Zero-truncated Count Data Research Articles

On Positive Inflated Geometric Distribution: Properties and Applications

One-inflation in zero-truncated count data has recently found considerable attention. In this regard, zero-truncated Geometric distribution and distribution to a point mass at one are used to create a one-inflated model, namely, one-inflated zero-truncated Geometric distribution. Its reliability characteristics, generating functions, and distributional properties are investigated in detail, which includes survival function, hazard rate function, reverse hazard rate function, probability generating function, characteristic function, variance, skewness, and kurtosis. Monte Carlo simulation have been undertaken to evaluate the effectiveness of the maximum likelihood estimators. To test the compatibility of our proposed model, the baseline model and the proposed model are distinguished by using the two different test procedures. The adaptability of the suggested model is demonstrated using two real-life datasets from separate domains by taking various performance measures into consideration.

A new class of zero-truncated counting models and its application

Count data is a type of data derived from the number of times an event occurs per unit of time, and zero-truncated count data refers to count data without zero, which often appears in various fields. In this paper, a new zero-truncated Bell (ZTBell) distribution is proposed on the basis of Bell distribution. We studied its statistical properties, exploring methods such as maximum likelihood estimation (MLE), expectation–maximization (EM) algorithm, and minimization–maximization (MM) algorithm for parameter estimation, as well as conducting likelihood ratio tests. In addition, we used the Bootstrap method to calculate the standard errors and confidence intervals of the parameters. The simulation results found that all of the MLE, MM algorithm and EM algorithm are effective. And, as the sample size increases, the estimates of the parameters are closer to the true values and the root mean square error is smaller. Finally, applying the model to a set of factory accident data, we found that the ZTBell distribution fits better than the other models and is close to the fitting results of the zero-truncated generalized Poisson distribution. But ZTBell distribution has only one parameter, so it’s even simpler compared to the latter. Therefore, the ZTBell distribution can be a good alternative to other zero-truncated distributions, which provides more options available for statistical analysis in this domain.

Regularized Models for Fitting Zero-Inflated and Zero-Truncated Count Data: A Comparative Analysis

Regularized Models for Fitting Zero-Inflated and Zero-Truncated Count Data: A Comparative Analysis

Mean regression model for the zero-truncated Poisson distribution and its generalization

Zero-truncated count data (e.g., days of staying in hospital; survival weeks of female patients with breast cancer) often arise in various fields such as medical studies. To model such data, the zero-truncated Poisson (ZTP) distribution is commonly utilized to investigate the relationship between the response counts and a set of covariates. For existing ZTP regression models, it is very hard to explain the regression coefficients β or it is quite difficult to perform a constrained optimization to calculate the maximum likelihood estimates (MLEs) of β. This paper aims to introduce a new mean regression model for the ZTP distribution with a clear interpretation about the regression coefficients. Because of a challenge that the original Poisson mean parameter λi cannot be expressed explicitly by the ZTP mean parameter μi, an embedded Newton–Raphson algorithm is developed to calculate the MLEs of regression coefficients. The construction of bootstrap confidence intervals is presented and three hypothesis tests (i.e., the likelihood ratio test, the Wald test and the score test) are considered. Furthermore, the ZTP mean regression model is generalized to the mean regression model for the k-truncated Poisson distribution. Simulation studies are conducted and two real data are analyzed to illustrate the proposed model and methods.

On the equivalence of one-inflated zero-truncated and zero-truncated one-inflated count data likelihoods.

One-inflation in zero-truncated count data has recently found considerable attention. There are currently two views in the literature. In the first approach, the untruncated model is considered as one-inflated whereas in the second approach the truncated model is viewed as one-inflated. Here, we show that both models have identical model spaces as well as identical maximum likelihoods. Consequences of population size estimation are illuminated, and the findings are illustrated at hand of two casestudies.

Novel Use of Capture-Recapture Methods to Estimate Completeness of Contact Tracing during an Ebola Outbreak, Democratic Republic of the Congo, 2018-2020.

