Summary. We consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined. Using an information theoretic argument we derive a measure pD for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest. In general pD approximately corresponds to the trace of the product of Fisher's information and the posterior covariance, which in normal models is the trace of the ‘hat’ matrix projecting observations onto fitted values. Its properties in exponential families are explored. The posterior mean deviance is suggested as a Bayesian measure of fit or adequacy, and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages. Adding pD to the posterior mean deviance gives a deviance information criterion for comparing models, which is related to other information criteria and has an approximate decision theoretic justification. The procedure is illustrated in some examples, and comparisons are drawn with alternative Bayesian and classical proposals. Throughout it is emphasized that the quantities required are trivial to compute in a Markov chain Monte Carlo analysis.

/pdf/bayesian-measures-of-model-complexity-and-fit-1ouzmtfu7x.pdf

Bayesian measures of model complexity and fit

The capabilities for visualization, rapid data retrieval, and manipulation in geographic information systems (GIS) have created the need for new techniques of exploratory data analysis that focus on the “spatial” aspects of the data. The identification of local patterns of spatial association is an important concern in this respect. In this paper, I outline a new general class of local indicators of spatial association (LISA) and show how they allow for the decomposition of global indicators, such as Moran's I, into the contribution of each observation. The LISA statistics serve two purposes. On one hand, they may be interpreted as indicators of local pockets of nonstationarity, or hot spots, similar to the Gi and G*i statistics of Getis and Ord (1992). On the other hand, they may be used to assess the influence of individual locations on the magnitude of the global statistic and to identify “outliers,” as in Anselin's Moran scatterplot (1993a). An initial evaluation of the properties of a LISA statistic is carried out for the local Moran, which is applied in a study of the spatial pattern of conflict for African countries and in a number of Monte Carlo simulations.

https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1538-4632.1995.tb00338.x

Local Indicators of Spatial Association—LISA

Statistics for Spatial Data.

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Modern Applied Statistics With S

Structured additive regression models are perhaps the most commonly used class of models in statistical applications. It includes, among others, (generalized) linear models, (generalized) additive models, smoothing spline models, state space models, semiparametric regression, spatial and spatiotemporal models, log-Gaussian Cox processes and geostatistical and geoadditive models. We consider approximate Bayesian inference in a popular subset of structured additive regression models, latent Gaussian models, where the latent field is Gaussian, controlled by a few hyperparameters and with non-Gaussian response variables. The posterior marginals are not available in closed form owing to the non-Gaussian response variables. For such models, Markov chain Monte Carlo methods can be implemented, but they are not without problems, in terms of both convergence and computational time. In some practical applications, the extent of these problems is such that Markov chain Monte Carlo sampling is simply not an appropriate tool for routine analysis. We show that, by using an integrated nested Laplace approximation and its simplified version, we can directly compute very accurate approximations to the posterior marginals. The main benefit of these approximations is computational: where Markov chain Monte Carlo algorithms need hours or days to run, our approximations provide more precise estimates in seconds or minutes. Another advantage with our approach is its generality, which makes it possible to perform Bayesian analysis in an automatic, streamlined way, and to compute model comparison criteria and various predictive measures so that models can be compared and the model under study can be challenged.

