Surveilling Spatially Concentrated Disease Patterns: A Bayesian Approach to Understanding Tuberculosis in the United States Open Access

Abstract

Despite advances in medical science, tuberculosis(TB) continues to be a significant health concern across the world. In the United States, local prevalence of TB is often highest in refugee communities due both to higher background prevalence among politically unstable populations and due to a long latent period between infection and active disease. In a public health surveillance setting, this pattern can result in relatively sharp increases in observed local prevalences. By pinpointing areas of concentrated TB prevalence and examining the factors contributing to these patterns, this research aims to enhance the strategic planning and implementation of TB control measures.

Bayesian hierarchical disease mapping models offer a robust means to borrow information across small areas in order to better interpret complex spatial patterns of disease prevalence, enabling public health professionals to identify high-risk areas and tailor interventions more effectively. However, such methods often result in strong

spatial smoothing that might dampen the ability to accurately estimate local spikes in prevalence. Here, we conduct a detailed comparison of Bayesian Hierarchical Models, customizing

approaches to better capture TB’s spatial heterogeneity.

Aim 1 explores the performance of hierarchical Bayesian disease mapping models in the context of spatially concentrated TB prevalence. Through a motivating study centered on TB prevalence in Metro Atlanta, we demonstrate the utility of these models in identifying high-risk areas and enhancing our understanding of TB’s spatial dynamics but also raise new questions regarding model performance. Aim 2 shifts the focus to model adequacy, using localized approaches to assess the fit of disease surveillance models, particularly in capturing sharp spatial transitions. We evaluate specific epidemiological characteristics of different regions. Finally, Aim 3 focuses on the development of an enhanced Bayesian Spatially Varying Coefficient Model (BSVCM) integrated with Locally Adaptive Smoothing (LAS). This model was specifically designed to address the limitations of over-smoothing in traditional models while also capturing spatial outliers. The LAS-enhanced BSVCM demonstrated significant improvements in modeling accuracy, particularly in regions with high spatial variability, making it a valuable tool for more accurate disease risk mapping and public health interventions. Taken together, the aims explore how customizing Bayesian hierarchical models can provide deeper insights into the epidemiology of TB in the presence of sharp changes in local prevalence.

Table of Contents

1 Introduction and Literature Review 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Spatial Modeling of Infectious Disease Surveillance Data . . . . . . . 3

1.3 Existing Methods and Estimation Approaches for Infectious Disease

Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Maximum Likelihood Estimation (MLE) . . . . . . . . . . . . 5

1.3.2 Hierarchical Bayesian Approximation . . . . . . . . . . . . . . 6

1.3.3 Bayesian Hierarchical Disease Mapping . . . . . . . . . . . . . 7

1.4 Motivating Example: Georgia Tuberculosis Surveillance Data . . . . . 10

1.4.1 Overview of Tuberculosis . . . . . . . . . . . . . . . . . . . . . 11

1.4.2 Tuberculosis Surveillance in the United States . . . . . . . . . 14

1.4.3 Tuberculosis Surveillance in Georgia . . . . . . . . . . . . . . 16

1.5 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Technical Background on Disease Mapping Topics 20

2.1 Theoretical Foundations of Linear Mixed Models . . . . . . . . . . . . 20

2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.2.1 Random Effects in Linear Models . . . . . . . . . . . 22

2.1.2.2 Prediction versus Estimation . . . . . . . . . . . . . 24

2.1.3 Best Linear Unbiased Prediction . . . . . . . . . . . . . . . . . 25

2.1.3.1 Links to Other Statistical Theory . . . . . . . . . . . 30

2.1.3.2 Further Commentary . . . . . . . . . . . . . . . . . . 32

2.1.4 “Old” versus “New” Style Random Effects . . . . . . . . . . . . 34

2.1.4.1 Restricted Maximum Likelihood (REML) . . . . . . 36

2.1.4.2 Penalized Maximum Likelihood . . . . . . . . . . . . 38

2.1.4.3 Bayesian Framework . . . . . . . . . . . . . . . . . . 41

2.1.4.4 Bayesian LASSO . . . . . . . . . . . . . . . . . . . . 44

2.1.4.5 Distinction Between Bayesian LASSO and L1 Norm

Random Intercept . . . . . . . . . . . . . . . . . . . 48

2.1.5 Contrasting Prediction and Estimation . . . . . . . . . . . . . 49

2.1.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.2 Bayesian Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.2.1 Prior Distributions . . . . . . . . . . . . . . . . . . . . . . . . 55

