A Solution to the Ecological Inference Problem (ebook)

Argumento de A Solution to the Ecological Inference Problem (ebook)

This book provides a solution to the ecological inference problem, which has plagued users of statistical methods for over seventy-five years: How can researchers reliably infer individual-level behavior from aggregate (ecological) data? In political science, this question arises when individual-level surveys are unavailable (for instance, local or comparative electoral politics), unreliable (racial politics), insufficient (political geography), or infeasible (political history). This ecological inference problem also confronts researchers in numerous areas of major significance in public policy, and other academic disciplines, ranging from epidemiology and marketing to sociology and quantitative history. Although many have attempted to make such cross-level inferences, scholars agree that all existing methods yield very inaccurate conclusions about the world. In this volume, Gary King lays out a unique--and reliable--solution to this venerable problem.

King begins with a qualitative overview, readable even by those without a statistical background. He then unifies the apparently diverse findings in the methodological literature, so that only one aggregation problem remains to be solved. He then presents his solution, as well as empirical evaluations of the solution that include over 16,000 comparisons of his estimates from real aggregate data to the known individual-level answer. The method works in practice.

King's solution to the ecological inference problem will enable empirical researchers to investigate substantive questions that have heretofore proved unanswerable, and move forward fields of inquiry in which progress has been stifled by this problem.

"This is a significant contribution to political methodology, and to statistical methodology throughout the social sciences. As always with Gary King's work, it is written with great flair and sophistication. This book will generate a good deal of excitement at the methodological frontier, and will also have a bracing impact on substantive research in a variety of fields."--Larry M. Bartels, Princeton University

"In this work, Gary King presents a number of new and important contributions to the field of statistical theory, and the practice of estimating choice probabilities from data aggregated into groups. An impressive statistical contribution."--Melvin J. Hinich, University of Texas-Austin

"For decades, market researchers and statisticians have lamented their inability to deduce individual behavior from data on groups. But Gary King . . .has come up with a formula that finally cracks this nut."--Business Week0List 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
4.2 Applying Goodman's Regression in 2 x 3 Tables 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
9.2.2 Unconditional Estimation: X_i as a Covariate 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
B.2 Heterogeneous Precincts 302
B.3 Heterogeneous Precincts 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