Analysis of Longitudinal Data of Epileptic Seizure Counts – A Two‐State Hidden Markov Regression Approach

This paper discusses a two‐state hidden Markov Poisson regression (MPR) model for analyzing longitudinal data of epileptic seizure counts, which allows for the rate of the Poisson process to depend on covariates through an exponential link function and to change according to the states of a two‐state Markov chain with its transition probabilities associated with covariates through a logit link function. This paper also considers a two‐state hidden Markov negative binomial regression (MNBR) model, as an alternative, by using the negative binomial instead of Poisson distribution in the proposed MPR model when there exists extra‐Poisson variation conditional on the states of the Markov chain. The two proposed models in this paper relax the stationary requirement of the Markov chain, allow for overdispersion relative to the usual Poisson regression model and for correlation between repeated observations. The proposed methodology provides a plausible analysis for the longitudinal data of epileptic seizure counts, and the MNBR model fits the data much better than the MPR model. Maximum likelihood estimation using the EM and quasi‐Newton algorithms is discussed. A Monte Carlo study for the proposed MPR model investigates the reliability of the estimation method, the choice of probabilities for the initial states of the Markov chain, and some finite sample behaviors of the maximum likelihood estimates, suggesting that (1) the estimation method is accurate and reliable as long as the total number of observations is reasonably large, and (2) the choice of probabilities for the initial states of the Markov process has little impact on the parameter estimates.     

Estimation of Poisson Rates with Misclassified Counts

The Poisson assumption is popular when data arises in the form of counts. In many applications such counts are fallible. Little research has been done on the Poisson distribution when both false positives and false negatives are present. We present a model in this paper that corrects for misclassification of count data. Bayesian estimators are developed. We provide the actual posterior distributions via integration. Markov Chain Monte Carlo results, which are more convenient for large sample sizes, are utilized for inference.     

On the generalized poisson regression mixture model for mapping quantitative trait loci with count data

Statistical methods for mapping quantitative trait loci (QTL) have been extensively studied. While most existing methods assume normal distribution of the phenotype, the normality assumption could be easily violated when phenotypes are measured in counts. One natural choice to deal with count traits is to apply the classical Poisson regression model. However, conditional on covariates, the Poisson assumption of mean-variance equality may not be valid when data are potentially under- or overdispersed. In this article, we propose an interval-mapping approach for phenotypes measured in counts. We model the effects of QTL through a generalized Poisson regression model and develop efficient likelihood-based inference procedures. This approach, implemented with the EM algorithm, allows for a genomewide scan for the existence of QTL throughout the entire genome. The performance of the proposed method is evaluated through extensive simulation studies along with comparisons with existing approaches such as the Poisson regression and the generalized estimating equation approach. An application to a rice tiller number data set is given. Our approach provides a standard procedure for mapping QTL involved in the genetic control of complex traits measured in counts.     

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.     

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.     

Zero-Inflated Poisson Models and C.A.MAN: A Tutorial Collection of Evidence

Analysis of count data is required in many areas of biometric interest. Often the simple Poisson distribution is not appropriate, since an extra-number of zero counts occur in the count data. Some current approaches for the problem at hand are reviewed. It will be argued that these situations can often be easily modeled using the zero-inflated Poisson distribution. A variety of applications are considered in which this occurs. Possibilities are outlined on how the validity of the zero-inflated Poisson can be validated including a comparison with the nonparametric Poisson mixture maximum likelihood estimator.     

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.     

Photon counting statistics analysis of biophotons from hands

The photon counting statistics of biophotons emitted from hands is studied with a view to test its agreement with the Poisson distribution. The moments of observed probability up to seventh order have been evaluated. The moments of biophoton emission from hands are in good agreement while those of dark counts of photomultiplier tube show large deviations from the theoretical values of Poisson distribution. The present results are consistent with the conventional delta-value analysis of the second moment of probability.     

Bayesian time-varying autoregressive models of COVID-19 epidemics

The COVID-19 pandemic has highlighted the importance of reliable statistical models which, based on the available data, can provide accurate forecasts and impact analysis of alternative policy measures. Here we propose Bayesian time-dependent Poisson autoregressive models that include time-varying coefficients to estimate the effect of policy covariates on disease counts. The model is applied to the observed series of new positive cases in Italy and in the United States. The results suggest that our proposed models are capable of capturing nonlinear growth of disease counts. We also find that policy measures and, in particular, closure policies and the distribution of vaccines, lead to a significant reduction in disease counts in both countries.     

Technical and biological variance structure in mRNA-Seq data: life in the real world

ABSTRACT: BACKGROUND: mRNA expression data from next generation sequencing platforms is obtained in the form of counts per gene or exon. Counts have classically been assumed to follow a Poisson distribution in which the variance is equal to the mean. The Negative Binomial distribution which allows for over-dispersion, i.e., for the variance to be greater than the mean, is commonly used to model count data as well. RESULTS: In mRNA-Seq data from 25 subjects, we found technical variation to generally follow a Poisson distribution as has been reported previously and biological variability was over-dispersed relative to the Poisson model. The mean-variance relationship across all genes was quadratic, in keeping with a Negative Binomial (NB) distribution. Over-dispersed Poisson and NB distributional assumptions demonstrated marked improvements in goodness-of-fit (GOF) over the standard Poisson model assumptions, but with evidence of over-fitting in some genes. Modeling of experimental effects improved GOF for high variance genes but increased the over-fitting problem. CONCLUSIONS: These conclusions will guide development of analytical strategies for accurate modeling of variance structure in these data and sample size determination which in turn will aid in the identification of true biological signals that inform our understanding of biological systems.     

