Hierarchical Poisson Model Research Articles

On the Correlation Structure of Gaussian Copula Models for Geostatistical Count Data

SummaryWe describe a class of random field models for geostatistical count data based on Gaussian copulas. Unlike hierarchical Poisson models often used to describe this type of data, Gaussian copula models allow a more direct modelling of the marginal distributions and association structure of the count data. We study in detail the correlation structure of these random fields when the family of marginal distributions is either negative binomial or zero‐inflated Poisson; these represent two types of overdispersion often encountered in geostatistical count data. We also contrast the correlation structure of one of these Gaussian copula models with that of a hierarchical Poisson model having the same family of marginal distributions, and show that the former is more flexible than the latter in terms of range of feasible correlation, sensitivity to the mean function and modelling of isotropy. An exploratory analysis of a dataset of Japanese beetle larvae counts illustrate some of the findings. All of these investigations show that Gaussian copula models are useful alternatives to hierarchical Poisson models, specially for geostatistical count data that display substantial correlation and small overdispersion.

Relative age and birthplace effect in Japanese professional sports: a quantitative evaluation using a Bayesian hierarchical Poisson model

Relative age effect (RAE) in sports has been well documented. Recent studies investigate the effect of birthplace in addition to the RAE. The first objective of this study was to show the magnitude of the RAE in two major professional sports in Japan, baseball and soccer. Second, we examined the birthplace effect and compared its magnitude with that of the RAE. The effect sizes were estimated using a Bayesian hierarchical Poisson model with the number of players as dependent variable. The RAEs were 9.0% and 7.7% per month for soccer and baseball, respectively. These estimates imply that children born in the first month of a school year have about three times greater chance of becoming a professional player than those born in the last month of the year. Over half of the difference in likelihoods of becoming a professional player between birthplaces was accounted for by weather conditions, with the likelihood decreasing by 1% per snow day. An effect of population size was not detected in the data. By investigating different samples, we demonstrated that using quarterly data leads to underestimation and that the age range of sampled athletes should be set carefully.

MCMC implementation of the optimal Bayesian classifier for non-Gaussian models: model-based RNA-Seq classification.

BackgroundSequencing datasets consist of a finite number of reads which map to specific regions of a reference genome. Most effort in modeling these datasets focuses on the detection of univariate differentially expressed genes. However, for classification, we must consider multiple genes and their interactions.ResultsThus, we introduce a hierarchical multivariate Poisson model (MP) and the associated optimal Bayesian classifier (OBC) for classifying samples using sequencing data. Lacking closed-form solutions, we employ a Monte Carlo Markov Chain (MCMC) approach to perform classification. We demonstrate superior or equivalent classification performance compared to typical classifiers for two synthetic datasets and over a range of classification problem difficulties. We also introduce the Bayesian minimum mean squared error (MMSE) conditional error estimator and demonstrate its computation over the feature space. In addition, we demonstrate superior or leading class performance over an RNA-Seq dataset containing two lung cancer tumor types from The Cancer Genome Atlas (TCGA).ConclusionsThrough model-based, optimal Bayesian classification, we demonstrate superior classification performance for both synthetic and real RNA-Seq datasets. A tutorial video and Python source code is available under an open source license at http://bit.ly/1gimnss.Electronic supplementary materialThe online version of this article (doi:10.1186/s12859-014-0401-3) contains supplementary material, which is available to authorized users.

Rat sightings in New York City are associated with neighborhood sociodemographics, housing characteristics, and proximity to open public space.

Rats are ubiquitous in urban environments and, as established reservoirs for infectious pathogens, present a control priority for public health agencies. New York City (NYC) harbors one of the largest rat populations in the United States, but surprising little study has been undertaken to define rat ecology across varied features of this urban landscape. More importantly, factors that may contribute to increased encounters between rats and humans have rarely been explored. Using city-wide records of rat sightings reported to the NYC Department of Health and Mental Hygiene, this investigation sought to identify sociodemographic, housing, and physical landscape characteristics that are associated with increased rat sightings across NYC census tracts. A hierarchical Bayesian conditional autoregressive Poisson model was used to assess these associations while accounting for spatial heterogeneity in the variance. Closer proximity to both subway lines and recreational public spaces was associated with a higher concentration of rat sightings, as was a greater presence of older housing, vacant housing units, and low education among the population. Moreover, these aspects of the physical and social landscape accurately predicted rat sightings across the city. These findings have identified specific features of the NYC urban environment that may help to provide direct control targets for reducing human–rat encounters.

