Evaluation of spatial Bayesian Empirical Likelihood models in analysis of small area data.

Bayesian empirical likelihood (BEL) models are becoming increasingly popular as an attractive alternative to fully parametric models. However, they have only recently been applied to spatial data analysis for small area estimation. This study considers the development of spatial BEL models using two popular conditional autoregressive (CAR) priors, namely BYM and Leroux priors. The performance of the proposed models is compared with their parametric counterparts and with existing spatial BEL models using independent Gaussian priors and generalised Moran basis priors. The models are applied to two benchmark spatial datasets, simulation study and COVID-19 data. The results indicate promising opportunities for these models to capture new insights into spatial data. Specifically, the spatial BEL models outperform the parametric spatial models when the underlying distributional assumptions of data appear to be violated.

Correction to 'Augmenting disease maps: a Bayesian meta-analysis approach'.

[This corrects the article DOI: 10.1098/rsos.192151.][This corrects the article DOI: 10.1098/rsos.192151.].

Multivariate Bayesian meta-analysis: joint modelling of multiple cancer types using summary statistics.

BACKGROUND: Cancer atlases often provide estimates of cancer incidence, mortality or survival across small areas of a region or country. A recent example of a cancer atlas is the Australian cancer atlas (ACA), that provides interactive maps to visualise spatially smoothed estimates of cancer incidence and survival for 20 different cancer types over 2148 small areas across Australia. METHODS: The present study proposes a multivariate Bayesian meta-analysis model, which can model multiple cancers jointly using summary measures without requiring access to the unit record data. This new approach is illustrated by modelling the publicly available spatially smoothed standardised incidence ratios for multiple cancers in the ACA divided into three groups: common, rare/less common and smoking-related. The multivariate Bayesian meta-analysis models are fitted to each group in order to explore any possible association between the cancers in three remoteness regions: major cities, regional and remote areas across Australia. The correlation between the pairs of cancers included in each multivariate model for a group was examined by computing the posterior correlation matrix for each cancer group in each region. The posterior correlation matrices in different remoteness regions were compared using Jennrich's test of equality of correlation matrices (Jennrich in J Am Stat Assoc. 1970;65(330):904-12. https://doi.org/10.1080/01621459.1970.10481133 ). RESULTS: Substantive correlation was observed among some cancer types. There was evidence that the magnitude of this correlation varied according to remoteness of a region. For example, there has been significant negative correlation between prostate and lung cancer in major cities, but zero correlation found in regional and remote areas for the same pair of cancer types. High risk areas for specific combinations of cancer types were identified and visualised from the proposed model. CONCLUSIONS: Publicly available spatially smoothed disease estimates can be used to explore additional research questions by modelling multiple cancer types jointly. These proposed multivariate meta-analysis models could be useful when unit record data are unavailable because of privacy and confidentiality requirements.

Augmenting disease maps: a Bayesian meta-analysis approach.

Analysis of spatial patterns of disease is a significant field of research. However, access to unit-level disease data can be difficult for privacy and other reasons. As a consequence, estimates of interest are often published at the small area level as disease maps. This motivates the development of methods for analysis of these ecological estimates directly. Such analyses can widen the scope of research by drawing more insights from published disease maps or atlases. The present study proposes a hierarchical Bayesian meta-analysis model that analyses the point and interval estimates from an online atlas. The proposed model is illustrated by modelling the published cancer incidence estimates available as part of the online Australian Cancer Atlas (ACA). The proposed model aims to reveal patterns of cancer incidence for the 20 cancers included in ACA in major cities, regional and remote areas. The model results are validated using the observed areal data created from unit-level data on cancer incidence in each of 2148 small areas. It is found that the meta-analysis models can generate similar patterns of cancer incidence based on urban/rural status of small areas compared with those already known or revealed by the analysis of observed data. The proposed approach can be generalized to other online disease maps and atlases.

Comparing Bayesian spatial models: Goodness-of-smoothing criteria for assessing under- and over-smoothing.

