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In areal unit data with missing or suppressed data, it desirable to create models that are able to predict observations that are not available. Traditional statistical methods achieve this through Bayesian hierarchical models that can capture the unexplained residual spatial autocorrelation through conditional autoregressive (CAR) priors, such that they can make predictions at geographically related spatial locations. In contrast, typical machine learning approaches such as random forests ignore this residual autocorrelation, and instead base predictions on complex non-linear feature-target relationships. In this paper, we propose CAR-Forest, a novel spatial prediction algorithm that combines the best features of both approaches by fusing them together. By iteratively refitting a random forest combined with a Bayesian CAR model in one algorithm, CAR-Forest can incorporate flexible feature-target relationships while still accounting for the residual spatial autocorrelation. Our results, based on a Scottish housing price data set, show that CAR-Forest outperforms Bayesian CAR models, random forests, and the state-of-the-art hybrid approach, geographically weighted random forest, providing a state-of-the-art framework for small-area spatial prediction.

Spatial correlation in areal unit count data is typically modelled by a set of random effects that are assigned a conditional autoregressive (CAR) prior distribution. The spatial correlation structure implied by this model depends on a binary neighbourhood matrix, where two random effects are assumed to be partially autocorrelated if their areal units share a common border, and are conditionally independent otherwise. This paper proposes a novel graph-based optimisation algorithm for estimating the neighbourhood matrix from the data, by viewing the areal units as the vertices of the graph and the neighbour relations as the set of edges. The superiority of our methodology compared to using the border sharing rule is comprehensively evidenced by simulation, before the method is applied to a new respiratory disease surveillance study in the Greater Glasgow and Clyde Health board in Scotland between 2011 and 2017.

Understanding spatial correlation is vital in many fields including epidemiology and social science. Lee, Meeks and Pettersson (Stat. Comput. 2021) recently demonstrated that improved inference for areal unit count data can be achieved by carrying out modifications to a graph representing spatial correlations; specifically, they delete edges of the planar graph derived from border-sharing between geographic regions in order to maximise a specific objective function. In this paper we address the computational complexity of the associated graph optimisation problem. We demonstrate that this problem cannot be solved in polynomial time unless P = NP; we further show intractability for two simpler variants of the problem. We follow these results with two parameterised algorithms that exactly solve the problem. Both of these solve not only the decision problem, but also enumerate all solutions with polynomial time precalculation, delay, and postcalculation time in respective restricted settings. For this problem, efficient enumeration allows the uncertainty in the spatial correlation to be utilised in the modelling. The first enumeration algorithm utilises dynamic programming on a tree decomposition, and has polynomial time precalculation and linear delay if both the treewidth and maximum degree are bounded. The second algorithm is restricted to problem instances with maximum degree three, as may arise from triangulations of planar surfaces, but can output all solutions with FPT precalculation time and linear delay when the maximum number of edges that can be removed is taken as the parameter., Comment: Added a number of enumeration results

The rates of respiratory prescriptions vary by GP surgery across Scotland, suggesting there are sizeable health inequalities in respiratory ill health across the country. The aim of this paper is to estimate the magnitude, spatial pattern and drivers of this spatial variation. Monthly data on respiratory prescriptions are available at the GP surgery level, which creates an interesting methodological challenge as these data are not the classical geostatistical, areal unit or point process data types. A novel process-convolution model is proposed, which extends existing methods by being an adaptive smoother via a random weighting scheme and using a tapering function to reduce the computational burden. The results show that particulate air pollution, poverty and ethnicity all drive the health inequalities, while there are additional regional inequalities in rates after covariate adjustment.

The health impact of long-term exposure to air pollution is now routinely estimated using spatial ecological studies, due to the recent widespread availability of spatial referenced pollution and disease data. However, this areal unit study design presents a number of statistical challenges, which if ignored have the potential to bias the estimated pollution-health relationship. One such challenge is how to control for the spatial autocorrelation present in the data after accounting for the known covariates, which is caused by unmeasured confounding. A second challenge is how to adjust the functional form of the model to account for the spatial misalignment between the pollution and disease data, which causes within-area variation in the pollution data. These challenges have largely been ignored in existing long-term spatial air pollution and health studies, so here we propose a novel Bayesian hierarchical model that addresses both challenges, and provide software to allow others to apply our model to their own data. The effectiveness of the proposed model is compared by simulation against a number of state of the art alternatives proposed in the literature, and is then used to estimate the impact of nitrogen dioxide and particulate matter concentrations on respiratory hospital admissions in a new epidemiological study in England in 2010 at the Local Authority level.

