One‐inflation and unobserved heterogeneity in population size estimation

We present the one‐inflated zero‐truncated negative binomial (OIZTNB) model, and propose its use as the truncated count distribution in Horvitz–Thompson estimation of an unknown population size. In the presence of unobserved heterogeneity, the zero‐truncated negative binomial (ZTNB) model is a natural choice over the positive Poisson (PP) model; however, when one‐inflation is present the ZTNB model either suffers from a boundary problem, or provides extremely biased population size estimates. Monte Carlo evidence suggests that in the presence of one‐inflation, the Horvitz–Thompson estimator under the ZTNB model can converge in probability to infinity. The OIZTNB model gives markedly different population size estimates compared to some existing truncated count distributions, when applied to several capture–recapture data that exhibit both one‐inflation and unobserved heterogeneity.     

Distribution of Polymorphus minutus among its intermediate hosts

Hirsch R. P. 1979. Distribution of Polymorphus minutus among its intermediate hosts. International journal for Parasitology10: 243–248. In 1971, Crofton investigated patterns of distribution of Polymorphus minutus in the intermediate host, Gammarus pulex. Among his conclusions were: (1) P. minutus populations occur in patterns similar to negative binomial distributions, and (2) parasite-induced host mortality results in patterns similar to truncated (high end) negative binomial distributions. Those conclusions, however, were not tested by statistical analyses. To test Crofton's observations, Chi-square goodness of fit tests were applied to data used by Crofton and an additional two stations sampled by Hynes & Nicholas in 1963. Analyses were expanded to include five theoretical distributions, four patterns of host mortality and various rates of host mortality. Truncated forms of negative binomial, positive binomial and Poisson distributions were also investigated where nontruncated distributions failed to fit observed distributions. It was found that negative binomial distributions most frequently describe patterns of P. minutus distribution with the exception of one population described by Poisson and another by positive binomial distributions. Crofton's assumption that truncated distributions result from parasite-induced host mortality seems unlikely in light of those analyses.     

Population dynamics of the chestnut gall-wasp,Dryocosmus kuriphilus Yasumatsu (Hymenoptera: Cynipidae) IV. Further analysis of the distribution of eggs and young larvae in buds using the truncated negative binomial series

Distribution pattern of eggs and the first instar larvae of the chestnut gall-wasp, Dryocosmus kuriphilus, per bud of the chestnut tree was re-examined using the truncatedPoisson and the truncated negative binomial series. The results are as follows:

1. About sixty percent of the distribution are approximated by the truncated negative binomial and another thirty percent by the truncatedPoisson . When the distribution has been approximated by the truncated negative binomial, the show scattered values, some of which are near thePoisson , but the mean value of is about 4 both in eggs and in the first instar larvae.
2. The number of buds which have not been infested by gall-wasps is resulted from various factors as follows: (a) Buds formed after the laying activity of gall-wasps has ceased; (b) Old and shrunk buds which were avoided in laying; (c) Buds in which eggs have died by the time of sampling; and (d) Buds escaping ovipositions by chance. Most of the gall-wasp-free buds located in the middle part of the shoot are accounted for the zero-class expected by the negative binomial series.
3. Brass (1958) moment method for estimating parameters of the truncated negative binomial give good precision within the range of from 1 to 10 and of from 1 to 6.
4. It is concluded that the truncated distributions used are useful for the purpose of analysis of the distribution pattern of the chestnut gall-wasp.

     

Some Approximations of the Borel-Tanner or the Generalized Poisson Distribution

This paper considers some approximations for the Borel-Tanner (Generalized Poisson) sums by using (i) Gram-Charlier Poisson expansion, (ii) Mixture of two Poisson distributions, (iii) Variance stabilizing technique, and (iv) negative binomial distribution. It has been found that the approximation obtained by using the negative binomial distribution seems to be more efficient than the other approximation.     

A probability model for the number of births in an equilibrium birth process

A probability model for the number of complete conceptions (that is, live births) taking into account foetal wastages, occurring in a couple during a specified period of time (T₀, T₀+T) is developed assuming that the data was collected starting a long time after marriage. A method of estimating some of the underlying parameters is given. The model is applied to data obtained in a Varanasi Survey in 1969–70.     

Use of compound-probability distributions in the study of induced post-implantation dominant lethals

The nature of the probability distribution of post-implantation dominant lethality was investigated in terms of the distribution of dead implants per female. It has been postulated that this distribution would be poisson in a control series of females but may follow a compound or a contagious distribution such as the beta binomial, negative binomial or Neyman type A in the treated series of females. The nature of these compound distributions for fitting mammalian mutagenicity has been examined. The implications of the results on the estimation of induced mutation rates are discussed.     

