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.     

Linear models of coregionalization for multivariate lattice data: a general framework for coregionalized multivariate CAR models

We present a general coregionalization framework for developing coregionalized multivariate Gaussian conditional autoregressive (cMCAR) models for Bayesian analysis of multivariate lattice data in general and multivariate disease mapping data in particular. This framework is inclusive of cMCARs that facilitate flexible modelling of spatially structured symmetric or asymmetric cross‐variable local interactions, allowing a wide range of separable or non‐separable covariance structures, and symmetric or asymmetric cross‐covariances, to be modelled. We present a brief overview of established univariate Gaussian conditional autoregressive (CAR) models for univariate lattice data and develop coregionalized multivariate extensions. Classes of cMCARs are presented by formulating precision structures. The resulting conditional properties of the multivariate spatial models are established, which cast new light on cMCARs with richly structured covariances and cross‐covariances of different spatial ranges. The related methods are illustrated via an in‐depth Bayesian analysis of a Minnesota county‐level cancer data set. We also bring a new dimension to the traditional enterprize of Bayesian disease mapping: estimating and mapping covariances and cross‐covariances of the underlying disease risks. Maps of covariances and cross‐covariances bring to light spatial characterizations of the cMCARs and inform on spatial risk associations between areas and diseases. Copyright © 2016 John Wiley & Sons, Ltd.     

Empirical Bayes estimates of cancer mortality rates using spatial models

We give results of empirical Bayes (EB) estimation of mortality rates designed to smooth observed SMR when random fluctuation of the observed deaths is important. We have specially studied the case where the prior distributions of the EB method have a spatial structure. The need for spatial modelling of cancer mortality rates in France is first shown with testing autocorrelation and fitting autoregressive spatial models, conditional (CAR) or simultaneous (SAR). A positive autocorrelation of the rates is shown for most cancer sites studied. As expected, EB estimates of mortality rates for common tumours are similar to SMRs. For rare tumours, the EB method identifies the extreme rates more clearly than SMRs by smoothing the SMRs with large variances. CAR or SAR models are adequate prior distributions for autocorrelated rates and produce quite similar rate estimates.     

Skew‐elliptical spatial random effect modeling for areal data with application to mapping health utilization rates

Mixed models incorporating spatially correlated random effects are often used for the analysis of areal data. In this setting, spatial smoothing is introduced at the second stage of a hierarchical framework, and this smoothing is often based on a latent Gaussian Markov random field. The Markov random field provides a computationally convenient framework for modeling spatial dependence; however, the Gaussian assumption underlying commonly used models can be overly restrictive in some applications. This can be a problem in the presence of outliers or discontinuities in the underlying spatial surface, and in such settings, models based on non‐Gaussian spatial random effects are useful. Motivated by a study examining geographic variation in the treatment of acute coronary syndrome, we develop a robust model for smoothing small‐area health service utilization rates. The model incorporates non‐Gaussian spatial random effects, and we develop a formulation for skew‐elliptical areal spatial models. We generalize the Gaussian conditional autoregressive model to the non‐Gaussian case, allowing for asymmetric skew‐elliptical marginal distributions having flexible tail behavior. The resulting new models are flexible, computationally manageable, and can be implemented in the standard Bayesian software WinBUGS. We demonstrate performance of the proposed methods and comparisons with other commonly used Gaussian and non‐Gaussian spatial prior formulations through simulation and analysis in our motivating application, mapping rates of revascularization for patients diagnosed with acute coronary syndrome in Quebec, Canada. Copyright © 2012 John Wiley & Sons, Ltd.     

Selecting a separable parametric spatiotemporal covariance structure for longitudinal imaging data

Longitudinal imaging studies allow great insight into how the structure and function of a subject's internal anatomy changes over time. Unfortunately, the analysis of longitudinal imaging data is complicated by inherent spatial and temporal correlation: the temporal from the repeated measures and the spatial from the outcomes of interest being observed at multiple points in a patient's body. We propose the use of a linear model with a separable parametric spatiotemporal error structure for the analysis of repeated imaging data. The model makes use of spatial (exponential, spherical, and Matérn) and temporal (compound symmetric, autoregressive‐1, Toeplitz, and unstructured) parametric correlation functions. A simulation study, inspired by a longitudinal cardiac imaging study on mitral regurgitation patients, compared different information criteria for selecting a particular separable parametric spatiotemporal correlation structure as well as the effects on types I and II error rates for inference on fixed effects when the specified model is incorrect. Information criteria were found to be highly accurate at choosing between separable parametric spatiotemporal correlation structures. Misspecification of the covariance structure was found to have the ability to inflate the type I error or have an overly conservative test size, which corresponded to decreased power. An example with clinical data is given illustrating how the covariance structure procedure can be performed in practice, as well as how covariance structure choice can change inferences about fixed effects. Copyright © 2014 John Wiley & Sons, Ltd.     