2.2.2 Markov Chain Monte Carlo (MCMC) . . . . . . . . . . . . . 56

2.2.2.1 Gibbs Sampling . . . . . . . . . . . . . . . . . . . . . 56

2.2.2.2 Metropolis-Hastings Algorithm . . . . . . . . . . . . 57

2.2.3 Efficient Posterior Sampling in Bayesian Inference . . . . . . . 57

2.2.3.1 Challenges with MCMC for GMRFs . . . . . . . . . 58

2.2.3.2 Hamiltonian Dynamics . . . . . . . . . . . . . . . . . 58

2.2.3.3 HMC in Bayesian Inference . . . . . . . . . . . . . . 59

2.2.3.4 Challenges and Advances in HMC . . . . . . . . . . 60

2.2.3.5 The No-U-Turn Sampler (NUTS) . . . . . . . . . . . 61

2.2.3.6 Applications of HMC in Hierarchical Models . . . . . 62

2.2.3.6.1 The Challenge of Centered Parameterization 63

2.2.3.6.2 Non-Centered Parameterization . . . . . . . 63

2.2.3.6.3 Benefits of HMC in Hierarchical Models . . 64

2.2.3.6.4 Practical Implementation in Stan . . . . . . 64

2.2.3.6.5 Case Studies and Applications . . . . . . . . 65

3 Hierarchical Bayesian Disease Mapping Model Performance in the

Presence of Spatially Concentrated Prevalence 66

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2.1 Study Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2.2 Bayesian Methodology for Disease Mapping . . . . . . . . . . 71

3.2.3 Hierarchical Bayesian Disease Mapping Models . . . . . . . . 71

3.2.4 Three-Stage Bayesian Hierarchical Model . . . . . . . . . . . . 73

3.2.4.1 Stage 1: Likelihood . . . . . . . . . . . . . . . . . . . 73

3.2.4.2 Stage 2: Random Effects Models . . . . . . . . . . . 75

3.2.5 Laplace Priors in Bayesian Disease Mapping . . . . . . . . . . 84

3.2.5.1 Choosing Between Laplace and Gaussian Priors . . . 85

3.2.6 Smoothing vs. Shrinkage in Disease Mapping . . . . . . . . . 85

3.2.6.1 The Horseshoe Model . . . . . . . . . . . . . . . . . 86

3.2.7 Implementation in Disease Mapping . . . . . . . . . . . . . . . 88

3.2.8 Comparison of Smoothing Techniques . . . . . . . . . . . . . . 88

3.2.8.1 Stage 3: Hyperpriors . . . . . . . . . . . . . . . . . . 89

3.3 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.3.1 Generating Simulated Data . . . . . . . . . . . . . . . . . . . 91

3.3.2 Hamiltonian Monte Carlo Settings . . . . . . . . . . . . . . . 92

3.3.3 Model Comparison . . . . . . . . . . . . . . . . . . . . . . . . 93

3.3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 95

3.3.5 Exploration of Random Effects Posterior Density . . . . . . . 95

3.3.6 Simulated Data Analysis . . . . . . . . . . . . . . . . . . . . . 98

3.3.7 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 102

3.4 Case Study: Analysis of TB Incidence in Metro Atlanta Using Real Data106

3.4.1 Model Application and Results . . . . . . . . . . . . . . . . . 106

3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4 A Localized Approach to Model Adequacy: Zooming in on Sharp

Spatial Transitions in Disease Surveillance Models 113

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.2.1 Overview of Global Measures of Model Fit . . . . . . . . . . . 117

