Applications of the proposed score test on several datasets are also illustrated.     

Functional Form for the Zero-Inflated Generalized Poisson Regression Model

The generalized Poisson (GP) regression is an increasingly popular approach for modeling overdispersed as well as underdispersed count data. Several parameterizations have been performed for the GP regression, and the two well known models, the GP-1 and the GP-2, have been applied. The GP-P regression, which has been recently proposed, has the advantage of nesting the GP-1 and the GP-2 parametrically, besides allowing the statistical tests of the GP-1 and the GP-2 against a more general alternative. In several cases, count data often have excessive number of zero outcomes than are expected in the Poisson. This zero-inflation phenomenon is a specific cause of overdispersion, and the zero-inflated Poisson (ZIP) regression model has been proposed. However, if the data continue to suggest additional overdispersion, the zero-inflated negative binomial (ZINB-1 and ZINB-2) and the zero-inflated generalized Poisson (ZIGP-1 and ZIGP-2) regression models have been considered as alternatives. This article proposes a functional form of the ZIGP which mixes a distribution degenerate at zero with a GP-P distribution. The suggested model has the advantage of nesting the ZIP and the two well known ZIGP (ZIGP-1 and ZIGP-2) regression models, besides allowing the statistical tests of the ZIGP-1 and the ZIGP-2 against a more general alternative. The ZIP and the functional form of the ZIGP regression models are fitted, compared and tested on two sets of count data; the Malaysian insurance claim data and the German healthcare data.     

Testing overdispersion in the zero-inflated Poisson model

The zero-inflated negative binomial (ZINB) model is used to account for commonly occurring overdispersion detected in data that are initially analyzed under the zero-inflated Poisson (ZIP) model. Tests for overdispersion (Wald test, likelihood ratio test [LRT], and score test) based on ZINB model for use in ZIP regression models have been developed. Due to similarity to the ZINB model, we consider the zero-inflated generalized Poisson (ZIGP) model as an alternate model for overdispersed zero-inflated count data. The score test has an advantage over the LRT and the Wald test in that the score test only requires that the parameter of interest be estimated under the null hypothesis. This paper proposes score tests for overdispersion based on the ZIGP model and illustrates that the derived score statistics are exactly the same as the score statistics under the ZINB model. A simulation study indicates the proposed score statistics are preferred to other tests for higher empirical power. In practice, based on the approximate mean–variance relationship in the data, the ZINB or ZIGP model can be considered, and a formal score test based on asymptotic standard normal distribution can be employed for assessing overdispersion in the ZIP model. We provide an example to illustrate the procedures for data analysis.     

Analyzing clustered count data with a cluster-specific random effect zero-inflated Conway–Maxwell–Poisson distribution

Count data analysis techniques have been developed in biological and medical research areas. In particular, zero-inflated versions of parametric count distributions have been used to model excessive zeros that are often present in these assays. The most common count distributions for analyzing such data are Poisson and negative binomial. However, a Poisson distribution can only handle equidispersed data and a negative binomial distribution can only cope with overdispersion. However, a Conway–Maxwell–Poisson (CMP) distribution [4] can handle a wide range of dispersion. We show, with an illustrative data set on next-generation sequencing of maize hybrids, that both underdispersion and overdispersion can be present in genomic data. Furthermore, the maize data set consists of clustered observations and, therefore, we develop inference procedures for a zero-inflated CMP regression that incorporates a cluster-specific random effect term. Unlike the Gaussian models, the underlying likelihood is computationally challenging. We use a numerical approximation via a Gaussian quadrature to circumvent this issue. A test for checking zero-inflation has also been developed in our setting. Finite sample properties of our estimators and test have been investigated by extensive simulations. Finally, the statistical methodology has been applied to analyze the maize data mentioned before.     

Comparisons of some bivariate regression models

The bivariate negative binomial regression (BNBR) and the bivariate Poisson log-normal regression (BPLR) models have been used to describe count data that are over-dispersed. In this paper, a new bivariate generalized Poisson regression (BGPR) model is defined. An advantage of the new regression model over the BNBR and BPLR models is that the BGPR can be used to model bivariate count data with either over-dispersion or under-dispersion. In this paper, we carry out a simulation study to compare the three regression models when the true data-generating process exhibits over-dispersion. In the simulation experiment, we observe that the bivariate generalized Poisson regression model performs better than the bivariate negative binomial regression model and the BPLR model.     

