Over-dispersed Poisson Model Research Articles

Log-Normal or Over-Dispersed Poisson?

Although both over-dispersed Poisson and log-normal chain-ladder models are popular in claim reserving, it is not obvious when to choose which model. Yet, the two models are obviously different. While the over-dispersed Poisson model imposes the variance to mean ratio to be common across the array, the log-normal model assumes the same for the standard deviation to mean ratio. Leveraging this insight, we propose a test that has the power to distinguish between the two models. The theory is asymptotic, but it does not build on a large size of the array and, instead, makes use of information accumulating within the cells. The test has a non-standard asymptotic distribution; however, saddle point approximations are available. We show in a simulation study that these approximations are accurate and that the test performs well in finite samples and has high power.

Development and assessment of management practices in a flock-specific lameness control plan: A stepped-wedge trial on 44 English sheep flocks

Lameness in sheep has economic and welfare implications, including loss of ewe body condition, lower lambing percentages, and poor lamb growth rates. It costs the UK sheep industry around £80 million per year. The majority of lameness is caused by the infectious diseases footrot and contagious ovine digital dermatitis, with white line separation, white line abscesses, and toe granulomas also reported by farmers. Most sheep farmers in the UK have other enterprises and care for their flock part-time. A lameness control plan (LCP) consisting of 37 management practices that covered all aspects of control of lameness was developed for part-time sheep farmers. Testing of the LCP was done using a stepped-wedge trial design with 7 visits to 44 flocks in England over 18 months. Flocks had 100–500 breeding ewes. Locomotion scoring was carried out at every visit, and farmers were interviewed every 6 months to record management practices. Clinical significance of changes in prevalence of lameness within each flock was analysed using the reliable change index (RCI). Management practices associated with prevalence of lameness were assessed using a multi-level multivariable over-dispersed Poisson model. The geometric mean (GM) prevalence of lameness at the start of the trial was 7.3% (95% confidence interval(CI) = 6.3–8.3%). Flocks with a clinically significant lameness reduction had a GM prevalence of lameness of 4.6% (95% CI = 4.1–5.2%), while flocks with a clinically significant increase in lameness had a GM prevalence of 10.5% (95% CI = 9.4–11.6%). Always separating lame sheep at treatment (relative risk (RR) = 0.60, 95% CI = 0.43-0.84) and culling sheep lame ≥2 occasions in a year (RR = 0.75, 95% CI = 0.61-0.92) were associated with a significant reduction in lameness. Compared with not footbathing at all, footbathing sheep when there were outbreaks of interdigital dermatitis reduced lameness (RR = 0.85, 95% CI = 0.75-0.96), however, ‘routine’ footbathing was not associated with a reduction in lameness. Housing during the previous lambing period was associated with a higher prevalence of lameness (RR = 1.23, 95% CI = 1.04–1.46). Vaccination in flocks with period prevalence ≥10% did not significantly reduce their risk of lameness compared to lower prevalence flocks. In conclusion, separating lame sheep at treatment, culling sheep lame ≥2 occasions per year, and only using a footbath to treat outbreaks of interdigital dermatitis are flock managements that contribute to improved control of lameness in flocks with part-time farmers.

Misspecification Tests for Log-Normal and Over-Dispersed Poisson Chain-Ladder Models

Despite the widespread use of chain-ladder models, so far no theory was available to test for model specification. The popular over-dispersed Poisson model assumes that the over-dispersion is common across the data. A further assumption is that accident year effects do not vary across development years and vice versa. The log-normal chain-ladder model makes similar assumptions. We show that these assumptions can easily be tested and that similar tests can be used in both models. The tests can be implemented in a spreadsheet. We illustrate the implementation in several empirical applications. While the results for the log-normal model are valid in finite samples, those for the over-dispersed Poisson model are derived for large cell mean asymptotics which hold the number of cells fixed. We show in a simulation study that the finite sample performance is close to the asymptotic performance.

Neural Network Embedding of the Over-Dispersed Poisson Reserving Model

The main idea of this paper is to embed a classical actuarial regression model into a neural network architecture. This nesting allows us to learn model structure beyond the classical actuarial regression model if we use as starting point of the neural network calibration exactly the classical actuarial model. Such models can be fitted efficiently which allows us to explore bootstrap methods for prediction uncertainty. As an explicit example we consider the cross-classified over-dispersed Poisson model for general insurance claims reserving. We demonstrate how this model can be improved by neural network features.

