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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
4.2 Applying Goodman's Regression in 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: 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: 's and 's	302
B.3 Heterogeneous Precincts: 's	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

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