Multilevel zero-inflated negative binomial regression modeling for over-dispersed count data with extra zeros

Count data with excess zeros often occurs in areas such as public health, epidemiology, psychology, sociology, engineering, and agriculture. Zero-inflated Poisson (ZIP) regression and zero-inflated negative binomial (ZINB) regression are useful for modeling such data, but because of hierarchical study design or the data collection procedure, zero-inflation and correlation may occur simultaneously. To overcome these challenges ZIP or ZINB may still be used. In this paper, multilevel ZINB regression is used to overcome these problems. The method of parameter estimation is an expectation-maximization algorithm in conjunction with the penalized likelihood and restricted maximum likelihood estimates for variance components. Alternative modeling strategies, namely the ZIP distribution are also considered. An application of the proposed model is shown on decayed, missing, and filled teeth of children aged 12 years old.     

Analysis of survival data by a Weibull-generalized Poisson distribution

In life-testing and survival analysis, sometimes the components are arranged in series or parallel system and the number of components is initially unknown. Thus, the number of components, say Z, is considered as random with an appropriate probability mass function. In this paper, we model the survival data with baseline distribution as Weibull and the distribution of Z as generalized Poisson, giving rise to four parameters in the model: increasing, decreasing, bathtub and upside bathtub failure rates. Two examples are provided and the maximum-likelihood estimation of the parameters is studied. Rao's score test is developed to compare the results with the exponential Poisson model studied by Kus [17] and the exponential-generalized Poisson distribution with baseline distribution as exponential and the distribution of Z as generalized Poisson. Simulation studies are carried out to examine the performance of the estimates.     

Analysis of survival data by an exponential-generalized Poisson distribution

In this paper, we consider the distribution of life length of a series system with random number of components, say Z. Considering the distribution of Z as generalized Poisson, an exponential-generalized Poisson (EGP) distribution is developed. The generalized Poisson distribution is a generalization of the Poisson distribution having one extra parameter. The structural properties of the resulting distribution are presented and the maximum likelihood estimation of the parameters is investigated. Extensive simulation studies are carried out to study the performance of the estimates. The score test is developed to test the importance of the extra parameter. For illustration, two real data sets are examined and it is shown that the EGP model, presented here, fits better than the exponential–Poisson distribution.     

Random effects regression models for count data with excess zeros in caries research

We extend the family of Poisson and negative binomial models to derive the joint distribution of clustered count outcomes with extra zeros. Two random effects models are formulated. The first model assumes a shared random effects term between the conditional probability of perfect zeros and the conditional mean of the imperfect state. The second formulation relaxes the shared random effects assumption by relating the conditional probability of perfect zeros and the conditional mean of the imperfect state to two different but correlated random effects variables. Under the conditional independence and the missing data at random assumption, a direct optimization of the marginal likelihood and an EM algorithm are proposed to fit the proposed models. Our proposed models are fitted to dental caries counts of children under the age of six in the city of Detroit.     

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.     

A bivariate Sarmanov regression model for count data with generalised Poisson marginals

We present a bivariate regression model for count data that allows for positive as well as negative correlation of the response variables. The covariance structure is based on the Sarmanov distribution and consists of a product of generalised Poisson marginals and a factor that depends on particular functions of the response variables. The closed form of the probability function is derived by means of the moment-generating function. The model is applied to a large real dataset on health care demand. Its performance is compared with alternative models presented in the literature. We find that our model is significantly better than or at least equivalent to the benchmark models. It gives insights into influences on the variance of the response variables.     

A Bayesian approach of joint models for clustered zero-inflated count data with skewness and measurement errors

Count data with excess zeros are widely encountered in the fields of biomedical, medical, public health and social survey, etc. Zero-inflated Poisson (ZIP) regression models with mixed effects are useful tools for analyzing such data, in which covariates are usually incorporated in the model to explain inter-subject variation and normal distribution is assumed for both random effects and random errors. However, in many practical applications, such assumptions may be violated as the data often exhibit skewness and some covariates may be measured with measurement errors. In this paper, we deal with these issues simultaneously by developing a Bayesian joint hierarchical modeling approach. Specifically, by treating intercepts and slopes in logistic and Poisson regression as random, a flexible two-level ZIP regression model is proposed, where a covariate process with measurement errors is established and a skew-t-distribution is considered for both random errors and random effects. Under the Bayesian framework, model selection is carried out using deviance information criterion (DIC) and a goodness-of-fit statistics is also developed for assessing the plausibility of the posited model. The main advantage of our method is that it allows for more robustness and correctness for investigating heterogeneity from different levels, while accommodating the skewness and measurement errors simultaneously. An application to Shanghai Youth Fitness Survey is used as an illustrate example. Through this real example, it is showed that our approach is of interest and usefulness for applications.     

A new count model generated from mixed Poisson transmuted exponential family with an application to health care data

In this article, a new mixed Poisson distribution is introduced. This new distribution is obtained by utilizing mixing process, with Poisson distribution as mixed distribution and Transmuted Exponential as mixing distribution. Distributional properties like unimodality, moments, over-dispersion, infinite divisibility are studied. Three methods viz. Method of moment, Method of moment and proportion, and Maximum-likelihood method are used for parameter estimation. Further, an actuarial application in context of aggregate claim distribution is presented. Finally, to show the applicability and superiority of proposed model, we discuss count data and count regression modeling and compare with some well established models.     

Score test for homogeneity of dispersion in generalized Poisson mixed models with excess zeros

In many applications, the clustered count data often contain excess zeros and the zero-inflated generalized Poisson mixed (ZIGPM) regression model may be suitable. However, dispersion in ZIGPM is often treated as fixed unknown parameter, and this assumption may be not appropriate in some situations. In this article, a score test for homogeneity of dispersion parameter in ZIGPM regression model is developed and corresponding test statistic is obtained. Sampling distribution and power of the score test statistic are investigated through Monte Carlo simulation. Finally, results from a biological example illustrate the usefulness of the diagnostic statistic.     