Despite its critical role in containing outbreaks, the efficacy of contact tracing, measured as the sensitivity of case detection, remains an elusive metric. We estimated the sensitivity of contact tracing by applying unilist capture-recapture methods on data from the 2018–2020 outbreak of Ebola virus disease in the Democratic Republic of the Congo. To compute sensitivity, we applied different distributional assumptions to the zero-truncated count data to estimate the number of unobserved case-patients with any contacts and infected contacts. Geometric distributions were the best-fitting models. Our results indicate that contact tracing efforts identified almost all (n = 792, 99%) of case-patients with any contacts but only half (n = 207, 48%) of case-patients with infected contacts, suggesting that contact tracing efforts performed well at identifying contacts during the listing stage but performed poorly during the contact follow-up stage. We discuss extensions to our work and potential applications for the ongoing coronavirus pandemic.

Application of zero-truncated count data regression models to air-pollution disease

Count data consist of non-negative integers that have many applications in various fields of studies. To handle count data, there are various statistical models that can be employed corresponding to the properties of the count data studied. Poisson regression model (PRM) is mostly used to model data with equidispersion, while negative binomial regression model (NBRM) is a model that is regularly employed to model over-dispersed count data. On the other hand, the usual count data regression models may not able to handle strictly positive counts. In this case, the appropriate model for the analysis of such data would be models truncated at zero. We are interested to study the relationship between pollution related disease with influential factors such as air pollution and climate variables in Johor Bahru, Malaysia, using these zero-truncated models, where the number of disease cases are strictly positive. In particular, the zero-truncated PRM and NBRM are used to determine the association between the number of dengue patients and their influential factors. From the study, zero-truncated NBRM is found to be the best model amongst the two models to model the relationship between the number dengue cases and air pollution and climate. Air pollution factors that significantly affect the number of cases for dengue are particulate matter (PM10) and sulfur dioxide. Also, humidity and temperature are the climate factors that significantly affect the number of dengue cases.

Power of Overdispersion Tests in Zero-Truncated Negative Binomial Regression Model

Poisson regression is the most extensively used model for modeling data that are measured as counts. The main characteristic of Poisson regression model is the equidispersion limitation in which the mean and variance of the count variable are the same. However, in many situations the variance of the count variable is greater than the mean which causes overdispersion, and hence, poor fit will be resulted when inference about regression parameters. Alternatively, the negative binomial regression is preferred when overdispersion is present. In addition, in particular cases, the zero counts are not observed in data which is known as zero-truncation. In the presence of overdispersion in zero-truncated count data, the zero-truncated negative binomial (ZTNB) regression model can be used as an alternative to zero-truncated Poisson (ZTP) regression model. In this paper, for testing overdispersion in ZTNB regression model against ZTP regression model, the likelihood ratio test (LRT), score test, and Wald test are proposed. A Monte-Carlo simulation is carried out in order to examine the empirical power for statistics of these tests under different levels of overdispersion and various sample sizes. The simulation results indicate that Wald test is more powerful than the LRT and score test for detecting the overdispersion parameter in ZTNB regression model against ZTP regression model, since it provides the highest statistical power. Thus, the Wald test is preferable for detecting the overdispersion problem in zero-truncated count data.

The identity of the zero-truncated, one-inflated likelihood and the zero-one-truncated likelihood for general count densities with an application to drink-driving in Britain

For zero-truncated count data, as they typically arise in capture-recapture modelling, we consider modelling under one-inflation. This is motivated by police data on drink-driving in Britain which shows high one-inflation. The data, which are used here, are from the years 2011 to 2015 and are based on DR10 endorsements. We show that inference for an arbitrary count density with one-inflation can be equivalently based upon the associated zero-one truncated count density. This simplifies inference considerably including maximum likelihood estimation and likelihood ratio testing. For the drink-driving application, we use the geometric distribution which shows a good fit. We estimate the total drink-driving as about $2{,}300{,}000$ drink-drivers in the observational period. As $227{,}578$ were observed, this means that only about 10% of the drink-driving population is observed with a bootstrap confidence interval of 9%–12%.