/pdf/approximate-bayesian-inference-for-latent-gaussian-models-by-2m3x22eic2.pdf

Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations

Preface and Acknowledgements to Second Edition Preface and Acknowledgements I: The Nature of Spatial Epidemiology 1 Definitions, Terminolgy and Data Sets 11 Map Hypotheses and Modelling Approaches 12 Definitions and Data Examples 13 Further definitions 14 Some Data Examples 2Scales of Measurement and Data Availability 21 Small Scale 22 Large Scale 23 Rate Dependence 24 DataQuality and the Ecological Fallacy 25 Edge Eects 3Geographical Representation and Mapping 31 Introduction and Definitions 32 Maps and Mapping 33 Statistical Accuracy 34 Aggregation 35 Mapping Issues related toAggregated Data 36 Conclusions 4Basic Models 41 Sampling Considerations 42 Likelihood-based and Bayesian Approaches 43 Point EventModels 44 CountModels 5Exploratory Approaches, Parametric Estimation and Inference 51 ExploratoryMethods 52 Parameter Estimation 53 Residual Diagnostics 54 Hypothesis Testing 55 Edge Eects II:Important Problems in Spatial Epidemiology 6Small Scale: Disease Clustering 61 Definition of Clusters and Clustering 62 Modelling Issues 63 Hypothesis Tests for Clustering 64 Space-Time Clustering 65 Clustering Examples 66 OtherMethods related to clustering 7Small Scale: Putative Sources of Hazard 71 Introduction 72 StudyDesign 73 Problems of Inference 74 Modelling the Hazard Exposure Risk 75 Models for Case Event Data 76 ACase Event Example 77 Models for CountData 78 ACountData Example 79 OtherDirections 8 Large Scale: Disease Mapping 81 Introduction 82 Simple Statistical Representation 83 BasicModels 84 AdvancedMethods 85 Model Variants and Extensions 86 ApproximateMethods 87 MultivariateMethods 88 Evaluation ofModel Performance 89 Hypothesis Testing in DiseaseMapping 810 Space-Time DiseaseMapping 811 Spatial Survival and longitudinal data 812 DiseaseMapping: Case Studies 9Ecological Analysis and Scale Change 91 Ecological Analysis: Introduction 92 Small-ScaleModelling Issues 93 Changes of Scale andMAUP 94 A Simple Example: Sudden Infant Death in North Carolina 95 ACase Study: Malaria and IDDM 10Infectious Disease Modelling 101 Introduction 102 GeneralModelDevelopment 103 SpatialModelDevelopment 104 Modelling Special Cases for Individual Level Data 105 Survival Analysis with spatial dependence 106 Individual level data example 107 Underascertainment and Censoring 108 Conclusions 11Large Scale: Surveillance 111 Process ControlMethodology 112 Spatio-Temporal Modelling 113 Spatio-TemporalMonitoring 114 Syndromic Surveillance 115 Multivariate-Mulitfocus Surveillance 116 Bayesian Approaches 117 Computational Considerations 118 Infectious Diseases 119 Conclusions Appendix A:Monte Carlo Testing, Parametric Bootstrap and Simulation Envelopes Appendix B:Markov ChainMonte Carlo Methods Appendix C:Algorithms and Software Appendix D: Glossary of Estimators Appendix E:Software Bibliography Index

Statistical methods in spatial epidemiology

BACKGROUND Introduction Data Sets Bayesian Inference and Modeling Likelihood Models Prior Distributions Posterior Distributions Predictive Distributions Bayesian Hierarchical Modeling Hierarchical Models Posterior Inference Exercises Computational Issues Posterior Sampling Markov Chain Monte Carlo Methods Metropolis and Metropolis-Hastings Algorithms Gibbs Sampling Perfect Sampling Posterior and Likelihood Approximations Exercises Residuals and Goodness-of-Fit Model Goodness-of-Fit Measures General Residuals Bayesian Residuals Predictive Residuals and the Bootstrap Interpretation of Residuals in a Bayesian Setting Exceedence Probabilities Exercises THEMES Disease Map Reconstruction and Relative Risk Estimation An Introduction to Case Event and Count Likelihoods Specification of the Predictor in Case Event and Count Models Simple Case and Count Data Models with Uncorrelated Random Effects Correlated Heterogeneity Models Convolution Models Model Comparison and Goodness-of-Fit Diagnostics Alternative Risk Models Edge Effects Exercises Disease Cluster Detection Cluster Definitions Cluster Detection using Residuals Cluster Detection using Posterior Measures Cluster Models Edge Detection and Wombling Ecological Analysis General Case of Regression Biases and Misclassification Error Putative Hazard Models Multiple Scale Analysis Modifiable Areal Unit Problem (MAUP) Misaligned Data Problem (MIDP) Multivariate Disease Analysis Notation for Multivariate Analysis Two Diseases Multiple Diseases Spatial Survival and Longitudinal Analyses General Issues Spatial Survival Analysis Spatial Longitudinal Analysis Extensions to Repeated Events Spatiotemporal Disease Mapping Case Event Data Count Data Alternative Models Infectious Diseases Appendix A: Basic R and WinBUGS Appendix B: Selected WinBUGS Code Appendix C: R Code for Thematic Mapping References Index

