Bivariate zero-inflated negative binomial regression model with applications

Count data often display excessive number of zero outcomes than are expected in the Poisson regression model. The zero-inflated Poisson regression model has been suggested to handle zero-inflated data, whereas the zero-inflated negative binomial (ZINB) regression model has been fitted for zero-inflated data with additional overdispersion. For bivariate and zero-inflated cases, several regression models such as the bivariate zero-inflated Poisson (BZIP) and bivariate zero-inflated negative binomial (BZINB) have been considered. This paper introduces several forms of nested BZINB regression model which can be fitted to bivariate and zero-inflated count data. The mean–variance approach is used for comparing the BZIP and our forms of BZINB regression model in this study. A similar approach was also used by past researchers for defining several negative binomial and zero-inflated negative binomial regression models based on the appearance of linear and quadratic terms of the variance function. The nested BZINB regression models proposed in this study have several advantages; the likelihood ratio tests can be performed for choosing the best model, the models have flexible forms of marginal mean–variance relationship, the models can be fitted to bivariate zero-inflated count data with positive or negative correlations, and the models allow additional overdispersion of the two dependent variables.     

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.     

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.     

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。     

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.     

Quasi-binomial zero-inflated regression model suitable for variables with bounded support

In recent years, a variety of regression models, including zero-inflated and hurdle versions, have been proposed to explain the case of a dependent variable with respect to exogenous covariates. Apart from the classical Poisson, negative binomial and generalised Poisson distributions, many proposals have appeared in the statistical literature, perhaps in response to the new possibilities offered by advanced software that now enables researchers to implement numerous special functions in a relatively simple way. However, we believe that a significant research gap remains, since very little attention has been paid to the quasi-binomial distribution, which was first proposed over fifty years ago. We believe this distribution might constitute a valid alternative to existing regression models, in situations in which the variable has bounded support. Therefore, in this paper we present a zero-inflated regression model based on the quasi-binomial distribution, taking into account the moments and maximum likelihood estimators, and perform a score test to compare the zero-inflated quasi-binomial distribution with the zero-inflated binomial distribution, and the zero-inflated model with the homogeneous model (the model in which covariates are not considered). This analysis is illustrated with two data sets that are well known in the statistical literature and which contain a large number of zeros.     

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.     

Distribution-free inference of zero-inflated binomial data for longitudinal studies

Count responses with structural zeros are very common in medical and psychosocial research, especially in alcohol and HIV research, and the zero-inflated Poisson (ZIP) and zero-inflated negative binomial models are widely used for modeling such outcomes. However, as alcohol drinking outcomes such as days of drinkings are counts within a given period, their distributions are bounded above by an upper limit (total days in the period) and thus inherently follow a binomial or zero-inflated binomial (ZIB) distribution, rather than a Poisson or ZIP distribution, in the presence of structural zeros. In this paper, we develop a new semiparametric approach for modeling ZIB-like count responses for cross-sectional as well as longitudinal data. We illustrate this approach with both simulated and real study data.     

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.     

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.     

零膨胀计数数据的联合建模及变量选择

零膨胀计数数据破坏了泊松分布的方差-均值关系，可由取值服从泊松分布的数据和取值为零(退化分布)的数据各占一定比例所构成的混合分布所解释。本文基于自适应弹性网技术， 研究了零膨胀计数数据的联合建模及变量选择问题.对于零膨胀泊松分布，引入潜变量，构造出零膨胀泊松模型的完全似然, 其中由零膨胀部分和泊松部分两项组成.考虑到协变量可能存在共线性和稀疏性，通过对似然函数加自适应弹性网惩罚得到目标函数,然后利用EM算法得到回归系数的稀疏估计量，并用贝叶斯信息准则BIC来确定最优调节参数.本文也给出了估计量的大样本性质的理论证明和模拟研究，最后把所提出的方法应用到实际问题中。     

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.     

Marginalized zero-inflated generalized Poisson regression

The generalized Poisson (GP) regression model has been used to model count data that exhibit over-dispersion or under-dispersion. The zero-inflated GP (ZIGP) regression model can additionally handle count data characterized by many zeros. However, the parameters of ZIGP model cannot easily be used for inference on overall exposure effects. In order to address this problem, a marginalized ZIGP is proposed to directly model the population marginal mean count. The parameters of the marginalized zero-inflated GP model are estimated by the method of maximum likelihood. The regression model is illustrated by three real-life data sets.     

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.     

