Disease mapping with errors in covariates

Bayesian disease mapping: Past, present, and future

On the occasion of the Spatial Statistics' 10th Anniversary, I reflect on the past and present of Bayesian disease mapping and look into its future. I focus on some key developments of models, and on recent evolution of multivariate and adaptive Gaussian Markov random fields and their impact and importance in disease mapping. I reflect on Bayesian disease mapping as a subject of spatial statistics that has advanced to date, and continues to grow, in scope and complexity alongside increasing needs of analytic tools for contemporary health science research, such as spatial epidemiology, population and public health, and medicine. I illustrate (potential) utility and impact of some of the disease mapping models and methods for analysing and monitoring communicable disease such as the COVID-19 infection risks during an ongoing pandemic.

Bayesian adjustment for measurement error in an offset variable in a Poisson regression model

Fatal car crashes are the leading cause of death among teenagers in the USA. The Graduated Driver Licensing (GDL) programme is one effective policy for reducing the number of teen fatal car crashes. Our study focuses on the number of fatal car crashes in Michigan during 1990–2004 excluding 1997, when the GDL started. We use Poisson regression with spatially dependent random effects to model the county level teen car crash counts. We develop a measurement error model to account for the fact that the total teenage population in the county level is used as a proxy for the teenage driver population. To the best of our knowledge, there is no existing literature that considers adjustment for measurement error in an offset variable. Furthermore, limited work has addressed the measurement errors in the context of spatial data. In our modelling, a Berkson measurement error model with spatial random effects is applied to adjust for the error-prone offset variable in a Bayesian paradigm. The Bayesian Markov chain Monte Carlo (MCMC) sampling is implemented in rstan. To assess the consequence of adjusting for measurement error, we compared two models with and without adjustment for measurement error. We found the effect of a time indicator becomes less significant with the measurement-error adjustment. It leads to our conclusion that the reduced number of teen drivers can help explain, to some extent, the effectiveness of GDL.

Vector Surveillance, Host Species Richness, and Demographic Factors as West Nile Disease Risk Indicators

West Nile virus (WNV) is the most common arthropod-borne virus (arbovirus) in the United States (US) and is the leading cause of viral encephalitis in the country. The virus has affected tens of thousands of US persons total since its 1999 North America introduction, with thousands of new infections reported annually. Approximately 1% of humans infected with WNV acquire neuroinvasive West Nile Disease (WND) with severe encephalitis and risk of death. Research describing WNV ecology is needed to improve public health surveillance, monitoring, and risk assessment. We applied Bayesian joint-spatiotemporal modeling to assess the association of vector surveillance data, host species richness, and a variety of other environmental and socioeconomic disease risk factors with neuroinvasive WND throughout the conterminous US. Our research revealed that an aging human population was the strongest disease indicator, but climatic and vector-host biotic interactions were also significant in determining risk of neuroinvasive WND. Our analysis also identified a geographic region of disproportionately high neuroinvasive WND disease risk that parallels the Continental Divide, and extends southward from the US-Canada border in the states of Montana, North Dakota, and Wisconsin to the US-Mexico border in western Texas. Our results aid in unraveling complex WNV ecology and can be applied to prioritize disease surveillance locations and risk assessment.

Bayesian spatial modelling of early childhood development in Australian regions

Background

Children’s early development plays a vital role for maintaining healthy lives and influences future outcomes. It is also heavily affected by community factors which vary geographically. Direct methods do not provide a comprehensive picture of this variation, especially for areas with sparse populations and low data coverage. In the context of Australia, the Australian Early Development Census (AEDC) provides a measure of early child development upon school entry. There are two primary aims of this study: (i) provide improved prevalence estimates of children who are considered as developmentally vulnerable in regions across Australia; (ii) ascertain how social-economic disadvantage partly explains the spatial variation.

Methods

We used Bayesian spatial hierarchical models with the Socio-economic Indexes for Areas (SEIFA) as a covariate to provide improved estimates of all 335 SA3 regions in Australia. The study included 308,953 children involved in the 2018 AEDC where 21.7% of them were considered to be developmentally vulnerable in at least one domain. There are five domains of developmental vulnerability—physical health and wellbeing; social competence; emotional maturity; language and cognitive skills; and communication and general knowledge.

Results

There are significant improvements in estimation of the prevalence of developmental vulnerability through incorporating the socio-economic disadvantage in an area. These improvements persist in all five domains—the largest improvements occurred in the Language and Cognitive Skills domain. In addition, our results reveal that there is an important geographical dimension to developmental vulnerability in Australia.

