Hurdle models for multilevel zero-inflated data via h-likelihood

Count data often exhibit overdispersion. One type of overdispersion arises when there is an excess of zeros in comparison with the standard Poisson distribution. Zero-inflated Poisson and hurdle models have been proposed to perform a valid likelihood-based analysis to account for the surplus of zeros. Further, data often arise in clustered, longitudinal or multiple-membership settings. The proper analysis needs to reflect the design of a study. Typically random effects are used to account for dependencies in the data. We examine the h-likelihood estimation and inference framework for hurdle models with random effects for complex designs. We extend the h-likelihood procedures to fit hurdle models, thereby extending h-likelihood to truncated distributions. Two applications of the methodology are presented.     

Marginalized multilevel hurdle and zero‐inflated models for overdispersed and correlated count data with excess zeros

Count data are collected repeatedly over time in many applications, such as biology, epidemiology, and public health. Such data are often characterized by the following three features. First, correlation due to the repeated measures is usually accounted for using subject‐specific random effects, which are assumed to be normally distributed. Second, the sample variance may exceed the mean, and hence, the theoretical mean–variance relationship is violated, leading to overdispersion. This is usually allowed for based on a hierarchical approach, combining a Poisson model with gamma distributed random effects. Third, an excess of zeros beyond what standard count distributions can predict is often handled by either the hurdle or the zero‐inflated model. A zero‐inflated model assumes two processes as sources of zeros and combines a count distribution with a discrete point mass as a mixture, while the hurdle model separately handles zero observations and positive counts, where then a truncated‐at‐zero count distribution is used for the non‐zero state. In practice, however, all these three features can appear simultaneously. Hence, a modeling framework that incorporates all three is necessary, and this presents challenges for the data analysis. Such models, when conditionally specified, will naturally have a subject‐specific interpretation. However, adopting their purposefully modified marginalized versions leads to a direct marginal or population‐averaged interpretation for parameter estimates of covariate effects, which is the primary interest in many applications. In this paper, we present a marginalized hurdle model and a marginalized zero‐inflated model for correlated and overdispersed count data with excess zero observations and then illustrate these further with two case studies. The first dataset focuses on the Anopheles mosquito density around a hydroelectric dam, while adolescents’ involvement in work, to earn money and support their families or themselves, is studied in the second example. Sub‐models, which result from omitting zero‐inflation and/or overdispersion features, are also considered for comparison's purpose. Analysis of the two datasets showed that accounting for the correlation, overdispersion, and excess zeros simultaneously resulted in a better fit to the data and, more importantly, that omission of any of them leads to incorrect marginal inference and erroneous conclusions about covariate effects. Copyright © 2014 John Wiley & Sons, Ltd.     

BLUP(REMQL) estimation of a correlated random effects negative binomial hurdle model

Bed days is a potentially useful metric of efficiency in clinical studies involving the hospital admission decision. However, this metric involves excess zeros, possible overdispersion, and possible clustering (in multi-site studies). A random effects negative binomial hurdle model can account for each of these issues. We extend this model to include site-level correlation between the two component parts and implement best linear unbiased prediction-type estimation with restricted maximum quasi-likelihood. This approach offers computational advantages over maximum likelihood in a generalized linear mixed model setting. Simulations show that the proposed approach performs well for fixed effects and variance components under a plausible range of bivariate correlation. The Emergency Department Community Acquired Pneumonia study motivates this work and illustrates the methods.     

Spatial Scan Statistics for Models with Excess Zeros and Overdispersion

Objective

To propose a more realistic model for disease cluster detection, through a modification of the spatial scan statistic to account simultaneously for inflated zeros and overdispersion.

Introduction

Spatial Scan Statistics [1] usually assume Poisson or Binomial distributed data, which is not adequate in many disease surveillance scenarios. For example, small areas distant from hospitals may exhibit a smaller number of cases than expected in those simple models. Also, underreporting may occur in underdeveloped regions, due to inefficient data collection or the difficulty to access remote sites. Those factors generate excess zero case counts or overdispersion, inducing a violation of the statistical model and also increasing the type I error (false alarms). Overdispersion occurs when data variance is greater than the predicted by the used model. To accommodate it, an extra parameter must be included; in the Poisson model, one makes the variance equal to the mean.