The association between cement type and the subsequent risk of revision surgery in primary total hip replacement

Background and purpose — To further improve the success of joint replacement surgery, attention needs to be paid to variations associated with improved or worsened outcomes. We investigated the association between the type of bone cement used and the risk of revision surgery after primary total hip replacement.Methods — We conducted a prospective study of data from the National Joint Registry for England and Wales between April 1, 2003 and December 31, 2013. 199,205 primary total hip replacements performed for osteoarthritis where bone cement was used were included. A multilevel over-dispersed piecewise Poisson model was used to estimate differences in the rate of revision by bone cement type adjusted for implant type, head size, age, sex, ASA grade, and surgical approach.Results — The rate of revision was higher in DePuy CMW3 medium viscosity with gentamicin (IRR 2.0, 95% CI 1.5–2.7) and DePuy SmartSet high viscosity plain (IRR 2.7, 95% CI 1.1–5.5), and lower in DePuy CMW1 high viscosity plain (IRR 0.44, 95% CI 0.19–0.89) bone cements compared with Heraeus Palacos high viscosity with gentamicin. Revision rates were similar between plain and antibiotic-loaded bone cement.Interpretation — The majority of bone cements performed similarly well, excluding DePuy SmartSet high viscosity and CMW3 high viscosity with gentamicin, which both had higher revision rates. We found no clear differences by viscosity or antibiotic content.

Association between Air Pollution and Emergency Room Visits for Atrial Fibrillation.

Despite the large prevalence in the population, possible factors responsible for the induction of atrial fibrillation (AF) events in susceptible individuals remain incompletely understood. We investigated the association between air pollution levels and emergency department admissions for AF in Rome. We conducted a 14 years’ time-series study to evaluate the association between the daily levels of air pollution (particulate matter, PM10 and PM2.5, and nitrogen dioxide, NO2) and the daily count of emergency accesses for AF (ICD-9 code: 427.31). We applied an over-dispersed conditional Poisson model to analyze the associations at different lags after controlling for time, influenza epidemics, holiday periods, temperature, and relative humidity. Additionally, we evaluated bi-pollutant models by including the other pollutant and the influence of several effect modifiers such as personal characteristics and pre-existing medical conditions. In the period of study, 79,892 individuals were admitted to the emergency departments of Rome hospitals because of AF (on average, 15.6 patients per day: min = 1, max = 36). Air pollution levels were associated with increased AF emergency visits within 24 h of exposure. Effect estimates ranged between 1.4% (0.7–2.3) for a 10 µg/m3 increase of PM10 to 3% (1.4–4.7) for a 10 µg/m3 increase of PM2.5 at lag 0–1 day. Those effects were higher in patients ≥75 years for all pollutants, male patients for PM10, and female patients for NO2. The presence of previous cardiovascular conditions, but not other effect modifiers, increase the pollution effects by 5–8% depending on the lag. This study found evidence that air pollution is associated with AF emergency visits in the short term.

Hierarchical probabilistic models for multiple gene/variant associations based on next-generation sequencing data.

MotivationThe identification of genetic variants influencing gene expression (known as expression quantitative trait loci or eQTLs) is important in unravelling the genetic basis of complex traits. Detecting multiple eQTLs simultaneously in a population based on paired DNA-seq and RNA-seq assays employs two competing types of models: models which rely on appropriate transformations of RNA-seq data (and are powered by a mature mathematical theory), or count-based models, which represent digital gene expression explicitly, thus rendering such transformations unnecessary. The latter constitutes an immensely popular methodology, which is however plagued by mathematical intractability.ResultsWe develop tractable count-based models, which are amenable to efficient estimation through the introduction of latent variables and the appropriate application of recent statistical theory in a sparse Bayesian modelling framework. Furthermore, we examine several transformation methods for RNA-seq read counts and we introduce arcsin, logit and Laplace smoothing as preprocessing steps for transformation-based models. Using natural and carefully simulated data from the 1000 Genomes and gEUVADIS projects, we benchmark both approaches under a variety of scenarios, including the presence of noise and violation of basic model assumptions. We demonstrate that an arcsin transformation of Laplace-smoothed data is at least as good as state-of-the-art models, particularly at small samples. Furthermore, we show that an over-dispersed Poisson model is comparable to the celebrated Negative Binomial, but much easier to estimate. These results provide strong support for transformation-based versus count-based (particularly Negative-Binomial-based) models for eQTL mapping.Availability and implementationAll methods are implemented in the free software eQTLseq: https://github.com/dvav/eQTLseqSupplementary information Supplementary data are available at Bioinformatics online.