A Bayesian Approach for Zero-Inflated Count Regression Models by Using the Reversible Jump Markov Chain Monte Carlo Method and an Application

In this study, estimation of the parameters of the zero-inflated count regression models and computations of posterior model probabilities of the log-linear models defined for each zero-inflated count regression models are investigated from the Bayesian point of view. In addition, determinations of the most suitable log-linear and regression models are investigated. It is known that zero-inflated count regression models cover zero-inflated Poisson, zero-inflated negative binomial, and zero-inflated generalized Poisson regression models. The classical approach has some problematic points but the Bayesian approach does not have similar flaws. This work points out the reasons for using the Bayesian approach. It also lists advantages and disadvantages of the classical and Bayesian approaches. As an application, a zoological data set, including structural and sampling zeros, is used in the presence of extra zeros. In this work, it is observed that fitting a zero-inflated negative binomial regression model creates no problems at all, even though it is known that fitting a zero-inflated negative binomial regression model is the most problematic procedure in the classical approach. Additionally, it is found that the best fitting model is the log-linear model under the negative binomial regression model, which does not include three-way interactions of factors.     

GEE-based zero-inflated generalized Poisson model for clustered over or under-dispersed count data

The zero-inflated regression models such as zero-inflated Poisson (ZIP), zero-inflated negative binomial (ZINB) or zero-inflated generalized Poisson (ZIGP) regression models can model the count data with excess zeros. The ZINB model can handle over-dispersed and the ZIGP model can handle the over or under-dispersed count data with excess zeros as well. Moreover, the count data may be correlated because of data collection procedure or special study design. The clustered sampling approach is one of the examples in which the correlation among subjects could be defined. In such situations, a marginal model using generalized estimating equation (GEE) approach can incorporate these correlations and lead up to the relationships at the population level. In this study, the GEE-based zero-inflated generalized Poisson regression model was proposed to fit over and under-dispersed clustered count data with excess zeros.     

Zero-inflated sum of Conway-Maxwell-Poissons (ZISCMP) regression

While excess zeros are often thought to cause data over-dispersion (i.e. when the variance exceeds the mean), this implication is not absolute. One should instead consider a flexible class of distributions that can address data dispersion along with excess zeros. This work develops a zero-inflated sum-of-Conway-Maxwell-Poissons (ZISCMP) regression as a flexible analysis tool to model count data that express significant data dispersion and contain excess zeros. This class of models contains several special case zero-inflated regressions, including zero-inflated Poisson (ZIP), zero-inflated negative binomial (ZINB), zero-inflated binomial (ZIB), and the zero-inflated Conway-Maxwell-Poisson (ZICMP). Through simulated and real data examples, we demonstrate class flexibility and usefulness. We further utilize it to analyze shark species data from Australia's Great Barrier Reef to assess the environmental impact of human action on the number of various species of sharks.     

Multivariate models for correlated count data

In this study, we deal with the problem of overdispersion beyond extra zeros for a collection of counts that can be correlated. Poisson, negative binomial, zero-inflated Poisson and zero-inflated negative binomial distributions have been considered. First, we propose a multivariate count model in which all counts follow the same distribution and are correlated. Then we extend this model in a sense that correlated counts may follow different distributions. To accommodate correlation among counts, we have considered correlated random effects for each individual in the mean structure, thus inducing dependency among common observations to an individual. The method is applied to real data to investigate variation in food resources use in a species of marsupial in a locality of the Brazilian Cerrado biome.     