Methods

Tools like the Generalized Poisson (GP) and the Double Poisson [2] may be a better option for this kind of problem, modeling separately the mean and variance, which could be easily adjusted by covariates. When excess zeros occur, the Zero Inflated Poisson (ZIP) model is used, although ZIP’s estimated parameters may be severely biased if nonzero counts are too dispersed, compared to the Poisson distribution. In this case the Inflated Zero models for the Generalized Poisson (ZIGP), Double Poisson (ZIDP) and Negative Binomial (ZINB) could be good alternatives to the joint modeling of excess zeros and overdispersion. By one hand, Zero Inflated Poisson (ZIP) models were proposed using the spatial scan statistic to deal with the excess zeros [3]. By the other hand, another spatial scan statistic was based on a Poisson-Gamma mixture model for overdispersion [4]. In this work we present a model which includes inflated zeros and overdispersion simultaneously, based on the ZIDP model. Let the parameter p indicate the zero inflation. As the the remaining parameters of the observed cases map and the parameter p are not independent, the likelihood maximization process is not straightforward; it becomes even more complicated when we include covariates in the analysis. To solve this problem we introduce a vector of latent variables in order to factorize the likelihood, and obtain a facilitator for the maximization process using the E-M (Expectation-Maximization) algorithm. We derive the formulas to maximize iteratively the likelihood, and implement a computer program using the E-M algorithm to estimate the parameters under null and alternative hypothesis. The p-value is obtained via the Fast Double Bootstrap Test [5].

Results

Numerical simulations are conducted to assess the effectiveness of the method. We present results for Hanseniasis surveillance in the Brazilian Amazon in 2010 using this technique. We obtain the most likely spatial clusters for the Poisson, ZIP, Poisson-Gamma mixture and ZIDP models and compare the results.

Conclusions

The Zero Inflated Double Poisson Spatial Scan Statistic for disease cluster detection incorporates the flexibility of previous models, accounting for inflated zeros and overdispersion simultaneously.The Hanseniasis study case map, due to excess of zero cases counts in many municipalities of the Brazilian Amazon and the presence of overdispersion, was a good benchmark to test the ZIDP model. The results obtained are easier to understand compared to each of the previous spatial scan statistic models, the Zero Inflated Poisson (ZIP) model and the Poisson-Gamma mixture model for overdispersion, taken separetely. The E-M algorithm and the Fast Double Bootstrap test are computationally efficient for this type of problem.     

Sensitivity of score tests for zero-inflation in count data

In many biomedical applications, count data have a large proportion of zeros and the zero-inflated Poisson regression (ZIP) model may be appropriate. A popular score test for zero-inflation, comparing the ZIP model to a standard Poisson regression model, was given by van den Broek. Similarly, for count data that exhibit extra zeros and are simultaneously overdispersed, a score test for testing the ZIP model against a zero-inflated negative binomial alternative was proposed by Ridout, Hinde and Demétrio. However, these test statistics are sensitive to anomalous cases in the data, and incorrect inferences concerning the choice of model may be drawn. In this paper, diagnostic measures are derived to assess the influence of observations on the score statistics. Two examples that motivated the application of zero-inflated regression models are considered to illustrate the importance of sensitivity analysis of the zero-inflation tests.     