Conclusion

Sparsely populated areas in sample surveys lead to unreliable direct estimates of the relatively small prevalence of child vulnerability. Bayesian spatial modelling can account for the spatial patterns in childhood vulnerability while including the impact of socio-economic disadvantage on geographic variation. Further investigation, using a broader range of covariates, could shed more light on explaining this spatial variation.

An intuitive Bayesian spatial model for disease mapping that accounts for scaling

In recent years, disease mapping studies have become a routine application within geographical epidemiology and are typically analysed within a Bayesian hierarchical model formulation. A variety of model formulations for the latent level have been proposed but all come with inherent issues. In the classical BYM (Besag, York and Mollié) model, the spatially structured component cannot be seen independently from the unstructured component. This makes prior definitions for the hyperparameters of the two random effects challenging. There are alternative model formulations that address this confounding; however, the issue on how to choose interpretable hyperpriors is still unsolved. Here, we discuss a recently proposed parameterisation of the BYM model that leads to improved parameter control as the hyperparameters can be seen independently from each other. Furthermore, the need for a scaled spatial component is addressed, which facilitates assignment of interpretable hyperpriors and make these transferable between spatial applications with different graph structures. The hyperparameters themselves are used to define flexible extensions of simple base models. Consequently, penalised complexity priors for these parameters can be derived based on the information-theoretic distance from the flexible model to the base model, giving priors with clear interpretation. We provide implementation details for the new model formulation which preserve sparsity properties, and we investigate systematically the model performance and compare it to existing parameterisations. Through a simulation study, we show that the new model performs well, both showing good learning abilities and good shrinkage behaviour. In terms of model choice criteria, the proposed model performs at least equally well as existing parameterisations, but only the new formulation offers parameters that are interpretable and hyperpriors that have a clear meaning.

Spatial smoothing in Bayesian models: a comparison of weights matrix specifications and their impact on inference

Background

When analysing spatial data, it is important to account for spatial autocorrelation. In Bayesian statistics, spatial autocorrelation is commonly modelled by the intrinsic conditional autoregressive prior distribution. At the heart of this model is a spatial weights matrix which controls the behaviour and degree of spatial smoothing. The purpose of this study is to review the main specifications of the spatial weights matrix found in the literature, and together with some new and less common specifications, compare the effect that they have on smoothing and model performance.

Methods

The popular BYM model is described, and a simple solution for addressing the identifiability issue among the spatial random effects is provided. Seventeen different definitions of the spatial weights matrix are defined, which are classified into four classes: adjacency-based weights, and weights based on geographic distance, distance between covariate values, and a hybrid of geographic and covariate distances. These last two definitions embody the main novelty of this research. Three synthetic data sets are generated, each representing a different underlying spatial structure. These data sets together with a real spatial data set from the literature are analysed using the models. The models are evaluated using the deviance information criterion and Moran’s I statistic.

Results

The deviance information criterion indicated that the model which uses binary, first-order adjacency weights to perform spatial smoothing is generally an optimal choice for achieving a good model fit. Distance-based weights also generally perform quite well and offer similar parameter interpretations. The less commonly explored options for performing spatial smoothing generally provided a worse model fit than models with more traditional approaches to smoothing, but usually outperformed the benchmark model which did not conduct spatial smoothing.

Conclusions

The specification of the spatial weights matrix can have a colossal impact on model fit and parameter estimation. The results provide some evidence that a smaller number of neighbours used in defining the spatial weights matrix yields a better model fit, and may provide a more accurate representation of the underlying spatial random field. The commonly used binary, first-order adjacency weights still appear to be a good choice for implementing spatial smoothing.

Electronic supplementary material

The online version of this article (10.1186/s12942-017-0120-x) contains supplementary material, which is available to authorized users.

Spatial scale in environmental risk mapping: A Valley fever case study

Background

Valley fever is a fungal infection occurring in desert regions of the U.S. and Central and South America. Environmental risk mapping for this disease is hampered by challenges with detection, case reporting, and diagnostics as well as challenges common to spatial data handling.

Design and methods.

Using 12,349 individual cases in Arizona from 2006 to 2009, we analyzed risk factors at both the individual and area levels.

Results.

Risk factors including elderly population, income status, soil organic carbon, and density of residential area were found to be positively associated with residence of Valley fever cases. A negative association was observed for distance to desert and pasture/hay land cover. The association between incidence and two land cover variables (shrub and cultivated crop lands) varied depending on the spatial scale of the analysis.

Conclusions

The consistence of age, income, population density, and proximity to natural areas supports that these are important predictors of Valley fever risk. However, the inconsistency of the land cover variables across scales highlights the importance of how scale is treated in risk mapping.

