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.     

The Structure of Compounded Trivariate Binomial Distributions

A family of trivariate binomial mixtures with respect to their exponent parameter is introduced and its structure is studied by the use of probability generating functions. Expressions for probabilities, factorial moments and factorial cumulants are given. Conditional distributions are also examined. Illustrative examples include the trivariate Poisson, binomial, negative binomial and modified logarithmic series distributions. In addition, properties of the compounded trivariate Poisson distribution are discussed. Finally biological, medical and ecological applications are indicated.     

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.     

On the Modified and Inflated Discrete Distributions

Firstly, a modified bivariate discrete distribution is considered where a set of counts are misreported as another set of counts with different modification rates. Variances and covariances are put in the closed form and for the case when all modification rates are the same, these variances and covariances are expressed as parabolic functions and they are actually evaluated for the bivariate negative binomial. Regarding the asymptotic distributions of the estimates, elements of variance-covariance matrix are obtained. Next, a multivariate inflated discrete distribution is taken up. For the case of inflated multivariate negative binomial, Bayesian estimates of inflation as well as those of parameters are given.     

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.     

Use of the beta-binomial distribution in dominant-lethal testing for "weak mutagenic activity".

When the results of biological experiments are tested statistically for a possible difference between a treatment and its control, the statistical inferences are valid only if the statistical procedure is derived from a model that fits the experimental results satisfactorily. In this paper it is shown that a beta-binomial distribution provided a better fit than a binomial distribution when the data used were based on a large number of counts of dead and total implants on dominant-lethal tests on mice. This suggests that the probability P that an implant will die is not constant over the experimental units. Tests derived from the beta-biomial distribution have been used and their results compared with those of the tests based on the binomial or constant P assumption. The tests based on the binomial model are erroneously too severe when P is not constant in a group. The problem of knowing whether the males or the females should be considered as the experimental units is considered. In this paper, calculations are carried out for the two situations. This problem will be further studied by computer simulation and the result will be presented in a next paper.It is also shown that a negative binomial distribution could be fitted to the dead implant counts. No test based on this model was used because it ignores the total implants. No familiar distribution could be fitted to the total implant counts.     

On Bivariate Cumulative Damage Models With Application

Bivariate cumulative damage models are proposed where the responses given the damages are independent random variables. The bivariate damage process can be either bivariate Poisson or bivariate gamma. A bivariate continuous cumulative damage model is investigated in which the responses given the damages have gamma distributions. In this case evaluation of the joint density function and bivariate tail probability function is facilitated by expanding the gamma distributions of the conditional responses by Laguerre polynomials. This approach also leads to evaluation of associated survival models. Moments and estimating equations are discussed. In addition, a bivariate discrete cumulative damage model is investigated in which the responses given the damages have a distribution chosen from a class that includes the negative binomial, the Neyman Type‐A, the Polya‐Aeppli, and the Lagrangian Poisson. Probabilities are obtained from recursive formulas which do not involve cancellation error as all quantities are non‐negative. Moments and estimating equations are presented for these models also. The continuous and the discrete models are applied to describe the rise of systolic and diastolic blood pressure with age.     

Analyzing Short-Term Noise Dependencies of Spike-Counts in Macaque Prefrontal Cortex Using Copulas and the Flashlight Transformation

Simultaneous spike-counts of neural populations are typically modeled by a Gaussian distribution. On short time scales, however, this distribution is too restrictive to describe and analyze multivariate distributions of discrete spike-counts. We present an alternative that is based on copulas and can account for arbitrary marginal distributions, including Poisson and negative binomial distributions as well as second and higher-order interactions. We describe maximum likelihood-based procedures for fitting copula-based models to spike-count data, and we derive a so-called flashlight transformation which makes it possible to move the tail dependence of an arbitrary copula into an arbitrary orthant of the multivariate probability distribution. Mixtures of copulas that combine different dependence structures and thereby model different driving processes simultaneously are also introduced. First, we apply copula-based models to populations of integrate-and-fire neurons receiving partially correlated input and show that the best fitting copulas provide information about the functional connectivity of coupled neurons which can be extracted using the flashlight transformation. We then apply the new method to data which were recorded from macaque prefrontal cortex using a multi-tetrode array. We find that copula-based distributions with negative binomial marginals provide an appropriate stochastic model for the multivariate spike-count distributions rather than the multivariate Poisson latent variables distribution and the often used multivariate normal distribution. The dependence structure of these distributions provides evidence for common inhibitory input to all recorded stimulus encoding neurons. Finally, we show that copula-based models can be successfully used to evaluate neural codes, e.g., to characterize stimulus-dependent spike-count distributions with information measures. This demonstrates that copula-based models are not only a versatile class of models for multivariate distributions of spike-counts, but that those models can be exploited to understand functional dependencies.     

A likelihood ratio test for equality of deviations from randomness

The use of the sample variance-to-mean ratio as a measure of deviation from randomness in spatial pattern is reviewed. The likelihood ratio method of constructing a statistical test for the equality of several population variance-to-mean ratios is described, and details are provided for the special case where counts are modelled as arising from a negative binomial distribution. This test is illustrated by application to example data sets in ecology. Likelihood ratio tests represent a general methodology whereby relationships among several indices of aggregation can be systematically investigated, provided one is able to specify a suitable parametric form for the underlying distributions.     