On the impact of residential history in the spatial analysis of diseases with a long latency period: A study of mesothelioma in Belgium

Mesothelioma is a rare cancer caused by exposure to asbestos. Belgium has a known long history of asbestos production, resulting in one of the highest mesothelioma mortality rates worldwide. While the production of asbestos has stopped completely, the long latency period of mesothelioma, which can fluctuate between 20 and 40 years after exposure, causes incidences still to be frequent. Mesothelioma's long incubation time affects our assessment of its geographical distribution as well. Since patients' residential locations are likely to change a number of times throughout their lives, the location where the patients develop the disease is often far from the location where they were exposed to asbestos. Using the residential history of patients, we propose the use of a convolution multiple membership model (MMM), which includes both a spatial conditional autoregressive and an unstructured random effect. Pancreatic cancer patients are used as a control population, reflecting the population at risk for mesothelioma. Results show the impact of the residential mobility on the geographical risk estimation, as well as the importance of acknowledging the latency period of a disease. A simulation study was conducted to investigate the properties of the convolution MMM. The robustness of the results for the convolution MMM is assessed via a sensitivity analysis.     

A score test for zero-inflation in correlated count data

To account for the preponderance of zero counts and simultaneous correlation of observations, a class of zero-inflated Poisson mixed regression models is applicable for accommodating the within-cluster dependence. In this paper, a score test for zero-inflation is developed for assessing correlated count data with excess zeros. The sampling distribution and the power of the test statistic are evaluated by simulation studies. The results show that the test statistic performs satisfactorily under a wide range of conditions. The test procedure is further illustrated using a data set on recurrent urinary tract infections.     

A comparison of conditional autoregressive models used in Bayesian disease mapping

Disease mapping is the area of epidemiology that estimates the spatial pattern in disease risk over an extended geographical region, so that areas with elevated risk levels can be identified. Bayesian hierarchical models are typically used in this context, which represent the risk surface using a combination of available covariate data and a set of spatial random effects. These random effects are included to model any overdispersion or spatial correlation in the disease data, that has not been accounted for by the available covariate information. The random effects are typically modelled by a conditional autoregressive (CAR) prior distribution, and a number of alternative specifications have been proposed. This paper critiques four of the most common models within the CAR class, and assesses their appropriateness via a simulation study. The four models are then applied to a new study mapping cancer incidence in Greater Glasgow, Scotland, between 2001 and 2005.     

Some findings on zero‐inflated and hurdle poisson models for disease mapping

Zero excess in the study of geographically referenced mortality data sets has been the focus of considerable attention in the literature, with zero‐inflation being the most common procedure to handle this lack of fit. Although hurdle models have also been used in disease mapping studies, their use is more rare. We show in this paper that models using particular treatments of zero excesses are often required for achieving appropriate fits in regular mortality studies since, otherwise, geographical units with low expected counts are oversmoothed. However, as also shown, an indiscriminate treatment of zero excess may be unnecessary and has a problematic implementation. In this regard, we find that naive zero‐inflation and hurdle models, without an explicit modeling of the probabilities of zeroes, do not fix zero excesses problems well enough and are clearly unsatisfactory. Results sharply suggest the need for an explicit modeling of the probabilities that should vary across areal units. Unfortunately, these more flexible modeling strategies can easily lead to improper posterior distributions as we prove in several theoretical results. Those procedures have been repeatedly used in the disease mapping literature, and one should bear these issues in mind in order to propose valid models. We finally propose several valid modeling alternatives according to the results mentioned that are suitable for fitting zero excesses. We show that those proposals fix zero excesses problems and correct the mentioned oversmoothing of risks in low populated units depicting geographic patterns more suited to the data.     

A Bayesian approach for analyzing zero‐inflated clustered count data with dispersion

In practice, count data may exhibit varying dispersion patterns and excessive zero values; additionally, they may appear in groups or clusters sharing a common source of variation. We present a novel Bayesian approach for analyzing such data. To model these features, we combine the Conway‐Maxwell‐Poisson distribution, which allows both overdispersion and underdispersion, with a hurdle component for the zeros and random effects for clustering. We propose an efficient Markov chain Monte Carlo sampling scheme to obtain posterior inference from our model. Through simulation studies, we compare our hurdle Conway‐Maxwell‐Poisson model with a hurdle Poisson model to demonstrate the effectiveness of our Conway‐Maxwell‐Poisson approach. Furthermore, we apply our model to analyze an illustrative dataset containing information on the number and types of carious lesions on each tooth in a population of 9‐year‐olds from the Iowa Fluoride Study, which is an ongoing longitudinal study on a cohort of Iowa children that began in 1991.     

