Mixture Of Negative Binomial Distributions Research Articles

Small Sample Inference for Two‐Way Capture‐Recapture Experiments

SummaryThe properties of the generalised Waring distribution defined on the non‐negative integers are reviewed. Formulas for its moments and its mode are given. A construction as a mixture of negative binomial distributions is also presented. Then we turn to the Petersen model for estimating the population size in a two‐way capture‐recapture experiment. We construct a Bayesian model for by combining a Waring prior with the hypergeometric distribution for the number of units caught twice in the experiment. Credible intervals for are obtained using quantiles of the posterior, a generalised Waring distribution. The standard confidence interval for the population size constructed using the asymptotic variance of Petersen estimator and 0.5 logit transformed interval are shown to be special cases of the generalised Waring credible interval. The true coverage of this interval is shown to be bigger than or equal to its nominal converage in small populations, regardless of the capture probabilities. In addition, its length is substantially smaller than that of the 0.5 logit transformed interval. Thus, the proposed generalised Waring credible interval appears to be the best way to quantify the uncertainty of the Petersen estimator for populations size.

The size of a Markovian SIR epidemic given only removal data

AbstractDuring an epidemic outbreak, typically only partial information about the outbreak is known. A common scenario is that the infection times of individuals are unknown, but individuals, on displaying symptoms, are identified as infectious and removed from the population. We study the distribution of the number of infectives given only the times of removals in a Markovian susceptible–infectious–removed (SIR) epidemic. Primary interest is in the initial stages of the epidemic process, where a branching (birth–death) process approximation is applicable. We show that the number of individuals alive in a time-inhomogeneous birth–death process at time $t \geq 0$ , given only death times up to and including time t, is a mixture of negative binomial distributions, with the number of mixing components depending on the total number of deaths, and the mixing weights depending upon the inter-arrival times of the deaths. We further consider the extension to the case where some deaths are unobserved. We also discuss the application of the results to control measures and statistical inference.

Beta–Negative Binomial Auto-Regressions for Modelling Integer-Valued Time Series with Extreme Observations

SummaryThe paper introduces a general class of heavy-tailed auto-regressions for modelling integer-valued time series with outliers. The specification proposed is based on a heavy-tailed mixture of negative binomial distributions that features an observation-driven dynamic equation for the conditional expectation. The existence of a stationary and ergodic solution for the class of auto-regressive processes is shown under general conditions. The estimation of the model can be easily performed by maximum likelihood given the closed form of the likelihood function. The strong consistency and the asymptotic normality of the estimator are formally derived. Two examples of specifications illustrate the flexibility of the approach and the relevance of the theoretical results. In particular, a linear dynamic equation and a score-driven equation for the conditional expectation are studied. The score-driven specification is shown to be particularly appealing as it delivers a robust filtering method that attenuates the effect of outliers. Empirical applications to the series of narcotics trafficking reports in Sydney and the euro–pound sterling exchange rate illustrate the effectiveness of the method in handling extreme observations.

Genomewide Mapping of Selection Signatures and Genes for Extreme Feather Pecking in Two Divergently Selected Laying Hen Lines.

Simple SummaryFeather pecking is a behavior frequently occurring in commercial layer flocks. It often leads to skin injuries and cannibalism. Besides economic losses, severe animal welfare problems cannot be ignored. Previous research has shown that the trait is heritable. Thus breeding against feather pecking is possible, but phenotyping in a commercial environment is economically unfeasible at the moment because of the lack of proper techniques. Therefore, understanding the genetic background of the trait is mandatory to establish a genomic breeding program. This would require genotypic information of the hens, which is feasible under practical conditions. In the present study, we used different methods to identify regions in the genome that influence feather pecking and extreme feather pecking. We found one trait associated with the genomic region. The use of genotypic information from this region in terms of selection against the undesired behavior may help to improve animal welfare in layer flocks.Feather pecking (FP) is a longstanding serious problem in commercial flocks of laying hens. It is a highly polygenic trait and the genetic background is still not completely understood. In order to find genomic regions influencing FP, selection signatures between laying hen lines divergently selected for high and low feather pecking were mapped using the intra-population iHS and the inter-population FST approach. In addition, the existence of an extreme subgroup of FP hens (EFP) across both selected lines has been demonstrated by fitting a mixture of negative binomial distributions to the data and calculating the posterior probability of belonging to the extreme subgroup (pEFP) for each hen. A genomewide association study (GWAS) was performed for the traits pEFP and FP delivered (FPD) with a subsequent post GWAS analysis. Mapping of selection signatures revealed no clear regions under selection. GWAS revealed a region on Chromosome 1, where the existence of a QTL influencing FP is likely. The candidate genes found in this region are a part of the GABAergic system, which has already been linked to FP in previous studies. Despite the polygenic nature of FP, selection on these candidate genes may reduce FP.

