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.     

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.     

Efficient Simulation of Multivariate Binomial and Poisson Distributions

Power investigations, for example, in statistical procedures for the assessment of agreement among multiple raters often require the simultaneous simulation of several dependent binomial or Poisson distributions to appropriately model the stochastical dependencies between the raters' results. Regarding the rather large dimensions of the random vectors to be generated and the even larger number of interactions to be introduced into the simulation scenarios to determine all necessary information on their distributions' dependence stucture, one needs efficient and fast algorithms for the simulation of multivariate Poisson and binomial distributions. Therefore two equivalent models for the multivariate Poisson distribution are combined to obtain an algorithm for the quick implementation of its multivariate dependence structure. Simulation of the multivariate Poisson distribution then becomes feasible by first generating and then convoluting independent univariate Poisson variates with appropriate expectations. The latter can be computed via linear recursion formulae. Similar means for simulation are also considered for the binomial setting. In this scenario it turns out, however, that exact computation of the probability function is even easier to perform; therefore corresponding linear recursion formulae for the point probabilities of multivariate binomial distributions are presented, which only require information about the index parameter and the (simultaneous) success probabilities, that is the multivariate dependence structure among the binomial marginals.     

Extended Poisson Process Modelling and Analysis of Count Data

It is shown that any discrete distribution with non-negative support has a representation in terms of an extended Poisson process (or pure birth process). A particular extension of the simple Poisson process is proposed: one that admits a variety of distributions; the equations for such processes may be readily solved numerically. An analytical approximation for the solution is given, leading to approximate mean-variance relationships. The resulting distributions are then applied to analyses of some biological data-sets.     

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.     

Cell survival and radiation induced chromosome aberrations

Summary Existing mathematical formulations to predict the frequency of radiation induced chromosome aberrations in 2nd post-irradiation division are based on the Poisson distribution [3, 4]. Meanwhile several studies have shown that intercellular distributions exist, deviating from Poisson. In the present study a modified model was developed which permits the application of empirical distributions. Transmission and survival parameters of aberrations can be iteratively computed. A general formula was derived for the calculation of cell survival from 1st to 2nd division.     

Applicability of the Poisson distribution to model the data of the German Children's Cancer Registry

Since 1980 the German Children's Cancer Registry has documented all childhood malignancies in the Federal Republic of Germany. Various statistical procedures have been proposed to identify municipalities or other geographic units with increased numbers of malignancies. Usually the Poisson distribution, which requires the malignancies to be distributed homogeneously and uncorrelated, is applied. Other discrete statistical distributions (so-called cluster distributions) like the generalized or compound Poisson distributions are applicable more generally. In this paper we present a first explorative approach to the question of whether it is necessary to use one of these cluster distributions to model the data of the German Children's Cancer Registry. In conclusion, we find no indication that the Poisson approach is insufficient.     

The Multiple Poisson Distribution,Its Characteristics and a Variety of Forms

The multiple Poisson distribution, known under different names, such as generalized Poisson, compound Poisson, composed Poisson, stuttering Poisson, Poisson power series, Poisson-stopped sum distribution, etc., plays an important role in discrete distribution theory. Here we want to show its basic characteristics, the variety of its forms and specify the generalizing distributions.     

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.     

Deviation from a Poisson distribution in a series of identical tests of E. coli cultures as a result of the effect of correlating factors of exo- and endogenous natures]

Vital cells number (VCN) in the sampling of E. coli populations was experimentally measured and distribution histograms were obtained. In most cases distributions show considerable deviation from Poisson model. VCN distribution histograms are polymodal, dispersion/arithmetical mean ratio may essentially differ from 1. The more essential differences from Poisson distribution were observed for populations with the higher cell concentration. Computer simulation of the VCN histograms indicated that additional parameters (such as those describing cellular interaction of different nature and/or other factors that influence random behaviour of cells) should be introduced into Poisson model to explain observed variations in VCN distribution histograms.     