Spatial regression with covariate measurement error: A semiparametric approach

Spatial data have become increasingly common in epidemiology and public health research thanks to advances in GIS (Geographic Information Systems) technology. In health research, for example, it is common for epidemiologists to incorporate geographically indexed data into their studies. In practice, however, the spatially defined covariates are often measured with error. Naive estimators of regression coefficients are attenuated if measurement error is ignored. Moreover, the classical measurement error theory is inapplicable in the context of spatial modeling because of the presence of spatial correlation among the observations. We propose a semiparametric regression approach to obtain bias-corrected estimates of regression parameters and derive their large sample properties. We evaluate the performance of the proposed method through simulation studies and illustrate using data on Ischemic Heart Disease (IHD). Both simulation and practical application demonstrate that the proposed method can be effective in practice.

Spatial Patterns of Ischemic Heart Disease in Shenzhen, China: A Bayesian Multi-Disease Modelling Approach to Inform Health Planning Policies

Incorporating the information of hypertension, this paper applies Bayesian multi-disease analysis to model the spatial patterns of Ischemic Heart Disease (IHD) risks. Patterns of harmful alcohol intake (HAI) and overweight/obesity are also modelled as they are common risk factors contributing to both IHD and hypertension. The hospitalization data of IHD and hypertension in 2012 were analyzed with three Bayesian multi-disease models at the sub-district level of Shenzhen. Results revealed that the IHD high-risk cluster shifted slightly north-eastward compared with the IHD Standardized Hospitalization Ratio (SHR). Spatial variations of overweight/obesity and HAI were found to contribute most to the IHD patterns. Identified patterns of IHD risk would benefit IHD integrated prevention. Spatial patterns of overweight/obesity and HAI could supplement the current disease surveillance system by providing information about small-area level risk factors, and thus benefit integrated prevention of related chronic diseases. Middle southern Shenzhen, where high risk of IHD, overweight/obesity, and HAI are present, should be prioritized for interventions, including alcohol control, innovative healthy diet toolkit distribution, insurance system revision, and community-based chronic disease intervention. Related health resource planning is also suggested to focus on these areas first.

Trends of breast cancer incidence in Iran during 2004-2008: a Bayesian space-time model

BACKGROUND

Breast cancer is the most frequently diagnosed cancer in women and estimating its relative risks and trends of incidence at the area-level is helpful for health policy makers. However, traditional methods of estimation which do not take spatial heterogeneity into account suffer from drawbacks and their results may be misleading, as the estimated maps of incidence vary dramatically in neighboring areas. Spatial methods have been proposed to overcome drawbacks of traditional methods by including spatial sources of variation in the model to produce smoother maps.

MATERIALS AND METHODS

In this study we analyzed the breast cancer data in Iran during 2004-2008. We used a method proposed to cover spatial and temporal effects simultaneously and their interactions to study trends of breast cancer incidence in Iran.

RESULTS

The results agree with previous studies but provide new information about two main issues regarding the trend of breast cancer in provinces of Iran. First, this model discovered provinces with high relative risks of breast cancer during the 5 years of the study. Second, new information was provided with respect to overall trend trends o. East-Azerbaijan, Golestan, North-Khorasan, and Khorasan-Razavi had the highest increases in rates of breast cancer incidence whilst Tehran, Isfahan, and Yazd had the highest incidence rates during 2004-2008.

CONCLUSIONS

Using spatial methods can provide more accurate and detailed information about the incidence or prevalence of a disease. These models can specify provinces with different health priorities in terms of needs for therapy and drugs or demands for efficient education, screening, and preventive policy into action.

Disease mapping via negative binomial regression M-quantiles

We introduce a semi-parametric approach to ecological regression for disease mapping, based on modelling the regression M-quantiles of a negative binomial variable. The proposed method is robust to outliers in the model covariates, including those due to measurement error, and can account for both spatial heterogeneity and spatial clustering. A simulation experiment based on the well-known Scottish lip cancer data set is used to compare the M-quantile modelling approach with a disease mapping approach based on a random effects model. This suggests that the M-quantile approach leads to predicted relative risks with smaller root mean square error. The paper concludes with an illustrative application of the M-quantile approach, mapping low birth weight incidence data for English Local Authority Districts for the years 2005-2010.

Semiparametric M-quantile regression for count data

Lung cancer incidence over 2005–2010 for 326 Local Authority Districts in England is investigated by ecological regression. Motivated from mis-specification of a Negative Binomial additive model, a semiparametric Negative Binomial M-quantile regression model is introduced. The additive part relates to those univariate or bivariate smoothing components, which are included in the model to capture nonlinearities in the predictor or to account for spatial dependence. All such components are estimated using penalized splines. The results show the capability of the semiparametric Negative Binomial M-quantile regression model to handle data with a strong spatial structure.

