Matched Sampling for Causal Effects

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Contributor Acknowledgments......Page 11
My Introduction to Matched Sampling......Page 15
PART I: THE EARLY YEARS AND THE INFLUENCE OF WILLIAM G. COCHRAN......Page 19
1. INTRODUCTION......Page 21
2. MAJOR THEMES OF ADVICE FOR OBSERVATIONAL STUDIES......Page 23
3. MATCHING IN ANALYTICAL STUDIES......Page 26
4. ANALYSIS OF COVARIANCE: ITS NATURE AND USES......Page 28
5. THE PLANNING OF OBSERVATIONAL STUDIES OF HUMAN POPULATIONS......Page 29
6. THE EFFECTIVENESS OF ADJUSTMENT BY SUBCLASSIFICATION IN REMOVING BIAS IN OBSERVATIONAL STUDIES......Page 33
7. THE USE OF COVARIANCE IN OBSERVATIONAL STUDIES......Page 34
8. OBSERVATIONAL STUDIES......Page 36
9. CONTROLLING BIAS IN OBSERVATIONAL STUDIES: A REVIEW – 1973......Page 37
10. RELATED WORK......Page 40
11. PLANNING AND ANALYSIS OF OBSERVATIONAL STUDIES......Page 41
12. STATISTICAL THEORY AND PRACTICE......Page 42
ACKNOWLEDGMENTS......Page 43
1. INTRODUCTION......Page 44
2.1. The Model......Page 45
2.3. Caliper Matching......Page 46
2.4. ‘Nearest Available’ Matching......Page 48
3.1. Regressions Linear But Not Parallel......Page 50
3.2. Regression Non-linear......Page 52
3.3. Regressions Parallel But Quadratic......Page 54
3.4. Errors of Measurement in x......Page 57
3.5. Omitted Confounding Variable......Page 58
4.2. Performance of the Two Methods......Page 60
5.1. Parallel Linear Regressions......Page 63
5.2. Non-parallel Linear Regressions......Page 64
5.4. Errors of Measurement in x......Page 65
6.1. Caliper Matching......Page 66
6.2. Categorical Matching......Page 67
6.3. Discriminant Matching......Page 68
6.4. Other Matching Methods......Page 69
7. SUMMARY COMMENTS......Page 70
PART II: UNIVARIATE MATCHING METHODS AND THE DANGERS OF REGRESSION ADJUSTMENT......Page 73
1. INTRODUCTION......Page 76
2. TERMINOLOGY; PERCENT REDUCTION IN BIAS......Page 77
3. THE MAXIMUM PERCENT REDUCTION IN BIAS GIVEN FIXED DISTRIBUTIONS AND FIXED SAMPLE SIZES......Page 81
4. MEAN-MATCHING......Page 83
5. PAIR-MATCHING......Page 85
6. ADVICE TO AN INVESTIGATOR......Page 88
(a) Choosing N......Page 89
(c) Choosing a Matching Method......Page 90
Proofs of Trends (3) and (4) in Section 3......Page 91
Notation used in the subroutines......Page 92
The Quality of Individual Matches:......Page 94
1.1. The Five Estimates of τ to be Considered Here......Page 95
1.2. The Percent Reduction in Bias Due to Matched Sampling and Regression Adjustment......Page 97
2.1. Conditions of the Monte Carlo Investigation......Page 98
2.2. Case I – Distribution of X Favorable to Matching…......Page 101
2.3. Case II…......Page 103
3. LINEAR REGRESSION ADJUSTMENT AND QUADRATIC RESPONSE SURFACES......Page 105
3.1. Bias of…......Page 107
3.2. Bias of …......Page 108
4.1. No Regression Adjustments…......Page 110
4.3. Other Estimates of…......Page 111
ACKNOWLEDGMENTS......Page 112
1. INTRODUCTION......Page 113
2. DEFINING THE EFFECTS OF TREATMENTS: ASSUMPTIONS AND NOTATION......Page 114
3. PRELIMINARY RESULTS......Page 116
4. ESTIMATING μ1(xij) AND μ2(xij) BY MODEL FITTING......Page 118
5. ESTIMATING μ1(xij) AND μ2(xij) BY BLOCKING ON X......Page 120
6. GENERALIZING TO A SUBPOPULATION OF P DEFINED BY X......Page 121
7. A SIMPLE EXAMPLE......Page 122