Bayesian age–period–cohort models with versatile interactions and long‐term predictions: mortality and population in Finland 1878–2050

Age–period–cohort (APC) models are widely used for studying time trends of disease incidence or mortality. Model identifiability has become less of a problem with Bayesian APC models. These models are usually based on random walk (RW1, RW2) smoothing priors. For long and complex time series and for long predicted periods, these models as such may not be adequate. We present two extensions for the APC models. First, we introduce flexible interactions between the age, period and cohort effects based on a two‐dimensional conditional autoregressive smoothing prior on the age/period plane. Our second extension uses autoregressive integrated (ARI) models to provide reasonable long‐term predictions. To illustrate the utility of our model framework, we provide stochastic predictions for the Finnish male and female population, in 2010–2050. For that, we first study and forecast all‐cause male and female mortality in Finland, 1878–2050, showing that using an interaction term is needed for fitting and interpreting the observed data. We then provide population predictions using a cohort component model, which also requires predictions for fertility and migration. As our main conclusion, ARI models have better properties for predictions than the simple RW models do, but mixing these prediction models with RW1 or RW2 smoothing priors for observed periods leads to a model that is not fully consistent. Further research with our model framework will concentrate on using a more consistent model for smoothing and prediction, such as autoregressive integrated moving average models with state‐space methods or Gaussian process priors. Copyright © 2013 John Wiley & Sons, Ltd.     

The bivariate combined model for spatial data analysis

To describe the spatial distribution of diseases, a number of methods have been proposed to model relative risks within areas. Most models use Bayesian hierarchical methods, in which one models both spatially structured and unstructured extra‐Poisson variance present in the data. For modelling a single disease, the conditional autoregressive (CAR) convolution model has been very popular. More recently, a combined model was proposed that ‘combines’ ideas from the CAR convolution model and the well‐known Poisson‐gamma model. The combined model was shown to be a good alternative to the CAR convolution model when there was a large amount of uncorrelated extra‐variance in the data. Less solutions exist for modelling two diseases simultaneously or modelling a disease in two sub‐populations simultaneously. Furthermore, existing models are typically based on the CAR convolution model. In this paper, a bivariate version of the combined model is proposed in which the unstructured heterogeneity term is split up into terms that are shared and terms that are specific to the disease or subpopulation, while spatial dependency is introduced via a univariate or multivariate Markov random field. The proposed method is illustrated by analysis of disease data in Georgia (USA) and Limburg (Belgium) and in a simulation study. We conclude that the bivariate combined model constitutes an interesting model when two diseases are possibly correlated. As the choice of the preferred model differs between data sets, we suggest to use the new and existing modelling approaches together and to choose the best model via goodness‐of‐fit statistics. Copyright © 2016 John Wiley & Sons, Ltd.     

A Bayesian multinomial model to analyse spatial patterns of childhood co-morbidity in Malawi

Children in less developed countries die from relatively small number of infectious disease, some of which epidemiologically overlap. Using self-reported illness data from the 2000 Malawi Demographic and Health Survey, we applied a random effects multinomial model to assess risk factors of childhood co-morbidity of fever, diarrhoea and pneumonia, and quantify area-specific spatial effects. The spatial structure was modelled using the conditional autoregressive prior. Various models were fitted and compared using deviance information criterion. Inference was Bayesian and was based on Markov Chain Monte Carlo simulation techniques. We found spatial variation in childhood co-morbidity and determinants of each outcome category differed. Specifically, risk factors associated with child co-morbidity included age of the child, place of residence, undernutrition, bednet use and Vitamin A. Higher residual risk levels were identified in the central and southern–eastern regions, particularly for fever, diarrhoea and pneumonia; fever and pneumonia; and fever and diarrhoea combinations. This linkage between childhood health and geographical location warrants further research to assess local causes of these clusters. More generally, although each disease has its own mechanism, overlapping risk factors suggest that integrated disease control approach may be cost-effective and should be employed.     