Estimation of malaria incidence in northern Namibia in 2009 using Bayesian conditional-autoregressive spatial-temporal models

As malaria transmission declines, it becomes increasingly important to monitor changes in malaria incidence rather than prevalence. Here, a spatio-temporal model was used to identify constituencies with high malaria incidence to guide malaria control. Malaria cases were assembled across all age groups along with several environmental covariates. A Bayesian conditional-autoregressive model was used to model the spatial and temporal variation of incidence after adjusting for test positivity rates and health facility utilisation. Of the 144,744 malaria cases recorded in Namibia in 2009, 134,851 were suspected and 9893 were parasitologically confirmed. The mean annual incidence based on the Bayesian model predictions was 13 cases per 1000 population with the highest incidence predicted for constituencies bordering Angola and Zambia. The smoothed maps of incidence highlight trends in disease incidence. For Namibia, the 2009 maps provide a baseline for monitoring the targets of pre-elimination.

On Bayesian shared component disease mapping and ecological regression with errors in covariates

Recent literature on Bayesian disease mapping presents shared component models (SCMs) for joint spatial modeling of two or more diseases with common risk factors. In this study, Bayesian hierarchical formulations of shared component disease mapping and ecological models are explored and developed in the context of ecological regression, taking into consideration errors in covariates. A review of multivariate disease mapping models (MultiVMs) such as the multivariate conditional autoregressive models that are also part of the more recent Bayesian disease mapping literature is presented. Some insights into the connections and distinctions between the SCM and MultiVM procedures are communicated. Important issues surrounding (appropriate) formulation of shared- and disease-specific components, consideration/choice of spatial or non-spatial random effects priors, and identification of model parameters in SCMs are explored and discussed in the context of spatial and ecological analysis of small area multivariate disease or health outcome rates and associated ecological risk factors. The methods are illustrated through an in-depth analysis of four-variate road traffic accident injury (RTAI) data: gender-specific fatal and non-fatal RTAI rates in 84 local health areas in British Columbia (Canada). Fully Bayesian inference via Markov chain Monte Carlo simulations is presented.

Risk of cancer in the vicinity of municipal solid waste incinerators: importance of using a flexible modelling strategy

Background

We conducted an ecological study in four French administrative departments and highlighted an excess risk in cancer morbidity for residents around municipal solid waste incinerators. The aim of this paper is to show how important are advanced tools and statistical techniques to better assess weak associations between the risk of cancer and past environmental exposures.

Methods

The steps to evaluate the association between the risk of cancer and the exposure to incinerators, from the assessment of exposure to the definition of the confounding variables and the statistical analysis carried out are detailed and discussed. Dispersion modelling was used to assess exposure to sixteen incinerators. A geographical information system was developed to define an index of exposure at the IRIS level that is the geographical unit we considered.

Population density, rural/urban status, socio-economic deprivation, exposure to air pollution from traffic and from other industries were considered as potential confounding factors and defined at the IRIS level. Generalized additive models and Bayesian hierarchical models were used to estimate the association between the risk of cancer and the index of exposure to incinerators accounting for the confounding factors.

Results

Modelling to assess the exposure to municipal solid waste incinerators allowed accounting for factors known to influence the exposure (meteorological data, point source characteristics, topography). The statistical models defined allowed modelling extra-Poisson variability and also non-linear relationships between the risk of cancer and the exposure to incinerators and the confounders.

Conclusion

In most epidemiological studies distance is still used as a proxy for exposure. This can lead to significant exposure misclassification. Additionally, in geographical correlation studies the non-linear relationships are usually not accounted for in the statistical analysis. In studies of weak associations it is important to use advanced methods to better assess dose-response relationships with disease risk.

Geographic variations in risk: adjusting for unmeasured confounders through joint modeling of multiple diseases

BACKGROUND

Chronic obstructive pulmonary disease (COPD) is an important cause of mortality with marked geographic variations in Great Britain. Additional factors beyond cigarette smoking are likely to influence these variations, but direct information on smoking by area is not readily available. We compared methods of jointly modeling the spatial distribution of mortality from COPD and lung cancer, using the latter as a proxy for smoking, to identify areas in which risk factors other than smoking may be important.

METHODS

We obtained district-level mortality and population data for men aged 45 years or older in 1981-1999 in Great Britain. Three models were compared: Bayesian ecological regression using observed (model 1) or spatially smoothed (model 2) lung cancer standardized mortality ratio (SMR) as a smoking proxy, and bivariate regression (model 3) treating smoking as a spatial latent variable common to both diseases.

RESULTS

Model selection criteria favored models 2 and 3 over model 1. Between 9% (model 3) and 25% (model 2) of spatial variation in COPD mortality was estimated to be unrelated to smoking. After adjustment for lung cancer as a proxy for smoking, both models showed similar geographic patterns of higher COPD mortality in conurbation and mining areas, historically associated with heavy industry and higher air pollution levels.

