Bayesian Analysis of RNA-Seq Data Using a Family of Negative Binomial Models.

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.

Methodologic Challenges in the Analysis of Count Data in Radiology Health Services Research

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

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.

A semiparametric negative binomial generalized linear model for modeling over-dispersed count data with a heavy tail: Characteristics and applications to crash data

Application of the Conway–Maxwell–Poisson generalized linear model for analyzing motor vehicle crashes

Identification of crash hot spots is the critical component of the highway safety management process. Errors in hot spot identification (HSID) may result in the inefficient use of resources for safety improvements. One HSID method that is based on the empirical Bayesian (EB) method has been widely used as an effective approach for identifying crash-prone sites. For the EB method, the negative binomial (NB) model is usually needed to obtain the EB estimates. Recently, some studies have shown that the Sichel (SI) model can be easily used in the EB modeling framework and potentially yield better EB estimates. The objective of this study was to compare the performance of the two crash prediction (SI and NB) models in identifying hot spots with the EB method. To accomplish the objective of this study, empirical crash data collected at highway segments in Texas were used to generate simulated crash counts. Three commonly used HSID methods (simple ranking, confidence interval, and EB) were applied with the use of simulated data. False positives, false negatives, and false identifications were calculated and compared across the methods. The simulation results in this study suggested that the SI-based EB method could consistently provide a better HSID result than the NB-based EB method. Moreover, EB methods yielded the lowest error percentage of the three HSID methods. This study confirmed that the EB technique was an effective method for identifying hazardous sites. On the basis of the findings in this study, it is recommended that transportation safety researchers consider the SI model as an alternative crash prediction model when the EB approach is used.

If X 1, X 2, … is a sequence of independent Bernoulli random variables, the number of successes in the first n trials has a binomial distribution and the number of failures before the rth success has a negative binomial distribution. From both the binomial and the negative binomial distributions, the Poisson distribution is obtainable as a limit. Moreover, gamma distributions (integer shape parameters) are limits of negative binomial distributions, and the normal distribution is a limit of negative binomial, Poisson, and gamma distributions. These basic facts from elementary probability have natural extensions to two dimensions because there is a unique natural bivariate Bernoulli distribution. In this article, such extensions yielding a family of bivariate distributions are obtained and studied.

Modeling over-dispersed crash data with a long tail: Examining the accuracy of the dispersion parameter in Negative Binomial models

Analyzing angle crashes at unsignalized intersections using machine learning techniques

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.

Developing a Random Parameters Negative Binomial-Lindley Model to analyze highly over-dispersed crash count data

Estimating Retail Demand with Poisson Mixtures and Out-of-Sample Likelihood

The question of how to characterize the bacterial density in a body of water when data are available as counts from a number of small-volume samples was examined for cases where either the Poisson or negative binomial probability distributions could be used to describe the bacteriological data. The suitability of the Poisson distribution when replicate analyses were performed under carefully controlled conditions and of the negative binomial distribution for samples collected from different locations and over time were illustrated by two examples. In cases where the negative binomial distribution was appropriate, a procedure was given for characterizing the variability by dividing the bacterial counts into homogeneous groups. The usefulness of this procedure was illustrated for the second example based on survey data for Lake Erie. A further illustration of the difference between results based on the Poisson and negative binomial distributions was given by calculating the probability of obtaining all samples sterile, assuming various bacterial densities and sample sizes.