Varieties of Count Data

The Poisson regression model is widely used for count data. This model assumes equidispersion. In practice, equidispersion is seldom reflected in data. However, in real-life data, the variance usually exceeds the mean. This situation is known as overdispersion. Negative binomial distribution and other Poisson mix models are often used to model overdispersion count data. Another extension of the negative binomial distribution in another model for count data is the univariate generalized Waring. In addition, the model developed by Famoye can be used in the analysis of count data. When the count data contains a large number of zeros, it is necessary to use zero-inflated models. In this study, different generalized regression models are emphasized for the analysis of excessive zeros count data. For this purpose, a real data set was analysed with the generalized Poisson model, generalized negative binomial model, generalized negative binomial Famoye, generalized Waring model, and the foregoing zero-inflated models. Log-likelihood, Akaike information criterion, Bayes information criterion, Vuong statistics were used for model comparisons.

Models of count with endogenous choices

Models of Count with Endogenous Choices

Paradigms for substance abuse cue-reactivity research involve pharmacological or stressful stimulation designed to elicit stress and craving responses in cocaine-dependent subjects. It is unclear as to whether stress induced from participation in such studies increases drug-seeking behavior. We propose a 2-state Hidden Markov model to model the number of cocaine abuses per week before and after participation in a stress-and cue-reactivity study. The hypothesized latent state corresponds to 'high' or 'low' use. To account for a preponderance of zeros, we assume a zero-inflated Poisson model for the count data. Transition probabilities depend on the prior week's state, fixed demographic variables, and time-varying covariates. We adopt a Bayesian approach to model fitting, and use the conditional predictive ordinate statistic to demonstrate that the zero-inflated Poisson hidden Markov model outperforms other models for longitudinal count data.

Count data are found in Ecological studies. The log-transformed strategy is commonly used in the count or rate data. The rate data are defined as the count data divided by a scale variable such as population at risk or an expected count. The log-transformed strategy is used to satisfy the parametric approach and simplify the model estimation. However, this strategy is not correct. The parameter estimation based on the log-transformed strategy could produce a biased estimate with a high standard error estimate. In this study, we are interested in evaluating the bias of parameter estimates based on the log-transformed strategy on the linear regression model. The generalized linear models have better performance in dealing with count data. However, some practitioners who are more familiar with the linear regression model prefer to use a log-transformed strategy and handle the zero cases by adding small values to zero observations. Simulation data from a Poisson distribution were used to compare the Poisson regression model and the linear regression model combined with the log-transformed strategy. The models were evaluated based on the bias and the root-mean-squared error statistics. We found that the linear regression with log-transformation strategy provided a high bias and a small value of root-mean-squared error, especially for small sample size and a small value of the count data. We also use real data set to explore more detail the uses of log-transformed strategy and compare it with Poisson regression.

Microorganisms play critical roles in human health and disease. They live in diverse communities in which they interact synergistically or antagonistically. Thus for estimating microbial associations with clinical covariates, such as treatment effects, joint (multivariate) statistical models are preferred. Multivariate models allow one to estimate and exploit complex interdependencies among multiple taxa, yielding more powerful tests of exposure or treatment effects than application of taxon-specific univariate analyses. Analysis of microbial count data also requires special attention because data commonly exhibit zero inflation, i.e., more zeros than expected from a standard count distribution. To meet these needs, we developed a Bayesian variable selection model for multivariate count data with excess zeros that incorporates information on the covariance structure of the outcomes (counts for multiple taxa), while estimating associations with the mean levels of these outcomes. Though there has been much work on zero-inflated models for longitudinal data, little attention has been given to high-dimensional multivariate zero-inflated data modeled via a general correlation structure. Through simulation, we compared performance of the proposed method to that of existing univariate approaches, for both the binary ("excess zero") and count parts of the model. When outcomes were correlated the proposed variable selection method maintained type I error while boosting the ability to identify true associations in the binary component of the model. For the count part of the model, in some scenarios the univariate method had higher power than the multivariate approach. This higher power was at a cost of a highly inflated false discovery rate not observed with the proposed multivariate method. We applied the approach to oral microbiome data from the Pediatric HIV/AIDS Cohort Oral Health Study and identified five (of 44) species associated with HIV infection.