Efficient regression modeling for correlated and overdispersed count data

Abstract

The objective of this paper is to propose an efficient estimation procedure in a marginal mean regression model for longitudinal count data and to develop a hypothesis test for detecting the presence of overdispersion. We extend the matrix expansion idea of quadratic inference functions to the negative binomial regression framework that entails accommodating both the within-subject correlation and overdispersion issue. Theoretical and numerical results show that the proposed procedure yields a more efficient estimator asymptotically than the one ignoring either the within-subject correlation or overdispersion. When the overdispersion is absent in data, the proposed method might hinder the estimation efficiency in practice, yet the Poisson regression based regression model is fitted to the data sufficiently well. Therefore, we construct the hypothesis test that recommends an appropriate model for the analysis of the correlated count data. Extensive simulation studies indicate that the proposed test can identify the effective model consistently. The proposed procedure is also applied to a transportation safety study and recommends the proposed negative binomial regression model.     

ESTIMATING A RATIO OF MEANS FROM BIVARIATE COUNTS WITH APPLICATIONS IN STEREOLOGY

In survey sampling and in stereology, it is often desirable to estimate the ratio of means θ= E(Y)/E(X) from bivariate count data (X, Y) with unknown joint distribution. We review methods that are available for this problem, with particular reference to stereological applications. We also develop new methods based on explicit statistical models for the data, and associated model diagnostics. The methods are tested on a stereological dataset. For point‐count data, binomial regression and bivariate binomial models are generally adequate. Intercept‐count data are often overdispersed relative to Poisson regression models, but adequately fitted by negative binomial regression.     

A comparison of count data models with an application to daily cigarette consumption of young persons

The objective of this study is providing a comparative assessment for researchers to deal with the challenges of analyzing count data and examining the factors associated with daily cigarette consumption among the young people in Turkey. We fitted Poisson (P), negative binomial (NB), zero-inflated Poisson (ZIP), zero-inflated negative binomial (ZINB), Poisson hurdle (PH) and negative binomial hurdle (NBH) regressions to cigarette consumption count data by using the 2014 Turkey Health Survey. Our results showed that the ZINB and NBH models should be preferred. We also found that, gender, employment and tobacco use at home are more effective factors for smokers and nonsmokers in the 15–24 age group in Turkey.     

Modelling count data with overdispersion and spatial effects

In this paper we consider regression models for count data allowing for overdispersion in a Bayesian framework. We account for unobserved heterogeneity in the data in two ways. On the one hand, we consider more flexible models than a common Poisson model allowing for overdispersion in different ways. In particular, the negative binomial and the generalized Poisson (GP) distribution are addressed where overdispersion is modelled by an additional model parameter. Further, zero-inflated models in which overdispersion is assumed to be caused by an excessive number of zeros are discussed. On the other hand, extra spatial variability in the data is taken into account by adding correlated spatial random effects to the models. This approach allows for an underlying spatial dependency structure which is modelled using a conditional autoregressive prior based on Pettitt et al. in Stat Comput 12(4):353–367, (2002). In an application the presented models are used to analyse the number of invasive meningococcal disease cases in Germany in the year 2004. Models are compared according to the deviance information criterion (DIC) suggested by Spiegelhalter et al. in J R Stat Soc B64(4):583–640, (2002) and using proper scoring rules, see for example Gneiting and Raftery in Technical Report no. 463, University of Washington, (2004). We observe a rather high degree of overdispersion in the data which is captured best by the GP model when spatial effects are neglected. While the addition of spatial effects to the models allowing for overdispersion gives no or only little improvement, spatial Poisson models with spatially correlated or uncorrelated random effects are to be preferred over all other models according to the considered criteria.     

Zero-inflated compound Poisson distributions in integer-valued GARCH models

In this paper we introduce a wide class of integer-valued stochastic processes that allows to take into consideration, simultaneously, relevant characteristics observed in count data namely zero inflation, overdispersion and conditional heteroscedasticity. This class includes, in particular, the compound Poisson, the zero-inflated Poisson and the zero-inflated negative binomial INGARCH models, recently proposed in literature. The main probabilistic analysis of this class of processes is here developed. Precisely, first- and second-order stationarity conditions are derived, the autocorrelation function is deduced and the strict stationarity is established in a large subclass. We also analyse in a particular model the existence of higher-order moments and deduce the explicit form for the first four cumulants, as well as its skewness and kurtosis.     