The distribution of extra-pair young within and among broods – a technique to calculate deviations from randomness

Molecular paternity tests show that extra-pair fertilizations are common in many socially monogamous bird species. However, the question of why females often seek extra-pair copulations is still controversial. Competing alternative explanations differ in their predictions on how extra-pair young should be distributed within and among broods. Analysing these distributions may therefore help to resolve this controversy. In several species broods without extra-pair young and those with total or nearly total extra-pair paternity have been claimed to be over-represented. Consequently, extra-pair nestlings would be distributed non-randomly among nests. To compute expected frequencies (i.e. the patterns of random distribution) the Poisson distribution has frequently been used. However, the Poisson distribution is not appropriate for two reasons. (1) Impossible configurations can receive positive probabilities. (2) The Poisson distribution approaches randomness only if extra-pair fertilizations are events of low probability, which often is not the case. We show how randomness can be computed appropriately by using the multivariate hypergeometric distribution. Re-analysing published data on distributions of extra-pair young, we show that the result that there are more broods containing either none or many extra-pair young than expected by chance is even more pronounced than previously thought.     

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.     

Conditionally in Bivariate Generalized Poisson Distributions

Three types of bivariate generalized Poisson distributions are defined and the structure of their conditional distributions is examined by using the Faa Di Bruno's formula. The resulting expressions involve Bell polynomials and can be interpreted in terms of convoluted random variables with one of the convolutes having the form of the marginal distribution. The three types of bivariate Neyman A distributions are used to illustrate the procedure.     

Some extensions of a linear model for categorical variables

The Grizzle-Starmer-Koch (GSK) model is extended to include the traditional log-linear model and a general class of Poisson and conditional Poisson distributions. Estimators of the model parameters are defined under general exact and stochastic linear constraints.     

Effect of offspring distribution on population survival

The survival probabilities of newly-formed colonies of organisms arising from the branching process formulation of individual reproduction are examined. Six types of 2-parameter discrete offspring distributions common to mathematical ecology are compared with respect to survival of newly formed colonies of offspring. It is found that the survival value of the distribution can be rank ordered in the following descending order: modified Poisson (highest) Neyman A, geometric Poisson, Pólya-Aeppli, negative binomial and the modified geometric. Causal factors for these differences and practical implications of these results are discussed.     

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.     

A two-stage test for ordered means in the poisson case with an example from mutagenicity testing

It is shown how two-stage methods developed for continuous distributions can be used to construct two-stage tests for certain hypotheses pertaining to discrete distributions. Specifically, a two-stage test for ordered means in the Poisson case is presented. An example from mutagenicity testing is discussed and data from it are used to illustrate the methodology. Results of a Monte Carlo power study are presented as well as a brief table of critical values.     

A clockwork hypothesis: synaptic release by rod photoreceptors must be regular

We can see at light intensities much lower than an average of one photon per rod photoreceptor, demonstrating that rods must be able to transmit a signal after absorption of a single photon. However, activation of one rhodopsin molecule (Rh*) hyperpolarizes a mammalian rod by just 1 mV. Based on the properties of the voltage-dependent Ca2+ channel and data on [Ca2+] in the rod synaptic terminal, the 1 mV hyperpolarization should reduce the rate of release of quanta of neurotransmitter by only 20%. If quantal release were Poisson, the distributions of quantal count in the dark and in response to one Rh* would overlap greatly. Depending on the threshold quantal count, the overlap would generate too frequent false positives in the dark, too few true positives in response to one Rh*, or both. Therefore, quantal release must be regular, giving narrower distributions of quantal count that overlap less. We model regular release as an Erlang process, essentially a mechanism that counts many Poisson events before release of a quantum of neurotransmitter. The combination of appropriately narrow distributions of quantal count and a suitable threshold can give few false positives and appropriate (e.g., 35%) efficiency for one Rh*.     

An intervened Poisson distribution and its medical application

Among probability distributions that are used to describe a chance mechanism whose observational apparatus becomes active only when at least one event occurs is the zero-truncated Poisson distribution (ZTPD). A modified version of the ZTPD, which we call an intervened Poisson distribution (IPD), is discussed in this paper. We give a genesis of IPD and obtain its statistical properties. A numerical example is included to illustrate the results.