CONCLUSIONS

Joint modeling of multiple diseases can be used to investigate geographic variations in risk. These models reveal patterns that are adjusted for the effects of shared area-level risk factors for which no direct data are available.

Polytomous Disease Mapping to Detect Uncommon Risk Factors for Related Diseases

A statistical model for jointly analysing the spatial variation of incidences of three (or more) diseases, with common and uncommon risk factors, is introduced. Deaths for different diseases are described by a logit model for multinomial responses (multinomial logit or polytomous logit model). For each area and confounding strata population (i.e. age-class, sex, race) the probabilities of death for each cause (the response probabilities) are estimated. A specic disease, the one having a common risk factor only, acts as the baseline category. The log odds are decomposed additively into shared (common to diseases different by the reference disease) and specic structured spatial variability terms, unstructured unshared spatial terms and confounders terms (such as age, race and sex) to adjust the crude observed data for their effects. Disease specic spatially structured effects are estimated; these are considered as latent variables denoting disease-specic risk factors. The model is presented with reference to a specic application. We considered the mortality data (from 1990 to 1994) relative to oral cavity, larynx and lung cancers in 13 age groups of males, in the 287 municipalities of Region of Tuscany (Italy). All these pathologies share smoking as a common risk factor; furthermore, two of them (oral cavity and larynx cancer) share alcohol consumption as a risk factor. All studies suggest that smoking and alcohol consumption are the major known risk factors for oral cavity and larynx cancers; nevertheless, in this paper, we investigate the possibility of other different risk factors for these diseases, or even the presence of an interaction effect (between smoking and alcohol risk factors) but with different spatial patterns for oral and larynx cancer. For each municipality and age-class the probabilities of death for each cause (the response probabilities) are estimated. Lung cancer acts as the baseline category. The log odds are decomposed additively into shared (common to oral cavity and larynx diseases) and specic structured spatial variability terms, unstructured unshared spatial terms and an age-group term. It turns out that oral cavity and larynx cancer have different spatial patterns for residual risk factors which are not the typical ones such as smoking habits and alcohol consumption. But, possibly, these patterns are due to different spatial interactions between smoking habits and alcohol consumption for the first and the second disease.

Outdoor NOx and stroke mortality: adjusting for small area level smoking prevalence using a Bayesian approach

There is increasing evidence, mainly from daily time series studies, linking air pollution and stroke. Small area level geographical correlation studies offer another means of examining the air pollution-stroke association. Populations within small areas may be more homogeneous than those within larger areal units, and census-based socioeconomic information may be available to adjust for confounding effects. Data on smoking from health surveys may be incorporated in spatial analyses to adjust for potential confounding effects but may be sparse at the small area level. Smoothing, using data from neighbouring areas, may be used to increase the precision of smoking prevalence estimates for small areas. We examined the effect of modelled outdoor NOx levels on stroke mortality using a Bayesian hierarchical spatial model to incorporate random effects, in order to allow for unmeasured confounders and to acknowledge sampling error in the estimation of smoking prevalence. We observed an association between NOx and stroke mortality after taking into account random effects at the small area level. We found no association between smoking prevalence and stroke mortality at the small area level after modelling took into account imprecision in estimating smoking prevalence. The approach we used to incorporate smoking as a covariate in a single large model is conceptually sound, though it made little difference to the substantive results.

Positional accuracy and geographic bias of four methods of geocoding in epidemiologic research

PURPOSE

We examined the geographic bias of four methods of geocoding addresses using ArcGIS, commercial firm, SAS/GIS, and aerial photography. We compared "point-in-polygon" (ArcGIS, commercial firm, and aerial photography) and the "look-up table" method (SAS/GIS) to allocate addresses to census geography, particularly as it relates to census-based poverty rates.

METHODS

We randomly selected 299 addresses of children treated for asthma at an urban emergency department (1999-2001). The coordinates of the building address side door were obtained by constant offset based on ArcGIS and a commercial firm and true ground location based on aerial photography.

RESULTS

Coordinates were available for 261 addresses across all methods. For 24% to 30% of geocoded road/door coordinates the positional error was 51 meters or greater, which was similar across geocoding methods. The mean bearing was -26.8 degrees for the vector of coordinates based on aerial photography and ArcGIS and 8.5 degrees for the vector based on aerial photography and the commercial firm (p < 0.0001). ArcGIS and the commercial firm performed very well relative to SAS/GIS in terms of allocation to census geography. For 20%, the door location based on aerial photography was assigned to a different block group compared to SAS/GIS. The block group poverty rate varied at least two standard deviations for 6% to 7% of addresses.

