The Asymptotic Covariance Matrix of the Maximum Likelihood Parameter Estimator in Conditional Poisson Log-linear Models

The asymptotic covariance matrix of the maximum likelihood estimator for the log-linear model is given for a general class of conditional Poisson distributions which include the unconditional Poisson, multinomial and product-multinomial, as special cases. The general conditions are given under which the maximum likelihood covariance matrix is equal to the covariance matrix of an equivalent closed-form weighted least squares estimator.     

Estimating divergence times in the presence of an overdispersed molecular clock

Molecular loci that fail relative-rate tests are said to be "overdispersed." Traditional molecular-clock approaches to estimating divergence times do not take this into account. In this study, a method was developed to estimate divergence times using loci that may be overdispersed. The approach was to replace the traditional Poisson process assumption with a more general stationary process assumption. A probability model was developed, and an accompanying computer program was written to find maximum-likelihood estimates of divergence times under both the Poisson process and the stationary process assumptions. In simulation, it was shown that confidence intervals under the traditional Poisson assumptions often vastly underestimate the true confidence limits for overdispersed loci. Both models were applied to two data sets: one from land plants, the other from the higher metazoans. In both cases, the traditional Poisson process model could be rejected with high confidence. Maximum-likelihood analysis of the metazoan data set under the more general stationary process suggested that their radiation occurred well over a billion years ago, but confidence intervals were extremely wide. It was also shown that a model consistent with a Cambrian (or nearly Cambrian) origination of the animal phyla, although significantly less likely than a much older divergence, fitted the data well. It is argued that without an a priori understanding of the variance in the time between substitutions, molecular data sets may be incapable of ever establishing the age of the metazoan radiation.     

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.     

Mapping quantitative trait loci using the MCMC procedure in SAS

The MCMC procedure in SAS (called PROC MCMC) is particularly designed for Bayesian analysis using the Markov chain Monte Carlo (MCMC) algorithm. The program is sufficiently general to handle very complicated statistical models and arbitrary prior distributions. This study introduces the SAS/MCMC procedure and demonstrates the application of the program to quantitative trait locus (QTL) mapping. A real life QTL mapping experiment in wheat female fertility trait was used as an example for the demonstration. The fertility trait phenotypes were described under three different models: (1) the Poisson model, (2) the Bernoulli model and (3) the zero-truncated Poisson model. One QTL was identified on the second chromosome. This QTL appears to control the switch of seed-producing ability of female plants but does not affect the number of seeds produced once the switch is turned on.     

Spatial variation of permittivity of an electrolyte solution in contact with a charged metal surface: a mini review

Contact between a charged metal surface and an electrolyte implies a particular ion distribution near the charged surface, i.e. the electrical double layer. In this mini review, different mean-field models of relative (effective) permittivity are described within a simple lattice model, where the orientational ordering of water dipoles in the saturation regime is taken into account. The Langevin–Poisson–Boltzmann (LPB) model of spatial variation of the relative permittivity for point-like ions is described and compared to a more general Langevin–Bikerman (LB) model of spatial variation of permittivity for finite-sized ions. The Bikerman model and the Poisson–Boltzmann model are derived as limiting cases. It is shown that near the charged surface, the relative permittivity decreases due to depletion of water molecules (volume-excluded effect) and orientational ordering of water dipoles (saturation effect). At the end, the LPB and LB models are generalised by also taking into account the cavity field.     

