Count Response Variables Research Articles

Approximate Gibbs sampler for efficient inference of hierarchical Bayesian models for grouped count data

Hierarchical Bayesian Poisson regression models (HBPRMs) provide a flexible modelling approach of the relationship between predictors and count response variables. The applications of HBPRMs to large-scale datasets require efficient inference algorithms due to the high computational cost of inferring many model parameters based on random sampling. Although Markov Chain Monte Carlo (MCMC) algorithms have been widely used for Bayesian inference, sampling using this class of algorithms is time-consuming for applications with large-scale data and time-sensitive decision-making, partially due to the non-conjugacy of many models. To overcome this limitation, this research develops an approximate Gibbs sampler (AGS) to efficiently learn the HBPRMs while maintaining the inference accuracy. In the proposed sampler, the data likelihood is approximated with Gaussian distribution such that the conditional posterior of the coefficients has a closed-form solution. Numerical experiments using real and synthetic datasets with small and large counts demonstrate the superior performance of AGS in comparison to the state-of-the-art sampling algorithm, especially for large datasets.

Copula deep learning control chart for multivariate zero inflated count response variables

With the increasing popularity of big data analysis, research on zero-inflated count data with the copula method has garnered significant attention because zero-inflated count data do not follow a normal distribution and have high correlation among variables. Within the domain of quality control, there has been limited emphasis on multivariate statistical process control (SPC) techniques that specifically address the challenge of multicollinearity within regression models for multivariate zero-inflated count responses. In this paper, we explain a computational challenge in handling big data with parametric generalized linear models, such as the zero-inflated Poisson model. This challenge motivates us to introduce a copula-based deep learning and neural network model involving multivariate zero-inflated count response variables with highly correlated explanatory variables. This approach builds upon existing deep learning and neural network models designed for univariate count responses, expanding them to encompass multivariate count response models through the integration of a copula regression framework. To evaluate the performance of our proposed methodology, we conduct a comparative analysis of accuracies using the zero-inflated Poisson model, univariate deep learning and neural network models, multivariate count response deep learning and neural network models, and our proposed copula deep learning and neural network models. This assessment involves employing metrics such as root mean square error (RMSE), weighted mean absolute percentage error (WMAPE), and mean absolute deviation (MAD). We include both copula-based asymmetrical zero-inflated simulated data and real-world data. We also propose a temporal dependence control chart for assessing the temporal dependence between bivariate zero-inflated count response variables. Our proposed copula deep learning and neural network temporal control charts can check time-varying dependence and outliers by leveraging the Shewhart statistical process control chart and t-copula ARMA-GARCH dynamic control correlation.

A flexible count regression model with varying precision

In this paper, we formulate mean and dispersion submodels for the class of Bell-Touchard regression models, and so we introduce the Bell-Touchard regression model with varying precision for count response variables. This regression model is, in some aspects, similar to the double generalized linear model framework, allowing for parameter interpretation in terms of the response variable in its original scale. We also provide an iterative process to estimate the regression coefficients in the mean and dispersion submodels, and develop diagnostic measures for this class of regression models. The methods developed are applied in a real data set for illustrative purposes.

High-dimensional covariate-augmented overdispersed poisson factor model.

The current Poisson factor models often assume that the factors are unknown, which overlooks the explanatory potential of certain observable covariates. This study focuses on high dimensional settings, where the number of the count response variables and/or covariates can diverge as the sample size increases. A covariate-augmented overdispersed Poisson factor model is proposed to jointly perform a high-dimensional Poisson factor analysis and estimate a large coefficient matrix for overdispersed count data. A group of identifiability conditions is provided to theoretically guarantee computational identifiability. We incorporate the interdependence of both response variables and covariates by imposing a low-rank constraint on the large coefficient matrix. To address the computation challenges posed by nonlinearity, two high-dimensional latent matrices, and the low-rank constraint, we propose a novel variational estimation scheme that combines Laplace and Taylor approximations. We also develop a criterion based on a singular value ratio to determine the number of factors and the rank of the coefficient matrix. Comprehensive simulation studies demonstrate that the proposed method outperforms the state-of-the-art methods in estimation accuracy and computational efficiency. The practical merit of our method is demonstrated by an application to the CITE-seq dataset. A flexible implementation of our proposed method is available in the R package COAP.