A hierarchical zero‐inflated model for species compositional data—from individual taxon responses to community response

Healthy macroinvertebrate communities are typically composed of a few abundant taxa and many relatively rare taxa. These rare taxa tend to be sensitive to pollution and, in aggregate, are important indicators of water quality (e.g., EPT taxa richness that combines Ephemeroptera, Plecoptera, and Trichoptera taxa). Rare taxa lead to community data sets that contain large numbers of zero values that are often a source of difficulty in data processing and analysis. A hierarchical zero‐inflated Poisson model is developed to study the responses to watershed urbanization of a group of macroinvertebrates taxa (EPT taxa) often used as indicators of good water quality and a healthy stream ecosystem. The zero‐inflated Poisson model is parameterized with a unimodal curve for the abundance model and a logit‐linear model for the zero model. The hierarchical model is used to integrate responses of individual taxa into a community level response. The emphasis of the model is on estimating the probability that an observed 0 represents the true absence of a taxon. By modeling the probability of true absence as a function of watershed level urbanization intensity both at individual taxon level and at the community level, we are able to develop a stressor‐specific tolerance indicator to (1) separate truly sensitive EPT taxa from more tolerant taxa, and (2) compare the responses of different communities to the same stressor or responses of the same community in different regions. We document the development of the model and the process of integrating individual taxon models into a community model. The model is applied to a data set collected for study of the effect of urbanization on stream ecosystems.

Prospective analysis of infectious disease surveillance data using syndromic information

In this paper, we describe a Bayesian hierarchical Poisson model for the prospective analysis of data for infectious diseases. The proposed model consists of two components. The first component describes the behavior of disease during nonepidemic periods and the second component represents the increase in disease counts due to the presence of an epidemic. A novelty of our model formulation is that the parameters describing the spread of epidemics are allowed to vary in both space and time. We also show how syndromic information can be incorporated into the model to provide a better description of the data and more accurate one-step-ahead forecasts. These real-time forecasts can be used to identify high-risk areas for outbreaks and, consequently, to develop efficient targeted surveillance. We apply the methodology to weekly emergency room discharges for acute bronchitis in South Carolina.

Multi-level Bayesian analyses for single- and multi-vehicle freeway crashes

This study presents multi-level analyses for single- and multi-vehicle crashes on a mountainous freeway. Data from a 15-mile mountainous freeway section on I-70 were investigated. Both aggregate and disaggregate models for the two crash conditions were developed. Five years of crash data were used in the aggregate investigation, while the disaggregate models utilized one year of crash data along with real-time traffic and weather data. For the aggregate analyses, safety performance functions were developed for the purpose of revealing the contributing factors for each crash type. Two methodologies, a Bayesian bivariate Poisson-lognormal model and a Bayesian hierarchical Poisson model with correlated random effects, were estimated to simultaneously analyze the two crash conditions with consideration of possible correlations. Except for the factors related to geometric characteristics, two exposure parameters (annual average daily traffic and segment length) were included. Two different sets of significant explanatory and exposure variables were identified for the single-vehicle (SV) and multi-vehicle (MV) crashes. It was found that the Bayesian bivariate Poisson-lognormal model is superior to the Bayesian hierarchical Poisson model, the former with a substantially lower DIC and more significant variables. In addition to the aggregate analyses, microscopic real-time crash risk evaluation models were developed for the two crash conditions. Multi-level Bayesian logistic regression models were estimated with the random parameters accounting for seasonal variations, crash-unit-level diversity and segment-level random effects capturing unobserved heterogeneity caused by the geometric characteristics. The model results indicate that the effects of the selected variables on crash occurrence vary across seasons and crash units; and that geometric characteristic variables contribute to the segment variations: the more unobserved heterogeneity have been accounted, the better classification ability. Potential applications of the modeling results from both analysis approaches are discussed.

Abstract TMP43: Ischemic Stroke Hospitalization Rates by State and Age

Background: Medicare data show regional differences in stroke hospitalizations for older patients, but data on regional patterns for younger patients are limited. We assessed geographic patterns of ischemic stroke hospitalizations rates by state for younger and older U.S. adults. Methods: Ischemic stroke hospitalizations (ICD-9 433, 434, 436) were identified using the Nationwide Inpatient Sample of the Healthcare Cost and Utilization Project (HCUP) for 2007-2008. Rates were expressed per 100,000 person-years (P-Y) with state-level age- and sex-specific Census data as a denominator. A hierarchical random effects Poisson model estimated state-specific risk-adjusted rates, adjusting for age-sex and applying HCUP sampling weights. Bootstrapping simulation was used to obtain interval estimates of rates for each state and age group (18-64 and 65+). Rates were categorized as being significantly higher, no different, or significantly lower than the national average based on their interval estimates. Results: More than half of strokes in older patients occurred in women (54.4%) compared to 43.5% in younger patients. The prevalence of many comorbidities (COPD, HF, HTN, and renal failure) was expectedly higher for older patients, but the prevalence of diabetes was higher in the younger group (36.2% vs. 30.5%, p<0.001). The overall stroke hospitalization rate was 93 per 100,000 P-Y in the younger group and 1134 per 100,000 P-Y for those 65+ yrs. Ten states’ classifications differed between age groups indicating moderate state-level variance, with rates above the national average tending to cluster in the southeast (figure). Conclusion: Geographic patterns of hospitalization were moderately different for younger and older patients with ischemic stroke. Understanding factors contributing to regional heterogeneity may help target areas in need of improved stroke prevention interventions.