A modification of Chao\u2019s lower bound estimator in the case of one-inflation

For zero-truncated count data, as they typically arise in capture-recapture modelling, the nonparametric lower bound estimator of Chao is a frequently used estimator of population size. It is a simple, nonparametric estimator involving only counts of one and counts of two. The estimator is asymptotically unbiased if the count distribution is a member of the power series family and is providing a lower bound estimator if the distribution is a mixture of a member of the power series family. However, if there is one-inflation Chao’s estimator can severely overestimate as we show here. This is also illustrated by routinely collected country-wide data on family violence in the Netherlands. A new lower bound estimator is developed which involves only counts of twos and threes, thus avoiding the overestimation caused by one-inflation. We show that the new estimator is asymptotically unbiased for a power series distribution with and without one-inflation and provides a lower bound estimator under a mixture of power series distributions with and without one-inflation. For all estimators bias-adjusted versions are developed that reduce the bias considerably when the sample size is small. A simulation study compares the modified Chao estimator with the conventional estimator as well as with an estimator suggested by Chiu and Chao more recently.

Type I multivariate zero-truncated/adjusted Poisson distributions with applications

Although there is substantial literature on the univariate zero-truncated Poisson (ZTP) distributions, few works have been done for developing multivariate ZTP distributions with dimension larger than two. In this paper, we first propose a new Type I multivariate ZTP distribution to model multivariate zero-truncated count data and develop its important distributional properties, while providing efficient statistical inference methods. The key idea for extending the univariate ZTP distribution to a multivariate version is to identify a new stochastic representation of the ZTP random variable, resulting in a novel expectation–maximization algorithm with a feature that the introduced latent variables are independent of the observed variables. Since the ZTP distribution is closely related with the zero-adjusted Poisson (ZAP) distribution, the second objective of this paper is to propose a new family of Type I multivariate ZAP distributions (including the Type I multivariate ZTP distribution, the Type I multivariate zero-inflated Poisson (ZIP) distribution, the Type I multivariate zero-deflated Poisson (ZDP) distribution as its special cases) by accounting for truncation, inflation and deflation at zero. The corresponding distributional properties are explored and the associated statistical methods are provided. Finally, three real data sets are used to illustrate the proposed distributions and corresponding analysis methods.

Modelling crash frequencies at signalized intersections with a truncated count data model

Modelling crash frequencies provides comprehensive insights into the safety effects of explanatory variables contributing to crash occurrence, and thus a variety of modelling techniques have been employed depending on what variable is of interest. Since accident frequencies are count data which are discrete and non-negative integers, Poisson and negative binomial regression models are used to fit count data with the entire distribution of counts. However, some of the crash counts might be omitted from a sample due to the limits of observability, hence resulting in the truncation of the sample. Without accounting for truncation, the parameters estimated will be biased and inconsistent, and thus careful attention should be paid to the estimation of count data models when a truncation problem exists. This paper describes the development of a truncated Poisson model with zero-truncated count data and the demonstration of differences in estimation results by comparing the results from truncated and untruncated Poisson models with truncated data. The major difference in estimation results is that significant factors included in the both models are not exactly the same. However, it is found that there is no difference in the directions of associations with safety of the significant variables included in the both models.

Determinants of cultural and popular celebration attendance: the case study of Seville Spring Fiestas

The Spring Fiestas in Seville (Spain) (SFS) are the most important cultural events in the city each year. The present paper pursues two aims. The first is to characterize the SFS as a new prototype of a complex cultural good that expresses the link between the people and the place in which they live based on material and immaterial cultural heritage represented through popular celebrations. The second goal is to conduct an empirical analysis of the determinants that shape attendance intensity by estimating a zero-truncated count data model using a unique dataset of attendees at the SFS in 2009. Findings indicate that attendance is strongly associated with variables reflecting knowledge, institutional links, past experiences, and the perceived external benefits generated by the existence of the SFS. The article contributes to the literature by exploring participation in popular celebrations, a field of inquiry that to date is extremely limited in cultural economics.