On Diagnostics in Bell--Touchard Regression Models

The Bell--Touchard regression model for count response variables was introduced recently in the statistical literature. This regression model for counts is, in some aspects, similar to the generalized linear model framework, and the mean response is related to covariates and unknown regression coefficients, which allows for parameter interpretation in terms of the response variable in its original scale. However, diagnostic methods for this class of regression models were not developed. In this paper, we fill this gap and propose a variety of diagnostic tools, such as leverage measures, global influence, normal curvatures of local influence under some perturbation schemes, and Pearson residuals. All diagnostic measures developed are applied in a real data set to analyze the fitted Bell--Touchard regression model in practice.

Kibria–Lukman estimator for the zero inflated negative binomial regression model: theory, simulation and applications

The zero inflated negative binomial model is an appropriate choice to model count response variables with excessive zeros and over-dispersion simultaneously. This article addresses the parameter estimation for the zero-inflated negative binomial model when there are many predictors and the problems of multicollinearity are present. Since, under multicollinearity, the widely used maximum likelihood estimator becomes unstable, this article proposes an alternative estimator, called Kibria–Lukman estimator for the zero inflated negative binomial model and provided some new biasing parameters. Then compared the performance of Kibria–Lukman estimator with some of the traditional biased estimators, that is, ridge and Liu estimators. The superiority of the proposed estimator is systematically scrutinized both theoretically and numerically. For numerical assessment, we conduct a Monte Carlo simulation study under different controlled factors. A numerical example is also applied to appraise the performance of proposed estimator. Based on the simulation and example results, we observed that the performance of the proposed estimator under different biasing parameters was better than that of the maximum likelihood estimator and other involved biased estimation methods when there exists a high but imperfect multicollinearity.

Bayesian Inference for a New Negative Binomial-Samade Model for Time Series Data Counts with Its Properties and Applications

A new distribution was developed that mixed the negative binomial (NB) and Samade distributions, called the negative binomial-Samade (NB-SA) distribution. The properties of this distribution were studied, and the newly created distribution was applied using the framework of generalized linear models to build a time series data count model. The characteristics of overdispersion and heavy-tailed distribution of the count response variables were applied in the actual dataset modeling. Distribution parameters and the regression coefficient were estimated using a Bayesian approach. Results showed that the NB-SA model had significantly the highest efficiency compared with the classical NB and Poisson models for analyzing factors influencing the daily number of COVID-19 deaths in Thailand.

Multivariate generalized linear mixed models for underdispersed count data

Researchers are often interested in understanding the relationship between a set of covariates and a set of response variables. To achieve this goal, the use of regression analysis, either linear or generalized linear models, is largely applied. However, such models only allow users to model one response variable at a time. Moreover, it is not possible to directly calculate from the regression model a correlation measure between the response variables. In this article, we employed the Multivariate Generalized Linear Mixed Models framework, which allows the specification of a set of response variables and calculates the correlation between them through a random effect structure that follows a multivariate normal distribution. We used the maximum likelihood estimation framework to estimate all model parameters using Laplace approximation to integrate out the random effects. The derivatives are provided by automatic differentiation. The outer maximization was made using a general-purpose algorithm such as PORT and Broyden–Fletcher–Goldfarb–Shanno algorithm (BFGS). We delimited this problem by studying count response variables with the following distributions: Poisson, negative binomial, Conway-Maxwell-Poisson (COM-Poisson), and double Poisson. While the first distribution can model only equidispersed data, the second models equi and overdispersed, and the third and fourth models all types of dispersion (i.e. including underdispersion). The models were implemented on software R with package TMB, based on C++ templates. Besides the full specification, models with simpler structures in the covariance matrix were considered (fixed and common variance, and ρ set to 0) and fixed dispersion. These models were applied to a dataset from the National Health and Nutrition Examination Survey, where two response variables are underdispersed and one can be considered equidispersed that were measured at 1281 subjects. The double Poisson full model specification overcame the other three competitors considering three goodness-of-fit measures: Akaike Information Criteria (AIC), Bayesian Information Criteria (BIC), and maximized log-likelihood. Consequently, it estimated parameters with smaller standard error and a greater number of significant correlation coefficients. Therefore, the proposed model can deal with multivariate count responses and measures the correlation between them taking into account the effects of the covariates.