Identifying high-risk areas for sporadic measles outbreaks: lessons from South Africa

To develop a model for identifying areas at high risk for sporadic measles outbreaks based on an analysis of factors associated with a national outbreak in South Africa between 2009 and 2011. Data on cases occurring before and during the national outbreak were obtained from the South African measles surveillance programme, and data on measles immunization and population size, from the District Health Information System. A Bayesian hierarchical Poisson model was used to investigate the association between the risk of measles in infants in a district and first-dose vaccination coverage, population density, background prevalence of human immunodeficiency virus (HIV) infection and expected failure of seroconversion. Model projections were used to identify emerging high-risk areas in 2012. A clear spatial pattern of high-risk areas was noted, with many interconnected (i.e. neighbouring) areas. An increased risk of measles outbreak was significantly associated with both the preceding build-up of a susceptible population and population density. The risk was also elevated when more than 20% of infants in a populous area had missed a first vaccine dose. The model was able to identify areas at high risk of experiencing a measles outbreak in 2012 and where additional preventive measures could be undertaken. The South African measles outbreak was associated with the build-up of a susceptible population (owing to poor vaccine coverage), high prevalence of HIV infection and high population density. The predictive model developed could be applied to other settings susceptible to sporadic outbreaks of measles and other vaccine-preventable diseases.

Disease mapping of Leishmaniasis outbreak in Afghanistan: spatial hierarchical Bayesian analysis

ObjectiveTo analyze the spatial pattern of Leishmaniasis disease in Afghanistan, using provincial level geo-referenced data. The disease is contracted through bites from sand flies and is the third most common vector-borne disease. Leishmaniasis is a serious health concern in Afghanistan with about 250 000 estimated new cases of cutaneous infection nationwide and 67,000 cases in Kabul. This makes Kabul the city with the largest incidence of the disease worldwide. MethodsWe use a Bayesian hierarchical Poisson model to estimate the influence of hypothesized risk factors on the relative risk of the disease. We use random components to take into account the lack of independence of the risk between adjacent areas. ResultsStatistical inference is carried out using Markov Chain Monte Carlo simulation. The final model specification includes altitude, two random components (intercept and slope) and utilizes a conditional autoregressive prior with a deviance information criterion of 247.761. Spatial scan statistics confirm disease clusters in the North-Eastern and South-Eastern regions of Afghanistan with a p-value of less than 0.0001. ConclusionsThe study confirms disease clusters in the North-Eastern and South-Eastern regions of Afghanistan. Our findings are robust with respect to the specification of the prior distribution and give important insights into the spatial dynamics of Leishmaniasis in Afghanistan.

Estimation of small area event rates and of the associated standard errors

We develop empirical best estimators for small area event rates based on the hierarchical Poisson model with log-normal mixing distribution, when the basic data consists of area level measurements. We derive an approximate expression to the mean squared error of the estimators and we provide a method for estimating this expression.

Bayesian hierarchical Poisson models with a hidden Markov structure for the detection of influenza epidemic outbreaks

Considerable effort has been devoted to the development of statistical algorithms for the automated monitoring of influenza surveillance data. In this article, we introduce a framework of models for the early detection of the onset of an influenza epidemic which is applicable to different kinds of surveillance data. In particular, the process of the observed cases is modelled via a Bayesian Hierarchical Poisson model in which the intensity parameter is a function of the incidence rate. The key point is to consider this incidence rate as a normal distribution in which both parameters (mean and variance) are modelled differently, depending on whether the system is in an epidemic or non-epidemic phase. To do so, we propose a hidden Markov model in which the transition between both phases is modelled as a function of the epidemic state of the previous week. Different options for modelling the rates are described, including the option of modelling the mean at each phase as autoregressive processes of order 0, 1 or 2. Bayesian inference is carried out to provide the probability of being in an epidemic state at any given moment. The methodology is applied to various influenza data sets. The results indicate that our methods outperform previous approaches in terms of sensitivity, specificity and timeliness.