On Mohan's Property of the Poisson Distribution

Two interesting results encountered in the literature concerning the Poisson and the negative binomial distributions are due to Moran (1952) and Patil & Seshadri (1964), respectively. Morans result provided a fundamental property of the Poisson distribution. Roughly speaking, he has shown that if Y, Z are independent, non-negative, integer-valued random variables with X = Y | Z then, under some mild restrictions, the conditional distribution of Y | X is binomial if and only if Y, Z are Poisson random variables. Motivated by Morans result Patil & Seshadri obtained a general characterization. A special case of this characterization suggests that, with conditions similar to those imposed by Moran, Y | X is negative hypergeometric if and only if Y, Z are negative binomials. In this paper we examine the results of Moran and Patil & Seshadri in the case where the conditional distribution of Y | X is truncated at an arbitrary point k – 1 (k = 1, 2, …). In fact we attempt to answer the question as to whether Morans property of the Poisson distribution, and subsequently Patil & Seshadris property of the negative binomial distribution, can be extended, in one form or another, to the case where Y | X is binomial truncated at k – 1 and negative hypergeometric truncated at k – 1 respectively.     

Bacterial Density in Water Determined by Poisson or Negative Binomial Distributions

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.     

One‐inflation and unobserved heterogeneity in population size estimation by Ryan T. Godwin

In this study, we would like to show that the one‐inflated zero‐truncated negative binomial (OIZTNB) regression model can be easily implemented in R via built‐in functions when we use mean‐parameterization feature of negative binomial distribution to build OIZTNB regression model. From the practitioners' point of view, we believe that this approach presents a computationally convenient way for implementation of the OIZTNB regression model.     

The Question of the Total Gene Number in DROSOPHILA MELANOGASTER

A statistical analysis has been carried out on the distribution and allelism of nearly 500 sex-linked, X-ray-induced, cytologically normal and rearranged lethal mutations in Drosophila melanogaster that were obtained by G. Lefevre. The mutations were induced in four different regions of the X chromosome: (1) 1A1-3E8, (2) 6D1-8A5, (3) 9E1-11A7 and (4) 19A1-20F4, which together comprise more than one-third of the entire chromosome.--The analysis shows that the number of alleles found at different loci does not fit a Poisson distribution, even when the proper procedures are taken to accommodate the truncated nature of the data. However, the allele distribution fits a truncated negative binomial distribution quite well, with cytologically normal mutations fitting better than rearrangement mutations. This indicates that genes are not equimutable, as required for the data to fit a Poisson distribution.--Using the negative binomial parameters to estimate the number of genes that did not produce a detectable lethal mutation in our experiment (n0) gave a larger number than that derived from the use of the Poisson parameter. Unfortunately, we cannot estimate the total numbers of nonvital loci, loci with undetectable phenotypes and loci having extremely low mutabilities. In any event, our estimate of the total vital gene number was far short of the total number of bands in the analyzed regions; yet, in several short intervals, we have found more vital genes than bands; in other intervals, fewer. We conclude that the one-band, one-gene hypothesis, in its literal sense, is not true; furthermore, it is difficult to support, even approximately.--The question of the total gene number in Drosophila will, not doubt, eventually be solved by molecular analyses, not by statistical analysis of mutation data or saturation studies.     

Generalized Poisson distribution: the property of mixture of Poisson and comparison with negative binomial distribution

We prove that the generalized Poisson distribution GP(theta, eta) (eta > or = 0) is a mixture of Poisson distributions; this is a new property for a distribution which is the topic of the book by Consul (1989). Because we find that the fits to count data of the generalized Poisson and negative binomial distributions are often similar, to understand their differences, we compare the probability mass functions and skewnesses of the generalized Poisson and negative binomial distributions with the first two moments fixed. They have slight differences in many situations, but their zero-inflated distributions, with masses at zero, means and variances fixed, can differ more. These probabilistic comparisons are helpful in selecting a better fitting distribution for modelling count data with long right tails. Through a real example of count data with large zero fraction, we illustrate how the generalized Poisson and negative binomial distributions as well as their zero-inflated distributions can be discriminated.     

Distributions of coital rates and of fecundability

Abstract

Attempts have been made to fit distributions of conception waits on the assumption that fecundability is distributed as a Pearson Type I beta distribution. Since the fits were unsatisfactory (Majumdar and Sheps, 1970), the question arises whether fecundability is distributed as a Type III. If it were, then the probability of coitus should also be distributed as a Type III, and distributions of coital rates should be better fitted by the negative binomial than by Skellam's distribution. The data available for such fits have been briefly reviewed here and fits attempted. There is some slight evidence in favor of the hypothesis, but more data are needed before the question can be decisively answered.     