Background stratified Poisson regression analysis of cohort data

Background stratified Poisson regression is an approach that has been used in the analysis of data derived from a variety of epidemiologically important studies of radiation-exposed populations, including uranium miners, nuclear industry workers, and atomic bomb survivors. We describe a novel approach to fit Poisson regression models that adjust for a set of covariates through background stratification while directly estimating the radiation-disease association of primary interest. The approach makes use of an expression for the Poisson likelihood that treats the coefficients for stratum-specific indicator variables as ‘nuisance’ variables and avoids the need to explicitly estimate the coefficients for these stratum-specific parameters. Log-linear models, as well as other general relative rate models, are accommodated. This approach is illustrated using data from the Life Span Study of Japanese atomic bomb survivors and data from a study of underground uranium miners. The point estimate and confidence interval obtained from this ‘conditional’ regression approach are identical to the values obtained using unconditional Poisson regression with model terms for each background stratum. Moreover, it is shown that the proposed approach allows estimation of background stratified Poisson regression models of non-standard form, such as models that parameterize latency effects, as well as regression models in which the number of strata is large, thereby overcoming the limitations of previously available statistical software for fitting background stratified Poisson regression models.     

Bringing consistency to simulation of population models--Poisson simulation as a bridge between micro and macro simulation

Population models concern collections of discrete entities such as atoms, cells, humans, animals, etc., where the focus is on the number of entities in a population. Because of the complexity of such models, simulation is usually needed to reproduce their complete dynamic and stochastic behaviour. Two main types of simulation models are used for different purposes, namely micro-simulation models, where each individual is described with its particular attributes and behaviour, and macro-simulation models based on stochastic differential equations, where the population is described in aggregated terms by the number of individuals in different states. Consistency between micro- and macro-models is a crucial but often neglected aspect. This paper demonstrates how the Poisson Simulation technique can be used to produce a population macro-model consistent with the corresponding micro-model. This is accomplished by defining Poisson Simulation in strictly mathematical terms as a series of Poisson processes that generate sequences of Poisson distributions with dynamically varying parameters. The method can be applied to any population model. It provides the unique stochastic and dynamic macro-model consistent with a correct micro-model. The paper also presents a general macro form for stochastic and dynamic population models. In an appendix Poisson Simulation is compared with Markov Simulation showing a number of advantages. Especially aggregation into state variables and aggregation of many events per time-step makes Poisson Simulation orders of magnitude faster than Markov Simulation. Furthermore, you can build and execute much larger and more complicated models with Poisson Simulation than is possible with the Markov approach.     

Extending Zelterman's approach for robust estimation of population size to zero-truncated clustered Data

Estimation of population size with missing zero-class is an important problem that is encountered in epidemiological assessment studies. Fitting a Poisson model to the observed data by the method of maximum likelihood and estimation of the population size based on this fit is an approach that has been widely used for this purpose. In practice, however, the Poisson assumption is seldom satisfied. Zelterman (1988) has proposed a robust estimator for unclustered data that works well in a wide class of distributions applicable for count data. In the work presented here, we extend this estimator to clustered data. The estimator requires fitting a zero-truncated homogeneous Poisson model by maximum likelihood and thereby using a Horvitz-Thompson estimator of population size. This was found to work well, when the data follow the hypothesized homogeneous Poisson model. However, when the true distribution deviates from the hypothesized model, the population size was found to be underestimated. In the search of a more robust estimator, we focused on three models that use all clusters with exactly one case, those clusters with exactly two cases and those with exactly three cases to estimate the probability of the zero-class and thereby use data collected on all the clusters in the Horvitz-Thompson estimator of population size. Loss in efficiency associated with gain in robustness was examined based on a simulation study. As a trade-off between gain in robustness and loss in efficiency, the model that uses data collected on clusters with at most three cases to estimate the probability of the zero-class was found to be preferred in general. In applications, we recommend obtaining estimates from all three models and making a choice considering the estimates from the three models, robustness and the loss in efficiency.     

Quasi Likelihood/Moment Method for Generalized and Restricted Generalized Poisson Regression Models and Its Application

This paper reviews the generalized Poisson regression model, the restricted generalized Poisson regression model and the mixed Poisson regression (negative binomial regression and Poisson inverse Gaussian regression) models which can be used for regression analysis of counts. The aim of this study is to demonstrate the quasi likelihood/moment method, which is used for estimation of the parameters of mixed Poisson regression models, also applicable to obtain the estimates of the parameters of the generalized Poisson regression and the restricted generalized Poisson regression models. Besides, at the end of this study an application related to this method for zoological data is given.     