4.2.1.1 Deviance Information Criterion (DIC) . . . . . . . . 117

4.2.1.2 Limitations of DIC . . . . . . . . . . . . . . . . . . . 118

4.2.2 Watanabe–Akaike information criterion (WAIC) . . . . . . . . 119

4.2.2.1 Calculation of WAIC . . . . . . . . . . . . . . . . . . 119

4.2.2.2 Advantages of WAIC . . . . . . . . . . . . . . . . . . 120

4.2.2.3 Limitations of WAIC . . . . . . . . . . . . . . . . . . 120

4.2.3 Localized Measures of Fit: Local DIC and Local WAIC . . . . 120

4.2.3.1 Local DIC for Administrative Units . . . . . . . . . . 121

4.2.3.2 Local WAIC for Administrative Units . . . . . . . . 121

4.3 Case Study: Local Measures of fit for TB in Clarkston, GA . . . . . . 122

4.3.1 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.3.2 Identifying Areas of Concern . . . . . . . . . . . . . . . . . . . 123

4.3.3 Leverage versus Model Adequacy Criteria . . . . . . . . . . . 127

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.4.1 Model Adequacy and Local Patterns of TB . . . . . . . . . . . 129

4.4.2 Implications for TB Surveillance in Clarkston . . . . . . . . . 129

4.4.3 Future Work: Spatially Varying Coefficient Models . . . . . . 130

5 Novel Insights in Spatial Epidemiology: Comparing Bayesian Spatially

Varying Coefficient Models 132

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.2.1 Bayesian Hierarchical Disease Mapping: Overview . . . . . . . 134

5.2.2 Extensions to the ICAR Model . . . . . . . . . . . . . . . . . 135

5.2.3 Limitations of Traditional ICAR Models and Their Extensions 136

5.2.4 Bayesian Spatially Varying Coefficient Models . . . . . . . . . 137

5.2.4.1 Model Structure . . . . . . . . . . . . . . . . . . . . 137

5.2.4.2 Key Advantages . . . . . . . . . . . . . . . . . . . . 138

5.2.5 Challenges Associated with Bayesian Spatially Varying Coefficient

Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.2.6 Proposed Method: Locally Adaptive Smoothing (LAS) . . . . 140

5.2.6.1 Dynamic Spatial Weights . . . . . . . . . . . . . . . 140

5.2.6.2 Adjusted Spatial Weights . . . . . . . . . . . . . . . 141

5.2.7 Restructured Spatially Varying Coefficient Model with LAS . 142

5.2.7.1 Model Specification . . . . . . . . . . . . . . . . . . . 142

5.2.8 Case Study: Application to Tuberculosis Mapping in Metro

Atlanta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.2.8.1 Model Implementation . . . . . . . . . . . . . . . . . 144

5.2.9 Model Validation and Evaluation . . . . . . . . . . . . . . . . 146

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.3.1 Posterior SVC Estimates . . . . . . . . . . . . . . . . . . . . . 147

5.3.2 Posterior Predictive Checks . . . . . . . . . . . . . . . . . . . 149

5.3.3 Predictive Interval Coverage . . . . . . . . . . . . . . . . . . . 149

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.4.1 Comparison to Existing Models . . . . . . . . . . . . . . . . . 152

5.4.2 Predictive Accuracy and Model Complexity . . . . . . . . . . 153

5.4.3 Implications for Public Health Interventions . . . . . . . . . . 153

6 Conclusions and Future Directions 155

6.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . 155

6.2 Future Directions and Conclusion . . . . . . . . . . . . . . . . . . . . 156

6.2.1 Application to Middle-Income Countries with Wealth Disparities156

6.2.2 Adjusting Spatial Weighting Schemes . . . . . . . . . . . . . . 158

6.2.3 Broadening Applicability to Other Infectious Diseases . . . . . 158

About this Dissertation

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School	Laney Graduate School
Department	Biostatistics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Statistics Health Sciences, Epidemiology Biology, Biostatistics
Keyword	Spatially Varying Coefficients Spatial Epidemiology Disease Mapping ICAR Spatial Outliers Deviance Information Criterion WAIC Locally Adaptive Smoothing BYM Horseshoe Model Tuberculosis (TB) Bayesian Hierarchical Models Public Health Surveillance
Committee Chair / Thesis Advisor	Lance Waller, Emory University
Committee Members	Neel Gandhi , Emory University Benjamin Risk, Emory University Natalie Dean, Emory University Robert Lyles, Emory University

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