Poisson Inverse Gaussian Regression

Background: Dengue Hemorrhagic Fever (DHF) is a major health problem in East Java Province due to population density, sanitation, availability of medical personnel, and altitude. DHF count data often experiences overdispersion, challenging the Poisson regression assumption in the Generalized Linear Model (GLM). Objective: To compare the performance of Poisson, Negative Binomial (NBR), and Poisson Inverse Gaussian (PIG) ​​models in modeling DHF cases. Method: A cross-sectional quantitative study analyzed 38 districts/cities in East Java (saturation sampling) using secondary data processed with . Analysis: Descriptive statistics, multicollinearity test (VIF<10), overdispersion test, and model comparison via AIC/deviance. Results: NBR achieved the lowest AIC (412.10) compared to Poisson (1466.60) and PIG (412.41), confirming significant overdispersion (deviance/df ≈37.16). Population density and altitude had a significant effect (p<0.05). Conclusion: NBR is optimal for DHF modeling, supporting targeted environmental and health interventions in East Java.

More than 60% of accidents in Iran occur within 30 kilometers of cities entrance roads. Therefore the number of accidents per kilometer in these regions, in contrast to the other parts of roads is very considerable. The city of Tehran, as the capital of Iran, is the cross point of major arterials of passenger and freight transportation. Thus the evaluation of road safety, entering and exiting the city in the 30 kilometers buffer, can be considered to improve countrywide road safety. To investigate the effects of roadway characteristics and traffic conditions, including number of traffic violations, on accident frequency within 30 km of Tehran entrance roads, Poisson and Negative Binomial regression model were calibrated in this study. Although the goodness of fit index in Poisson model is much greater than the Negative Binomial model, the accident data dispersion in this study implies that Negative Binomial model is more appropriate to show the influences of effective variables. According to results of modeling, the variables including average daily volume, share of heavy vehicles, average daily speed violations and number of direct accesses and interchanges have effective influence on accident frequencies. Controlling access to roads and using effective actions to reduce the speed and number of speed violation can be accounted as effective policies to reduce accident frequencies within 30 km of Tehran entrance roads.

Statistical modelling for falls count data

The foliage in a fully-leafed deciduous forest canopy is clumped. Consequently, theory indicates that the probability of beam penetration will be estimated more accurately with a model based on the negative binomial distribution than with a model based on the Poisson distribution, incorporating an assumption of a spherical leaf inclination angle distribution. Flux densities of photosynthetically active radiation (PAR) were measured in and above a deciduous forest canopy and were computed with the canopy radiative transfer models based on the negative binomial and Poisson distributions. These radiation values were used to compute canopy photosynthesis and stomatal conductance. Canopy photosynthesis and stomatal conductance, based on the negative binomial model, overestimated values computed from measured PAR profiles by 8 and 9%. respectively. The canopy photosynthesis and stomatal conductance values computed with the spherical Poisson model under-estimated measured values by 17 and 10%, respectively. Thus, the negative binomial radiative transfer model improves estimates of canopy photosynthesis and, to a lesser extent, stomatal conductance, inside a deciduous forest.

The scaling relation between the size of a galaxy’s globular cluster (GC) population (N GC) and the galaxy’s stellar mass (M *) is usually described with a continuous, linear model, but in reality it is a count relationship that should be modeled as such. For massive galaxies, a negative binomial (NB) model has been shown to describe the data well, but it is unclear how the scaling relation behaves at low galaxy masses where a substantial portion of galaxies have N GC = 0. In this work, we test the utility of Poisson and NB models for describing the low-mass end of the N GC−M * scaling relation. We introduce the use of zero-inflated versions of these models, which allow for larger zero populations (e.g., galaxies without GCs) than would otherwise be predicted. We evaluate our models with a variety of predictive model comparison methods, including predictive intervals, the leave-one-out cross-validation criterion, and posterior predictive comparisons. We find that the NB model is consistent with our data, but the naive Poisson is not. Moreover, we find that zero inflation of the models is not necessary to describe the population of low-mass galaxies that lack GCs, suggesting that a single formation and evolutionary process acts over all galaxy masses. Under the NB model, there does not appear to be anything unique about the lack of GCs in many low-mass galaxies; they are simply the low-mass extension of the larger N GC−M * scaling relation.

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.