CONCLUSION

We found important differences in distance and bearing between geocoding relative to aerial photography. Allocation of locations based on aerial photography to census-based geographic areas could lead to substantial errors.

Space-time models with time-dependent covariates for the analysis of the temporal lag between socioeconomic factors and lung cancer mortality

The relationship between socioeconomic factors and mortality for lung cancer is investigated. To identify the proper lag time between socioeconomic factors and lung cancer mortality, a space-time hierarchical Bayesian model with time-dependent covariates is adopted. A real example on lung cancer mortality, males, in Tuscany (Italy) during the period 1971-1999, is provided. Results confirm the presence of an association between mortality for lung cancer and socioeconomic factors with a temporal lag (latency time) of at least 10 years.

Residual Spatial Correlation Between Geographically Referenced Observations

BACKGROUND

Analytic methods commonly used in epidemiology do not account for spatial correlation between observations. In regression analyses, this omission can bias parameter estimates and yield incorrect standard error estimates. We present a Bayesian hierarchical model (BHM) approach that accounts for spatial correlation, and illustrate its strengths and weaknesses by applying this modeling approach to data on Wuchereria bancrofti infection in Haiti.

METHODS

A program to eliminate lymphatic filariasis in Haiti assessed prevalence of W. bancrofti infection in 57 schools across Leogane Commune. We analyzed the spatial pattern in the prevalence data using semi-variograms and correlograms. We then modeled the data using (1) standard logistic regression (GLM); (2) non-Bayesian logistic generalized linear mixed models (GLMMs) with school-specific nonspatial random effects; (3) BHMs with school-specific nonspatial random effects; and (4) BHMs with spatial random effects.

RESULTS

An exponential semi-variogram with an effective range of 2.15 km best fit the data. GLMM and nonspatial BHM point estimates were comparable and also were generally similar with the marginal GLM point estimates. In contrast, compared with the nonspatial mixed model results, spatial BHM point estimates were markedly attenuated.

DISCUSSION

The clear spatial pattern evident in the Haitian W. bancrofti prevalence data and the observation that point estimates and standard errors differed depending on the modeling approach indicate that it is important to account for residual spatial correlation in analyses of W. bancrofti infection data. Bayesian hierarchical models provide a flexible, readily implementable approach to modeling spatially correlated data. However, our results also illustrate that spatial smoothing must be applied with care.

Towards joint disease mapping

This article discusses and extends statistical models to jointly analyse the spatial variation of rates of several diseases with common risk factors. We start with a review of methods for separate analyses of diseases, then move to ecological regression approaches, where the rates from one of the diseases enter as surrogate covariates for exposure. Finally, we propose a general framework for jointly modelling the variation of two or more diseases, some of which share latent spatial fields, but with possibly different risk gradients. In our application, we consider mortality data on oral, oesophagus, larynx and lung cancers for males in Germany, which all share smoking as a common risk factor. Furthermore, the first three cancers are also known to be related to excessive alcohol consumption. An empirical comparison of the different models based on a formal model criterion as well as on the posterior precision of the relative risk estimates strongly suggests that the joint modelling approach is a useful and valuable extension over individual analyses.

Area-level risks for BSE in British cattle before and after the July 1988 meat and bone meal feed ban

In this paper we investigate area-level risk factors for BSE for the cattle population present in Great Britain between 1986 and 1997. By dividing this population into two birth cohorts, those born before the July 1988 ban on feeding ruminant-derived meat and bone meal to ruminants and those born after, second-order regional influences are distinguished from the strong first-order south-to-north gradient of area-level BSE risk using Bayesian hierarchical models that account for structured (spatially correlated) and unstructured heterogeneity in the data. For both cohorts area-level risk of BSE was increased by a more southerly location and greater numbers of dairy cattle, relative to non-dairy cattle. For the cohort of cattle born after the July 1988 ban on feeding ruminant-derived meat and bone meal area-level BSE risk was additionally associated with greater numbers of pigs, relative to cattle. These findings support the role of low level cross-contamination of cattle feed by pig feed as an influence on BSE incidence risk as the epidemic evolved. Prior to the 1988 meat and bone meal ban unexplained BSE risk was relatively uniformly distributed across the country whereas after the ban there were spatially aggregated areas of unexplained risk in the northern and eastern regions of England suggesting that local influences allowed BSE control measures to be less-successfully applied in these areas, compared with the rest of the country. We conclude that spatially localised influences were operating in divergent ways during the two phases of the epidemic.