Estimation of the within-herd transmission parameter of bovine leukemia virus

In this study, we estimated the transmission parameter of bovine leukemia virus (BLV) infection within herds by following up the serological status of cattle in beef and dairy herds. Two consecutive serological tests using ELISA were conducted for cattle at four beef breeding farms and nine dairy farms with an interval of 5 months. Assuming that sero-converted animals were newly infected, transmission parameters were estimated using a hierarchical Bayesian Poisson model. All tested herds had at least one positive animal at the first testing, but sero-prevalence considerably varied between herds. It is estimated that 0.62 animals were infected by one infected animal introduced into a fully susceptible population for both beef breeding and dairy farms during the 5-month interval. The 95% credible interval of the transmission parameter for the combined herds was 0.37–0.89 over this period. Asymptomatically infected animals with BLV can become the source of lifelong infection in herds. These estimated transmission parameters indicated that the early removal of infected animals with periodical tests could reduce the dissemination of BLV infection within herds.

Empirical Evaluation of Alternative Approaches in Identifying Crash Hot Spots

This study proposes a framework of a model-based hot spot identification method by applying full Bayes (FB) technique. In comparison with the state-of-the-art approach [i.e., empirical Bayes method (EB)], the advantage of the FB method is the capability to seamlessly integrate prior information and all available data into posterior distributions on which various ranking criteria could be based. With intersection crash data collected in Singapore, an empirical analysis was conducted to evaluate the following six approaches for hot spot identification: ( a) naive ranking using raw crash data, ( b) standard EB ranking, ( c) FB ranking using a Poisson-gamma model, ( d) FB ranking using a Poisson-lognormal model, ( e) FB ranking using a hierarchical Poisson model, and ( f) FB ranking using a hierarchical Poisson (AR-1) model. The results show that ( a) when using the expected crash rate–related decision parameters, all model-based approaches perform significantly better in safety ranking than does the naive ranking method, and ( b) the FB approach using hierarchical models significantly outperforms the standard EB approach in correctly identifying hazardous sites.

Robustness of the BYM model in absence of spatial variation in the residuals

BackgroundIn the context of ecological studies, the Bayesian hierarchical Poisson model is of prime interest when studying the association between environmental exposure and rare diseases. However, adding spatially structured extra-variability in the model fitted to the data when such extra-variability does not exist conditionally on the covariates included in the model (over-fitting) may bias the estimation of the ecological association between covariates and relative risks toward the null. In order to investigate that possibility, a simulation study of the impact of introducing unnecessary residual spatial structure in the estimation model was conducted.ResultsIn the case where no underlying extra-variability from the Poisson process exists, the simulation results show that models accounting for structured and unstructured residuals do not underestimate the ecological association, unless covariates have a very strong autocorrelation structure, i.e., 0.98 at 100 km on a territory of diameter 1000 km."

Hierarchical overdispersed Poisson model with macrolevel autocorrelation

We review Bayesian analysis of hierarchical non-standard Poisson regression models with an emphasis on microlevel heterogeneity and macrolevel autocorrelation. For the former case, we confirm that negative binomial regression usually accounts for microlevel heterogeneity (overdispersion) satisfactorily; for the latter case, we apply the simple first-order Markov transition model to conveniently capture the macrolevel autocorrelation which often arises from temporal and/or spatial count data, rather than attaching complex random effects directly to the regression parameters. Specifically, we extend the hierarchical (multilevel) Poisson model into negative binomial models with macrolevel autocorrelation using restricted gamma mixture with unit mean and Markov transition covariate created from preceding residuals. We prove a mild sufficient condition for posterior propriety under flat prior for the interesting fixed effects. Our methodology is implemented by analyzing the Baltic sea peracarids diurnal activity data published in the marine biology and ecology literature.

Modeling Sage data with a truncated gamma-Poisson model.