Extended Poisson process modelling and analysis of grouped binary data

A simple extension of the Poisson process results in binomially distributed counts of events in a time interval. A further extension generalises this to probability distributions under‐ or over‐dispersed relative to the binomial distribution. Substantial levels of under‐dispersion are possible with this modelling, but only modest levels of over‐dispersion – up to Poisson‐like variation. Although simple analytical expressions for the moments of these probability distributions are not available, approximate expressions for the mean and variance are derived, and used to re‐parameterise the models. The modelling is applied in the analysis of two published data sets, one showing under‐dispersion and the other over‐dispersion. More appropriate assessment of the precision of estimated parameters and reliable model checking diagnostics follow from this more general modelling of these data sets.     

The spatial distribution of the wheat-bulb fly,Leptohylemyia coarctata (fall.) (Diptera: Anthomyiidae)

The spatial distribution of the eggs, larvae, pupae and adults of the wheat-bulb fly was investigated by fitting 42 sets of data comprising 1334 samples to the Poisson and negative binomial distributions, and by using the power law (S²=am^b). In general, the tests indicated that all stages were aggregated and fitted the negative binomial model.     

A score test for testing a zero-inflated Poisson regression model against zero-inflated negative binomial alternatives

Count data often show a higher incidence of zero counts than would be expected if the data were Poisson distributed. Zero-inflated Poisson regression models are a useful class of models for such data, but parameter estimates may be seriously biased if the nonzero counts are overdispersed in relation to the Poisson distribution. We therefore provide a score test for testing zero-inflated Poisson regression models against zero-inflated negative binomial alternatives.     

Multivariate Polya and Inverse Polya Distributions of Order k

Multivariate Polya and inverse Polya distributions of order k are derived by means of generalized urn models and by compounding the type II multinomial and multivariate negative binomial distributions of order k of PHILIPPOU , ANTZOULAKOS and TRIPSIANNIS (1990, 1988), respectively, with the Dirichlet distribution. It is noted that the above two distributions include as special cases a multivariate hypergeometric distribution of order k, a negative one, an inverse one, a negative inverse one and a discrete uniform of the same order. The probability generating functions, means, variances and covariances of the new distributions are obtained and five asymptotic results are established relating them to the above-mentioned multinomial and multivariate negative binomial distributions of order k, and to the type II negative binomial and the type I multivariate Poisson distributions of order k of PHILIPPOU (1983), and PHILIPPOU , ANTZOULAKOS and TRIPSIAN-NIS (1988), respectively. Potential applications are also indicated. The present paper extends to the multivariate case the work of PHILIPPOU , TRIPSIANNIS and ANTZOULAKOS (1989) on Polya and inverse Polya distributions of order k..     

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     

An estimator of the mutant frequency in assays using transgenic animals

The Poisson distribution is a fundamental probability model for count data, and is a natural model for the observed plaque counts in mutation assays using animals with lambda or PhiX174 transgenes. The Poisson likelihood for observed counts is a function of the mutant fraction, and it is straightforward to derive the associated maximum likelihood estimate of the mutant fraction and its variance. The estimate is easy to calculate, and if not the same, very similar to ad hoc estimates in current use. The model indicates the proper way to combine data from a number of plates, possibly prepared with different sample dilutions. The estimator of the mutant fraction is biased as a consequence of dividing by a random variable, the plaque count used to calculate the total recovered plaque-forming units. Fortunately, the bias becomes negligible as this count becomes large. On the other hand, increasing this count can increase the variance by decreasing the amount of sample assayed for mutant phages. Concurrent heed to the bias and the variance provides some guidance as to the optimum allocation of a sample into portions assayed for mutant phages and total recovered phages. The distribution of the estimate of the mutant fraction is related to the binomial distribution. This relationship implies a binomial distribution for the mutant count conditional on an overall count (either the sum of mutant and counted total plaques or the sum of counted mutant and non-mutant plaques). A special but important case occurs when each plate can be evaluated for mutant plaques and non-mutant plaques. Then, the observed proportion of mutants estimates the mutant fraction. More generally, the relationship to a binomial distribution provides a procedure for calculating a confidence interval.     

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.     

Regression models for overdispersed jejunal surviving crypts data

The dose-response model concerns to establish a relationship between a dose and the magnitude of the response produced by the dose. A common complication in the dose-response model for jejunal crypts cell surviving data is overdispersion, where the observed variation exceeds that predicted from the binomial distribution. In this study, two different methods for analyzing jejunal crypts cell survival after regimens of several fractions are contrasted and compared. One method is the logistic regression approach, where the numbers of surviving crypts are predicted by the logistic function of a single dose of radiation. The other one is the transform-both-sides approach, where the arcsine transformation family is applied based on the first-order variance-stabilizing transformation. This family includes the square root, arcsine, and hyperbolic arcsine transformations, which have been used for Poisson, binomial, and negative binomial count data, as special cases. These approaches are applied to a data set from radiobiology. Simulation study indicates that the arcsine transformation family is more efficient than the logistic regression when there exists moderate overdispersion.     

Poisson adjacency distributions in genome comparison: multichromosomal, circular, signed and unsigned cases

The number of common adjacencies of genetic markers, as a measure of the similarity of two genomes, has been widely used as indicator of evolutionary relatedness and as the basis for inferring phylogenetic relationships. Its probability distribution enables statistical tests in detecting whether significant evolutionary signal remains in the marker order. In this article, we derive the probability distributions of the number of adjacencies for a number of types of genome--signed or unsigned, circular or linear, single-chromosome or multichromosomal. Generating functions are found for singlechromosome cases, from which exact counts can be calculated. Probability approaches are adopted for multichromosomal cases, where we.nd the exact values for expectations and variances. In both cases, the limiting distributions are derived in term of numbers of adjacencies. For all unsigned cases, the limiting distribution is Poisson with parameter 2; for all signed cases, the limiting distribution is Poisson with parameter (1/2).