The Gamma-count distribution in the analysis of experimental underdispersed data

Event counts are response variables with non-negative integer values representing the number of times that an event occurs within a fixed domain such as a time interval, a geographical area or a cell of a contingency table. Analysis of counts by Gaussian regression models ignores the discreteness, asymmetry and heteroscedasticity and is inefficient, providing unrealistic standard errors or possibly negative predictions of the expected number of events. The Poisson regression is the standard model for count data with underlying assumptions on the generating process which may be implausible in many applications. Statisticians have long recognized the limitation of imposing equidispersion under the Poisson regression model. A typical situation is when the conditional variance exceeds the conditional mean, in which case models allowing for overdispersion are routinely used. Less reported is the case of underdispersion with fewer modeling alternatives and assessments available in the literature. One of such alternatives, the Gamma-count model, is adopted here in the analysis of an agronomic experiment designed to investigate the effect of levels of defoliation on different phenological states upon the number of cotton bolls. Data set and code for analysis are available as online supplements. Results show improvements over the Poisson model and the semi-parametric quasi-Poisson model in capturing the observed variability in the data. Estimating rather than assuming the underlying variance process leads to important insights into the process.     

Count data and treatment heterogeneity in 2×2 crossover trials

Count data are routinely assumed to have a Poisson distribution, especially when there are no straightforward diagnostic procedures for checking this assumption. We reanalyse two data sets from crossover trials of treatments for angina pectoris , in which the outcomes are counts of anginal attacks. Standard analyses focus on treatment effects, averaged over subjects; we are also interested in the dispersion of these effects (treatment heterogeneity). We set up a log-Poisson model with random coefficients to estimate the distribution of the treatment effects and show that the analysis is very sensitive to the distributional assumption; the population variance of the treatment effects is confounded with the (variance) function that relates the conditional variance of the outcomes, given the subject's rate of attacks, to the conditional mean. Diagnostic model checks based on resampling from the fitted distribution indicate that the default choice of the Poisson distribution for the analysed data sets is poorly supported. We propose to augment the data sets with observations of the counts, made possibly outside the clinical setting, so that the conditional distribution of the counts could be established.     

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.     

Efficient Bayesian inference for COM-Poisson regression models

COM-Poisson regression is an increasingly popular model for count data. Its main advantage is that it permits to model separately the mean and the variance of the counts, thus allowing the same covariate to affect in different ways the average level and the variability of the response variable. A key limiting factor to the use of the COM-Poisson distribution is the calculation of the normalisation constant: its accurate evaluation can be time-consuming and is not always feasible. We circumvent this problem, in the context of estimating a Bayesian COM-Poisson regression, by resorting to the exchange algorithm, an MCMC method applicable to situations where the sampling model (likelihood) can only be computed up to a normalisation constant. The algorithm requires to draw from the sampling model, which in the case of the COM-Poisson distribution can be done efficiently using rejection sampling. We illustrate the method and the benefits of using a Bayesian COM-Poisson regression model, through a simulation and two real-world data sets with different levels of dispersion.     

A percentile regression model with an application to error frequency in group conversation tests

In many applications, estimates of percentile curves interest investigators more than the mean regression line. We consider the situation where the dependent variable is a compound Poisson variable. Explicit parametric assumptions are made using a linear model for scale and heterogeneous variance. The model is applied to the data on group conversation tests in spoken English.     

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.     

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.     

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.     

A mixture model with Poisson and zero-truncated Poisson components to analyze road traffic accidents in Turkey

The analysis of traffic accident data is crucial to address numerous concerns, such as understanding contributing factors in an accident''s chain-of-events, identifying hotspots, and informing policy decisions about road safety management. The majority of statistical models employed for analyzing traffic accident data are logically count regression models (commonly Poisson regression) since a count – like the number of accidents – is used as the response. However, features of the observed data frequently do not make the Poisson distribution a tenable assumption. For example, observed data rarely demonstrate an equal mean and variance and often times possess excess zeros. Sometimes, data may have heterogeneous structure consisting of a mixture of populations, rather than a single population. In such data analyses, mixtures-of-Poisson-regression models can be used. In this study, the number of injuries resulting from casualties of traffic accidents registered by the General Directorate of Security (Turkey, 2005–2014) are modeled using a novel mixture distribution with two components: a Poisson and zero-truncated-Poisson distribution. Such a model differs from existing mixture models in literature where the components are either all Poisson distributions or all zero-truncated Poisson distributions. The proposed model is compared with the Poisson regression model via simulation and in the analysis of the traffic data.     

The zero-inflated Poisson model and the decayed, missing and filled teeth index in dental epidemiology

For frequency counts, the situation of extra zeros often arises in biomedical applications. This is demonstrated with count data from a dental epidemiological study in Belo Horizonte (the Belo Horizonte caries prevention study) which evaluated various programmes for reducing caries. Extra zeros, however, violate the variance–mean relationship of the Poisson error structure. This extra-Poisson variation can easily be explained by a special mixture model, the zero-inflated Poisson (ZIP) model. On the basis of the ZIP model, a graphical device is presented which not only summarizes the mixing distribution but also provides visual information about the overall mean. This device can be exploited to evaluate and compare various groups. Ways are discussed to include covariates and to develop an extension of the conventional Poisson regression. Finally, a method to evaluate intervention effects on the basis of the ZIP regression model is described and applied to the data of the Belo Horizonte caries prevention study.     