Hierarchical additive modeling of nonlinear association with spatial correlations—An application to relate alcohol outlet density and neighborhood assault rates

Previous studies have suggested a link between alcohol outlets and assaults. In this paper, we explore the effects of alcohol availability on assaults at the census tract level over time. In addition, we use a natural experiment to check whether a sudden loss of alcohol outlets is associated with deeper decreasing in assault violence. Several features of the data raise statistical challenges: (1) the association between covariates (for example, the alcohol outlet density of each census tract) and the assault rates may be complex and therefore cannot be described using a linear model without covariates transformation, (2) the covariates may be highly correlated with each other, (3) there are a number of observations that have missing inputs, and (4) there is spatial association in assault rates at the census tract level. We propose a hierarchical additive model, where the nonlinear correlations and the complex interaction effects are modeled using the multiple additive regression trees and the residual spatial association in the assault rates that cannot be explained in the model are smoothed using a conditional autoregressive (CAR) method. We develop a two‐stage algorithm that connects the nonparametric trees with CAR to look for important covariates associated with the assault rates, while taking into account the spatial association of assault rates in adjacent census tracts. The proposed method is applied to the Los Angeles assault data (1990–1999). To assess the efficiency of the method, the results are compared with those obtained from a hierarchical linear model. Copyright © 2009 John Wiley & Sons, Ltd.     

Bayesian Markov switching models for the early detection of influenza epidemics

The early detection of outbreaks of diseases is one of the most challenging objectives of epidemiological surveillance systems. In this paper, a Markov switching model is introduced to determine the epidemic and non-epidemic periods from influenza surveillance data: the process of differenced incidence rates is modelled either with a first-order autoregressive process or with a Gaussian white-noise process depending on whether the system is in an epidemic or in a non-epidemic phase. The transition between phases of the disease is modelled as a Markovian process. Bayesian inference is carried out on the former model to detect influenza epidemics at the very moment of their onset. Moreover, the proposal provides the probability of being in an epidemic state at any given moment. In order to validate the methodology, a comparison of its performance with other alternatives has been made using influenza illness data obtained from the Sanitary Sentinel Network of the Comunitat Valenciana, one of the 17 autonomous regions in Spain. Copyright (c) 2008 John Wiley & Sons, Ltd.     

Multilevel time series models with applications to repeated measures data

The analysis of repeated measures data can be conducted efficiently using a two-level random coefficients model. A standard assumption is that the within-individual (level 1) residuals are uncorrelated. In some cases, especially where measurements are made close together in time, this may not be reasonable and this additional correlation structure should also be modelled. A time series model for such data is proposed which consists of a standard multilevel model for repeated measures data augmented by an autocorrelation model for the level 1 residuals. First- and second-order autoregressive models are considered in detail, together with a seasonal component. Both discrete and continuous time are considered and it is shown how the autocorrelation parameters can themselves be structured in terms of further explanatory variables. The models are fitted to a data set consisting of repeated height measurements on children.     

A bayesian analysis for spatial processes with application to disease mapping

In epidemiology, maps of disease rates and disease risk provide a spatial perspective for researching disease aetiology. For rare diseases or when the population base is small, the rate and risk estimates may be unstable. We propose using a Bayesian analysis based on the conditional autoregressive (CAR) process that will spatially smooth disease rates or risk estimates by allowing each site to 'borrow strength' from its neighbours. Covariates may be included in the model in such a way as to establish a possible association between risk factors and disease incidence. Bayesian inferences are implemented from a direct resampling scheme where large samples are generated from the various posterior distributions. The methodology is demonstrated with a simulation that assesses the effect of sample size and the model parameters on inferences for the parameters. Our approach is also used to spatially smooth district lip cancer rates in Scotland using the CAR model with a covariate that allows for exposure to sunlight.     

