Table of Contents

List of Figures xi

List of Tables xiii

Preface xv

PART I: INTRODUCTION 1

1. Qualitative Overview 3

1.1 The Necessity of Ecological Inferences 7

1.2 The Problem 12

1.3 The Solution 17

1.4 The Evidence 22

1.5 The Method 26

2. Formal Statement of the Problem 28

PART II: CATALOG OF PROBLEMS TO FIX 35

3. Aggregation Problems 37

3.1 Goodman's Regression: A Definition 37

3.2 The Indeterminacy Problem 39

3.3 The Grouping Problem 46

3.4 Equivalence of the Grouping and Indeterminacy Problems 53

3.5 A Concluding Definition 54

4. Non-Aggregation Problems 56

4.1 Goodman Regression Model Problems 56 68

4.3 Double Regression Problems 71

4.4 Concluding Remarks 73

PART III: THE PROPOSED SOLUTION 75

5. The Data: Generalizing the Method of Bounds 77

5.1 Homogeneous Precincts: No Uncertainty 78

5.2 Heterogeneous Precincts: Upper and Lower Bounds 79

5.2.1 Precinct-Level Quantities of Interest 79

5.2.2 District-Level Quantities of Interest 83

5.3 An Easy Visual Method for Computing Bounds 85

6. The Model 91

6.1 The Basic Model 92

6.2 Model Interpretation 94

6.2.1 Observable Implications of Model Parameters 96

6.2.2 Parameterizing the Truncated Bivariate Normal 102

6.2.3 Computing 2p Parameters from Only p Observations 106

6.2.4 Connections to the Statistics of Medical and Seismic Imaging 112

6.2.5 Would a Model of Individual-Level Choices Help? 119

7. Preliminary Estimation 123

7.1 A Visual Introduction 124

7.2 The Likelihood Function 132

7.3 Parameterizations 135

7.4 Optional Priors 138

7.5 Summarizing Information about Estimated Parameters 139

8. Calculating Quantities of Interest 141

8.1 Simulation Is Easier than Analytical Derivation 141

8.1.1 Definitions and Examples 142

8.1.2 Simulation for Ecological Inference 144

8.2 Precinct-Level Quantities 145

8.3 District-Level Quantities 149

8.4 Quantities of Interest from Larger Tables 151

8.4.1 A Multiple Imputation Approach 151

8.4.2 An Approach Related to Double Regression 153

8.5 Other Quantities of Interest 156

9. Model Extensions 158

9.1 What Can Go Wrong? 158

9.1.1 Aggregation Bias 159

9.1.2 Incorrect Distributional Assumptions 161

9.1.3 Spatial Dependence 164

9.2 Avoiding Aggregation Bias 168

9.2.1 Using External Information 169 174

9.2.3 Tradeoffs and Priors for the Extended Model 179

9.2.4 Ex Post Diagnostics 183

9.3 Avoiding Distributional Problems 184

9.3.1 Parametric Approaches 185

9.3.2 A Nonparametric Approach 191

PART IV: VERIFICATION 197

10. A Typical Application Described in Detail: Voter Registration by Race 199

10.1 The Data 199

10.2 Likelihood Estimation 200

10.3 Computing Quantities of Interest 207

10.3.1 Aggregate 207

10.3.2 County Level 209

10.3.3 Other Quantities of Interest 215

11. Robustness to Aggregation Bias: Poverty Status by Sex 217

11.1 Data and Notation 217

11.2 Verifying the Existence of Aggregation Bias 218

11.3 Fitting the Data 220

11.4 Empirical Results 222

12. Estimation without Information: Black Registration in Kentucky 226

12.1 The Data 226

12.2 Data Problems 227

12.3 Fitting the Data 228

12.4 Empirical Results 232

13. Classic Ecological Inferences 235

13.1 Voter Transitions 235

13.1.1 Data 235

13.1.2 Estimates 238

13.2 Black Literacy in 1910 241

PART V: GENERALIZATIONS AND CONCLUDING SUGGESTIONS 247

14. Non-Ecological Aggregation Problems 249

14.1 The Geographer's Modifiable Areal Unit Problem 249

14.1.1 The Problem with the Problem 250

14.1.2 Ecological Inference as a Solution to the Modifiable Areal Unit Problem 252

14.2 The Statistical Problem of Combining Survey and Aggregate Data 255

14.3 The Econometric Problem of Aggregating Continuous Variables 258

14.4 Concluding Remarks on Related Aggregation Research 262

15. Ecological Inference in Larger Tables 263

15.1 An Intuitive Approach 264

15.2 Notation for a General Approach 267

15.3 Generalized Bounds 269

15.4 The Statistical Model 271

15.5 Distributional Implications 273

15.6 Calculating the Quantities of Interest 276

15.7 Concluding Suggestions 276

16. A Concluding Checklist 277

PART VI: APPENDICES 293

A. Proof That All Discrepancies Are Equivalent 295

B Parameter Bounds 301

B.1 Homogeneous Precincts 301 302 303

C Conditional Posterior Distribution 304

C.1 Using Bayes Theorem 305

C.2 Using Properties of Normal Distributions 306

D The Likelihood Function 307

E The Details of Nonparametric Estimation 309

F Computational Issues 311

Glossary of Symbols 313

References 317

Index 337