Statistical models used to estimate the spatio-temporal pattern in disease risk from areal unit data represent the risk surface for each time period with known covariates and a set of spatially smooth random effects. The latter act as a proxy for unmeasured spatial confounding, whose spatial structure is often characterised by a spatially smooth evolution between some pairs of adjacent areal units while other pairs exhibit large step changes. This spatial heterogeneity is not consistent with existing global smoothing models, in which partial correlation exists between all pairs of adjacent spatial random effects. Therefore we propose a novel space-time disease model with an adaptive spatial smoothing specification that can identify step changes. The model is motivated by a new study of respiratory and circulatory disease risk across the set of Local Authorities in England, and is rigorously tested by simulation to assess its efficacy. Results from the England study show that the two diseases have similar spatial patterns in risk, and exhibit a number of common step changes in the unmeasured component of risk between neighbouring local authorities.

In epidemiological disease mapping one aims to estimate the spatio-temporal pattern in disease risk and identify high-risk clusters, allowing health interventions to be appropriately targeted. Bayesian spatio-temporal models are used to estimate smoothed risk surfaces, but this is contrary to the aim of identifying groups of areal units that exhibit elevated risks compared with their neighbours. Therefore, in this paper we propose a new Bayesian hierarchical modelling approach for simultaneously estimating disease risk and identifying high-risk clusters in space and time. Inference for this model is based on Markov chain Monte Carlo simulation, using the freely available R package CARBayesST that has been developed in conjunction with this paper. Our methodology is motivated by two case studies, the first of which assesses if there is a relationship between Public health Districts and colon cancer clusters in Georgia, while the second looks at the impact of the smoking ban in public places in England on cardiovascular disease clusters.

Disease mapping is the field of spatial epidemiology interested in estimating the spatial pattern in disease risk across $n$ areal units. One aim is to identify units exhibiting elevated disease risks, so that public health interventions can be made. Bayesian hierarchical models with a spatially smooth conditional autoregressive prior are used for this purpose, but they cannot identify the spatial extent of high-risk clusters. Therefore we propose a two stage solution to this problem, with the first stage being a spatially adjusted hierarchical agglomerative clustering algorithm. This algorithm is applied to data prior to the study period, and produces $n$ potential cluster structures for the disease data. The second stage fits a separate Poisson log-linear model to the study data for each cluster structure, which allows for step-changes in risk where two clusters meet. The most appropriate cluster structure is chosen by model comparison techniques, specifically by minimising the Deviance Information Criterion. The efficacy of the methodology is established by a simulation study, and is illustrated by a study of respiratory disease risk in Glasgow, Scotland.

Estimation of the long-term health effects of air pollution is a challenging task, especially when modelling small-area disease incidence data in an ecological study design. The challenge comes from the unobserved underlying spatial correlation structure in these data, which is accounted for using random effects modelled by a globally smooth conditional autoregressive model. These smooth random effects confound the effects of air pollution, which are also globally smooth. To avoid this collinearity a Bayesian localised conditional autoregressive model is developed for the random effects. This localised model is flexible spatially, in the sense that it is not only able to model step changes in the random effects surface, but also is able to capture areas of spatial smoothness in the study region. This methodological development allows us to improve the estimation performance of the covariate effects, compared to using traditional conditional auto-regressive models. These results are established using a simulation study, and are then illustrated with our motivating study on air pollution and respiratory ill health in Greater Glasgow, Scotland in 2010. The model shows substantial health effects of particulate matter air pollution and income deprivation, whose effects have been consistently attenuated by the currently available globally smooth models.

