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.

Modeling total predation to avoid perverse outcomes from cat control in a data-poor island ecosystem.

Data hungry, complex ecosystem models are often used to predict the consequences of threatened species management, including perverse outcomes. Unfortunately, this approach is impractical in many systems, which have insufficient data to parameterize ecosystem interactions or reliably calibrate or validate such models. Here we demonstrate a different approach, using a minimum realistic model to guide decisions in data- and resource-scarce systems. We illustrate our approach with a case-study in an invaded ecosystem from Christmas Island, Australia, where there are concerns that cat eradication to protect native species, including the red-tailed tropicbird, could release meso-predation by invasive rats. We use biophysical constraints (metabolic demand) and observable parameters (e.g. prey preferences) to assess the combined cat and rat abundances which would threaten the tropicbird population. We find that the population of tropicbirds cannot be sustained if predated by 1607 rats (95% credible interval (CI) [103, 5910]) in the absence of cats, or 21 cats (95% CI [2, 82]) in the absence of rats. For every cat removed from the island, the bird's net population growth rate improves, provided that the rats do not increase by more than 77 individuals (95% CI [30, 174]). Thus, in this context, one cat is equivalent to 30-174 rats. Our methods are especially useful for on-the-ground predator control in the absence of knowledge of predator-predator interactions, to assess whether 1) the current abundance of predators threatens the prey population of interest, 2) managing one predator species alone is sufficient to protect the prey species given potential release of another predator, and 3) control of multiple predator species is needed to meet the conservation goal. Our approach demonstrates how to use limited information for maximum value in data-poor systems, by shifting the focus from predicting future trajectories, to identifying conditions which threaten the conservation goal. This article is protected by copyright. All rights reserved.

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.

Informing management decisions for ecological networks, using dynamic models calibrated to noisy time-series data.

Well-intentioned environmental management can backfire, causing unforeseen damage. To avoid this, managers and ecologists seek accurate predictions of the ecosystem-wide impacts of interventions, given small and imprecise datasets, which is an incredibly difficult task. We generated and analysed thousands of ecosystem population time series to investigate whether fitted models can aid decision-makers to select interventions. Using these time-series data (sparse and noisy datasets drawn from deterministic Lotka-Volterra systems with two to nine species, of known network structure), dynamic model forecasts of whether a species' future population will be positively or negatively affected by rapid eradication of another species were correct > 70% of the time. Although 70% correct classifications is only slightly better than an uninformative prediction (50%), this classification accuracy can be feasibly improved by increasing monitoring accuracy and frequency. Our findings suggest that models may not need to produce well-constrained predictions before they can inform decisions that improve environmental outcomes.

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.

Evaluating health facility access using Bayesian spatial models and location analysis methods.

BACKGROUND: Floating catchment methods have recently been applied to identify priority regions for Automated External Defibrillator (AED) deployment, to aid in improving Out of Hospital Cardiac Arrest (OHCA) survival. This approach models access as a supply-to-demand ratio for each area, targeting areas with high demand and low supply for AED placement. These methods incorporate spatial covariates on OHCA occurrence, but do not provide precise AED locations, which are critical to the initial intent of such location analysis research. Exact AED locations can be determined using optimisation methods, but they do not incorporate known spatial risk factors for OHCA, such as income and demographics. Combining these two approaches would evaluate AED placement impact, describe drivers of OHCA occurrence, and identify areas that may not be appropriately covered by AED placement strategies. There are two aims in this paper. First, to develop geospatial models of OHCA that account for and display uncertainty. Second, to evaluate the AED placement methods using geospatial models of accessibility. We first identify communities with the greatest gap between demand and supply for allocating AEDs. We then use this information to evaluate models for precise AED location deployment. METHODS: Case study data set consisted of 2802 OHCA events and 719 AEDs. Spatial OHCA occurrence was described using a geospatial model, with possible spatial correlation accommodated by introducing a conditional autoregressive (CAR) prior on the municipality-level spatial random effect. This model was fit with Integrated Nested Laplacian Approximation (INLA), using covariates for population density, proportion male, proportion over 65 years, financial strength, and the proportion of land used for transport, commercial, buildings, recreation, and urban areas. Optimisation methods for AED locations were applied to find the top 100 AED placement locations. AED access was calculated for current access and 100 AED placements. Priority rankings were then given for each area based on their access score and predicted number of OHCA events. RESULTS: Of the 2802 OHCA events, 64.28% occurred in rural areas, and 35.72% in urban areas. Additionally, over 70% of individuals were aged over 65. Supply of AEDs was less than demand in most areas. Priority regions for AED placement were identified, and access scores were evaluated for AED placement methodology by ranking the access scores and the predicted OHCA count. AED placement methodology placed AEDs in areas with the highest priority, but placed more AEDs in areas with more predicted OHCA events in each grid cell. CONCLUSION: The methods in this paper incorporate OHCA spatial risk factors and OHCA coverage to identify spatial regions most in need of resources. These methods can be used to help understand how AED allocation methods affect OHCA accessibility, which is of significant practical value for communities when deciding AED placements.