Bayesian Disease Mapping: Hierarchical Modeling in Spatial Epidemiology

Objective To characterize the built nutritional environment in terms of types and number of food stores, availability, and cost of selected food items in a rural area. Design A cross-sectional survey of food stores conducted in 2004. Subjects/setting We selected a rural county (population 91,582; 1,106 square miles). Food stores identified from a database were mapped and presence, location, and store type verified by ground-truthing. Stores were surveyed for availability and cost of selected foods. Main outcome measures Price and availability of a limited number of staple foods representing the main food groups. Statistical analyses performed Availability comparisons used least square means models and price comparisons used t tests. Results Of 77 stores identified, 16% were supermarkets, 10% grocery stores, and 74% convenience stores. There were seven stores per 100 square miles and eight stores per 10,000 residents. Availability of more healthful foods was substantially higher at supermarkets and grocery stores. For instance, low-fat/nonfat milk, apples, high-fiber bread, eggs, and smoked turkey were available in 75% to 100% of supermarkets and groceries and at 4% to 29% of convenience stores. Foods that were available at both supermarkets and convenience stores tended to be substantially more expensive at convenience stores. The healthful version of a food was typically more expensive than the less healthful version. Conclusions In this rural environment, stores offering more healthful and lower-cost food selections were outnumbered by convenience stores offering lower availability of more healthful foods. Our findings underscore the challenges of shopping for healthful and inexpensive foods in rural areas.

Food store types, availability, and cost of foods in a rural environment

Preface. Notation. 0.1 Standard notation for multilevel modelling. 0.2 Spatial multiple-membership models and the MMMC notation. 0.3 Standard notation for WinBUGS models. 1. Disease mapping basics. 1.1 Disease mapping and map reconstruction. 1.2 Disease map restoration. 2. Bayesian hierarchical modelling. 2.1 Likelihood and posterior distributions. 2.2 Hierarchical models. 2.3 Posterior inference. 2.4 Markov chain Monte Carlo methods. 2.5 Metropolis and Metropolis-Hastings algorithms. 2.6 Residuals and goodness of fit. 3. Multilevel modelling. 3.1 Continuous response models. 3.2 Estimation procedures for multilevel models. 3.3 Poisson response models. 3.4 Incorporating spatial information. 3.5 Discussion. 4. WinBUGS basics. 4.1 About WinBUGS. 4.2 Start using WinBUGS. 4.3 Specification of the model. 4.4 Model fitting. 4.5 Scripts. 4.6 Checking convergence. 4.7 Spatial modelling: GeoBUGS. 4 .8 Conclusions. 5. MLwiN basics. 5.1 About MLwiN. 5.2 Getting started. 5.3 Fitting statistical models. 5.4 MCMC estimation in MLwiN. 5.5 Spatial modelling. 5.6 Conclusions. 6. Relative risk estimation. 6.1 Relative risk estimation using WinBUGS. 6.2 Spatial prediction. 6.3 An analysis of the Ohio dataset using MLwiN. 7. Focused clustering: the analysis of putative health hazards. 7.1 Introduction. 7.2 Study design. 7.3 Problems of inference. 7.4 Modelling the hazard exposure risk. 7.5 Models for count data. 7.6 Bayesian models. 7.7 Focused clustering in WinBUGS. 7.8 Focused clustering in MLwiN. 8. Ecological analysis. 8.1 Introduction. 8.2 Statistical models. 8.3 WinBUGS analyses of ecological datasets. 8.4 MLwiN analyses of ecological datasets. 9. Spatially-correlated survival analysis. 9.1 Survival analysis in WinBUGS. 9.2 Survival analysis in MLwiN. 10. Epilogue. Appendix 1: WinBUGS code for focused clustering models. A.1: Falkirk example. A.2: Ohio example. Appendix 2: S-Plus function for conversion to GeoBUGS format. Bibliography. Index.

Andrew B. Lawson

Papers

Statistics for Spatial Data.

Statistical methods in spatial epidemiology

Bayesian Disease Mapping: Hierarchical Modeling in Spatial Epidemiology

Food store types, availability, and cost of foods in a rural environment

Disease Mapping with WinBUGS and MLwiN