Spatial heterogeneity of the risk of BSE in France following the ban of meat and bone meal in cattle feed

In France, meat-and-bone meal (MBM) has been prohibited for cattle feeding since 1990, but bovine spongiform encephalopathy (BSE) cases, called 'NAIF', appeared in animals born after this feed ban. Furthermore, in 1996 a new measure was taken: removal of cadavers and specified risk materials (SRM) from the processing of MBM dedicated to animal feed. Nevertheless, BSE cases (called 'super-NAIF') appeared in cattle born after this measure was in force. We analysed the spatial distribution of 445 'NAIF' and 58 'super-NAIF' cases detected in France from July 1, 2001 to July 31, 2003. The detection of BSE was based both on the mandatory reporting system (MRS) and the systematic test screening of cattle at the abattoir and at the fallen-animal plant with rapid tests. The background population was based on the adult-cow census. The disease mapping of the BSE risk was based on the standardised incidence ratio (stochastic Poisson process). A spatial component, which takes into account the spatial dependence between the geographical units by a notion of adjacency was used to eliminate the over-dispersion in the risk assessment. The geographical units were defined by hexagons with a side of 23km (France had 1264 hexagons). The parameters were estimated by a Metropolis Gibbs sampling algorithm using the Markov-chain Monte Carlo methods. The BSE cases were not randomly distributed. Furthermore, the areas at risk for the 'super-NAIF' matched part of the areas at risk for the 'NAIF' cases-which suggests that it might be a common source of contamination.

Spatial epidemiology: current approaches and future challenges

Spatial epidemiology is the description and analysis of geographic variations in disease with respect to demographic, environmental, behavioral, socioeconomic, genetic, and infectious risk factors. We focus on small-area analyses, encompassing disease mapping, geographic correlation studies, disease clusters, and clustering. Advances in geographic information systems, statistical methodology, and availability of high-resolution, geographically referenced health and environmental quality data have created unprecedented new opportunities to investigate environmental and other factors in explaining local geographic variations in disease. They also present new challenges. Problems include the large random component that may predominate disease rates across small areas. Though this can be dealt with appropriately using Bayesian statistics to provide smooth estimates of disease risks, sensitivity to detect areas at high risk is limited when expected numbers of cases are small. Potential biases and confounding, particularly due to socioeconomic factors, and a detailed understanding of data quality are important. Data errors can result in large apparent disease excess in a locality. Disease cluster reports often arise nonsystematically because of media, physician, or public concern. One ready means of investigating such concerns is the replication of analyses in different areas based on routine data, as is done in the United Kingdom through the Small Area Health Statistics Unit (and increasingly in other European countries, e.g., through the European Health and Environment Information System collaboration). In the future, developments in exposure modeling and mapping, enhanced study designs, and new methods of surveillance of large health databases promise to improve our ability to understand the complex relationships of environment to health.

Bayesian statistics for parasitologists

Bayesian statistical methods are increasingly being used in the analysis of parasitological data. Here, the basis of differences between the Bayesian method and the classical or frequentist approach to statistical inference is explained. This is illustrated with practical implications of Bayesian analyses using prevalence estimation of strongyloidiasis and onchocerciasis as two relevant examples. The strongyloidiasis example addresses the problem of parasitological diagnosis in the absence of a gold standard, whereas the onchocerciasis case focuses on the identification of villages warranting priority mass ivermectin treatment. The advantages and challenges faced by users of the Bayesian approach are also discussed and the readers pointed to further directions for a more in-depth exploration of the issues raised. We advocate collaboration between parasitologists and Bayesian statisticians as a fruitful and rewarding venture for advancing applied research in parasite epidemiology and the control of parasitic infections.

Spatial analysis of disease

In this chapter, we have reviewed the history of the spatial analysis of disease and the statistical methods used for the exploratory analysis, testing and modeling of spatial patterns. In the next chapter, the principles described here will be illustrated.

On the precision of the conditionally autoregressive prior in spatial models

Bayesian analyses of spatial data often use a conditionally autoregressive (CAR) prior, which can be written as the kernel of an improper density that depends on a precision parameter tau that is typically unknown. To include tau in the Bayesian analysis, the kernel must be multiplied by tau(k) for some k. This article rigorously derives k = (n - I)/2 for the L2 norm CAR prior (also called a Gaussian Markov random field model) and k = n - I for the L1 norm CAR prior, where n is the number of regions and I the number of "islands" (disconnected groups of regions) in the spatial map. Since I = 1 for a spatial structure defining a connected graph, this supports Knorr-Held's (2002, in Highly Structured Stochastic Systems, 260-264) suggestion that k = (n - 1)/2 in the L2 norm case, instead of the more common k = n/2. We illustrate the practical significance of our results using a periodontal example.