A Multinomial Model for the Quality Control of Colony Counting Procedures

The so‐called good‐laboratory‐practice (GLP) test provides an experimental design and appropriate statistical analysis for the problem of analyst performance assessment in microbiological laboratories. For a given sample material multiple dilution series are generated yielding colony counts from several dilution levels. Statistical evaluation is based on the assumption of Poisson‐distributed colony forming units. In this paper a new model based on conditional binomial and multinomial distributions is presented and it is shown how it is related to the standard model which assumes Poisson‐distributed colony counts. The effects of common working errors on the statistical evaluation of the GLP‐test are investigated.     

Statistical evaluations of viral clearance studies for biological products

Since most biological products are derived from living cell culture, it is possible that viral contaminants be transmitted to the final product. Regulatory guidance requires that viral clearance studies be conducted to demonstrate the capacity of the production process in viral removal and inactivation. The key is accurate estimation of viral titer and reduction factor (RF), defined as the difference in log₁₀ virus titers before and after each step of purification. Darling et al. (1998) [1] suggested a method for analysis of clearance studies. However it is unable to establish an estimate of RF when the post-process viral counts are zero. In this paper, we provide theoretical justification of the method based on normal distribution and discuss the caveats regarding the degrees of freedom. We propose two alternative methods under the assumption that the number of plaques follows a Poisson distribution. Through simulation studies, the Poisson-based methods are shown to provide better estimates of viral titers. Under the Poisson model, we also derive a method to calculate the exact confidence limits for the viral titer and reduction factor even if the post-process viral counts are zero. The use of the methods is illustrated through numerical examples.     

Improving removal-based estimates of abundance by sampling a population of spatially distinct subpopulations

A statistical modeling framework is described for estimating the abundances of spatially distinct subpopulations of animals surveyed using removal sampling. To illustrate this framework, hierarchical models are developed using the Poisson and negative-binomial distributions to model variation in abundance among subpopulations and using the beta distribution to model variation in capture probabilities. These models are fitted to the removal counts observed in a survey of a federally endangered fish species. The resulting estimates of abundance have similar or better precision than those computed using the conventional approach of analyzing the removal counts of each subpopulation separately. Extension of the hierarchical models to include spatial covariates of abundance is straightforward and may be used to identify important features of an animal's habitat or to predict the abundance of animals at unsampled locations.     

Assessing the Exceptionality of Coloured Motifs in Networks

Various methods have been recently employed to characterise the structure of biological networks. In particular, the concept of network motif and the related one of coloured motif have proven useful to model the notion of a functional/evolutionary building block. However, algorithms that enumerate all the motifs of a network may produce a very large output, and methods to decide which motifs should be selected for downstream analysis are needed. A widely used method is to assess if the motif is exceptional, that is, over- or under-represented with respect to a null hypothesis. Much effort has been put in the last thirty years to derive -values for the frequencies of topological motifs, that is, fixed subgraphs. They rely either on (compound) Poisson and Gaussian approximations for the motif count distribution in Erdös-Rényi random graphs or on simulations in other models. We focus on a different definition of graph motifs that corresponds to coloured motifs. A coloured motif is a connected subgraph with fixed vertex colours but unspecified topology. Our work is the first analytical attempt to assess the exceptionality of coloured motifs in networks without any simulation. We first establish analytical formulae for the mean and the variance of the count of a coloured motif in an Erdös-Rényi random graph model. Using simulations under this model, we further show that a Pólya-Aeppli distribution better approximates the distribution of the motif count compared to Gaussian or Poisson distributions. The Pólya-Aeppli distribution, and more generally the compound Poisson distributions, are indeed well designed to model counts of clumping events. Altogether, these results enable to derive a -value for a coloured motif, without spending time on simulations.     

Universality in neural networks: the importance of the ‘mean firing rate’

We present a general analysis of highly connected recurrent neural networks which are able to learn and retrieve a finite number of static patterns. The arguments are based on spike trains and their interval distribution and require no specific model of a neuron. In particular, they apply to formal two-state neurons as well as to more refined models like the integrate-and-fire neuron or the Hodgkin-Huxley equations. We show that the mean firing rate defined as the inverse of the mean interval length is the only relevant parameter (apart from the synaptic weights) that determines the existence of retrieval solutions with a large overlap with one of the learnt patterns. The statistics of the spiking noise (Gaussian, Poisson or other) and hence the shape of the interval distribution does not matter. Thus our unifying approach explains why, and when, all the different associative networks which treat static patterns yield basically the same results, i.e., belong to the same universality class.     

The mode of spontaneous transmitter release at the insect neuromuscular junction.

The time intervals between miniature excitatory postsynaptic potentials and the counts of them in the cockroach, Periplaneta americana, were analyzed, using a computer program to test for properties of a Poisson process. The miniature potentials occurred basically in random manner at this neuromuscular junction. Although the distribution of the potentials did not fit the criteria for a Poisson process when the muscle fiber exhibited the short burst of high-frequency discharges, it was suggested that the primary process of such a distribution is Poisson, which is occasionally contaminated by the burst phase of the release rates.     

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.