Epidemiology of mesothelioma of the pericardium and tunica vaginalis testis

PurposeMalignant mesothelioma most commonly arises in the pleura and peritoneum but also occurs rarely at other anatomical sites with mesothelial tissue, namely, the pericardium and tunica vaginalis testis (TVT). This review provides a better understanding of the epidemiology of mesothelioma of these extrapleural sites.MethodsWe conducted a systematic review of the epidemiologic and clinical literature on pericardial mesothelioma and mesothelioma of the TVT. We also analyzed U.S. Surveillance, Epidemiology, and End Results cancer registry data to describe incidence patterns of these malignancies.ResultsAn etiologic role of asbestos exposure has been hypothesized for pericardial and TVT mesotheliomas, but no analytical case-control epidemiologic studies exist to test this relationship. A substantial proportion of cases with these malignancies report no known asbestos exposure. In large occupational cohorts with heavy asbestos exposures, no cases of pericardial or TVT mesothelioma have been reported. Trends in the incidence of these malignancies do not match those of pleural mesothelioma, which correspond to historical trends of commercial asbestos use. A male preponderance of pericardial mesothelioma is not evident.ConclusionsIn the absence of analytic epidemiologic studies, the etiologic role of environmental risk factors for mesothelioma of the pericardium and TVT remains elusive.     

Local influence measure of zero‐inflated generalized Poisson mixture regression models

In many practical applications, count data often exhibit greater or less variability than allowed by the equality of mean and variance, referred to as overdispersion/underdispersion, and there are several reasons that may lead to the overdispersion/underdispersion such as zero inflation and mixture. Moreover, if the count data are distributed as a generalized Poisson or a negative binomial distribution that accommodates extra variation not explained by a simple Poisson or a binomial model, then the dispersion occurs too. In this paper, we deal with a class of two‐component zero‐inflated generalized Poisson mixture regression models to fit such data and propose a local influence measure procedure for model comparison and statistical diagnostics. At first, we formally develop a general model framework that unifies zero inflation, mixture as well as overdispersion/underdispersion simultaneously, and then we mainly investigate two types of perturbation schemes, the global and individual perturbation schemes, for perturbing various model assumptions and detecting influential observations. Also, we obtain the corresponding local influence measures. Our method is novel for count data analysis and can be used to explore these essential issues such as zero inflation, mixture, and dispersion related to zero‐inflated generalized Poisson mixture models. On the basis of the results of model comparison, we could further conduct the sensitivity analysis of perturbation as well as hypothesis test with more accuracy. Finally, we employ here a simulation study and a real example to illustrate the proposed local influence measures. Copyright © 2012 John Wiley & Sons, Ltd.     

Modeling heaping in self-reported cigarette counts

In studies of smoking behavior, some subjects report exact cigarette counts, whereas others report rounded-off counts, particularly multiples of 20, 10 or 5. This form of data reporting error, known as heaping, can bias the estimation of parameters of interest such as mean cigarette consumption. We present a model to describe heaped count data from a randomized trial of bupropion treatment for smoking cessation. The model posits that the reported cigarette count is a deterministic function of an underlying precise cigarette count variable and a heaping behavior variable, both of which are at best partially observed. To account for an excess of zeros, as would likely occur in a smoking cessation study where some subjects successfully quit, we model the underlying count variable with zero-inflated count distributions. We study the sensitivity of the inference on smoking cessation by fitting various models that either do or do not account for heaping and zero inflation, comparing the models by means of Bayes factors. Our results suggest that sufficiently rich models for both the underlying distribution and the heaping behavior are indispensable to obtaining a good fit with heaped smoking data. The analyses moreover reveal that bupropion has a significant effect on the fraction abstinent, but not on mean cigarette consumption among the non-abstinent.     

Risk assessment of lung cancer and mesothelioma in people living near asbestos-related factories in Taiwan.