Nonlinear poisson equation for heterogeneous media

The Poisson equation is a widely accepted model for electrostatic analysis. However, the Poisson equation is derived based on electric polarizations in a linear, isotropic, and homogeneous dielectric medium. This article introduces a nonlinear Poisson equation to take into consideration of hyperpolarization effects due to intensive charges and possible nonlinear, anisotropic, and heterogeneous media. Variational principle is utilized to derive the nonlinear Poisson model from an electrostatic energy functional. To apply the proposed nonlinear Poisson equation for the solvation analysis, we also construct a nonpolar solvation energy functional based on the nonlinear Poisson equation by using the geometric measure theory. At a fixed temperature, the proposed nonlinear Poisson theory is extensively validated by the electrostatic analysis of the Kirkwood model and a set of 20 proteins, and the solvation analysis of a set of 17 small molecules whose experimental measurements are also available for a comparison. Moreover, the nonlinear Poisson equation is further applied to the solvation analysis of 21 compounds at different temperatures. Numerical results are compared to theoretical prediction, experimental measurements, and those obtained from other theoretical methods in the literature. A good agreement between our results and experimental data as well as theoretical results suggests that the proposed nonlinear Poisson model is a potentially useful model for electrostatic analysis involving hyperpolarization effects.     

Bayesian latent variable models for mixed discrete outcomes

In studies of complex health conditions, mixtures of discrete outcomes (event time, count, binary, ordered categorical) are commonly collected. For example, studies of skin tumorigenesis record latency time prior to the first tumor, increases in the number of tumors at each week, and the occurrence of internal tumors at the time of death. Motivated by this application, we propose a general underlying Poisson variable framework for mixed discrete outcomes, accommodating dependency through an additive gamma frailty model for the Poisson means. The model has log-linear, complementary log-log, and proportional hazards forms for count, binary and discrete event time outcomes, respectively. Simple closed form expressions can be derived for the marginal expectations, variances, and correlations. Following a Bayesian approach to inference, conditionally-conjugate prior distributions are chosen that facilitate posterior computation via an MCMC algorithm. The methods are illustrated using data from a Tg.AC mouse bioassay study.     

Smooth tests for the zero-inflated poisson distribution

In this article we construct three smooth goodness-of-fit tests for testing for the zero-inflated Poisson (ZIP) distribution against general smooth alternatives in the sense of Neyman. We apply our tests to a data set previously claimed to be ZIP distributed, and show that the ZIP is not a good model to describe the data. At rejection of the null hypothesis of ZIP, the individual components of the test statistic, which are directly related to interpretable parameters in a smooth model, may be used to gain insight into an alternative distribution.     

Mapping species abundance by a spatial zero‐inflated Poisson model: a case study in the Wadden Sea,the Netherlands

The objective of the study was to provide a general procedure for mapping species abundance when data are zero‐inflated and spatially correlated counts. The bivalve species Macoma balthica was observed on a 500×500 m grid in the Dutch part of the Wadden Sea. In total, 66% of the 3451 counts were zeros. A zero‐inflated Poisson mixture model was used to relate counts to environmental covariates. Two models were considered, one with relatively fewer covariates (model “small”) than the other (model “large”). The models contained two processes: a Bernoulli (species prevalence) and a Poisson (species intensity, when the Bernoulli process predicts presence). The model was used to make predictions for sites where only environmental data are available. Predicted prevalences and intensities show that the model “small” predicts lower mean prevalence and higher mean intensity, than the model “large”. Yet, the product of prevalence and intensity, which might be called the unconditional intensity, is very similar. Cross‐validation showed that the model “small” performed slightly better, but the difference was small. The proposed methodology might be generally applicable, but is computer intensive.     