Smoke-Free Legislation in Spain and Prematurity.

Spain implemented a partial smoking ban in 2006 followed by a comprehensive ban in 2011. The objective was to examine the association between these smoke-free policies and different perinatal complications. Cross-sectional study including all live births between 2000 and 2013. Selected adverse birth outcomes were: preterm births (<37 gestational weeks), small for gestational age (SGA; <10th weight percentile according to Spanish reference tables), and low birth weight (<2500 g). We estimated immediate and gradual rate changes after smoking bans by using overdispersed Poisson models with different linear trends for 2000 to 2005 (preban), 2006 to 2010 (partial ban), and 2011 to 2013 (comprehensive ban). Models were adjusted for maternal sociodemographics, health care during the delivery, and smoking prevalence during pregnancy. The comprehensive ban was associated with preterm birth rate reductions of 4.5% (95% confidence interval [CI]: 2.9%-6.1%) and 4.1% (95% CI: 2.5%-5.6%) immediately and 1 year after implementation, respectively. The low birth weight rate also dropped immediately (2.3%; 95% CI: 0.7%-3.8%) and 1 year after the comprehensive ban implementation (3.5%; 95% CI: 2.1%-5.0%). There was an immediate reduction in the SGA rate at the onset of the partial ban (4.9%; 95% CI: 3.5%-6.2%), which was sustained 1 year postimplementation. Although not associated with the comprehensive ban at the onset, the SGA rate declined by 1.7% (95% CI: 0.3%-3.1%) 1 year postimplementation. The implementation of the Spanish smoke-free policies was associated with a risk reduction for preterm births and low birth weight infants, especially with the introduction of the more restrictive ban.

Variability in Metagenomic Count Data and Its Influence on the Identification of Differentially Abundant Genes.

Metagenomics is the study of microorganisms in environmental and clinical samples using high-throughput sequencing of random fragments of their DNA. Since metagenomics does not require any prior culturing of isolates, entire microbial communities can be studied directly in their natural state. In metagenomics, the abundance of genes is quantified by sorting and counting the DNA fragments. The resulting count data are high-dimensional and affected by high levels of technical and biological noise that make the statistical analysis challenging. In this article, we introduce an hierarchical overdispersed Poisson model to explore the variability in metagenomic data. By analyzing three comprehensive data sets, we show that the gene-specific variability varies substantially between genes and is dependent on biological function. We also assess the power of identifying differentially abundant genes and show that incorrect assumptions about the gene-specific variability can lead to unacceptable high rates of false positives. Finally, we evaluate shrinkage approaches to improve the variance estimation and show that the prior choice significantly affects the statistical power. The results presented in this study further elucidate the complex variance structure of metagenomic data and provide suggestions for accurate and reliable identification of differentially abundant genes.

On Comparison of Stochastic Reserving Methods with Bootstrapping

We consider the well-known stochastic reserve estimation methods on the basis of generalized linear models, such as the (over-dispersed) Poisson model, the gamma model and the log-normal model. For the likely variability of the claims reserve, bootstrap method is considered. In the bootstrapping framework, we discuss the choice of residuals, namely the Pearson residuals, the deviance residuals and the Anscombe residuals. In addition, several possible residual adjustments are discussed and compared in a case study. We carry out a practical implementation and comparison of methods using real-life insurance data to estimate reserves and their prediction errors. We propose to consider proper scoring rules for model validation, and the assessments will be drawn from an extensive case study.

Temperature Variability and Occurrence of Diarrhoea in Children under Five-Years-Old in Cape Town Metropolitan Sub-Districts.

This paper describes the relationship between temperature change and diarrhoea in under five-year-old children in the Cape Town Metropolitan Area (CTMA) of South Africa. The study used climatic and aggregated surveillance diarrhoea incidence data of two peak periods of seven months each over two consecutive years. A Poisson regression model and a lagged Poisson model with autocorrelation was performed to test the relationship between climatic parameters (minimum and maximum temperature) and incidence of diarrhoea. In total, 58,617 cases of diarrhoea occurred in the CTMA, which is equivalent to 8.60 cases per 100 population under five years old for the study period. The mixed effect overdispersed Poisson model showed that a cluster adjusted effect of an increase of 5 °C in minimum and maximum temperature results in a 40% (Incidence risk ratio IRR: 1.39, 95% CI 1.31–1.48) and 32% (IRR: 1.32, 95% CI: 1.22–1.41) increase in incident cases of diarrhoea, respectively, for the two periods studied. Autocorrelation of one-week lag (Autocorrelation AC 1) indicated that a 5 °C increase in minimum and maximum temperature led to 15% (IRR: 1.46, 95% CI: 1.09–1.20) and 6% (IRR: 1.06, 95% CI: 1.01–1.12) increase in diarrhoea cases, respectively. In conclusion, there was an association between an increase in minimum and maximum temperature, and the rate at which diarrhoea affected children under the age of five years old in the Cape Town Metropolitan Area. This finding may have implications for the effects of global warming and requires further investigation.

