A Generalized Negative Binomial Distribution

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.

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.

Negative binomial group distribution was proposed in the literature which was motivated by inverse sampling when considering group inspection: products are inspected group by group, and the number of non-conforming items of a group is recorded only until the inspection of the whole group is finished. The non-conforming probability p of the population is thus the parameter of interest. In this paper, the confidence interval construction for this parameter is investigated. The common normal approximation and exact method are applied. To overcome the drawbacks of these commonly used methods, a composite method that is based on the confidence intervals of the negative binomial distribution is proposed, which benefits from the relationship between negative binomial distribution and negative binomial group distribution. Simulation studies are carried out to examine the performances of our methods. A real data example is also presented to illustrate the application of our method.

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

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

There are several parametric models to describe the charged particle multiplicity distributions in literature. Among them, log-normal distribution and negative binomial distribution are the most well-known parametric models. In this study, negative binomial and log-normal distributions are compared and tested in neutrino interactions produced in DONUT experiment. This analysis was carried out using the real experimental results of the DONUT Collaboration. The results of the analysis show that the neutrino data, reported by DONUT collaboration, is well described by negative binomial and log-normal distribution. In terms of fit parameters, the χ/ndf value of the both distribution NBD and LND are close to unity but log-normal distribution data slightly better than negative binomial distribution.

We estimate the quantum state of a qubit and a quantized radiation field yielding a generalized negative binomial distribution (GNBD). We give an explicit form for various generalized negative binomial states associated to superposition, even, odd, and q-deformed states. We investigate the dynamical properties of the Mandel parameter as a quantifier of the statistical properties for the radiation field corresponding to its dynamics. We obtain the quantum Fisher information based on the estimation of the atomic state and compare it with the Mandel parameter for different instances of the GNBD. The link between the statistical quantities for different parameters of the GNBD is explored.

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.

BackgroundRNA sequencing (RNA-seq) is a powerful tool for genome-wide expression profiling of biological samples with the advantage of high-throughput and high resolution. There are many existing algorithms nowadays for quantifying expression levels and detecting differential gene expression, but none of them takes the misaligned reads that are mapped to non-exonic regions into account. We developed a novel algorithm, XBSeq, where a statistical model was established based on the assumption that observed signals are the convolution of true expression signals and sequencing noises. The mapped reads in non-exonic regions are considered as sequencing noises, which follows a Poisson distribution. Given measureable observed and noise signals from RNA-seq data, true expression signals, assuming governed by the negative binomial distribution, can be delineated and thus the accurate detection of differential expressed genes.ResultsWe implemented our novel XBSeq algorithm and evaluated it by using a set of simulated expression datasets under different conditions, using a combination of negative binomial and Poisson distributions with parameters derived from real RNA-seq data. We compared the performance of our method with other commonly used differential expression analysis algorithms. We also evaluated the changes in true and false positive rates with variations in biological replicates, differential fold changes, and expression levels in non-exonic regions. We also tested the algorithm on a set of real RNA-seq data where the common and different detection results from different algorithms were reported.ConclusionsIn this paper, we proposed a novel XBSeq, a differential expression analysis algorithm for RNA-seq data that takes non-exonic mapped reads into consideration. When background noise is at baseline level, the performance of XBSeq and DESeq are mostly equivalent. However, our method surpasses DESeq and other algorithms with the increase of non-exonic mapped reads. Only in very low read count condition XBSeq had a slightly higher false discovery rate, which may be improved by adjusting the background noise effect in this situation. Taken together, by considering non-exonic mapped reads, XBSeq can provide accurate expression measurement and thus detect differential expressed genes even in noisy conditions.

Background: In multiple sclerosis (MS) studies, the most appropriate model for the distribution of the number of relapses was shown to be the negative binomial (NB) distribution. Objective: To determine whether the sample-size estimation (SSE) and the analysis of annualized relapse rates (ARRs) in randomized controlled trials (RCTs) were aligned and compare the SSE between normal and NB distributions. Methods: Systematic review of phase 3 and 4 RCTs for which the primary endpoint was ARR in relapsing remitting MS published since 2008 in pre-selected major medical journals. A PubMed search was performed on 30 November 2020. We checked whether the SSE and ARR analyses were congruent. We also performed standardized (fixed α/β, number of arms and overdispersion) SSEs using data collected from the studies. Results: Twenty articles (22 studies) were selected. NB distribution (or quasi-Poisson) was used for SSE in only 7/22 studies, whereas 21/22 used it for ARR analyses. SSE relying on NB regression necessitated a smaller sample size in 21/22 of our calculations. Conclusion: SSE was rarely performed using the most appropriate model. However, the use of an NB model is recommended to optimize the number of included patients and to be congruent with the final analysis.