The purpose of this chapter is to present recent advances in models for genomic-enabled prediction developed for ordinal categorical and count data. For both models we provide details of their corresponding derivation and then apply them to a real data set. The proposed models were derived using a Bayesian framework. Bayesian logistic ordinal regression (BLOR) and Bayesian negative binomial regression (BNBR) make use of the Polya-Gamma distribution to produce an analytic Gibbs, a sampler with similar full conditional distributions of a model with Gaussian response and can be used for complex data sets as those that arise in the context of genomic selection where the sample size usually is smaller than the number of covariates (markers). We illustrate the proposed models using simulation and a real data set. Results indicate that our models for ordinal categorical and count data are a good alternative for analyzing ordinal and count data in the context of genomic-enabled prediction.

Count data, including zero counts arise in a wide variety of application, hence models for counts have become widely popular in many fields. In the statistics field, one may define the count data as that type of observation which takes only the non-negative integers value. Sometimes researchers may Counts more zeros than the expected. Excess zero can be defined as Zero-Inflation. Data with abundant zeros are especially popular in health, marketing, finance, econometric, ecology, statistics quality control, geographical, and environmental fields when counting the occurrence of certain behavioral and natural events, such as frequency of alcohol use, take drugs, number of cigarettes smoked, the occurrence of earthquakes, rainfall, and etc. Some models have been used to analyzing count data such as the zero-inflated Poisson (ZIP) model and the negative binomial model. In this paper, the models, Poisson, Negative Binomial, ZIP, and ZINB were been used to analyze rainfall data.

The Poisson regression model is an important analytic tool that should be used in count data modeling. When the value of excess dispersion of variables, then the model is not appropriate to apply in case the value of the mean is not equal to the value of the variance of the Poisson distribution. The results are compatible with data when there is the use of Bell regression model. The number of zeros in the count data that is seen is very high. In this case, the Zero-Inflated Bell regression model is an alternative to the Bell regression model. Parameters of the Zero-Inflated Bell regression model are estimated mostly through the approach of maximum likelihood. In an extended linear model, in which the response variable is modeled by two or more explanatory variables, as in the Zero-Inflated Bell regression model, linear dependence is a threat in a real-life analysis. It reduced the maximum likelihood estimator in its effectiveness. In a bid to solve this issue, this paper explores the performance of the generalized shrinkage estimator in the zero-inflated Bell regression model. The superiority of the proposed approaches over the traditional maximum likelihood estimator is validated by results of the simulations and implementations.

Overdispersion is common in models of count data in ecology and evolutionary biology, and can occur due to missing covariates, non-independent (aggregated) data, or an excess frequency of zeroes (zero-inflation). Accounting for overdispersion in such models is vital, as failing to do so can lead to biased parameter estimates, and false conclusions regarding hypotheses of interest. Observation-level random effects (OLRE), where each data point receives a unique level of a random effect that models the extra-Poisson variation present in the data, are commonly employed to cope with overdispersion in count data. However studies investigating the efficacy of observation-level random effects as a means to deal with overdispersion are scarce. Here I use simulations to show that in cases where overdispersion is caused by random extra-Poisson noise, or aggregation in the count data, observation-level random effects yield more accurate parameter estimates compared to when overdispersion is simply ignored. Conversely, OLRE fail to reduce bias in zero-inflated data, and in some cases increase bias at high levels of overdispersion. There was a positive relationship between the magnitude of overdispersion and the degree of bias in parameter estimates. Critically, the simulations reveal that failing to account for overdispersion in mixed models can erroneously inflate measures of explained variance (r2), which may lead to researchers overestimating the predictive power of variables of interest. This work suggests use of observation-level random effects provides a simple and robust means to account for overdispersion in count data, but also that their ability to minimise bias is not uniform across all types of overdispersion and must be applied judiciously.

This paper discusses the problem of variance specification in models for event count data. Event counts are dependent variables that can take on only nonnegative integer values, such as the number of wars or coups d'etat in a year. I discuss several generalizations of the Poisson regression model, presented in King (1988), to allow for substantively interesting stochastic processes that do not fit into the Poisson framework. Individual models that cope with, and help analyze, heterogeneity, contagion. and negative contagion are each shown to lead to specific statistical models for event count data. In addition. I derive a new generalized event count (GEC) model that enables researchers to extract significant amounts of new information from existing data by estimating features of these unobserved substantive processes. Applications of this model to congressional challenges of presidential vetoes and superpower conflict demonstrate the dramatic advantages of this approach.