Association between air pollution and emergency room admissions for atrial fibrillation

INTRODUCTION: Recent studies have pointed out also the possible association of air pollution exposure with atrial fibrillation (AF), the most frequent cardiac arrhythmia diagnosed in clinical practice. We investigated the association between air pollution levels and emergency room visits for AF. METHODS: We conducted a 14 years time-series study to evaluate the association between daily levels of air pollution (particulate matter, PM10 and PM2.5 and nitrogen dioxide, NO2) and daily count of emergency access for AF (ICD-9 code: 427.31) for the city of Rome (Italy). We applied over-dispersed conditional Poisson model to analyze the associations at different lags after controlling for seasonality, day of the week, influenza epidemics, holiday periods, temperature and relative humidity. Additionally, we evaluated bi-pollutant models by including the other pollutant in turn and the influence of several effect modifiers such as personal characteristics and pre-existing medical conditions on a subset of data. RESULTS: In total 79,892 admissions were recorded in emergency room departments of Rome hospitals in the study period. At lag of 0-1 days, the increased risk of admission for 10 ug/m3 increase in PM10 was 1.45% (0.65, 2.26), 3.00% (1.35-4.67) for PM25 and 1.19% (0.27, 2.13) for NO2.The NO2 effect is consistent when also PM10 concentration is high, showing an increase in AF emergency admissions between 0.6% and 1.8%, depending on the lag. Presence of previous cardiovascular conditions, but not other effect modifiers, increase pollution effects by 5-8% depending on the lag. CONCLUSION: We found that high concentrations of air pollution may trigger AF emergency room visits in the short term

Exposure to Bankers: Networks and Stock Market Participation

In this paper we investigate the importance of acquaintance networks for financial decisions by households. We construct a variable capturing the expected proximity or social closeness to a subpopulation of financially savvy people using an overdispersed Poisson model. This measure captures the exposure to people with financial knowledge in an investor's acquaintance network. We find that investors with a higher exposure to financial savvy people are more likely to invest in stocks. This holds after controlling for a wide range of known stock market participation determinants. Moreover when restricting to households with elevated levels of trust or wealth the impact of exposure to financial savvy people increases. Our main findings continue to hold in several analyses to uncover an exogenous effect from proximity on stock market participation. Furthermore we show that the importance of acquaintance networks extends to other financial decisions as well.

Modeling spatiotemporal bacterial variability with meteorological and watershed land-use characteristics

Bacteria are a primary contaminant in natural surface water. The instream concentration of fecal coliform, a potential indicator of pathogens, is influenced by meteorological conditions and land-use characteristics. However, the relationships between these conditions and fecal coliforms are not fully understood. Furthermore, the sources of large variability in fecal coliform counts, e.g., temporal or spatial sources, remain unexplained, especially at large scales. This study proposes the use of Bayesian overdispersed Poisson models, whereby the combined effects of temperature, rainfall, and land-use characteristics on fecal coliform concentration are quantified with predictive uncertainty, and the sources of variability in fecal coliform concentration are assessed. The models were developed using 8-year weekly observations of fecal coliforms obtained from the Wachusett Reservoir watershed in Massachusetts, USA. The results highlight the importance of interactions among meteorological and land-use characteristics in controlling the instream fecal coliform concentration; the increase in fecal coliform concentration with temperature increase was more drastic when rainfall occurred. Also, the responses of fecal coliforms to temperature increases were more pronounced in forest-dominated than in urban-dominated areas. In contrast, the fecal coliform concentration increased more rapidly with rainfall increases in urban-dominated than in forest-dominated areas. The models also demonstrate that among the sources of variability, the monthly component made the most significant contribution to the variability in fecal coliform concentrations. Our results suggest that seasonally dependent processes, including surface runoff, are critical factors that regulate fecal coliform concentration in streams.

Examining Change in Adolescent Street Efficacy and Its Association With Violent Outcomes.