Count data are most commonly modeled using the Poisson model, or by one of its many extensions. Such extensions are needed for a variety of reasons: (1) a hierarchical structure in the data, e.g., due to clustering, the collection of repeated measurements of the outcome, etc.; (2) the occurrence of overdispersion (or underdispersion), meaning that the variability encountered in the data is not equal to the mean, as prescribed by the Poisson distribution; and (3) the occurrence of extra zeros beyond what a Poisson model allows. The first issue is often accommodated through the inclusion of random subject-specific effects. Though not always, one conventionally assumes such random effects to be normally distributed. Overdispersion is often dealt with through a model developed for this purpose, such as, for example, the negative-binomial model for count data. This can be conceived through a random Poisson parameter. Excess zeros are regularly accounted for using so-called zero-inflated models, which combine either a Poisson or negative-binomial model with an atom at zero. The novelty of this article is that it combines all these features. The work builds upon the modelling framework defined by Molenberghs et al. ( 2010 ) in which clustering and overdispersion are accommodated for through two separate sets of random effects in a generalized linear model.

Generalized Extreme Value models for count data: Application to worker telecommuting frequency choices

Crash prediction based on random effect negative binomial model considering data heterogeneity

Modeling Techniques for Fitting Healthcare Resource Use Costs Derived From Patient Reported Counts of Hospital, ER, And PCP Visits

Analysis of road accidents in two mixed industrial urban zones, using nested Poisson and Negative Binomial models

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.

This study aimed to estimate the economic value of the recreational fishing in the Jeonnam marine ranching area as a part of the total socioeconomic evaluation of the Jeonnam marine ranching program. A travel cost method was applied to the estimation of economic value of the recreational fishing in the Jeonnam marine ranching area and input variables included annual fishing trip days, average travel cost per trip, average catch amount, monthly income, marriage, age, and personal perception on the marine ranching program. In the analysis, due to its characteristic of count data, both poisson model and negative binomial model were used. Model results indicated that a negative binomial model was statistically more suitable than the poisson model as the overdispersion problem occurred in the poisson model. All signs of the estimated parameters were estimated as previous studies showed. Based on the results, the economic value per trip of the recreational fishing in the Jeonnam marine ranching area was estimated to be 145,000 won and the annual total economic value of the recreational fishing in the Jeonnam marine ranching area was analyzed to be 2,514,000 won. In addition, the change of total value by catch rate showed that the economic value could be increased by 180,900 won as the catch increased by one kilogram.

I previously indicated that extended Poisson and negative binomial models have generally been developed to solve either a distributional or variance problem arising in the base Poisson and NB2 models. We shall discover later that some extended negative binomial models rely on the NB1 parameterization rather than on the NB2, e.g. fixed-effects negative binomial. But, except for a few models, most negative binomial models used in actual research are based on the NB2 model. Changes to the negative binomial variance function were considered in the last chapter. In this chapter, we address the difficulties arising when the zero counts in Poisson and NB2 models differ considerably from model distributional assumptions. In the first section we address models that cannot have zero counts. Zero-truncated count models Many times we are asked to model count data that structurally exclude zero counts. Hospital length-of-stay data are an example of this type of data. When a patient first enters the hospital, the count begins upon registration, with the length of stay given as 1. There can be no 0 days – unless we are describing patients who do not enter the hospital. This latter situation describes a different type of model where there may be two generating processes – one, for example, for patients who may or may not enter the hospital, and another for patients who are admitted. This type of model will be discussed later.

The development of models and methods for cure rate estimation has recently burgeoned into an important subfield of survival analysis. Much of the literature focuses on the standard mixture model. Recently, process-based models have been suggested. We focus on several models based on first passage times for Wiener processes. Whitmore and others have studied these models in a variety of contexts. Lee and Whitmore (Stat Sci 21(4):501-513, 2006) give a comprehensive review of a variety of first hitting time models and briefly discuss their potential as cure rate models. In this paper, we study the Wiener process with negative drift as a possible cure rate model but the resulting defective inverse Gaussian model is found to provide a poor fit in some cases. Several possible modifications are then suggested, which improve the defective inverse Gaussian. These modifications include: the inverse Gaussian cure rate mixture model; a mixture of two inverse Gaussian models; incorporation of heterogeneity in the drift parameter; and the addition of a second absorbing barrier to the Wiener process, representing an immunity threshold. This class of process-based models is a useful alternative to the standard model and provides an improved fit compared to the standard model when applied to many of the datasets that we have studied. Implementation of this class of models is facilitated using expectation-maximization (EM) algorithms and variants thereof, including the gradient EM algorithm. Parameter estimates for each of these EM algorithms are given and the proposed models are applied to both real and simulated data, where they perform well.