IntroductionCount data are often collected in chronic disease research, and sometimes these data have a skewed distribution. The number of unhealthy days reported in the Behavioral Risk Factor Surveillance System (BRFSS) is an example of such data: most respondents report zero days. Studies have either categorized the Healthy Days measure or used linear regression models. We used alternative regression models for these count data and examined the effect on statistical inference.MethodsUsing responses from participants aged 35 years or older from 12 states that included a homeownership question in their 2009 BRFSS, we compared 5 multivariate regression models — logistic, linear, Poisson, negative binomial, and zero-inflated negative binomial — with respect to 1) how well the modeled data fit the observed data and 2) how model selections affect inferences.ResultsMost respondents (66.8%) reported zero mentally unhealthy days. The distribution was highly skewed (variance = 58.7, mean = 3.3 d). Zero-inflated negative binomial regression provided the best-fitting model, followed by negative binomial regression. A significant independent association between homeownership and number of mentally unhealthy days was not found in the logistic, linear, or Poisson regression model but was found in the negative binomial model. The zero-inflated negative binomial model showed that homeowners were 24% more likely than nonowners to have excess zero mentally unhealthy days (adjusted odds ratio, 1.24; 95% confidence interval, 1.08–1.43), but it did not show an association between homeownership and the number of unhealthy days.ConclusionOur comparison of regression models indicates the importance of examining data distribution and selecting models with appropriate assumptions. Otherwise, statistical inferences might be misleading.

The Conway–Maxwell–Poisson (COMP) model is one of the count data regression models used in over- and underdispersion cases. Thus, COMP regression is a flexible model in the count data models. In the regression analysis, when the explanatory variables are correlated with each other, multicollinearity exists; this inflates the standard error of the maximum likelihood estimates. To handle the effect of multicollinearity, we proposed a new modified Liu estimator for the COMP regression model based on two shrinkage parameters. This estimator is proposed to reduce the effect of multicollinearity on the standard error of the estimates. To evaluate the performance of the proposed estimator, the mean squared error (MSE) criterion is employed. Theoretical comparison of the proposed estimator with existing estimators (ridge, Liu, and modified one-parameter Liu estimators) is made. The results of the simulation study and real-life application indicate the superiority of the proposed estimator.

The community may contain a large percentage of zero values that cause the community distribution to move away from zero, and this group is referred to as not following a normal distribution, so one of the conditions of the linear regression models is permeated. This type of society can be seen in many applications such as insurance, meteorology, auditing, environment, and manufacturing. The zero-community number is often analyzed via a two-part admixture model: The first part is probabilistic from zero and the second part is regular with a specific probability distribution. Problems of confidence estimation of the zero-classifier population mean under normal models have been present in research. Regression models have also been developed for the zero population groups. However, many of these models are aimed at counting data, although regression models with responses of a continuous type can be seen in application quite often. Moreover, these regression models for homeless populations do not address situations in which the data available for analysis were obtained through complex probability sampling designs. Different statistical methods and models have been developed for the statistical analysis of such population. Based on the current research, most of the special studies focus on estimating the population mean and developing regression models. This dissertation will also focus on developing regression models. This dissertation will also focus on developing regression models. Most of the regression models developed for the null population found in research have given more attention to population data in which observations can take only nonnegative integer values that arise from counting rather than ordering. They also use maximum possibility methods and pseudo greatest possibility methods to estimate expected responses in Value Variable/ future variables.

BackgroundDiabetes is one of the top four non-communicable diseases that cause death and illness to many people around the world. This study aims to use an efficient count data model to estimate socio-environmental factors associated with diabetes incidences in Tanzania mainland, addressing lack of evidence on the efficient count data model for estimating factors associated with disease incidences disparities.MethodsThis study analyzed diabetes counts in 184 Tanzania mainland councils collected in 2020. The study applied generalized Poisson, negative binomial, and Poisson count data models and evaluated their adequacy using information criteria and Pearson chi-square values.ResultsThe data were over-dispersed, as evidenced by the mean and variance values and the positively skewed histograms. The results revealed uneven distribution of diabetes incidence across geographical locations, with northern and urban councils having more cases. Factors like population, GDP, and hospital numbers were associated with diabetes counts. The GP model performed better than NB and Poisson models.ConclusionThe occurrence of diabetes can be attributed to geographical locations. To address this public health issue, environmental interventions can be implemented. Additionally, the generalized Poisson model is an effective tool for analyzing health information system count data across different population subgroups.