Score Tests for Zero-Inflation in Overdispersed Count Data

The negative binomial (NB) model and the generalized Poisson (GP) model are common alternatives to Poisson models when overdispersion is present in the data. Having accounted for initial overdispersion, we may require further investigation as to whether there is evidence for zero-inflation in the data. Two score statistics are derived from the GP model for testing zero-inflation. These statistics, unlike Wald-type test statistics, do not require that we fit the more complex zero-inflated overdispersed models to evaluate zero-inflation. A simulation study illustrates that the developed score statistics reasonably follow a χ² distribution and maintain the nominal level. Extensive simulation results also indicate the power behavior is different for including a continuous variable than a binary variable in the zero-inflation (ZI) part of the model. These differences are the basis from which suggestions are provided for real data analysis. Two practical examples are presented in this article. Results from these examples along with practical experience lead us to suggest performing the developed score test before fitting a zero-inflated NB model to the data.     

Generalized Linear Factor Models: A New Local EM Estimation Algorithm

In this article, a general approach to latent variable models based on an underlying generalized linear model (GLM) with factor analysis observation process is introduced. We call these models Generalized Linear Factor Models (GLFM). The observations are produced from a general model framework that involves observed and latent variables that are assumed to be distributed in the exponential family. More specifically, we concentrate on situations where the observed variables are both discretely measured (e.g., binomial, Poisson) and continuously distributed (e.g., gamma). The common latent factors are assumed to be independent with a standard multivariate normal distribution. Practical details of training such models with a new local expectation-maximization (EM) algorithm, which can be considered as a generalized EM-type algorithm, are also discussed. In conjunction with an approximated version of the Fisher score algorithm (FSA), we show how to calculate maximum likelihood estimates of the model parameters, and to yield inferences about the unobservable path of the common factors. The methodology is illustrated by an extensive Monte Carlo simulation study and the results show promising performance.     

A joint model for hierarchical continuous and zero-inflated overdispersed count data

Many applications in public health, medical and biomedical or other studies demand modelling of two or more longitudinal outcomes jointly to get better insight into their joint evolution. In this regard, a joint model for a longitudinal continuous and a count sequence, the latter possibly overdispersed and zero-inflated (ZI), will be specified that assembles aspects coming from each one of them into one single model. Further, a subject-specific random effect is included to account for the correlation in the continuous outcome. For the count outcome, clustering and overdispersion are accommodated through two distinct sets of random effects in a generalized linear model as proposed by Molenberghs et al. [A family of generalized linear models for repeated measures with normal and conjugate random effects. Stat Sci. 2010;25:325–347]; one is normally distributed, the other conjugate to the outcome distribution. The association among the two sequences is captured by correlating the normal random effects describing the continuous and count outcome sequences, respectively. An excessive number of zero counts is often accounted for by using a so-called ZI or hurdle model. ZI models combine either a Poisson or negative-binomial model with an atom at zero as a mixture, while the hurdle model separately handles the zero observations and the positive counts. This paper proposes a general joint modelling framework in which all these features can appear together. We illustrate the proposed method with a case study and examine it further with simulations.     

Zero-spiked regression models generated by gamma random variables with application in the resin oil production

Zero-inflated data are more frequent when the data represent counts. However, there are practical situations in which continuous data contain an excess of zeros. In these cases, the zero-inflated Poisson, binomial or negative binomial models are not suitable. In order to reduce this gap, we propose the zero-spiked gamma-Weibull (ZSGW) model by mixing a distribution which is degenerate at zero with the gamma-Weibull distribution, which has positive support. The model attempts to estimate simultaneously the effects of explanatory variables on the response variable and the zero-spiked. We consider a frequentist analysis and a non-parametric bootstrap for estimating the parameters of the ZSGW regression model. We derive the appropriate matrices for assessing local influence on the model parameters. We illustrate the performance of the proposed regression model by means of a real data set (copaiba oil resin production) from a study carried out at the Department of Forest Science of the Luiz de Queiroz School of Agriculture, University of São Paulo. Based on the ZSGW regression model, we determine the explanatory variables that can influence the excess of zeros of the resin oil production and identify influential observations. We also prove empirically that the proposed regression model can be superior to the zero-adjusted inverse Gaussian regression model to fit zero-inflated positive continuous data.     

Markov regression models for count time series with excess zeros: A partial likelihood approach

Count data with excess zeros are common in many biomedical and public health applications. The zero-inflated Poisson (ZIP) regression model has been widely used in practice to analyze such data. In this paper, we extend the classical ZIP regression framework to model count time series with excess zeros. A Markov regression model is presented and developed, and the partial likelihood is employed for statistical inference. Partial likelihood inference has been successfully applied in modeling time series where the conditional distribution of the response lies within the exponential family. Extending this approach to ZIP time series poses methodological and theoretical challenges, since the ZIP distribution is a mixture and therefore lies outside the exponential family. In the partial likelihood framework, we develop an EM algorithm to compute the maximum partial likelihood estimator (MPLE). We establish the asymptotic theory of the MPLE under mild regularity conditions and investigate its finite sample behavior in a simulation study. The performances of different partial-likelihood based model selection criteria are compared in the presence of model misspecification. Finally, we present an epidemiological application to illustrate the proposed methodology.