Using the Value of Information to improve conservation decision making.

Conservation decisions are challenging, not only because they often involve difficult conflicts among outcomes that people value, but because our understanding of the natural world and our effects on it is fraught with uncertainty. Value of Information (VoI) methods provide an approach for understanding and managing uncertainty from the standpoint of the decision maker. These methods are commonly used in other fields (e.g. economics, public health) and are increasingly used in biodiversity conservation. This decision-analytical approach can identify the best management alternative to select where the effectiveness of interventions is uncertain, and can help to decide when to act and when to delay action until after further research. We review the use of VoI in the environmental domain, reflect on the need for greater uptake of VoI, particularly for strategic conservation planning, and suggest promising areas for new research. We also suggest common reporting standards as a means of increasing the leverage of this powerful tool. The environmental science, ecology and biodiversity categories of the Web of Knowledge were searched using the terms 'Value of Information,' 'Expected Value of Perfect Information,' and the abbreviation 'EVPI.' Google Scholar was searched with the same terms, and additionally the terms decision and biology, biodiversity conservation, fish, or ecology. We identified 1225 papers from these searches. Included studies were limited to those that showed an application of VoI in biodiversity conservation rather than simply describing the method. All examples of use of VOI were summarised regarding the application of VoI, the management objectives, the uncertainties, the models used, how the objectives were measured, and the type of VoI. While the use of VoI appears to be on the increase in biodiversity conservation, the reporting of results is highly variable, which can make it difficult to understand the decision context and which uncertainties were considered. Moreover, it was unclear if, and how, the papers informed management and policy interventions, which is why we suggest a range of reporting standards that would aid the use of VoI. The use of VoI in conservation settings is at an early stage. There are opportunities for broader applications, not only for species-focussed management problems, but also for setting local or global research priorities for biodiversity conservation, making funding decisions, or designing or improving protected area networks and management. The long-term benefits of applying VoI methods to biodiversity conservation include a more structured and decision-focused allocation of resources to research.

Spatial variation in cancer incidence and survival over time across Queensland, Australia.

Interpreting changes over time in small-area variation in cancer survival, in light of changes in cancer incidence, aids understanding progress in cancer control, yet few space-time analyses have considered both measures. Bayesian space-time hierarchical models were applied to Queensland Cancer Registry data to examine geographical changes in cancer incidence and relative survival over time for the five most common cancers (colorectal, melanoma, lung, breast, prostate) diagnosed during 1997-2004 and 2005-2012 across 516 Queensland residential small-areas. Large variation in both cancer incidence and survival was observed. Survival improvements were fairly consistent across the state, although small for lung cancer. Incidence changes varied by location and cancer type, ranging from lung and colorectal cancers remaining relatively constant over time, to prostate cancer dramatically increasing across the entire state. Reducing disparities in cancer-related outcomes remains a health priority, and space-time modelling of different measures provides an important mechanism by which to monitor progress.

Concentrations of organochlorine pesticides in pooled human serum by age and gender.

Organochlorine pesticides (OCPs) have been used for many decades in Australia with cessation of selected persistent and bioaccumulative OCPs ranging from the 1970s to as recently as 2007. The specific aims of this study were to use samples representative of an Australian population to assess age and gender differences in the concentration of OCPs in human blood sera and to investigate temporal trends in these chemicals. Serum was collected from de-identified, surplus pathology samples over five time periods (2002/03, 2006/07, 2008/09, 2010/11 and 2012/13), with 183 serum pools made from 12,175 individual samples; 26 pools in 2002/03, 85 pools in 2006/07 and 24 pools each in 2008/09, 2010/11 and 2012/13. Samples were analyzed for hexachlorobenzene (HCB), ß-hexachlorocyclohexane (ß-HCH), γ -hexachlorocyclohexane (lindane) (γ-HCH), oxy-chlordane, trans-nonachlor, p,p'-DDE, o,p'-DDT, p,p'-DDT and Mirex. Stratification criteria included gender and age (0-4; 5-15; 16-30; 31-45; 46-60; and >60 years) with age additionally stratified by adults >16 years and children 0-4 and 5-15 years. All pools from all collection periods had detectable concentrations of OCPs with a detection frequency of >60% for HCB, ß-HCH, trans-nonachlor, p,p'-DDT and p,p'-DDE. The overall OCP concentrations increased with age with the highest concentrations in the >60 years groups. Females did not have higher mean OCP concentrations than males except for HCB concentrations (p=0.0006). Temporal trends showed overall decreasing serum concentrations by collection period with the exception of an increase in OCP concentrations between 2006/07 and 2008/09. Excluding this data point, HCB decreased from year to year by 7-76%; ß-HCH concentrations decreased by 14 - 38%; trans-nonachlor concentrations decreased by 10 - 65%; p,p'-DDE concentrations decreased by 6 - 52%; and p,p'-DDT concentrations decreased by 7 - 30%. The results indicate that OCP concentrations have decreased over time as is to be expected following the phase out of these chemicals in Australia.