A bivariate mixture of negative binomial distributions and its applications

We construct a new bivariate mixture of negative binomial distributions which represents over-dispersed data more efficiently. This is an extension of a univariate mixture of beta and negative binomial distributions. Characteristics of this joint distribution are studied including conditional distributions. Some properties of the correlation coefficient are explored. We demonstrate the applicability of our proposed model by fitting to three real data sets with correlated count data. A comparison is made with some previously used models to show the effectiveness of the new model.

Bayesian negative binomial mixture regression models for the analysis of sequence count and methylation data.

In this article, we develop a Bayesian hierarchical mixture regression model for studying the association between a multivariate response, measured as counts on a set of features, and a set of covariates. We have available RNA-Seq and DNA methylation data measured on breast cancer patients at different stages of the disease. We account for the heterogeneity and over-dispersion of count data (here, RNA-Seq data) by considering a mixture of negative binomial distributions and incorporate the covariates (here, methylation data) into the model via a linear modeling construction on the mean components. Our modeling construction includes several innovative characteristics. First, it employs selection techniques that allow the identification of a small subset of features that best discriminate the samples while simultaneously selecting a set of covariates associated to each feature. Second, it incorporates known dependencies into the feature selection process via the use of Markov random field (MRF) priors. On simulated data, we show how incorporating existing information via the prior model can improve the accuracy of feature selection. In the analysis of RNA-Seq and DNA methylation data on breast cancer, we incorporate knowledge on relationships among genes via a gene-gene network, which we extract from the KEGG database. Our data analysis identifies genes which are discriminatory of cancer stages and simultaneously selects significant associations between those genes and DNA methylation sites. A biological interpretation of our findings reveals several biomarkers that can help understanding the effect of DNA methylation on gene expression transcription across cancer stages.

Assignment of endogenous retrovirus integration sites using a mixture model

Structural variation occurs in the genomes of individuals because of the different positions occupied by repetitive genome elements like endogenous retroviruses, or ERVs. The presence or absence of ERVs can be determined by identifying the junction with the host genome using high-throughput sequence technology and a clustering algorithm. The resulting data give the number of sequence reads assigned to each ERV-host junction sequence for each sampled individual. Variability in the number of reads from an individual integration site makes it difficult to determine whether a site is present for low read counts. We present a novel two-component mixture of negative binomial distributions to model these counts and assign a probability that a given ERV is present in a given individual. We explain how our approach is superior to existing alternatives, including another form of two-component mixture model and the much more common approach of selecting a threshold count for declaring the presence of an ERV. We apply our method to a data set of ERV integrations in mule deer (Odocoileus hemionus), a species for which no genomic resources are available, and demonstrate that the discovered patterns of shared integration sites contain information about animal relatedness.

A Financial Market Model Incorporating Herd Behaviour.

Herd behaviour in financial markets is a recurring phenomenon that exacerbates asset price volatility, and is considered a possible contributor to market fragility. While numerous studies investigate herd behaviour in financial markets, it is often considered without reference to the pricing of financial instruments or other market dynamics. Here, a trader interaction model based upon informational cascades in the presence of information thresholds is used to construct a new model of asset price returns that allows for both quiescent and herd-like regimes. Agent interaction is modelled using a stochastic pulse-coupled network, parametrised by information thresholds and a network coupling probability. Agents may possess either one or two information thresholds that, in each case, determine the number of distinct states an agent may occupy before trading takes place. In the case where agents possess two thresholds (labelled as the finite state-space model, corresponding to agents’ accumulating information over a bounded state-space), and where coupling strength is maximal, an asymptotic expression for the cascade-size probability is derived and shown to follow a power law when a critical value of network coupling probability is attained. For a range of model parameters, a mixture of negative binomial distributions is used to approximate the cascade-size distribution. This approximation is subsequently used to express the volatility of model price returns in terms of the model parameter which controls the network coupling probability. In the case where agents possess a single pulse-coupling threshold (labelled as the semi-infinite state-space model corresponding to agents’ accumulating information over an unbounded state-space), numerical evidence is presented that demonstrates volatility clustering and long-memory patterns in the volatility of asset returns. Finally, output from the model is compared to both the distribution of historical stock returns and the market price of an equity index option.