Disease maps display the spatial pattern in disease risk, so that high-risk clusters can be identified. The spatial structure in the risk map is typically represented by a set of random effects, which are modelled with a conditional autoregressive (CAR) prior. Such priors include a global spatial smoothing parameter, whereas real risk surfaces are likely to include areas of smooth evolution as well as discontinuities, the latter of which are known as risk boundaries. Therefore, this paper proposes an extension to the class of CAR priors, which can identify both areas of localised spatial smoothness and risk boundaries. However, allowing for this localised smoothing requires large numbers of correlation parameters to be estimated, which are unlikely to be well identified from the data. To address this problem we propose eliciting an informative prior about the locations of such boundaries, which can be combined with the information from the data to provide more precise posterior inference. We test our approach by simulation, before applying it to a study of the risk of emergency admission to hospital in Greater Glasgow, Scotland.

Conditional autoregressive (CAR) models are commonly used to capture spatial correlation in areal unit data, and are typically specified as a prior distribution for a set of random effects, as part of a hierarchical Bayesian model. The spatial correlation structure induced by these models is determined by geographical adjacency, so that two areas have correlated random effects if they share a common border. However, this correlation structure is too simplistic for real data, which are instead likely to include sub-regions of strong correlation as well as locations at which the response exhibits a step-change. Therefore this paper proposes an extension to CAR priors, which can capture such localised spatial correlation. The proposed approach takes the form of an iterative algorithm, which sequentially updates the spatial correlation structure in the data as well as estimating the remaining model parameters. The efficacy of the approach is assessed by simulation, and its utility is illustrated in a disease mapping context, using data on respiratory disease risk in Greater Glasgow, Scotland.

In disease mapping, the aim is to estimate the spatial pattern in disease risk over an extended geographical region, so that areas with elevated risks can be identified. A Bayesian hierarchical approach is typically used to produce such maps, which models the risk surface with a set of spatially smooth random effects. However, in complex urban settings there are likely to be boundaries in the risk surface, which separate populations that are geographically adjacent but have very different risk profiles. Therefore this paper proposes an approach for detecting such risk boundaries, and tests its effectiveness by simulation. Finally, the model is applied to lung cancer incidence data in Greater Glasgow, Scotland, between 2001 and 2005.

This paper presents an approach to estimating the health effects of an environmental hazard. The approach is general in nature, but is applied here to the case of air pollution. It uses a computer model involving ambient pollution and temperature inputs, to simulate the exposures experienced by individuals in an urban area, whilst incorporating the mechanisms that determine exposures. The output from the model comprises a set of daily exposures for a sample of individuals from the population of interest. These daily exposures are approximated by parametric distributions, so that the predictive exposure distribution of a randomly selected individual can be generated. These distributions are then incorporated into a hierarchical Bayesian framework (with inference using Markov Chain Monte Carlo simulation) in order to examine the relationship between short-term changes in exposures and health outcomes, whilst making allowance for long-term trends, seasonality, the effect of potential confounders and the possibility of ecological bias. The paper applies this approach to particulate pollution (PM$_{10}$) and respiratory mortality counts for seniors in greater London ($\geq$65 years) during 1997. Within this substantive epidemiological study, the effects on health of ambient concentrations and (estimated) personal exposures are compared., Comment: 21 pages, 4 figures

The relationship between short-term exposure to air pollution and mortality or morbidity has been the subject of much recent research, in which the standard method of analysis uses Poisson linear or additive models. In this paper we use a Bayesian dynamic generalised linear model (DGLM) to estimate this relationship, which allows the standard linear or additive model to be extended in two ways: (i) the long-term trend and temporal correlation present in the health data can be modelled by an autoregressive process rather than a smooth function of calendar time; (ii) the effects of air pollution are allowed to evolve over time. The efficacy of these two extensions are investigated by applying a series of dynamic and non-dynamic models to air pollution and mortality data from Greater London. A Bayesian approach is taken throughout, and a Markov chain monte carlo simulation algorithm is presented for inference. An alternative likelihood based analysis is also presented, in order to allow a direct comparison with the only previous analysis of air pollution and health data using a DGLM., Comment: Accepted for publication in Environmetrics, October 2nd 2007