Prediction error variance and expected response to selection,when selection is based on the best predictor – for Gaussian and threshold characters,traits following a Poisson mixed model and survival traits

In this paper, we consider selection based on the best predictor of animal additive genetic values in Gaussian linear mixed models, threshold models, Poisson mixed models, and log normal frailty models for survival data (including models with time-dependent covariates with associated fixed or random effects). In the different models, expressions are given (when these can be found – otherwise unbiased estimates are given) for prediction error variance, accuracy of selection and expected response to selection on the additive genetic scale and on the observed scale. The expressions given for non Gaussian traits are generalisations of the well-known formulas for Gaussian traits – and reflect, for Poisson mixed models and frailty models for survival data, the hierarchal structure of the models. In general the ratio of the additive genetic variance to the total variance in the Gaussian part of the model (heritability on the normally distributed level of the model) or a generalised version of heritability plays a central role in these formulas.     

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.     

Mixed Poisson regression models with covariate dependent rates

This paper studies a class of Poisson mixture models that includes covariates in rates. This model contains Poisson regression and independent Poisson mixtures as special cases. Estimation methods based on the EM and quasi-Newton algorithms, properties of these estimates, a model selection procedure, residual analysis, and goodness-of-fit test are discussed. A Monte Carlo study investigates implementation and model choice issues. This methodology is used to analyze seizure frequency and Ames salmonella assay data.     

Complementary responses to mean and variance modulations in the perfect integrate-and-fire model

In the perfect integrate-and-fire model (PIF), the membrane voltage is proportional to the integral of the input current since the time of the previous spike. It has been shown that the firing rate within a noise free ensemble of PIF neurons responds instantaneously to dynamic changes in the input current, whereas in the presence of white noise, model neurons preferentially pass low frequency modulations of the mean current. Here, we prove that when the input variance is perturbed while holding the mean current constant, the PIF responds preferentially to high frequency modulations. Moreover, the linear filters for mean and variance modulations are complementary, adding exactly to one. Since changes in the rate of Poisson distributed inputs lead to proportional changes in the mean and variance, these results imply that an ensemble of PIF neurons transmits a perfect replica of the time-varying input rate for Poisson distributed input. A more general argument shows that this property holds for any signal leading to proportional changes in the mean and variance of the input current.     

Comparison of Bayesian methods for flexible modeling of spatial risk surfaces in disease mapping

Bayesian hierarchical models usually model the risk surface on the same arbitrary geographical units for all data sources. Poisson/gamma random field models overcome this restriction as the underlying risk surface can be specified independently to the resolution of the data. Moreover, covariates may be considered as either excess or relative risk factors. We compare the performance of the Poisson/gamma random field model to the Markov random field (MRF)‐based ecologic regression model and the Bayesian Detection of Clusters and Discontinuities (BDCD) model, in both a simulation study and a real data example. We find the BDCD model to have advantages in situations dominated by abruptly changing risk while the Poisson/gamma random field model convinces by its flexibility in the estimation of random field structures and by its flexibility incorporating covariates. The MRF‐based ecologic regression model is inferior. WinBUGS code for Poisson/gamma random field models is provided.     

Zero‐modified Poisson model: Bayesian approach,influence diagnostics,and an application to a Brazilian leptospirosis notification data

In this paper, a Bayesian method for inference is developed for the zero‐modified Poisson (ZMP) regression model. This model is very flexible for analyzing count data without requiring any information about inflation or deflation of zeros in the sample. A general class of prior densities based on an information matrix is considered for the model parameters. A sensitivity study to detect influential cases that can change the results is performed based on the Kullback–Leibler divergence. Simulation studies are presented in order to illustrate the performance of the developed methodology. Two real datasets on leptospirosis notification in Bahia State (Brazil) are analyzed using the proposed methodology for the ZMP model.