A rich learning lesson using the Poisson distribution

In this note, we explore the rich information about inference that the Poisson distribution has. The source of this information is mainly the fact that the mean and variance of this distribution are equal.     

The cosine geometric distribution with count data modeling

In this paper, a new two-parameter discrete distribution is introduced. It belongs to the family of the weighted geometric distribution (GD), with the feature of using a particular trigonometric weight. This configuration adds an oscillating property to the former GD which can be helpful in analyzing the data with over-dispersion, as developed in this study. First, we present the basic statistical properties of the new distribution, including the cumulative distribution function, hazard rate function and moment generating function. Estimation of the related model parameters is investigated using the maximum likelihood method. A simulation study is performed to illustrate the convergence of the estimators. Applications to two practical datasets are given to show that the new model performs at least as well as some competitors.     

Hybrid-based confidence intervals for the ratio of two treatment means in the over-dispersed Poisson data

In many clinical trials and epidemiological studies, comparing the mean count response of an exposed group to a control group is often of interest. This type of data is often over-dispersed with respect to Poisson variation, and previous studies usually compared groups using confidence intervals (CIs) of the difference between the two means. However, in some situations, especially when the means are small, interval estimation of the mean ratio (MR) is preferable. Moreover, Cox and Lewis [4 D.R. Cox and P.A.W. Lewis, The Statistical Analysis of Series of Events, Methuen, London, 1966.[Crossref] , [Google Scholar]] pointed out many other situations where the MR is more relevant than the difference of means. In this paper, we consider CI construction for the ratio of means between two treatments for over-dispersed Poisson data. We develop several CIs for the situation by hybridizing two separate CIs for two individual means. Extensive simulations show that all hybrid-based CIs perform reasonably well in terms of coverage. However, the CIs based on the delta method using the logarithmic transformation perform better than other intervals in the sense that they have slightly shorter interval lengths and show better balance of tail errors. These proposed CIs are illustrated with three real data examples.     

The empirical Bayes estimators of the parameter of the Poisson distribution with a conjugate gamma prior under Stein's loss function

For the hierarchical Poisson and gamma model, we calculate the Bayes posterior estimator of the parameter of the Poisson distribution under Stein's loss function which penalizes gross overestimation and gross underestimation equally and the corresponding Posterior Expected Stein's Loss (PESL). We also obtain the Bayes posterior estimator of the parameter under the squared error loss and the corresponding PESL. Moreover, we obtain the empirical Bayes estimators of the parameter of the Poisson distribution with a conjugate gamma prior by two methods. In numerical simulations, we have illustrated: The two inequalities of the Bayes posterior estimators and the PESLs; the moment estimators and the Maximum Likelihood Estimators (MLEs) are consistent estimators of the hyperparameters; the goodness-of-fit of the model to the simulated data. The numerical results indicate that the MLEs are better than the moment estimators when estimating the hyperparameters. Finally, we exploit the attendance data on 314 high school juniors from two urban high schools to illustrate our theoretical studies.     

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.     

Characterizations of discrete compound Poisson distributions

ABSTRACT

The aim of this paper is to give some new characterizations of discrete compound Poisson distributions. Firstly, we give a characterization by the Lévy–Khintchine formula of infinitely divisible distributions under some conditions. The second characterization need to present by row sum of random triangular arrays converges in distribution. And we give an application in probabilistic number theory, the strongly additive function converging to a discrete compound Poisson in distribution. The next characterization, is an extension of Watanabe’s theorem of characterization of homogeneous Poisson process. The last characterization will be illustrated by waiting time distributions, especially the matrix-exponential representation.     

Analysis of count data with covariate dependence in both mean and variance

Extended Poisson process modelling is generalised to allow for covariate-dependent dispersion as well as a covariate-dependent mean response. This is done by a re-parameterisation that uses approximate expressions for the mean and variance. Such modelling allows under- and over-dispersion, or a combination of both, in the same data set to be accommodated within the same modelling framework. All the necessary calculations can be done numerically, enabling maximum likelihood estimation of all model parameters to be carried out. The modelling is applied to re-analyse two published data sets, where there is evidence of covariate-dependent dispersion, with the modelling leading to more informative analyses of these data and more appropriate measures of the precision of any estimates.     

Recent and classical goodness-of-fit tests for the Poisson distribution

We give a critical synopsis of classical and recent tests for Poissonity, our emphasis being on procedures which are consistent against general alternatives. Two classes of weighted Cramér–von Mises type test statistics, based on the empirical probability generating function process, are studied in more detail. Both of them generalize already known test statistics by introducing a weighting parameter, thus providing more flexibility with regard to power against specific alternatives. In both cases, we prove convergence in distribution of the statistics under the null hypothesis in the setting of a triangular array of rowwise independent and identically distributed random variables as well as consistency of the corresponding test against general alternatives. Therefore, a sound theoretical basis is provided for the parametric bootstrap procedure, which is applied to obtain critical values in a large-scale simulation study. Each of the tests considered in this study, when implemented via the parametric bootstrap method, maintains a nominal level of significance very closely, even for small sample sizes. The procedures are applied to four well-known data sets.     

Mixture of the Riesz Distribution with Respect to a Multivariate Poisson

The aim of this article is to study a statistical model obtained by the mixture of the Riesz probability distribution on symmetric matrices with respect to a multivariate Poisson distribution. We show that this distribution is related to the modified Bessel function of the first kind. We then determine the domain of the means and the variance function of the generated natural exponential family.