Promotion time models with time-changing exposure and heterogeneity: application to infectious diseases

Promotion time models have been recently adapted to the context of infectious diseases to take into account discrete and multiple exposures. However, Poisson distribution of the number of pathogens transmitted at each exposure was a very strong assumption and did not allow for inter-individual heterogeneity. Bernoulli, the negative binomial, and the compound Poisson distributions were proposed as alternatives to Poisson distribution for the promotion time model with time-changing exposure. All were derived within the frailty model framework. All these distributions have a point mass at zero to take into account non-infected people. Bernoulli distribution, the two-component cure rate model, was extended to multiple exposures. Contrary to the negative binomial and the compound Poisson distributions, Bernoulli distribution did not enable to connect the number of pathogens transmitted to the delay between transmission and infection detection. Moreover, the two former distributions enable to account for inter-individual heterogeneity. The delay to surgical site infection was an example of single exposure. The probability of infection was very low; thus, estimation of the effect of selected risk factors on that probability obtained with Bernoulli and Poisson distributions were very close. The delay to nosocomial urinary tract infection was a multiple exposure example. The probabilities of pathogen transmission during catheter placement and catheter presence were estimated. Inter-individual heterogeneity was very high, and the fit was better with the compound Poisson and the negative binomial distributions. The proposed models proved to be also mechanistic. The negative binomial and the compound Poisson distributions were useful alternatives to account for inter-individual heterogeneity.     

Point and Interval Estimation of the Population Size Using a Zero‐Truncated Negative Binomial Regression Model

This paper presents the zero‐truncated negative binomial regression model to estimate the population size in the presence of a single registration file. The model is an alternative to the zero‐truncated Poisson regression model and it may be useful if the data are overdispersed due to unobserved heterogeneity. Horvitz–Thompson point and interval estimates for the population size are derived, and the performance of these estimators is evaluated in a simulation study. To illustrate the model, the size of the population of opiate users in the city of Rotterdam is estimated. In comparison to the Poisson model, the zero‐truncated negative binomial regression model fits these data better and yields a substantially higher population size estimate. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

负二项分布与昆虫种群空间格局分析的研究现状

对农业有害生物及其天敌种群密度的正确估计是实施IPM(有害生物综合治理)方案的先决条件,因此,抽样方法一直被列为昆虫学,生态学和植物保护科学中最重要的基本     

The Negative Binomial Distribution of Order k and Some of Its Properties

The negative binomial distribution of order k is introduced and briefly studied. First it is shown that it is a proper probability distribution. Then its probability generating function, mean and variance are derived. Finally it is shown that the number of trials until the rth kth consecutive success (r ≧ 1, k ≧ 1) in independent trials with constant success probability p (0 < p < 1) is distributed as negative binomial distribution of order k. The present paper generalizes results of SHANE (1973), PHILIPPOU and MUWAFI (1982), and PHILIPPOU, GEORGHIOU and PHILIPPOU (1982).     

Negative binomial additive models

The generalized additive model is extended to handle negative binomial responses. The extension is complicated by the fact that the negative binomial distribution has two parameters and is not in the exponential family. The methodology is applied to data involving DNA adduct counts and smoking variables among ex-smokers with lung cancer. A more detailed investigation is made of the parametric relationship between the number of adducts and years since quitting while retaining a smooth relationship between adducts and the other covariates.     

Calculating census efficiency for river birds: a case study with the White-throated Dipper Cinclus cinclus in the Pyrénées

Using the binomial law we modelled field data to estimate the probability ( ̂ ) of detecting pairs of breeding White-throated Dippers, and the population size ( ̂ ± confidence limits). The model was divided into two parts according to whether the actual size of the population under study was known or not; in the latter case the truncated binomial model was used. Dipper abundance data were collected from three 4-km-long river tracts in the Pyrénées (France) during the breeding seasons of different years. Goodness-of-fit tests indicated that the binomial model fitted the data well. For a given visit during the survey, the estimated probability of detecting any pair of Dippers if they were present was always high (0.63–0.94) and constant from year to year but not between sites. Estimations ( ̂ ) of the size of the population provided by the binomial model were very close to that derived from mapping techniques. This study provides the first ever quantification of the number of visits required to detect birds on linear territories: three visits were necessary to detect the whole breeding population.     

Generalized Binomial Regression Model

The generalized negative binomial distribution has been found useful in fitting over-dispersed as well as under-dispersed count data. We define and study the generalized binomial regression model which is used to predict a count response variable affected by one or more explanatory variables. The methods of maximum likelihood and moments are given for estimating the model parameters. Approximate tests for the adequacy of the model are considered. The generalized binomial regression model has been applied to two observed data sets to which binomial regression model was applied earlier.