Research suggests that street efficacy-the perceived ability to avoid dangerous situations in one's neighborhood-is related to violent outcomes. We investigated change in street efficacy using data from the Project on Human Development in Chicago Neighborhoods. Measures of street efficacy and violence (offending, victimization, secondary exposure) were constructed at Waves 2 (1997-2000) and 3 (2000-2002) from youth interviews and parent reports. Results from hierarchical overdispersed Poisson models indicated that street efficacy was a malleable construct with stronger contemporaneous than enduring effects. Furthermore, rates of all three violent outcomes were reduced as individuals increased their levels of street efficacy. Findings support (a) teaching youths to safely navigate violent neighborhood situations and (b) continuously reinforcing youths' perceived ability to navigate danger.

Effect of the 13-valent pneumococcal conjugate vaccine on invasive pneumococcal disease in England and Wales 4 years after its introduction: an observational cohort study

The 13-valent pneumococcal conjugate vaccine (PCV13) protects against key serotypes that increased after routine immunisation with the seven-valent vaccine (PCV7), but its potential for herd protection and serotype replacement is uncertain. The aim of this study was to analyse the effect of the 13-valent pneumococcal conjugate vaccine on invasive pneumococcal disease in England and Wales 4 years after its introduction. We used a national dataset of electronically reported and serotyped invasive pneumococcal disease cases in England and Wales to estimate incidence rate ratios (IRRs) for vaccine and non-vaccine type invasive pneumococcal disease between July, 2013, and June, 2014, versus the pre-PCV13 and pre-PCV7 baseline. Incidence rates were corrected for missing serotype data and changes in surveillance sensitivity over time. An over-dispersed Poisson model was used to estimate IRRs and confidence intervals. Incidence of invasive pneumococcal disease in the epidemiological year 2013/14 decreased by 32% compared with the pre-PCV13 baseline (incidence 10·14 per 100,000 in 2008-10 vs 6·85 per 100,000 in 2013/14; IRR 0·68, 95% CI 0·64-0·72). This was due to an 86% reduction of the serotypes covered by PCV7 (1·46 vs 0·20 per 100,000; IRR 0·14, 0·10-0·18) and a 69% reduction of the additional six serotypes covered by PCV13 (4·48 vs 1·40 per 100,000; IRR 0·31, 0·28-0·35). When compared with the pre-PCV7 baseline, there was a 56% overall reduction in invasive pneumococcal disease (15·63 vs 6·85 per 100,000; IRR 0·44, 95% CI 0·43-0·47). Compared with the pre-PCV13 baseline, the incidence of non-PCV13 serotypes increased (incidence all ages 4·19 vs 5·25 per 100,000; IRR 1·25, 95% CI 1·17-1·35) due to increases across a broad range of serotypes in children younger than 5 years and in people aged 45 years or more. In children younger than 5 years, incidence of non-PCV13 serotypes in 2013/14 was higher than in 2012/13 (age <2 years: 12·03 vs 10·83 per 100,000; age 2-4 years: 4·08 vs 3·63 per 100,000). 8 years of PCV use in England and Wales has reduced the overall incidence of invasive pneumococcal disease by more than 50%. The herd protection induced by PCV7 is continuing, and similar indirect protection is occurring from the additional serotypes covered by PCV13. There is, however, evidence of increasing invasive pneumococcal disease due to non-PCV13 serotypes, particularly in children younger than 5 years in 2014. If this increase continues, the maximum benefit of the PCV13 programme in children might already have been achieved. Public Health England funds national surveillance of invasive pneumococcal disease.

Timing and effect of a safe routes to school program on child pedestrian injury risk during school travel hours: Bayesian changepoint and difference-in-differences analysis.

BackgroundIn 2005, the US Congress allocated $612 million for a national Safe Routes to School (SRTS) program to encourage walking and bicycling to schools. We evaluated the effectiveness of a SRTS in controlling pedestrian injuries among school-age children.MethodsBayesian changepoint analysis was applied to model the quarterly counts of pedestrian injuries among 5- to 19-year old children in New York City between 2001 and 2010 during school-travel hours in census tracts with and without SRTS. Overdispersed Poisson model was used to estimate difference-in-differences in injury risk between census tracts with and without SRTS following the changepoint.ResultsIn SRTS-intervention census tracts, a change point in the quarterly counts of injuries was identified in the second quarter of 2008, which was consistent with the timing of the implementation of SRTS interventions. In census tracts with SRTS interventions, the estimated quarterly rates of pedestrian injury per 10,000 population among school-age children during school-travel hours were 3.47 (95% Credible Interval [CrI] 2.67, 4.39) prior to the changepoint, and 0.74 (95% CrI 0.30, 1.50) after the changepoint. There was no change in the average number of quarterly injuries in non-SRTS census tracts. Overdispersed Poisson modeling revealed that SRTS implementation was associated with a 44% reduction (95% Confidence Interval [CI] 87% decrease to 130% increase) in school-age pedestrian injury risk during school-travel hours.ConclusionsBayesian changepoint analysis of quarterly counts of school-age pedestrian injuries successfully identified the timing of SRTS intervention in New York City. Implementation of the SRTS program in New York City appears to be effective in reducing school-age pedestrian injuries during school-travel hours.Electronic supplementary materialThe online version of this article (doi:10.1186/s40621-014-0017-0) contains supplementary material, which is available to authorized users.