Factors influencing the intention to revisit a cultural attraction: The case study of the Museum of Modern and Contemporary Art in Rovereto

Abstract This paper analyses the different factors influencing the intention to revisit a cultural attraction with an application to the Museum for Modern and Contemporary Art (MART) in Rovereto, Italy. The empirical data were obtained from a survey undertaken in 2009 and a zero-truncated count data model is estimated. The findings reveal that sociodemographic characteristics positively influence the probability to return to the museum. Also, as reported in other studies, the temporary exhibitions offered by the museum have a significant impact with an incidence rate ratio almost twice as high. No matter how much visitors spend on accommodation, they are less likely to revisit if they travel in groups, by train or on foot, are far from their town of origin and have spent a long time visiting the museum.

Differential Measurement Errors in Zero-Truncated Regression Models for Count Data

Measurement errors in covariates may result in biased estimates in regression analysis. Most methods to correct this bias assume nondifferential measurement errors-i.e., that measurement errors are independent of the response variable. However, in regression models for zero-truncated count data, the number of error-prone covariate measurements for a given observational unit can equal its response count, implying a situation of differential measurement errors. To address this challenge, we develop a modified conditional score approach to achieve consistent estimation. The proposed method represents a novel technique, with efficiency gains achieved by augmenting random errors, and performs well in a simulation study. The method is demonstrated in an ecology application.

An extension of an over-dispersion test for count data

While over-dispersion in capture–recapture studies is well known to lead to poor estimation of population size, current diagnostic tools to detect the presence of heterogeneity have not been specifically developed for capture–recapture studies. To address this, a simple and efficient method of testing for over-dispersion in zero-truncated count data is developed and evaluated. The proposed method generalizes an over-dispersion test previously suggested for un-truncated count data and may also be used for testing residual over-dispersion in zero-inflation data. Simulations suggest that the asymptotic distribution of the test statistic is standard normal and that this approximation is also reasonable for small sample sizes. The method is also shown to be more efficient than an existing test for over-dispersion adapted for the capture–recapture setting. Studies with zero-truncated and zero-inflated count data are used to illustrate the test procedures.

Equivalence of Truncated Count Mixture Distributions and Mixtures of Truncated Count Distributions

This article is about modeling count data with zero truncation. A parametric count density family is considered. The truncated mixture of densities from this family is different from the mixture of truncated densities from the same family. Whereas the former model is more natural to formulate and to interpret, the latter model is theoretically easier to treat. It is shown that for any mixing distribution leading to a truncated mixture, a (usually different) mixing distribution can be found so that the associated mixture of truncated densities equals the truncated mixture, and vice versa. This implies that the likelihood surfaces for both situations agree, and in this sense both models are equivalent. Zero-truncated count data models are used frequently in the capture-recapture setting to estimate population size, and it can be shown that the two Horvitz-Thompson estimators, associated with the two models, agree. In particular, it is possible to achieve strong results for mixtures of truncated Poisson densities, including reliable, global construction of the unique NPMLE (nonparametric maximum likelihood estimator) of the mixing distribution, implying a unique estimator for the population size. The benefit of these results lies in the fact that it is valid to work with the mixture of truncated count densities, which is less appealing for the practitioner but theoretically easier. Mixtures of truncated count densities form a convex linear model, for which a developed theory exists, including global maximum likelihood theory as well as algorithmic approaches. Once the problem has been solved in this class, it might readily be transformed back to the original problem by means of an explicitly given mapping. Applications of these ideas are given, particularly in the case of the truncated Poisson family.

MODELLING TRUNCATED AND CLUSTERED COUNT DATA

Count response data often exhibit departures from the assumptions of standard Poisson generalized linear models (McCullagh & Nelder 1989). In particular, cluster level correlation of the data and truncation at zero are two common characteristics of such data. In this paper we describe a random components truncated Poisson model that can be applied to clustered and zero-truncated count data. Residual maximum likelihood method estimators for the parameters of this model are developed and their use illustrated using a data set of non-zero counts of sheets with edge strain defects in iron sheets produced by the Mobarekeh Steel Complex, Iran. We also report on a small scale simulation study that supports the estimation procedure.