Bacterial Density in Water Determined by Poisson or Negative Binomial Distributions

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.

Effectiveness of sampling plans by attributes based on mixture distributions characterising microbial clustering in food

Forest tree ingrowth is a highly variable and largely stochastic process. Consequently, predicting occurrence, frequency, and composition of ingrowth is a challenging task but of great importance in long-term forest growth and yield model projections. However, ingrowth data often require different statistical techniques other than traditional Gaussian regression, because these data are often bounded, skewed, and non-normal and commonly contain a large fraction of zeros. This study presents a set of regression models based on discrete Poisson and negative binomial probability distributions for ingrowth data collected from permanent sample plots in the Acadian Forest Region of North America. Models considered here include regular Poisson, zero-inflated Poisson (ZIP), zero-altered Poisson (ZAP; hurdle Poisson), regular negative binomial (NB), zero-inflated negative binomial (ZINB), and zero-altered negative binomial (ZANB; hurdle NB). Plot-level random effects were incorporated into each of these models. The ZINB model with random effects was found to provide the best fit statistics for modeling annualized occurrence and frequency of ingrowth. The key explanatory variables were stand basal area per hectare, percentage of hardwood basal area, number of trees per hectare, a measure of site quality, and the minimum measured diameter at breast height of each plot. A similar model was developed to predict species composition. All models showed logical behavior despite the high variability observed in the original data.

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

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

Limited information is available about the abundance of Atlantic Cod Gadus morhua and Cusk Brosme brosme in the coastal Gulf of Maine because the presence of lobster traps limits commercial fishing and surveys for these species. We developed abundance indices for Atlantic Cod and Cusk from bycatch data obtained in a lobster sea sampling program. We applied generalized linear models (GLMs) to standardize Atlantic Cod and Cusk bycatch rates. The CPUE data, measured as the count of Atlantic Cod and Cusk observed per trap haul, are characterized by an extremely skewed distribution with a high percentage of zero observations. Two general approaches were applied to tackle the zero-dominated data: modeling with different error distributions and aggregating data over spatial scales. We evaluated eight models: binomial, Poisson, negative binomial, Tweedie model, hurdle model (also referred to as the delta approach) with Poisson and negative binomial distribution, and zero-inflated model with Poisson and negative binomial distribution. The data were aggregated using six grids ranging from 0.01 to 0.5 geological degrees (0.6–30 nautical miles). The standardized CPUE showed a gradual decline from 2006 to 2011, except for 2009 when a slight increase occurred for Atlantic Cod, and a general decline from 2006 to 2010 followed by an increase in 2011 for Cusk. The standardized CPUEs were consistent, in general, among the models and among different spatial aggregation scenarios, suggesting that the CPUE standardization is robust with respect to choices of data aggregation and statistical models. This study highlights the feasibility of developing abundance indices based on bycatch data for monitoring fish stock dynamics in data-limited regions. Received September 17, 2014; accepted April 16, 2015

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.

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

The Poisson-lognormal distribution was suggested as a model for commonness of species by Preston. The analysis leading him to suggest this distribution was as follows: Suppose that the number of each species caught in a trap is assumed to be a single sample from a Poisson distribution with mean λ. R. S. Uhler and P. G. Bradley proposed two models to describe a phenomenon in petroleum exploration. These two models used the sum of lognormal random variables as a compounder of Poisson and negative binomial distributions. P. Holgate noted that a clear distinction between the P-LN and the lognormal has not always been maintained. R. M. Cassie indicated that for most plankton populations, samples with μ = 100 would not depart significantly from a lognormal distribution.

Sample size and power determination is the first step in the experimental design of a successful study. Sample size and power calculation is required for applications for National Institutes of Health (NIH) funding. Sample size and power calculation is well established for traditional biological studies such as mouse model, genome wide association study (GWAS), and microarray studies. Recent developments in high-throughput sequencing technology have allowed RNAseq to replace microarray as the technology of choice for high-throughput gene expression profiling. However, the sample size and power analysis of RNAseq technology is an underdeveloped area. Here, we present RNAseqPS, an advanced online RNAseq power and sample size calculation tool based on the Poisson and negative binomial distributions. RNAseqPS was built using the Shiny package in R. It provides an interactive graphical user interface that allows the users to easily conduct sample size and power analysis for RNAseq experimental design. RNAseqPS can be accessed directly at http://cqs.mc.vanderbilt.edu/shiny/RNAseqPS/.

In reinsurance, Poisson and Negative binomial distributions are employed for modeling frequency. However, the incomplete data regarding reported incurred claims above a priority level presents challenges in estimation. This paper focuses on frequency estimation using Schnieper's framework [Schnieper, R. (1991). Separating true ibnr and ibner claims. ASTIN Bulletin, 21(1), 111–127.] for claim numbering. We demonstrate that Schnieper's model is consistent with a Poisson distribution for the total number of claims above a priority at each year of development, providing a robust basis for parameter estimation. Additionally, we explain how to build an alternative assumption based on a Negative binomial distribution, which yields similar results. The study includes a bootstrap procedure to manage uncertainty in parameter estimation and a case study comparing assumptions and evaluating the impact of the bootstrap approach.

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.

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.

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.