Multivariate mixed Poisson Generalized Inverse Gaussian INAR(1) regression

In this paper, we present a novel family of multivariate mixed Poisson-Generalized Inverse Gaussian INAR(1), MMPGIG-INAR(1), regression models for modelling time series of overdispersed count response variables in a versatile manner. The statistical properties associated with the proposed family of models are discussed and we derive the joint distribution of innovations across all the sequences. Finally, for illustrative purposes different members of the MMPGIG-INAR(1) class are fitted to Local Government Property Insurance Fund data from the state of Wisconsin via maximum likelihood estimation.

On the mean-parameterized Bell–Touchard regression model for count data

The two-parameter Bell–Touchard distribution corresponds to a quite flexible yet tractable parametric family of discrete distributions, which arises prominently as an empirical model for count data. In this paper, we introduce a novel parametric regression model for count response variables on the basis of the mean-parameterized Bell–Touchard distribution, which allows interpretation of the regression coefficients in terms of the expectation of the count response variable, like the generalized linear model framework. The unknown parameters of the mean-parameterized Bell–Touchard regression model are estimated using the traditional maximum likelihood estimation procedure. We also consider the deviance residuals to assess departures from model assumptions. Empirical applications are provided to illustrate the mean-parameterized Bell–Touchard regression model in practice, and comparisons with some popular existing count regression models are made.

Improved shrinkage estimators in zero-inflated negative binomial regression model

‎Zero-inflated negative binomial model is an appropriate choice to model count response variables with excessive zeros and over-dispersion simultaneously. ‎This paper addressed parameter estimation in the zero-inflated negative binomial model when there are many parameters, ‎so that some of them have not influence on the response variable. ‎We proposed parameter estimation based on the linear shrinkage, ‎pretest, ‎shrinkage pretest, ‎Stein-type, ‎and positive Stein-type estimators. ‎We obtained the asymptotic distributional biases and risks of the suggested estimators theoretically. ‎‎We also conducted a Monte Carlo simulation study ‎to compare the performance of each estimator with the unrestricted estimator using simulated relative efficiency (SRE) criterion.‎‎ ‎The results reveal that the SREs of proposed estimators are higher than the unrestricted estimator. ‎The suggested estimators were applied to the wildlife fish data to appraise their performance.

Joint modeling of two count variables using a shared random effect model in the presence of clusters for complex data

In epidemiology, it is often the case that two or more correlated count response variables are encountered. Under this scenario, it is more efficient to model the data using a joint model. Besides, if one of these count variables has an excess of zeros (spike at zero) the log link cannot be used in general. The situation is more complicated when the data is grouped into clusters. A Generalized Linear Mixed Model (GLMM) is used to accommodate this cluster covariance. The objective of this research is to develop a new modeling approach that can handle this situation. The method is illustrated on a global data set of Covid 19 patients. The important conclusions are that the new model was successfully implemented both in theory and practice. A plot of the residuals indicated a well-fitting model to the data.

A zero altered Poisson random forest model for genomic-enabled prediction.

In genomic selection choosing the statistical machine learning model is of paramount importance. In this paper, we present an application of a zero altered random forest model with two versions (ZAP_RF and ZAPC_RF) to deal with excess zeros in count response variables. The proposed model was compared with the conventional random forest (RF) model and with the conventional Generalized Poisson Ridge regression (GPR) using two real datasets, and we found that, in terms of prediction performance, the proposed zero inflated random forest model outperformed the conventional RF and GPR models.