Sequence count data are commonly modelled using the negative binomial (NB) distribution. Several empirical studies, however, have demonstrated that methods based on the NB-assumption do not always succeed in controlling the false discovery rate (FDR) at its nominal level. In this paper, we propose a dedicated statistical goodness of fit test for the NB distribution in regression models and demonstrate that the NB-assumption is violated in many publicly available RNA-Seq and 16S rRNA microbiome datasets. The zero-inflated NB distribution was not found to give a substantially better fit. We also show that the NB-based tests perform worse on the features for which the NB-assumption was violated than on the features for which no significant deviation was detected. This gives an explanation for the poor behaviour of NB-based tests in many published evaluation studies. We conclude that nonparametric tests should be preferred over parametric methods.

Characterizing the empirical distribution of the frequency of n-mers is a vital step in understanding the entire genome. This will allow for researchers to examine how complex the genome really is, and move beyond simple, traditional modeling frameworks that are often biased in the presence of abundant and/or extremely rare words. We hypothesize that models based on the negative binomial distribution and its zero-inflated counterpart will characterize the n-mer distributions of genomes better than the Poisson. Our study examined the empirical distribution of the frequency of n-mers (6 ≤ n ≤ 11) in 2,199 genomes. We considered four distributions: Poisson, negative binomial, zero-inflated Poisson, and zero-inflated negative binomial (ZINB). The number of genomes that have nullomers in 6-, 7-, and 8-mers was 150, 602 and 2,012, respectively, whereas all of the genomes for the 9-, 10-, and 11-mers had nullomers. In each n-mer considered, the negative binomial model performed the best for at least 93% of the 2,199 genomes; however, a small percentage (i.e., <7%) of the genomes did prefer the ZINB. The negative binomial and zero-inflation distributions extend the traditional Poisson setting and are more flexible in handling overdispersion that can be caused by an increase in nullomers. In an effort to characterize the distribution of the frequency of n-mers, researchers should also consider other discrete distributions that are more flexible and adjust for possible overdispersion.

This paper introduced a new mixed distribution, namely the negative binomial-generalized exponential (NB-GE) distribution, which it obtained by mixing the negative binomial (NB) distribution with a generalized exponential (GE) distribution. The NB-GE distribution offers the advantage of being able to handle this kind of data sets, while still maintaining similar characteristics as the traditional negative binomial. The closed form and the factorial moment of the NB-GE distribution are derived. The NB-GE distribution appropriate when we find the one related to count data which have a large number of zeros and overdispersion in Poisson distribution. In addition, we present the basic properties of the new distribution such as mean, variance, skewness and kurtosis. Including, the negative binomial-exponential distributions are presented as special cases of this NB-GE distribution. Parameters estimation is also implemented using maximum likelihood method and the usefulness of the NB-GE distribution is illustrated by real data set. When using a goodness of fit test, the results show that the NB-GE distribution can provide a better fit than the NB distribution and Poisson distribution for count data that contain a large number of zeros.

&amp;lt;p&amp;gt;Modeling seismicity data is challenging and it remains a subject of ongoing research. Assumptions about the distribution of earthquake numbers play an important role in seismic hazard and risk analysis. The most common distribution that has been widely used in modeling earthquake numbers is the Poisson distribution because of its simplicity and easy to use. However, the heterogeneity in earthquake data and temporal dependencies that are often present in many real earthquake sequences make the use of the Poisson distribution inadequate. So, we propose the use of a Hidden Markov model (HMM) with state-specific Negative Binomial distributions in which some states are allowed to approach the Poisson distribution. A HMM is a generalization of a mixture model where the different unobservable (hidden) states are related through a Markov process rather than being independent of each other. We parameterize the Negative Binomial distribution in terms of the mean and dispersion (clustering) parameter. Maximum likelihood estimates of the models&amp;amp;#8217; parameters are obtained through an Expectation-Maximization algorithm (EM-algorithm).&amp;lt;/p&amp;gt;&amp;lt;p&amp;gt;We apply the model to real earthquake data. We have selected the area of Killini Western Greece to test the proposed hypothesis. The area of Killini has been selected based on the fact that in a time window of 17 years three clusters of seismicity associated with strong mainshocks are included in the catalog. Application of the model to the data resulted in three states, representing different levels of seismicity (low, medium, high). The state that corresponds to the low seismicity level approaches the Poisson distribution while the other two states (medium and high) are following the Negative Binomial distribution. This result complies with the nature of the data. The variation within each state that is introduced to the model by the Negative Binomial distribution is greater in the states of medium and high seismicity.&amp;amp;#160;&amp;lt;/p&amp;gt;

Statistical modelling for falls count data