Estimates from environmental risk assessments are criticized by professionals who indicate that inaccuracies occur in exposure assessment, model selection, and determination of the population at risk. In the current study, we tackled the aforementioned issues and estimated the risks of lung cancer and mesothelioma caused by airborne asbestos among individuals who lived near asbestos factories in Taiwan. We conducted 8-h full-period samplings upwind and downwind from each factory, and we used transmission-electronic microscopy (10,000x) and phase-contrast microscopy to determine asbestos concentrations in and around each factory. We estimated the numbers of residents who lived in concentric circles of 200-m, 400-m, and 600-m diameters around each factory. A dose-response model for asbestos-induced lung cancer was adopted from a summary of seven epidemiological studies. The asbestos-mesothelioma models were patterned after the first-exposure-effect models developed by Peto and Finkelstein. The data obtained from phase-contrast microscopy significantly overestimated the risk, compared with transmission-electronic microscopy. The estimates we calculated from adopting the arithmetic mean were approximately 2-fold higher than those we calculated with the geometric mean. There were relatively low concentrations of asbestos in the study areas, thus causing an absence of a significant difference in risk estimates between different models for mesothelioma. Among the more than 20,000 residents who lived near 41 asbestos factories in Taiwan, we found that the numbers of expected excess deaths from lung cancer and mesothelioma were 5 and less than 1, respectively. We concluded that in future risk assessments for ambient asbestos exposure, investigators should adopt transmission-electronic microscopy and the geometric mean estimate. Moreover, Taiwan should enhance asbestos-control programs to assure the safety of residents who live near asbestos factories.     

A proportional study of cancer registrations of dockyard workers.

Evidence of an occupational effect due to asbestos exposure was sought by comparing cancer registrations for dockyard workers with those for Plymouth men in the same age groups for 1960-69. The results show a significant excess of pleural tumours in the dockyard group but no significant excesses at other sites. Registrations for pleural mesothelioma were recorded for men with many dockyard occupations and the observed mean interval between first exposure to asbestos and registration for a pleural tumour (37.6 years) indicates that cases of pleural mesothelioma may be expected to occur among these workers for many years, even though crocidolite is no longer used in naval ships. A trend of increasing annual excess of stomach and gastrointestinal registrations was observed in the dockyard group. No cases of peritoneal mesothelioma were recorded but it is likely that some did occur which were diagnosed as cancers arising from other sites. This may account for some of the excess of gastrointestinal registrations.     

A proportional study of cancer registrations of dockyard workers.

Evidence of an occupational effect due to asbestos exposure was sought by comparing cancer registrations for dockyard workers with those for Plymouth men in the same age groups for 1960-69. The results show a significant excess of pleural tumours in the dockyard group but no significant excesses at other sites. Registrations for pleural mesothelioma were recorded for men with many dockyard occupations and the observed mean interval between first exposure to asbestos and registration for a pleural tumour (37.6 years) indicates that cases of pleural mesothelioma may be expected to occur among these workers for many years, even though crocidolite is no longer used in naval ships. A trend of increasing annual excess of stomach and gastrointestinal registrations was observed in the dockyard group. No cases of peritoneal mesothelioma were recorded but it is likely that some did occur which were diagnosed as cancers arising from other sites. This may account for some of the excess of gastrointestinal registrations.     

Asbestos and Peritoneal Mesothelioma among College-educated Men

Abstract

The proportion of peritoneal mesotheliomas among all mesotheliomas has been decreasing, leading some to suggest that peritoneal mesothelioma occurs only after high levels of exposure to asbestos. To investigate the relationship between asbestos exposure and the development of peritoneal mesothelioma, a case–control study examined 40 cases of primary peritoneal mesothelioma from a single institution. This series differed from previous reports in that 75% of the cases and controls had attended college. Results show an odds ratio of 6.6 for asbestos exposure among this group of primary peritoneal mesothelioma cases with relatively slight asbestos exposures.     

Preface

Abstract

Background: Out of 143 countries that consumed asbestos between 2003 and 2007, only 44 have banned asbestos. This study tried to explain why some countries have banned asbestos while others have not, based on a synthesis that asbestos ban policy of a country will rely on a process of cognition of threats and exploration of safer alternatives.

