A Solution to the Ecological Inference Problem: Reconstructing Individual Behavior from Aggregate Data

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Princeton University Press, 1997 - Political Science - 342 pages
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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.

 

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Contents

Qualitative Overview
3
11 The Necessity of Ecological Inferences
7
12 The Problem
12
13 The Solution
17
14 The Evidence
22
15 The Method
26
Formal Statement of the Problem
28
Catalog of Problems to Fix
35
924 Ex Post Diagnostics
183
93 Avoiding Distributional Problems
184
933 Parametric Approaches
185
932 A Nonparametric Approach
191
Verification
197
A Typical Application Described in Detail Voter Registration by Race
199
102 Likelihood Estimation
200
103 Computing Quantities of Interest
206

Aggregation Problems
37
32 The Indeterminacy Problem
39
33 The Grouping Problem
46
34 Equivalence of the Grouping and Indeterminacy Problems
53
35 A Concluding Definition
54
NonAggregation Problems
56
42 Applying Goodmans Regression in 2 x 3 Tables
68
43 Double Regression Problems
71
44 Concluding Remarks
73
The Proposed Solution
75
The Data Generalizing the Method of Bounds
77
No Uncertainly
78
Upper and Lower Bounds
79
522 DistrictLevel Quantities of Interest
83
53 An Easy Visual Method for Computing Bounds
85
The Model
91
61 The Basic Model
92
62 Model Interpretation
94
621 Observable Implications of Model Parameters
96
622 Parameterizing the Truncated Bivariate Normal
102
623 Computing 2p Parameters from Only p Observations
106
624 Connections to the Statistics of Medical and Seismic Imaging
112
625 Would a Model of IndividualLevel Choices Help?
119
Preliminary Estimation
123
71 A Visual Introduction
124
72 The Likelihood Function
132
73 Parameterizations
135
74 Optional Priors
138
75 Summarizing Information about Estimated Parameters
139
Calculating Quantities of Interest
141
811 Definitions and Examples
142
812 Simulation for Ecological Inference
144
82 PrecinctLevel Quantities
145
83 DistrictLevel Quantities
149
84 Quantities of Interest from Larger Tables
151
842 An Approach Related to Double Regression
153
85 Other Quantities of Interest
156
Model Extensions
158
971 Aggregation Bias
159
912 Incorrect Distributional Assumptions
161
913 Spatial Dependence
164
92 Avoiding Aggregation Bias
168
921 Using External Information
169
X as a Covariate
174
923 Tradeoffs and Priors for the Extended Model
179
1031 Aggregate
207
1032 County Level
209
1033 Other Quantities of Interest
215
Robustness to Aggregation Bias Poverty Status by Sex
217
112 Verifying the Existence of Aggregation Bias
218
113 Fitting the Data
220
114 Empirical Results
222
Estimation without Information Black Registration in Kentucky
226
122 Data Problems
227
123 Fitting the Data
228
124 Empirical Results
232
Classic Ecological Inferences
235
1312 Estimates
238
132 Black Literacy in 1910
241
Generalizations and Concluding Suggestions
247
NonEcological Aggregation Problems
249
1411 The Problem with the Problem
250
1412 Ecological Inference as a Solution to the Modifiable Areal Unit Problem
252
142 The Statistical Problem of Combining Survey and Aggregate Data
255
143 The Econometric Problem of Aggregation Continuous Variables
258
144 Concluding Remarks on Related Aggregation Research
262
Ecological Inference in Larger Tables
263
151 An Intuitive Approach
264
152 Notation for a General Approach
267
153 Generalized Bounds
269
154 The Statistical Model
271
155 Distributional Implication
273
156 Calculating the Quantities of Interest
276
A Concluding Checklist
277
Appendices
293
Proof That All Discrepancies Are Equivalent
295
Parameter Bounds
301
𝜷s and 𝜽s
302
𝜆s
303
Conditional Posterior Distribution
304
C1 Using Bayes Theorem
305
C2 Using Properties of Normal Distributions
306
The Likelihood Function
307
The Details of Nonparametric Estimation
309
Computational Issues
311
Glossary of Symbols
313
References
317
Index
337
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About the author (1997)

Gary King is Professor of Government at Harvard University. He has authored and coauthored numerous journal articles and books in the field of political methodology, including "Designing Social Inquiry: Scientific Inference in Qualitative Research" (Princeton).

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