7.1. Assignment on the Basis of Pretest......Page 123
7.2. Assignment Not on the Basis of Pretest......Page 125
8. DISCUSSION OF NEEDED INVESTIGATIONS......Page 127
ACKNOWLEDGMENTS......Page 128
PART III: BASIC THEORY OF MULTIVARIATE MATCHING......Page 129
1. INTRODUCTION......Page 131
2. EQUAL PERCENT BIAS REDUCING MATCHING METHODS......Page 132
3. SPECIAL CASES IN WHICH UNIVARIATE MATCHING IS EPBR......Page 133
4. SPECIAL CASES IN WHICH “EQUIBIAS INTERSECTION” MATCHING ON X IS EPBR......Page 134
5. LARGE SAMPLE EPBR MATCHING METHODS FOR X NORMALLY DISTRIBUTED......Page 135
6. SPECIAL CASES IN WHICH EXCHANGEABLE MATCHING ON X IS EPBR......Page 136
8. DISCRIMINANT AND INNER PRODUCT MATCHING......Page 138
APPENDIX A......Page 140
APPENDIX C......Page 142
1. INTRODUCTION......Page 143
2. THE MAXIMUM PERCENT REDUCTION IN BIAS GIVEN FIXED DISTRIBUTIONS AND FIXED SAMPLE SIZES......Page 144
3. ATTAINING THE “OPTIMAL” PERCENT REDUCTION IN BIAS, MIN {100, θMAX}......Page 146
4. TRENDS IN THE MAXIMUM PERCENT REDUCTION IN BIAS......Page 147
5. SUGGESTIONS FOR MINIMAL SAMPLE SIZES......Page 148
5.3. Estimating the Functions…......Page 150
5.4. Estimating the Minimum r2 Needed to Obtain a Given Percent Reduction in Bias When r1 = 1......Page 151
5.5. Estimating the Minimum r2 and r2 Needed to Obtain a Given Percent Reduction in Bias......Page 152
APPENDIX A......Page 153
APPENDIX B......Page 154
The Ratio to Total Bias to Average Bias......Page 155
1. INTRODUCTION......Page 156
2.1. Matching Methods to be Studied......Page 157
2.2. The Treatment Effect......Page 158
2.3. Estimators of…......Page 159
3. MONTE CARLO COMPARISONS OF PROCEDURES – CONDITIONS......Page 160
4. REGRESSION ADJUSTMENTS WITH RANDOM SAMPLES......Page 162
5. RESULTS WITH r > 1......Page 163
5.1. Two Specific Estimators......Page 164
5.2. An ANOVA of the Results When r > 1......Page 166
5.3. Metric Matching vs. Discriminant Matching......Page 168
5.4. Comparing Regression Adjustments......Page 169
5.5. The Effect of Using Different Ratios for Matching......Page 170
6. DISCUSSION......Page 171
Discussion of Monte Carlo Study......Page 172
2. NOTATION......Page 174
3. MONTE CARLO PERCENT REDUCTIONS IN BIAS FOR BIVARIATE NORMAL X......Page 175
4. WHY NOT 100% BIAS REDUCTION?......Page 177
APPENDIX......Page 178
PART IV: FUNDAMENTALS OF PROPENSITY SCORE MATCHING......Page 181
1.1. The Structure of Studies for Causal Effects......Page 184
1.2. Balancing Scores and the Propensity Score......Page 185
2.1. Outline......Page 186
2.2. Large-Sample Theory......Page 187
2.3. Some Small-Sample Theory......Page 190
3.1. Techniques for Adjustment in Observational Studies......Page 191
3.2. Use of Propensity Scores to Construct Matched Samples from Treatment Groups......Page 192
3.3. Subclassification on Propensity Scores......Page 195
3.4. Propensity Scores and Covariance Adjustment......Page 197
ACKNOWLEDGMENTS......Page 198
1. INTRODUCTION AND NOTATION......Page 199
2. THE STRUCTURE OF THE SENSITIVITY ANALYSIS......Page 201
3. AN EXAMPLE USING DATA ON CORONARY ARTERY DISEASE......Page 202
4. CALCULATING THE MAXIMUM LIKELIHOOD ESTIMATES OF…AND…......Page 204
Subclass Specific Results......Page 205
1.1. Adjustment by Subclassification in Observational Studies......Page 207
1.2. The Propensity Score in Observational Studies......Page 208