Parameter Estimation and Hypothesis Testing of Multivariate Poisson Inverse Gaussian Regression

Multivariate Poisson regression is used in order to model two or more count response variables. The Poisson regression has a strict assumption, that is the mean and the variance of response variables are equal (equidispersion). Practically, the variance can be larger than the mean (overdispersion). Thus, a suitable method for modelling these kind of data needs to be developed. One alternative model to overcome the overdispersion issue in the multi-count response variables is the Multivariate Poisson Inverse Gaussian Regression (MPIGR) model, which is extended with an exposure variable. Additionally, a modification of Bessel function that contain factorial functions is proposed in this work to make it computable. The objective of this study is to develop the parameter estimation and hypothesis testing of the MPIGR model. The parameter estimation uses the Maximum Likelihood Estimation (MLE) method, followed by the Newton–Raphson iteration. The hypothesis testing is constructed using the Maximum Likelihood Ratio Test (MLRT) method. The MPIGR model that has been developed is then applied to regress three response variables, i.e., the number of infant mortality, the number of under-five children mortality, and the number of maternal mortality on eight predictors. The unit observation is the cities and municipalities in Java Island, Indonesia. The empirical results show that three response variables that are previously mentioned are significantly affected by all predictors.

Three-level zero-inflated Conway–Maxwell–Poisson regression model for analyzing dispersed clustered count data with extra zeros

The count response variables are usually included of extra zeros. A useful tool for modeling such data is zero-inflated regression models. In the last decade, the Conway–Maxwell–Poisson model is applied for analyzing count data that can handle under- and over-dispersed data, besides that can encompass the Poisson and negative binomial. Sometimes, due to the sampling design or the data collection procedure, the data simultaneously are clustered or correlated with extra zeros and under- or over-dispersion. We applied a three-level zero-inflated Conway–Maxwell–Poisson regression model to overcome these problems. An expectation-maximization algorithm is used to estimate the model parameters of an appropriate penalized log-likelihood function. Model flexibility and finite-sample properties of this methodology have been investigated by extensive simulation study. The method has been illustrated with an application on real data in the health survey. Furthermore, we compared the results of the model with a three-level zero-inflated negative binomial regression model.

Analysis of mixed correlated bivariate zero-inflated count and (k, l)-inflated beta responses with application to social network datasets

ABSTRACTThis paper presents a new model that monitors the basic network formation mechanisms via the attributes through time. It considers the issue of joint modeling of longitudinal inflated (0, 1)-support continuous and inflated count response variables. For joint model of mentioned response variables, a correlated generalized linear mixed model is studied. The fraction response is inflated in two points k and l (k < l) and a k and l inflated beta distribution is introduced to use as its distribution. Also, the count response is inflated in zero and we use some members of zero-inflated power series distributions, hurdle-at-zero, members of zero-inflated double power series distributions and zero-inflated generalized Poisson distribution as our count response distribution. A full likelihood-based approach is used to yield maximum likelihood estimates of the model parameters and the model is applied to a real social network obtained from an observational study where the rate of the ith node’s responsiveness to the jth node and the number of arrows or edges with some specific characteristics from the ith node to the jth node are the correlated inflated (0, 1)-support continuous and inflated count response variables, respectively. The effect of the sender and receiver positions in an office environment on the responses are investigated simultaneously.

Survey of equine castration techniques, preferences and outcomes among Australian veterinarians.