Spatio-temporal relative survival of breast and colorectal cancer in Queensland, Australia 2001-2011.

Despite improvements in cancer survival across many developed countries, it is unclear how survival is changing over time in small areas. This study investigated changes in breast and colorectal cancer survival across 478 areas over 11 years (2001-2011), and the influence of early diagnosis on changes. Queensland Cancer Registry data were analysed using an introduced Bayesian spatio-temporal flexible parametric relative survival model. All areas showed survival improvements between 2001-2003 and 2008-2011. The median absolute 5-year survival improvement for localised breast cancer was small (1.8%), compared to advanced (4.8%) and unknown (7.9%) breast cancer, as well as localised (2.6%), advanced (5.0%) and unknown (4.8%) colorectal cancers. Improvements in non-diagnostic factors, such as patient treatment and management, appear to be the main influence on recent survival increases for breast and colorectal cancers. Important inequalities in cancer survival between small areas remain.

Bayesian adaptive design: improving the effectiveness of monitoring of the Great Barrier Reef.

Monitoring programs are essential for understanding patterns, trends, and threats in ecological and environmental systems. However, such programs are costly in terms of dollars, human resources, and technology, and complex in terms of balancing short- and long-term requirements. In this work, We develop new statistical methods for implementing cost-effective adaptive sampling and monitoring schemes for coral reef that can better utilize existing information and resources, and which can incorporate available prior information. Our research was motivated by developing efficient monitoring practices for Australia's Great Barrier Reef. We develop and implement two types of adaptive sampling schemes, static and sequential, and show that they can be more informative and cost-effective than an existing (nonadaptive) monitoring program. Our methods are developed in a Bayesian framework with a range of utility functions relevant to environmental monitoring. Our results demonstrate the considerable potential for adaptive design to support improved management outcomes in comparison to set-and-forget styles of surveillance monitoring.

A flexible parametric approach to examining spatial variation in relative survival.

Most of the few published models used to obtain small-area estimates of relative survival are based on a generalized linear model with piecewise constant hazards under a Bayesian formulation. Limitations of these models include the need to artificially split the time scale, restricted ability to include continuous covariates, and limited predictive capacity. Here, an alternative Bayesian approach is proposed: a spatial flexible parametric relative survival model. This overcomes previous limitations by combining the benefits of flexible parametric models: the smooth, well-fitting baseline hazard functions and predictive ability, with the Bayesian benefits of robust and reliable small-area estimates. Both spatially structured and unstructured frailty components are included. Spatial smoothing is conducted using the intrinsic conditional autoregressive prior. The model was applied to breast, colorectal, and lung cancer data from the Queensland Cancer Registry across 478 geographical areas. Advantages of this approach include the ease of including more realistic complexity, the feasibility of using individual-level input data, and the capacity to conduct overall, cause-specific, and relative survival analysis within the same framework. Spatial flexible parametric survival models have great potential for exploring small-area survival inequalities, and we hope to stimulate further use of these models within wider contexts. Copyright © 2016 John Wiley & Sons, Ltd.

Making the most of spatial information in health: a tutorial in Bayesian disease mapping for areal data.

Disease maps are effective tools for explaining and predicting patterns of disease outcomes across geographical space, identifying areas of potentially elevated risk, and formulating and validating aetiological hypotheses for a disease. Bayesian models have become a standard approach to disease mapping in recent decades. This article aims to provide a basic understanding of the key concepts involved in Bayesian disease mapping methods for areal data. It is anticipated that this will help in interpretation of published maps, and provide a useful starting point for anyone interested in running disease mapping methods for areal data. The article provides detailed motivation and descriptions on disease mapping methods by explaining the concepts, defining the technical terms, and illustrating the utility of disease mapping for epidemiological research by demonstrating various ways of visualising model outputs using a case study. The target audience includes spatial scientists in health and other fields, policy or decision makers, health geographers, spatial analysts, public health professionals, and epidemiologists.

Spatially Varying Coefficient Inequalities: Evaluating How the Impact of Patient Characteristics on Breast Cancer Survival Varies by Location.