Regression analysis of zero-inflated time-series counts: application to air pollution related emergency room visit data

Time-series count data with excessive zeros frequently occur in environmental, medical and biological studies. These data have been traditionally handled by conditional and marginal modeling approaches separately in the literature. The conditional modeling approaches are computationally much simpler, whereas marginal modeling approaches can link the overall mean with covariates directly. In this paper, we propose new models that can have conditional and marginal modeling interpretations for zero-inflated time-series counts using compound Poisson distributed random effects. We also develop a computationally efficient estimation method for our models using a quasi-likelihood approach. The proposed method is illustrated with an application to air pollution-related emergency room visits. We also evaluate the performance of our method through simulation studies.     

Conditional inference in finite population sampling under a calibration model

Several estimators, including the classical and the regression estimators of finite population mean, are compared, both theoretically and empirically, under a calibration model, where the dependent variable(y), and not the independent variable(x), can be observed for all units of the finite population. It is shown asymptotically that when conditioned on x, the bias of the classical estimator may be much smaller than that of the regression estimators; whereas when conditioned on y, the regression estimator may have much smaller conditional bias than the classical estimator. Since all the y's(not x's) can be observed, it seems appropriate to make comparison under the conditional distribution of each estimator with y fixed. In this case, the regression estimator has smaller variance, smaller conditional bias, and the conditional coverage probability closer to its nominal level     

Linear models with an inverse gaussian poisson error distribution

The inverse Gaussian-Poisson (two-parameter Sichel) distribution is useful in fitting overdispersed count data. We consider linear models on the mean of a response variable, where the response is in the form of counts exhibiting extra-Poisson variation, and assume an IGP error distribution. We show how maximum likelihood estimation may be carried out using iterative Newton-Raphson IRLS fitting, where GLIM is used for the IRLS part of the maximization. Approximate likelihood ratio tests are given.     

LOG-LINEAR MODELS FOR MEAN AND DISPERSION IN MIXED POISSON REGRESSION MODELS

This paper is concerned with the analysis of repeated measures count data overdispersed relative to a Poisson distribution, with the overdispersion possibly heterogeneous. To accommodate the overdispersion, the Poisson random variable is compounded with a gamma random variable, and both the mean of the Poisson and the variance of the gamma are modelled using log linear models. Maximum likelihood estimates (MLE) are then obtained. The paper also gives extended quasi-likelihood estimates for a more general class of compounding distributions which are shown to be approximations to the MLEs obtained for the gamma case. The theory is illustrated by modelling the determination of asbestos fibre intensity on membrane filters mounted on microscope slides.     

Regression for doubly inflated multivariate Poisson distributions

Dependent multivariate count data occur in several research studies. These data can be modelled by a multivariate Poisson or Negative binomial distribution constructed using copulas. However, when some of the counts are inflated, that is, the number of observations in some cells are much larger than other cells, then the copula-based multivariate Poisson (or Negative binomial) distribution may not fit well and it is not an appropriate statistical model for the data. There is a need to modify or adjust the multivariate distribution to account for the inflated frequencies. In this article, we consider the situation where the frequencies of two cells are higher compared to the other cells and develop a doubly inflated multivariate Poisson distribution function using multivariate Gaussian copula. We also discuss procedures for regression on covariates for the doubly inflated multivariate count data. For illustrating the proposed methodologies, we present real data containing bivariate count observations with inflations in two cells. Several models and linear predictors with log link functions are considered, and we discuss maximum likelihood estimation to estimate unknown parameters of the models.     

Finite population prediction under a linear function superpopulation model:a bayesian perspective

Exact and approximate Bayesian inference is developed for the prediction problem in finite populations under a linear functional superpopulation model. The models considered are the usual regression models involving two variables, X and Y, where the independent variable X is measured with error. The approach is based on the conditional distribution of Y given X and our predictor is the posterior mean of the quantity of interest (population total and population variance) given the observed data. Empirical investigations about optimal purposive samples and possible model misspecifications based on comparisons with the corresponding models where X is measured without error are also reported.     

A poisson-gamma model for two-stage cluster sampling data

We propose a model for count data from two-stage cluster sampling, where observations within each cluster are subjected simultaneously to internal influences and external factors at the cluster level. This model can be seen as a two-stage hierarchical model with local and global predictors. This parameter-driven model causes the counts within a cluster to share a common latent factor and to be correlated. Maximum likelihood (ml) estimation based on an EM algorithm for the model is discussed. Simulation study is carried out to assess the benefit of using ml estimates compared to a standard Poisson regression analysis that ignores the within cluster correlation.     

A parametric model for heterogeneity in paired Poisson counts

We present a model for data in the form of matched pairs of counts. Our work is motivated by a problem in fission-track analysis, where the determination of a crystal's age is based on the ratio of counts of spontaneous and induced tracks. It is often reasonable to assume that the counts follow a Poisson distribution, but typically they are overdispersed and there exists a positive correlation between the numbers of spontaneous and induced tracks in the same crystal. We propose a model that allows for both overdispersion and correlation by assuming that the mean densities follow a bivariate Wishart distribution. Our model is quite general, having the usual negative-binomial and Poisson models as special cases. We propose a maximum-likelihood estimation method based on a stochastic implementation of the EM algorithm, and we derive the asymptotic standard errors of the parameter estimates. We illustrate the method with a data set of fission-track counts in matched areas of zircon crystals.     

The Touchard distribution

We present a novel model, which is a two-parameter extension of the Poisson distribution. Its normalizing constant is related to the Touchard polynomials, hence the name of this model. It is a flexible distribution that can account for both under- or overdispersion and concentration of zeros that are frequently found in non-Poisson count data. In contrast to some other generalizations, the Hessian matrix for maximum likelihood estimation of the Touchard parameters has a simple form. We exemplify with three data sets, showing that our suggested model is a competitive candidate for fitting non-Poisson counts.     

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.