随机效应零膨胀索赔次数回归模型

索赔频率预测是非寿险费率厘定的重要组成部分。最常使用的索赔频率预测模型是泊松回归和负二项回归,以及与它们相对应的零膨胀回归模型。但是,当索赔次数观察值既具有零膨胀特征,又存在组内相依结构时,上述模型都不能很好地拟合实际数据。为此,本文在泊松分布、负二项分布、广义泊松分布、P型负二项分布等条件下分别建立了随机效应零膨胀损失次数回归模型。为了改进模型的预测效果,对于连续型的解释变量,还引入了二次平滑项,并建立了结构性零比例与解释变量之间的回归关系。基于一组实际索赔次数数据的实证分析结果表明,该模型可以显著改进现有模型的拟合效果。     

Geographically Weighted Negative Binomial Regression—incorporating overdispersion

Global regression assumes that a single model adequately describes all parts of a study region. However, the heterogeneity in the data may be sufficiently strong that relationships between variables can not be spatially constant. In addition, the factors involved are often sufficiently complex that it is difficult to identify them in the form of explanatory variables. As a result Geographically Weighted Regression (GWR) was introduced as a tool for the modeling of non-stationary spatial data. Using kernel functions, the GWR methodology allows the model parameters to vary spatially and produces non-parametric surfaces of their estimates. To model count data with overdispersion, it is more appropriate to use a negative binomial distribution instead of a Poisson distribution. Therefore, we propose the Geographically Weighted Negative Binomial Regression (GWNBR) method for the modeling of data with overdispersion. The results obtained using simulated and real data show the superiority of this method for the modeling of non-stationary count data with overdispersion compared with competing models, such as global regressions, e.g., Poisson and negative binomial and Geographically Weighted Poisson Regression (GWPR). Moreover, we illustrate that these competing models are special cases of the more robust model GWNBR.     

Score tests for heterogeneity and overdispersion in zero‐inflated Poisson and binomial regression models

Hall (2000) has described zero‐inflated Poisson and binomial regression models that include random effects to account for excess zeros and additional sources of heterogeneity in the data. The authors of the present paper propose a general score test for the null hypothesis that variance components associated with these random effects are zero. For a zero‐inflated Poisson model with random intercept, the new test reduces to an alternative to the overdispersion test of Ridout, Demério & Hinde (2001). The authors also examine their general test in the special case of the zero‐inflated binomial model with random intercept and propose an overdispersion test in that context which is based on a beta‐binomial alternative.     

Bayesian estimation and case influence diagnostics for the zero-inflated negative binomial regression model

In recent years, there has been considerable interest in regression models based on zero-inflated distributions. These models are commonly encountered in many disciplines, such as medicine, public health, and environmental sciences, among others. The zero-inflated Poisson (ZIP) model has been typically considered for these types of problems. However, the ZIP model can fail if the non-zero counts are overdispersed in relation to the Poisson distribution, hence the zero-inflated negative binomial (ZINB) model may be more appropriate. In this paper, we present a Bayesian approach for fitting the ZINB regression model. This model considers that an observed zero may come from a point mass distribution at zero or from the negative binomial model. The likelihood function is utilized to compute not only some Bayesian model selection measures, but also to develop Bayesian case-deletion influence diagnostics based on q-divergence measures. The approach can be easily implemented using standard Bayesian software, such as WinBUGS. The performance of the proposed method is evaluated with a simulation study. Further, a real data set is analyzed, where we show that ZINB regression models seems to fit the data better than the Poisson counterpart.     

Poisson regression models with errors-in-variables: implication and treatment

Overdispersion has been a common phenomenon in count data and usually treated with the negative binomial model. This paper shows that measurement errors in covariates in general also lead to overdispersion on the observed data if the true data generating process is indeed the Poisson regression. This kind of overdispersion cannot be treated using the negative binomial model, as otherwise, biases will occur. To provide consistent estimates, we propose a new type of corrected score estimator assuming that the distribution of the latent variables is known. The consistency and asymptotic normality of the proposed estimator are established. Simulation results show that this estimator has good finite sample performance. We also illustrate that the Akaike information criterion and Bayesian information criterion work well for selecting the correct model if the true model is the errors-in-variables Poisson regression.