Bayesian semiparametric model with spatially–temporally varying coefficients selection

In spatiotemporal analysis, the effect of a covariate on the outcome usually varies across areas and time. The spatial configuration of the areas may potentially depend on not only the structured random intercept but also spatially varying coefficients of covariates. In addition, the normality assumption of the distribution of spatially varying coefficients could lead to potential biases of estimations. In this article, we proposed a Bayesian semiparametric space–time model where the spatially–temporally varying coefficient is decomposed as fixed, spatially varying, and temporally varying coefficients. We nonparametrically modeled the spatially varying coefficients of space–time covariates by using the area‐specific Dirichlet process prior with weights transformed via a generalized transformation. We modeled the temporally varying coefficients of covariates through the dynamic model. We also took into account the uncertainty of inclusion of the spatially–temporally varying coefficients by variable selection procedure through determining the probabilities of different effects for each covariate. The proposed semiparametric approach shows its improvement compared with the Bayesian spatial–temporal models with normality assumption on spatial random effects and the Bayesian model with the Dirichlet process prior on the random intercept. We presented a simulation example to evaluate the performance of the proposed approach with the competing models. We used an application to low birth weight data in South Carolina as an illustration. Copyright © 2013 John Wiley & Sons, Ltd.     

Rising to the challenge: acute stress appraisals and selection centre performance in applicants to postgraduate specialty training in anaesthesia

The ability to work under pressure is a vital non-technical skill for doctors working in acute medical specialties. Individuals who evaluate potentially stressful situations as challenging rather than threatening may perform better under pressure and be more resilient to stress and burnout. Training programme recruitment processes provide an important opportunity to examine applicants’ reactions to acute stress. In the context of multi-station selection centres for recruitment to anaesthesia training programmes, we investigated the factors influencing candidates’ pre-station challenge/threat evaluations and the extent to which their evaluations predicted subsequent station performance. Candidates evaluated the perceived stress of upcoming stations using a measure of challenge/threat evaluation—the cognitive appraisal ratio (CAR)—and consented to release their demographic details and station scores. Using regression analyses we determined which candidate and station factors predicted variation in the CAR and whether, after accounting for these factors, the CAR predicted candidate performance in the station. The CAR was affected by the nature of the station and candidate gender, but not age, ethnicity, country of training or clinical experience. Candidates perceived stations involving work related tasks as more threatening. After controlling for candidates’ demographic and professional profiles, the CAR significantly predicted station performance: ‘challenge’ evaluations were associated with better performance, though the effect was weak. Our selection centre model can help recruit prospective anaesthetists who are able to rise to the challenge of performing in stressful situations but results do not support the direct use of challenge/threat data for recruitment decisions.     

Spatial analysis of socioeconomic determinants of homicide in Brazil

OBJECTIVE: To investigate the association between homicide rates and socio-economic variables taking into account the spatial site of the indicators. METHODS: An ecological study was conducted. The dependent variable was the rate of homicides among the male population aged 15 to 49 years, residing in the districts of the State of Pernambuco from 1995 to 1998. The independent variables were an index of the living conditions, per capita family income, Theil inequality index, Gini index, average income of the head of the family, poverty index, rate of illiteracy, and demographic density. The following techniques were used in the analysis: a spatial autocorrelation test determined by the Moran index, multiple linear regression, a spatial regression model (CAR) and a generalized additive model for the detection of spatial trend (LOESS). RESULTS: The illiteracy and the poverty index explained 24.6% of the total variability of the homicide rates and there was an inverse relationship. Moran's I statistics indicated spatial autocorrelation between municipalities. The multiple linear regression model best fitted for the purposes of this study was the Conditional Auto Regressive (CAR) model. The latter confirmed the association between the poverty index, illiteracy and homicide rates. CONCLUSIONS: The inverse association observed between socio-economic indicators and homicides may be expressing a process that propitiates improvement in living conditions and that is linked predominantly to conditions that generate violence, such as drug traffic.     

Permutation tests for detecting and estimating mixtures in task performance within groups

We propose a two-sample permutation test incorporating mixture models as a general tool for detecting and quantifying effects on task performance. We illustrate the proposed method with examples where the dependent measures under investigation are recorded for normal controls and relatives of patients with schizophrenia on a delayed response, spatial and object working memory task. Our mixture modelling in relatives allows the component distributions to arise from different continuous parametric families. We also investigate the effects of the within-family correlation and the prior distribution of the mixing proportion on the test results. The power of the test depends on sample sizes, the mixing proportion, the difference in component means and the ratio of component variances.     