Method: As we hypothesized that increased social cost of mesothelioma, capacity of health-related infrastructures, and policy diffusion from adjacent countries were related to asbestos ban adoption, published databases of asbestos ban years, mesothelioma mortality, country rankings in health care and human rights standings, and distribution of banning countries over 14 regions were analyzed accordingly.

Results: The average mesothelioma death rate was significantly higher for countries with asbestos bans than in those with no ban (4·59 versus 1·83/million). No-ban countries had less well-developed health-related infrastructures. Among European countries, there was a tendency toward geographical diffusion of asbestos ban policy from Nordic to Western and then other European countries over the years. Even though aberrant cases were also noted where bans were instituted even without mesothelioma database, these were rather exceptions than rules.

Conclusion: Risk cognition is a complex process, but the presence of well-functioning health infrastructures, as well as the increased social cost of mesothelioma, that can make the plight of asbestos victims visible to the eyes of public and policy makers, may have contributed to this process. Asbestos ban policy from adjacent countries might have facilitated the adoption of alternative solutions.     

Environmental exposure to asbestos in the area around Goor has been established as the cause of pleural mesothelioma in women

OBJECTIVE: To determine whether a disease cluster of 22 additional cases of pleural mesothelioma among women could be attributed to environmental asbestos exposure due to asbestos fibers from waste material on roads and property yards. The women studied were observed in an area with substantial environmental exposure to asbestos during the period 1989-2002. DESIGN: Ecological study. METHOD: In the study period of 1989-2002, all cases of mesothelioma among women, based on a strict histopathologic definition, occurring in the region of Twente, The Netherlands (n = 59) were provided by the regional cancer register. Additional information was collected on the occupational histories of the cases and their partners and addresses of residence through medical records, general practitioners, and next-of-kin. Environmental asbestos exposure was assigned to all cases that had had a long-term stay in a house in the area around Goor with demonstrated local environmental asbestos pollution and where any contact with asbestos through occupation or in the household had been excluded. RESULTS: In the risk area around Goor, out ofa total of 28 cases ofwomen with pleural mesothelioma, asbestos in the environment was found to be the only source of asbestos exposure for to women. In a further 4 women, environmental asbestos exposure was found to be the most likely cause of pleural mesothelioma. The average cumulative exposure was around 0.11 fiber/ml x exposure years. The observed extra incidence of 22 cases was attributed to the environmental exposure to asbestos in 64% (14/22) of cases. CONCLUSION: The environmental pollution to asbestos waste materials in the area around Goor was the main cause of the strongly increased incidence of pleural mesothelioma among women in this area. Taking into account an equal risk among men, the consequences of asbestos exposure in the area around Goor in the next 25 years are likely to result in 2 cases of pleural mesothelioma each year.     

Mortality of asbestos workers in England and Wales 1971-81.

A national study of British asbestos workers is briefly described and the mortality experience of 31 150 male asbestos workers in England and Wales who had been medically examined at least once as part of that survey is presented. The survey population is divided into workers with occupational exposure to asbestos before the inception of the 1969 Asbestos Regulations and those who worked with asbestos only after 1969. Of the 1128 who had died, 897 had worked before 1969; 34 of the death certificates received for these men mentioned mesothelioma and for another nine asbestosis was reported in the absence of mesothelioma or lung cancer. A statistically significant excess of lung cancer (SMR 136) was found. For the post-1969 workers, one case of asbestosis and one case of mesothelioma were reported, but further investigation of these cases showed probable occupational exposure to asbestos many years before 1969. The time from first exposure for this section of the population is too short to exclude an excess of asbestos related disease. The most noticeable excess of asbestos related disease was seen among the insulation workers who had more than twice (SMR 256) the expected number of deaths from lung cancer, and for whom almost 10% of all death certificates mentioned mesothelioma. No excess of any alimentary tract cancer was found and the population showed a significant deficit of large bowel cancer mortality (SMR 54).