The negative binomial-generalized exponential (NB-GE) distribution

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.

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.

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

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.

With the advance of new sequencing technologies producing massive short reads data, metagenomics is rapidly growing, especially in the fields of environmental biology and medical science. The metagenomic data are not only high dimensional with large number of features and limited number of samples but also complex with a large number of zeros and skewed distribution. Efficient computational and statistical tools are needed to deal with these unique characteristics of metagenomic sequencing data. In metagenomic studies, one main objective is to assess whether and how multiple microbial communities differ under various environmental conditions. We propose a two-stage statistical procedure for selecting informative features and identifying differentially abundant features between two or more groups of microbial communities. In the functional analysis of metagenomes, the features may refer to the pathways, subsystems, functional roles and so on. In the first stage of the proposed procedure, the informative features are selected using elastic net as reducing the dimension of metagenomic data. In the second stage, the differentially abundant features are detected using generalized linear models with a negative binomial distribution. Compared with other available methods, the proposed approach demonstrates better performance for most of the comprehensive simulation studies. The new method is also applied to two real metagenomic datasets related to human health. Our findings are consistent with those in previous reports. R code and two example datasets are available at http://cals.arizona.edu/∼anling/software.htm. Supplementary file is available at Bioinformatics online.

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

Sequential negative binomial problems and statistical ecology: A selected review with new directions

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.

Some Thoughts About Data Type, Distribution, and Statistical Significance

Multinomial -mixture models are commonly used to fit data from a removal sampling protocol. If the mixing distribution is negative binomial, the distribution of the counts does not appear to have been identified, and practitioners approximate the requisite likelihood by placing an upper bound on the embedded infinite sum. In this paper, the distribution which underpins the multinomial -mixture model with a negative binomial mixing distribution is shown to belong to the broad class of multivariate negative binomial distributions. Specifically, the likelihood can be expressed in closed form as the product of conditional and marginal likelihoods and the information matrix shown to be block diagonal. As a consequence, the nature of the maximum likelihood estimates of the unknown parameters and their attendant standard errors can be examined and tests of the hypothesis of the Poisson against the negative binomial mixing distribution formulated. In addition, appropriate multinomial -mixture models for data sets which include zero site totals can also be constructed. Two illustrative examples are provided.

The Negative Binomial distribution (NBD) is used for modeling many types of count data, including gene expression counts obtained by RNA sequencing technologies (RNA-Seq). Finding outliers in this type of data has been shown in recent research to help in identifying rare genetic disorders in humans. Existing Bayesian approaches to detecting outliers in data following the NBD are either computationally inefficient or too general and hence do not leverage the NBD’s specificities in an optimal way. We present a novel Bayesian outlier score for data following the NBD, relying on recent advances in the inference of its dispersion parameter through a special method of Gibbs sampling. The novel Bayesian model on which our score is based — OutPyRX (Outlier detection in Python for RNA-Seq, eXtended version) — improves the model of its predecessor OutPyR by introducing novel parameters that are derived from OutPyR’s. These novel parameters allow more than 6 times faster convergence of the novel outlier score compared to OutPyR’s while having a negligible computational impact on the Gibbs sampling procedure. We show that, in terms of area under precision-recall curve (AUC) values, the novel score outcompetes existing scores on 21 out of 24 datasets that we derived from 4 real datasets by injecting artificial outliers. However, OutPyRX does not perform confounder control which is required for some datasets containing biological outliers. The model is general and can be applied to other similar count data. The code for our model is available at https://github.com/esalkovic/outpyrx .

The extended Poisson process model is a generalisation of the simple Poisson process. It allows for the construction of distributions that can be over-dispersed or under-dispersed relative to the Poisson distribution. The modelling includes the Poisson and negative binomial distributions as special cases. Generally, a broad class of dispersion models can be characterised in an intuitive and appealing way by a simple parameterisation. By incorporating transition modelling techniques the extended Poisson process model can be used to model longitudinal count data. The technique is illustrated by analysing the epilepsy data in Thall and Vail (1990). Comparing a number of analyses of the data shows that some covariates have a more significant effect using this modelling than from using mixed Poisson models. It is argued that this could be due to the mixed Poisson models used in the other analyses not providing an appropriate description of the residual variation, with the greater flexibility of the extended Poisson process model generally enabling more critical assessment of covariate effects than more standard mixed Poisson modelling. The extended Poisson process model is also applied to data from a repeated trawl study in the Great Barrier Reef Marine Park conducted by CSIRO Marine Research and the Queensland Department of Primary Industries. Prawn trawling is a major fishing activity in the Great Barrier Reef Marine Park, generating millions of dollars for the Australian economy. It is important that appropriate management decisions are made to ensure the sustainability of the industry. The study was conducted to quantify the recovery of seabed fauna after repeated trawling. Data from the study were composed of counts of observed benthic organisms on the seabed. It was found that the extended Poisson process model more faithfully described the data than more standard models such as those based on Poisson or negative binomial distribution. It is argued that the use of inappropriate models can result in unreliable inferences, which in turn could lead to poor management decisions. Fitting the extended Poisson process model with covariate dependent mean can be problematic because of the computing burden. A Fisher scoring algorithm is developed for the extended Poisson process model that considerably reduces the computation required. Quine's absence from school data (see Aitkin, 1978) are analysed using this Fisher scoring algorithm for the extended Poisson process model, and the results contrasted with an analysis of the data by Venables and Ripley (2002). It is found that the extended Poisson process model sharpens the statistical tests for regression terms in the model, resulting in different inferences from those based on the negative binomial distribution. In addition, the Fisher scoring algorithm reduced the computation time for parameter estimation from over seven days to less than one hour for each model fitted. It is also shown that the Fisher scoring algorithm for the extended Poisson process model can be expressed as an iterative re-weighted least squares algorithm. This puts parameter estimation for the extended Poisson process model under the same umbrella as standard generalised linear models, making it simpler to implement in existing statistical packages such as GLIM which use iterative re-weighted least squares to estimate parameters.n

A new class of distributions with increasing, decreasing, bathtub-shaped and unimodal hazard rate forms called generalized quadratic hazard rate-power series distribution is proposed. The new distribution is obtained by compounding the generalized quadratic hazard rate and power series distributions. This class of distributions contains several important distributions appeared in the literature, such as generalized quadratic hazard rate-geometric, -Poisson, -logarithmic, -binomial and -negative binomial distributions as special cases. We provide comprehensive mathematical properties of the new distribution. We obtain closed-form expressions for the density function, cumulative distribution function, survival and hazard rate functions, moments, mean residual life, mean past lifetime, order statistics and moments of order statistics; certain characterizations of the proposed distribution are presented as well. The special distributions are studied in some details. The maximum likelihood method is used to estimate the unknown parameters. We propose to use EM algorithm to compute the maximum likelihood estimators of the unknown parameters. It is observed that the proposed EM algorithm can be implemented very easily in practice. One data set has been analyzed for illustrative purposes. It is observed that the proposed model and the EM algorithm work quite well in practice.

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.