Parametric bootstrap and penalized quasi-likelihood inference in conditional autoregressive models

This paper discusses a variety of conditional autoregressive (CAR) models for mapping disease rates, beyond the usual first-order intrinsic CAR model. We illustrate the utility and scope of such models for handling different types of data structures. To encourage their routine use for map production at statistical and health agencies, a simple algorithm for fitting such models is presented. This is derived from penalized quasi-likelihood (PQL) inference which uses an analogue of best-linear unbiased estimation for the regional risk ratios and restricted maximum likelihood for the variance components. We offer the practitioner here the use of the parametric bootstrap for inference. It is more reliable than standard maximum likelihood asymptotics for inference purposes since relevant hypotheses for the mapping of rates lie on the boundary of the parameter space. We illustrate the parametric bootstrap test of the practically relevant and important simplifying hypothesis that there is no spatial autocorrelation. Although the parametric bootstrap requires computational effort, it is straightforward to implement and offers a wealth of information relating to the estimators and their properties. The proposed methodology is illustrated by analysing infant mortality in the province of British Columbia in Canada.     

Bayesian random effects selection in mixed accelerated failure time model for interval‐censored data

In many medical problems that collect multiple observations per subject, the time to an event is often of interest. Sometimes, the occurrence of the event can be recorded at regular intervals leading to interval‐censored data. It is further desirable to obtain the most parsimonious model in order to increase predictive power and to obtain ease of interpretation. Variable selection and often random effects selection in case of clustered data become crucial in such applications. We propose a Bayesian method for random effects selection in mixed effects accelerated failure time (AFT) models. The proposed method relies on the Cholesky decomposition on the random effects covariance matrix and the parameter‐expansion method for the selection of random effects. The Dirichlet prior is used to model the uncertainty in the random effects. The error distribution for the accelerated failure time model has been specified using a Gaussian mixture to allow flexible error density and prediction of the survival and hazard functions. We demonstrate the model using extensive simulations and the Signal Tandmobiel Study^®. Copyright © 2013 John Wiley & Sons, Ltd.     

Parameter clustering in Bayesian functional principal component analysis of neuroscientific data

The extraordinary advancements in neuroscientific technology for brain recordings over the last decades have led to increasingly complex spatiotemporal data sets. To reduce oversimplifications, new models have been developed to be able to identify meaningful patterns and new insights within a highly demanding data environment. To this extent, we propose a new model called parameter clustering functional principal component analysis (PCl‐fPCA) that merges ideas from functional data analysis and Bayesian nonparametrics to obtain a flexible and computationally feasible signal reconstruction and exploration of spatiotemporal neuroscientific data. In particular, we use a Dirichlet process Gaussian mixture model to cluster functional principal component scores within the standard Bayesian functional PCA framework. This approach captures the spatial dependence structure among smoothed time series (curves) and its interaction with the time domain without imposing a prior spatial structure on the data. Moreover, by moving the mixture from data to functional principal component scores, we obtain a more general clustering procedure, thus allowing a higher level of intricate insight and understanding of the data. We present results from a simulation study showing improvements in curve and correlation reconstruction compared with different Bayesian and frequentist fPCA models and we apply our method to functional magnetic resonance imaging and electroencephalogram data analyses providing a rich exploration of the spatiotemporal dependence in brain time series.     

Measuring spatial effects in time to event data: a case study using months from angiography to coronary artery bypass graft (CABG)

The application of Bayesian hierarchical models to measure spatial effects in time to event data has not been widely reported. This case study aims to estimate the effect of area of residence on waiting times to coronary artery bypass graft (CABG) and to assess the role of important individual specific covariates (age, sex and disease severity). The data involved all patients with definite coronary artery disease who were referred to one cardiothoracic unit from five contiguous health authorities covering 488 electoral wards (areas). Time to event was the waiting time in months from angiography (diagnosis) to CABG (event). A number of discrete time survival models were fitted to the data. A discrete baseline hazard was estimated by fitting waiting time non-parametrically into the models. Ward was fitted as a spatial effect using a Gaussian Markov random field prior. Individual specific covariates considered were age, sex and number of diseased vessels. The recently proposed DIC criteria was used for comparing models. Results showed a marked spatial effect on time to bypass surgery after including age, sex and disease severity in the model. Notably this spatial effect was not apparent when these covariates were not included in the model. The observed small area spatial variation in time to CABG warrants further investigation.