An increasing number of studies have identified spatial differences in breast cancer survival. However little is known about whether the structure and dynamics of this spatial inequality are consistent across a region. This study aims to evaluate the spatially varying nature of predictors of spatial inequality in relative survival for women diagnosed with breast cancer across Queensland, Australia. All Queensland women aged less than 90 years diagnosed with invasive breast cancer from 1997 to 2007 and followed up to the end of 2008 were extracted from linked Queensland Cancer Registry and BreastScreen Queensland data. Bayesian relative survival models were fitted using various model structures (a spatial regression model, a varying coefficient model and a finite mixture of regressions model) to evaluate the relative excess risk of breast cancer, with the use of Markov chain Monte Carlo computation. The spatially varying coefficient models revealed that some covariate effects may not be constant across the geographic regions of the study. The overall spatial patterns showed lower survival among women living in more remote areas, and higher survival among the urbanised south-east corner. Notwithstanding this, the spatial survival pattern for younger women contrasted with that for older women as well as single women. This complex spatial interplay may be indicative of different factors impacting on survival patterns for these women.

Bayesian Estimation of Small Effects in Exercise and Sports Science.

The aim of this paper is to provide a Bayesian formulation of the so-called magnitude-based inference approach to quantifying and interpreting effects, and in a case study example provide accurate probabilistic statements that correspond to the intended magnitude-based inferences. The model is described in the context of a published small-scale athlete study which employed a magnitude-based inference approach to compare the effect of two altitude training regimens (live high-train low (LHTL), and intermittent hypoxic exposure (IHE)) on running performance and blood measurements of elite triathletes. The posterior distributions, and corresponding point and interval estimates, for the parameters and associated effects and comparisons of interest, were estimated using Markov chain Monte Carlo simulations. The Bayesian analysis was shown to provide more direct probabilistic comparisons of treatments and able to identify small effects of interest. The approach avoided asymptotic assumptions and overcame issues such as multiple testing. Bayesian analysis of unscaled effects showed a probability of 0.96 that LHTL yields a substantially greater increase in hemoglobin mass than IHE, a 0.93 probability of a substantially greater improvement in running economy and a greater than 0.96 probability that both IHE and LHTL yield a substantially greater improvement in maximum blood lactate concentration compared to a Placebo. The conclusions are consistent with those obtained using a 'magnitude-based inference' approach that has been promoted in the field. The paper demonstrates that a fully Bayesian analysis is a simple and effective way of analysing small effects, providing a rich set of results that are straightforward to interpret in terms of probabilistic statements.

Geographic variation in the intended choice of adjuvant treatments for women diagnosed with screen-detected breast cancer in Queensland.

BACKGROUND: Although early diagnosis and improved treatment can reduce breast cancer mortality, there still appears to be a geographic differential in patient outcomes. This study aims to determine and quantify spatial inequalities in intended adjuvant (radio-, chemo- and hormonal) therapy usage among women with screen-detected breast cancer in Queensland, Australia. METHODS: Linked population-based datasets from BreastScreen Queensland and the Queensland Cancer Registry during 1997-2008 for women aged 40-89 years were used. We adopted a Bayesian shared spatial component model to evaluate the relative intended use of each adjuvant therapy across 478 areas as well as common spatial patterns between treatments. RESULTS: Women living closer to a cancer treatment facility were more likely to intend to use adjuvant therapy. This was particularly marked for radiotherapy when travel time to the closest radiation facility was 4 + h (OR =0.41, 95 % CrI: [0.23, 0.74]) compared to <1 h. The shared spatial effect increased towards the centres with concentrations of radiotherapy facilities, in north-east (Townsville) and south-east (Brisbane) regions of Queensland. Moreover, the presence of residual shared spatial effects indicates that there are other unmeasured geographical barriers influencing women's treatment choices. CONCLUSIONS: This highlights the need to identify the additional barriers that impact on treatment intentions among women diagnosed with screen-detected breast cancer, particularly for those women living further away from cancer treatment centers.

A Four Dimensional Spatio-Temporal Analysis of an Agricultural Dataset.

While a variety of statistical models now exist for the spatio-temporal analysis of two-dimensional (surface) data collected over time, there are few published examples of analogous models for the spatial analysis of data taken over four dimensions: latitude, longitude, height or depth, and time. When taking account of the autocorrelation of data within and between dimensions, the notion of closeness often differs for each of the dimensions. Here, we consider a number of approaches to the analysis of such a dataset, which arises from an agricultural experiment exploring the impact of different cropping systems on soil moisture. The proposed models vary in their representation of the spatial correlation in the data, the assumed temporal pattern and choice of conditional autoregressive (CAR) and other priors. In terms of the substantive question, we find that response cropping is generally more effective than long fallow cropping in reducing soil moisture at the depths considered (100 cm to 220 cm). Thus, if we wish to reduce the possibility of deep drainage and increased groundwater salinity, the recommended cropping system is response cropping.