BACKGROUND: Many methods of spatial smoothing have been developed, for both point data as well as areal data. In Bayesian spatial models, this is achieved by purposefully designed prior(s) or smoothing functions which smooth estimates towards a local or global mean. Smoothing is important for several reasons, not least of all because it increases predictive robustness and reduces uncertainty of the estimates. Despite the benefits of smoothing, this attribute is all but ignored when it comes to model selection. Traditional goodness-of-fit measures focus on model fit and model parsimony, but neglect "goodness-of-smoothing", and are therefore not necessarily good indicators of model performance. Comparing spatial models while taking into account the degree of spatial smoothing is not straightforward because smoothing and model fit can be viewed as opposing goals. Over- and under-smoothing of spatial data are genuine concerns, but have received very little attention in the literature. METHODS: This paper demonstrates the problem with spatial model selection based solely on goodness-of-fit by proposing several methods for quantifying the degree of smoothing. Several commonly used spatial models are fit to real data, and subsequently compared using the goodness-of-fit and goodness-of-smoothing statistics. RESULTS: The proposed goodness-of-smoothing statistics show substantial agreement in the task of model selection, and tend to avoid models that over- or under-smooth. Conversely, the traditional goodness-of-fit criteria often don't agree, and can lead to poor model choice. In particular, the well-known deviance information criterion tended to select under-smoothed models. CONCLUSIONS: Some of the goodness-of-smoothing methods may be improved with modifications and better guidelines for their interpretation. However, these proposed goodness-of-smoothing methods offer researchers a solution to spatial model selection which is easy to implement. Moreover, they highlight the danger in relying on goodness-of-fit measures when comparing spatial models.

Development of the Australian Cancer Atlas: spatial modelling, visualisation, and reporting of estimates.

BACKGROUND: It is well known that the burden caused by cancer can vary geographically, which may relate to differences in health, economics or lifestyle. However, to date, there was no comprehensive picture of how the cancer burden, measured by cancer incidence and survival, varied by small geographical area across Australia. METHODS: The Atlas consists of 2148 Statistical Areas level 2 across Australia defined by the Australian Statistical Geography Standard which provide the best compromise between small population and small area. Cancer burden was estimated for males, females, and persons separately, with 50 unique sex-specific (males, females, all persons) cancer types analysed. Incidence and relative survival were modelled with Bayesian spatial models using the Leroux prior which was carefully selected to provide adequate spatial smoothing while reflecting genuine geographic variation. Markov Chain Monte Carlo estimation was used because it facilitates quantifying the uncertainty of the posterior estimates numerically and visually. RESULTS: The results of the statistical model and visualisation development were published through the release of the Australian Cancer Atlas ( https://atlas.cancer.org.au ) in September, 2018. The Australian Cancer Atlas provides the first freely available, digital, interactive picture of cancer incidence and survival at the small geographical level across Australia with a focus on incorporating uncertainty, while also providing the tools necessary for accurate estimation and appropriate interpretation and decision making. CONCLUSIONS: The success of the Atlas will be measured by how widely it is used by key stakeholders to guide research and inform decision making. It is hoped that the Atlas and the methodology behind it motivates new research opportunities that lead to improvements in our understanding of the geographical patterns of cancer burden, possible causes or risk factors, and the reasons for differences in variation between cancer types, both within Australia and globally. Future versions of the Atlas are planned to include new data sources to include indicators such as cancer screening and treatment, and extensions to the statistical methods to incorporate changes in geographical patterns over time.

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.

Bayesian spatiotemporal modelling for identifying unusual and unstable trends in mammography utilisation.

OBJECTIVES: To compare two Bayesian models capable of identifying unusual and unstable temporal patterns in spatiotemporal data. SETTING: Annual counts of mammography screening users from each statistical local area (SLA) in Brisbane, Australia, recorded between 1997 and 2008 inclusive. PRIMARY OUTCOME MEASURES: Mammography screening counts. RESULTS: The temporal trends of 91 SLAs (58%) were dissimilar from the overall common temporal trend. SLAs that followed the common temporal trend also tended to have stable temporal trends. SLAs with unstable temporal trends tended to be situated farther from the city and farther from mammography screening facilities. CONCLUSIONS: This paper demonstrates the usefulness of the two models in identifying unusual and unstable temporal trends, and the synergy obtained when both models are applied to the same data set. An analysis of these models has provided interesting insights into the temporal trends of mammography screening counts and has shown several possible avenues for further research, such as extending the models to allow for multiple common temporal trends and accounting for additional spatiotemporal heterogeneity.