(1) To collect the perceptions of veterinarians performing equine castrations in Australia on techniques, preferences and outcomes, (2) to investigate veterinarian use and experience with the Henderson castrating instrument and (3) to investigate potential associations between demographics, castration methods and techniques, and complications. Online survey of members of the Australian Veterinary Association's Special Interest Group, Equine Veterinarians Australia (EVA). A link to the survey was included in the EVA e-newsletter and practices on the EVA website were contacted by telephone and follow-up email. Fisher's exact test was used to determine associations between ligation and complications. A generalised linear model with a negative binomial family was used to determine associations between count response variables and categorical independent variables. Responses were obtained from 138 veterinarians (response rate, 13.1%) who performed 5330 castrations over 12 months. Castrations were most commonly performed in the field, on anaesthetised horses, using emasculators, via an open approach and without ligation of the spermatic cord. Estimated complications after use of emasculators were swelling (25%), haemorrhage (5%) and infection (5%). The Henderson instrument was used by approximately 10% of respondents and its use for castration was associated with fewer reports of postoperative swelling compared with emasculators (P = 0.002). Rates of evisceration with the Henderson and emasculator methods were comparable (0.43% and 0.9%, respectively). Castration preferences varied widely among survey participants. Reported complication types and rates were comparable to those reported previously in other countries. Perceptions that the Henderson instrument was associated with less swelling should be investigated further via a prospective controlled investigation.

Review of Zero-Inflated Models with Missing Data

The literature of count regression models covers a large scope of studies and applications that implemented simple and standard models for count response variables by using Poisson regression models, binomial regression models, negative binomial regression models, geometric regression models, or generalized Poisson regression models. These regression models have received considerable attention in various situations. Nevertheless in many fields, the distribution of the count response variable may display a feature of excess zeros for which the aforementioned regression models may fail to provide an adequate fit. To remedy this handicap, a class of distributions known as zero-inflated models is considered as the most appropriate approach for dealing properly with this issue of excess zeros. In addition to the zero-inflated problem, it happens quite often that the sample data sets under investigation are not completely observed. This refers to the missing data problem. In this study, our primary interest is in reviewing studies that considered simultaneously the missing data problem and the zero-inflated feature in modeling zero-inflated data. Moreover, we discuss their methodologies and results and some potential directions of the future research.

Multivariate Pascal Mixture Regression Models for Correlated Claim Frequencies

In this article, we propose a multivariate Pascal mixture regression model as an alternative to understand the association between multivariate count response variables and their covariates. When compared to the copula approach, this proposed class of regression models is not only less complex but can account for more versatile dependence structures and still allow for an intuitive explanation. We examine some of the properties possessed by this class of regression models and show its connections to several other models. For fitting purposes, we use the expectation-maximization (EM) algorithm which we find to be more effective and efficient. A by-product of this algorithm is that it provides for more reliable estimated standard errors of the regression coefficients useful for inference. Four different simulation studies are conducted to examine the performance of the fitting algorithm and the versatility of the proposed model while its applicability is additionally demonstrated by fitting an automobile insurance claim count dataset. All results are satisfactory and show that the proposed model can be a promising candidate for multivariate count regression modeling.

Modelling count response variables in informetric studies: Comparison among count, linear, and lognormal regression models

Abstract The purpose of the study is to compare the performance of count regression models to those of linear and lognormal regression models in modelling count response variables in informetric studies. Identified count response variables in informetric studies include the number of authors, the number of references, the number of views, the number of downloads, and the number of citations received by an article. Also of a count nature are the number of links from and to a website. Data were collected from the United States Patent and Trademark Office ( www.uspto.gov ), an open access journal ( www.informationr.net/ir/ ), Web of Science, and Maclean's magazine. The datasets were then used to compare the performance of linear and lognormal regression models with those of Poisson, negative binomial, and generalized Poisson regression models. It was found that due to over-dispersion in most response variables, the negative binomial regression model often seems to be more appropriate for informetric datasets than the Poisson and generalized Poisson regression models. Also, the regression analyses showed that linear regression model predicted some negative values for five of the nine response variables modelled, and for all the response variables, it performed worse than both the negative binomial and lognormal regression models when either Akaike's Information Criterion (AIC) or Bayesian Information Criterion (BIC) was used as the measure of goodness of fit statistics. The negative binomial regression model performed significantly better than the lognormal regression model for four of the response variables while the lognormal regression model performed significantly better than the negative binomial regression model for two of the response variables but there was no significant difference in the performance of the two models for the remaining three response variables.