Approximate cross-validatory predictive checks in disease mapping models

When fitting complex hierarchical disease mapping models, it can be important to identify regions that diverge from the assumed model. Since full leave-one-out cross-validatory assessment is extremely time-consuming when using Markov chain Monte Carlo (MCMC) estimation methods, Stern and Cressie consider an importance sampling approximation. We show that this can be improved upon through replication of both random effects and data. Our approach is simple to apply, entirely generic, and may aid the criticism of any Bayesian hierarchical model.

Spatial statistical methods in health

The study of the geographical distribution of disease incidence and its relationship to potential risk factors (referred to here as "geographical epidemiology") has provided, and continues to provide, rich ground for the application and development of statistical methods and models. In recent years increasingly powerful and versatile statistical tools have been developed in this application area. This paper discusses the general classes of problem in geographical epidemiology and reviews the key statistical methods now being employed in each of the application areas identified. The paper does not attempt to exhaustively cover all possible methods and models, but extensive references are provided to further details and to additional approaches. The overall aim is to provide a picture of the "current state of the art" in the use of spatial statistical methods in epidemiological and public health research. Following the review of methods, the main software environments which are available to implement such methods are discussed. The paper concludes with some brief general reflections on the epidemiological and public health implications of the use of spatial statistical methods in health and on associated benefits and problems.

Modelling categorical covariates in Bayesian disease mapping by partition structures

We consider the problem of mapping the risk from a disease using a series of regional counts of observed and expected cases, and information on potential risk factors. To analyse this problem from a Bayesian viewpoint, we propose a methodology which extends a spatial partition model by including categorical covariate information. Such an extension allows detection of clusters in the residual variation, reflecting further, possibly unobserved, covariates. The methodology is implemented by means of reversible jump Markov chain Monte Carlo sampling. An application is presented in order to illustrate and compare our proposed extensions with a purely spatial partition model. Here we analyse a well-known data set on lip cancer incidence in Scotland.

Probabilistic small area risk assessment using GIS-based data: a case study on Finnish childhood diabetes. Geographic information systems

A Bayesian hierarchical spatial model is constructed to describe the regional incidence of insulin dependent diabetes mellitus (IDDM) among the under 15-year-olds in Finland. The model exploits aggregated pixel-wise locations for both the cases and the population at risk. Typically such data arise from combining geographic information systems (GIS) with large databases. The dates of diagnosis and locations of the cases are observed from 1987 to 1996. The population at risk counts are available for every second year during the same period. A hierarchical model is suggested for the pixel wise case counts, including a population model to account for the uncertainty of the population at risk over the years. The model is applied in the construction of disease maps (aggregated 100 km(2) pixels), and spatial posterior predictive distributions are computed to study whether there can be found a statistically exceptional number of cases in a small area of interest.

An empirical evaluation of various priors in the empirical Bayes estimation of small area disease risks

Empirical and fully Bayes estimation of small area disease risks places a prior distribution on area-specific risks. Several forms of priors have been used for this purpose including gamma, log-normal and non-parametric priors. Spatial correlation among area-specific risks can be incorporated in log-normal priors using Gaussian Markov random fields or other models of spatial dependence. However, the criterion for choosing one prior over others has been mostly logical reasoning. In this paper, we evaluate empirically the various priors used in the empirical Bayes estimation of small area disease risks. We utilize a Spanish mortality data set of a 12-year period to give the underlying true risks, and estimate the true risks using only a 3-year portion of the data set. Empirical Bayes estimates are shown to have substantially smaller mean squared errors than Poisson likelihood-based estimates. However, relative performances of various priors differ across a variety of mortality outcomes considered. In general, the non-parametric prior provides good estimates for lower-risk areas, while spatial priors provide good estimates for higher-risk areas. Ad hoc composite estimates averaging the estimates from the non-parametric prior and those from a spatial log-normal prior appear to perform well overall. This suggests that an empirical Bayes prior that strikes a balance between these two priors, if one can construct such a prior, may prove to be useful for the estimation of small area disease risks.

Issues in the statistical analysis of small area health data

The availability of geographically indexed health and population data, with advances in computing, geographical information systems and statistical methodology, have opened the way for serious exploration of small area health statistics based on routine data. Such analyses may be used to address specific questions concerning health in relation to sources of pollution, to investigate clustering of disease or for hypothesis generation. We distinguish four types of analysis: disease mapping; geographic correlation studies; the assessment of risk in relation to a prespecified point or line source, and cluster detection and disease clustering. A general framework for the statistical analysis of small area studies will be considered. This framework assumes that populations at risk arise from inhomogeneous Poisson processes. Disease cases are then realizations of a thinned Poisson process where the risk of disease depends on the characteristics of the person, time and spatial location. Difficulties of analysis and interpretation due to data inaccuracies and aggregation will be addressed with particular reference to ecological bias and confounding. The use of errors-in-variables modelling in small area analyses will be discussed.