Emergency department visits and hospitalizations by tube-fed nursing home residents with varying degrees of cognitive impairment: a national study

BackgroundNumerous studies indicate that the use of feeding tubes (FT) in persons with advanced cognitive impairment (CI) does not improve clinical outcomes or survival, and results in higher rates of hospitalization and emergency department (ED) visits. It is not clear, however, whether such risk varies by resident level of CI and whether these ED visits and hospitalizations are potentially preventable. The objective of this study was to determine the rates of ED visits, hospitalizations and potentially preventable ambulatory care sensitive (ACS) ED visits and ACS hospitalizations for long-stay NH residents with FTs at differing levels of CI.MethodsWe linked Centers for Medicare and Medicaid Services inpatient & outpatient administrative claims and beneficiary eligibility data with Minimum Data Set (MDS) resident assessment data for nursing home residents with feeding tubes in a 5% random sample of Medicare beneficiaries residing in US nursing facilities in 2006 (n = 3479). Severity of CI was measured using the Cognitive Performance Scale (CPS) and categorized into 4 groups: None/Mild (CPS = 0-1, MMSE = 22-25), Moderate (CPS = 2-3, MMSE = 15-19), Severe (CPS = 4-5, MMSE = 5-7) and Very Severe (CPS = 6, MMSE = 0-4). ED visits, hospitalizations, ACS ED visits and ACS hospitalizations were ascertained from inpatient and outpatient administrative claims. We estimated the risk ratio of each outcome by CI level using over-dispersed Poisson models accounting for potential confounding factors.ResultsTwenty-nine percent of our cohort was considered “comatose” and “without any discernible consciousness”, suggesting that over 20,000 NH residents in the US with feeding tubes are non-interactive. Approximately 25% of NH residents with FTs required an ED visit or hospitalization, with 44% of hospitalizations and 24% of ED visits being potentially preventable or for an ACS condition. Severity of CI had a significant effect on rates of ACS ED visits, but little effect on ACS hospitalizations.ConclusionsED visits and hospitalizations are common in cognitively impaired tube-fed nursing home residents and a substantial proportion of ED visits and hospitalizations are potentially preventable due to ACS conditions.

Poverty and mental health in Indonesia

Community and facility studies in developing countries have generally demonstrated an inverse relationship between poverty and mental health. However, recent population-based studies contradict this. In India and Indonesia the poor and non-poor show no difference in mental health. We revisit the relationship between poverty and mental health using a validated measure of depressive symptoms (CES-D) and a new national sample from Indonesia – a country where widespread poverty and deep inequality meet with a neglected mental health service sector. Results from three-level overdispersed Poisson models show that a 1% decrease in per capita household expenditure was associated with a 0.05% increase in CES-D score (depressive symptoms), while using a different indicator (living on less than $2 a day) it was estimated that the poor had a 5% higher CES-D score than the better off. Individual social capital and religiosity were found to be positively associated with mental health while adverse events were negatively associated. These findings provide support for the established view regarding the deleterious association between poverty and mental health in developed and developing countries.

Computation of marginal likelihoods with data-dependent support for latent variables

Several Monte Carlo methods have been proposed for computing marginal likelihoods in Bayesian analyses. Some of these involve sampling from a sequence of intermediate distributions between the prior and posterior. A difficulty arises if the support in the posterior distribution is a proper subset of that in the prior distribution. This can happen in problems involving latent variables whose support depends upon the data and can make some methods inefficient and others invalid. The correction required for models of this type is derived and its use is illustrated by finding the marginal likelihoods in two examples. One concerns a model for competing risks. The other involves a zero-inflated over-dispersed Poisson model for counts of centipedes, using latent Gaussian variables to capture spatial dependence.