BackgroundSerial Analysis of Gene Expressions (SAGE) produces gene expression measurements on a discrete scale, due to the finite number of molecules in the sample. This means that part of the variance in SAGE data should be understood as the sampling error in a binomial or Poisson distribution, whereas other variance sources, in particular biological variance, should be modeled using a continuous distribution function, i.e. a prior on the intensity of the Poisson distribution. One challenge is that such a model predicts a large number of genes with zero counts, which cannot be observed.ResultsWe present a hierarchical Poisson model with a gamma prior and three different algorithms for estimating the parameters in the model. It turns out that the rate parameter in the gamma distribution can be estimated on the basis of a single SAGE library, whereas the estimate of the shape parameter becomes unstable. This means that the number of zero counts cannot be estimated reliably. When a bivariate model is applied to two SAGE libraries, however, the number of predicted zero counts becomes more stable and in approximate agreement with the number of transcripts observed across a large number of experiments. In all the libraries we analyzed there was a small population of very highly expressed tags, typically 1% of the tags, that could not be accounted for by the model. To handle those tags we chose to augment our model with a non-parametric component. We also show some results based on a log-normal distribution instead of the gamma distribution.ConclusionBy modeling SAGE data with a hierarchical Poisson model it is possible to separate the sampling variance from the variance in gene expression. If expression levels are reported at the gene level rather than at the tag level, genes mapped to multiple tags must be kept separate, since their expression levels show a different statistical behavior. A log-normal prior provided a better fit to our data than the gamma prior, but except for a small subpopulation of tags with very high counts, the two priors are similar.

Iowa's Experience with Road Diet Measures: Use of Bayesian Approach to Assess Impacts on Crash Frequencies and Crash Rates

A before-and-after study implemented from a Bayesian perspective to assess crash history reduction due to road diets in Iowa was conducted by the Iowa State University Department of Statistics in cooperation with the Iowa Department of Transportation's Office of Traffic and Safety. The study used both monthly crash data and estimated volumes obtained from TAS for 30 sites—15 treatment and 15 comparison-over 23 years (1982 to 2004). The sites had volumes ranging from 2,030 to 15, 350 during this period and were largely located in smaller urbanized areas. The research objective was to assess whether road diets appear to result in crash reductions on Iowa roads. Crash data were analyzed at each site before and after the conversions were completed. Given the random, rare nature of crash events, a hierarchical Poisson model was fitted, such that log mean rate was expressed as a piecewise linear function of time period, seasonal effects, and a random effect corresponding to each site. Monthly traffic volumes we...

Iowa's Experience with Road Diet Measures

A before-and-after study implemented from a Bayesian perspective to assess crash history reduction due to road diets in Iowa was conducted by the Iowa State University Department of Statistics in cooperation with the Iowa Department of Transportation's Office of Traffic and Safety. The study used both monthly crash data and estimated volumes obtained from TAS for 30 sites—15 treatment and 15 comparison—over 23 years (1982 to 2004). The sites had volumes ranging from 2,030 to 15,350 during this period and were largely located in smaller urbanized areas. The research objective was to assess whether road diets appear to result in crash reductions on Iowa roads. Crash data were analyzed at each site before and after the conversions were completed. Given the random, rare nature of crash events, a hierarchical Poisson model was fitted, such that log mean rate was expressed as a piecewise linear function of time period, seasonal effects, and a random effect corresponding to each site. Monthly traffic volumes were incorporated as exposures. Estimation of model parameters was conducted within a Bayesian framework. Results indicate a 25.2% reduction in crash frequency per mile and an 18.8% reduction in crash rate. This differs from a previously publicized study that reported a 6% reduction in crash frequency per mile and an insignificant indication for crash rate effects. The results from the Iowa study fit practitioner experience and agree with another Iowa study that used a simple before-and-after approach on the same sites.

Benchmarking patient outcomes.

To examine the usefulness of three types of benchmarking for interpreting patient outcome data. This study was part of a multiyear, multihospital longitudinal survey of 10 patient outcomes. The patient outcome used for this methodologic presentation was central line infections (CLI). The sample included eight hospitals in an integrated healthcare system, with a range in size from 144 to 861 beds. The unit of analysis for CLI was the number of line days, with the CLI rate defined as the number of infections per 1,000 patient-line days per month. Data on each outcome were collected at the unit level according to standardized protocols. Results were submitted via standardized electronic forms to a central data management center. Data for this presentation were analyzed using a Bayesian hierarchical Poisson model. Results are presented for each hospital and the system as a whole. In comparison to published benchmarks, hospital performances were mixed with regard to CLI. Five of the 8 hospitals exceeded 2.2 infections per 1,000 patient-line days. When benchmarks were established for each hospital using 95% credible intervals, hospitals did reasonably well with only isolated months reaching or going beyond the benchmark limits. When the entire system was used to establish benchmarks with the 95% credible intervals, the hospitals that reached or exceeded the benchmark limits remained the same, but some hospitals had CLI rates more frequently in the upper 50% of the benchmarking limits. Benchmarking of quality indicators can be accomplished in a variety of ways as a means to quantify patient care and identify areas needing attention and improvement. Hospital-specific and system-wide benchmarks provide relevant feedback for improving performance at individual hospitals.