DEXUS: identifying differential expression in RNA-Seq studies with unknown conditions

Detection of differential expression in RNA-Seq data is currently limited to studies in which two or more sample conditions are known a priori. However, these biological conditions are typically unknown in cohort, cross-sectional and nonrandomized controlled studies such as the HapMap, the ENCODE or the 1000 Genomes project. We present DEXUS for detecting differential expression in RNA-Seq data for which the sample conditions are unknown. DEXUS models read counts as a finite mixture of negative binomial distributions in which each mixture component corresponds to a condition. A transcript is considered differentially expressed if modeling of its read counts requires more than one condition. DEXUS decomposes read count variation into variation due to noise and variation due to differential expression. Evidence of differential expression is measured by the informative/noninformative (I/NI) value, which allows differentially expressed transcripts to be extracted at a desired specificity (significance level) or sensitivity (power). DEXUS performed excellently in identifying differentially expressed transcripts in data with unknown conditions. On 2400 simulated data sets, I/NI value thresholds of 0.025, 0.05 and 0.1 yielded average specificities of 92, 97 and 99% at sensitivities of 76, 61 and 38%, respectively. On real-world data sets, DEXUS was able to detect differentially expressed transcripts related to sex, species, tissue, structural variants or quantitative trait loci. The DEXUS R package is publicly available from Bioconductor and the scripts for all experiments are available at http://www.bioinf.jku.at/software/dexus/.

A full likelihood procedure of exchangeable negative binomials for modelling correlated and overdispersed count data

This paper introduces an exchangeable negative binomial distribution resulting from relaxing the independence of the Bernoulli sequence associated with a negative binomial distribution to exchangeability. It is demonstrated that the introduced distribution is a mixture of negative binomial distributions and can be characterized by infinitely many parameters that form a completely monotone sequence. The moments of the distribution are derived and a small simulation is conducted to illustrate the distribution. For data analytic purposes, two methods, truncation and completely-monotone links, are given for converting the saturated distribution of infinitely many parameters to parsimonious distributions of finitely many parameters. A full likelihood procedure is described which can be used to investigate correlated and overdispersed count data common in biomedical sciences and teratology. In the end, the introduced distribution is applied to analyze a real clinical data of burn wounds on patients.

Relationship between prevalence and intensity of Plasmodium falciparum infection in natural populations of Anopheles mosquitoes.

Wild-caught Anopheles gambiae s. l. and An. funestus were dissected and their midguts were examined for the presence of Plasmodium falciparum oocyst infections. The mean intensity of infection and the prevalence of infected mosquitoes were determined for each sample, with one sample representing the mosquitoes caught in a single house at any given time. The patterns of infection were investigated using the relationships between prevalence, intensity, and variance within samples, and were found to be consistent with laboratory infections. The overall distribution of oocysts is characterized by a mixture of negative binomial distributions with means determined by the infectiousness of the human hosts, and a constant degree of aggregation (k = 0.0767) presumably determined by the development of oocysts within mosquitoes. The prevalence/intensity relationship was treated as a bivariate distribution to ascertain the effect of sample size on the accuracy of estimation, and to allow inference of intensity from prevalence. In mathematical models fitted to the collected data, sample size affected directly the minimum possible prevalence of infection, and the accuracy of both mean and prevalence estimations. Based on minimum possible positive prevalence rates, data from samples of less than 20-25 mosquitoes would provide unacceptable errors in prevalence estimations. However, natural oocyst rates are consistently higher than the minimum prevalence, and it is suggested that any interpretations from samples of less than approximately 40 mosquitoes must be treated with some caution. Such variation in natural samples means that prediction of intensity of infection from prevalence (or vice versa) is extremely inaccurate.

Estimating a Market Equilibrium Search Model from Panel Data on Individuals

from Albrecht and Axell (1984) and is based on workers who are homogeneous in terms of market productivity and heterogeneous in terms of nonmarket productivity, and on firms which are heterogeneous in terms of productive efficiency. The equilibrium model is contrasted in terms of its fit to the data with an unrestricted version of the model which is based on a mixture of negative binomial distributions. The equilibrium model fails to conform to the data in exactly the dimension of its major focus, namely it implies that measurement error accounts for almost all of the dispersion in observed wages. The equilibrium model also does not do well in fitting the unemployment duration distribution compared to the unrestricted model. The problem is that the duration distribution itself does not support the existence of significant heterogeneity, as evidenced by the estimates of the unrestricted model. The paper also illustrates the use of such models for policy analysis by simulating the welfare effects of a minimum wage.