2.1. The First Fit and Subclassification......Page 209
2.2. Refinement of the Fitted Propensity Score and the Balance Obtained in the Final Subclassification......Page 211
2.3. The Fitted Propensity Score: Overlap of Treated and Control Groups......Page 212
2.4. Incomplete Covariate Information......Page 213
3.2. Subclass-Specific Estimates; Direct Adjustment......Page 214
3.3. Adjustment and Estimation Within Subpopulations Defined by x......Page 216
The Effectiveness of Subclassification on the Propensity Score in Removing Bias......Page 217
Balancing Properties of the Propensity Score with Incomplete Data......Page 219
ACKNOWLEDGMENTS......Page 220
1. INTRODUCTION: BACKGROUND; WHY MATCH?......Page 221
2.2. Estimating the Propensity Score......Page 224
2.3. Matching Methods That Are Equal-Percent Bias Reducing......Page 225
3.2. Nearest Available Matching on the Estimated Propensity Score......Page 226
3.3. Mahalanobis Metric Matching Including the Propensity Score......Page 227
3.4. Nearest Available Mahalanobis Metric Matching Within Calipers Defined by the Propensity Score......Page 228
3.5. Nonlinear Response Surfaces......Page 229
ACKNOWLEDGMENT......Page 230
1.1. The Effects Caused by Treatments......Page 231
1.2. Matching as a Method for Subsampling a Large Reservoir of Potential Controls......Page 232
1.4. Strongly Ignorable Treatment Assignment......Page 233
1.5. Matching on the Basis of x Alone......Page 234
2.1. Overview: The Three Components of the Bias......Page 235
2.3. Exact, Incomplete Matching on x......Page 236
2.5. Addressing the Three Components of Bias......Page 238
3.1. Introduction: Purpose; Background......Page 239
3.2. Methods: Incomplete Categorical Matching; The Standardized Difference in Covariate Means as a Measure of Covariate Imbalance......Page 242
3.3. Results: Substantial Differences Between Matched and Unmatched Treated Children......Page 243
3.4. Complete Matching Using the Propensity Score......Page 244
ACKNOWLEDGMENT......Page 245
PART V: AFFINELY INVARIANT MATCHING METHODS WITH ELLIPSOIDALLY SYMMETRIC DISTRIBUTIONS, THEORY AND METHODOLOGY......Page 247
1. BACKGROUND......Page 249
2.1. Ellipsoidal Distributions......Page 250
2.3. Affinely Invariant Matching Methods......Page 251
2.4. The Best Linear Discriminant......Page 252
3.1. Results Based on Symmetry......Page 253
3.2. Corollaries That Decompose the Effect of Matching......Page 254
3.3. Proofs of the Corollaries......Page 255
4.1. Conditionally Affinely Invariant Matching Methods......Page 256
4.3. Canonical Form for Conditionally Ellipsoidal Distributions......Page 257
4.4. Representation for a Linear Combination Y......Page 258
4.5. Extensions of Results in Section 3.3......Page 259
5. CONCLUSIONS......Page 261
1.1. Matched Sampling......Page 263
1.3. Linear Propensity Score Matching Methods......Page 264
1.4. Application of Matching Results from Rubin & Thomas (1992a)......Page 265
2.1. Matching on the Population Discriminant, Z......Page 266
2.2. Matching on Estimated Linear Propensity Scores......Page 267
2.3. Summarized Results......Page 268
3.3. The Distribution of W......Page 269
3.4. A Corollary for W......Page 271
3.6. Proof of Theorem 2.2......Page 272
4.1. Derivation of Approximation for h......Page 273
4.2. Derivation of Approximation for g......Page 274
4.3. Derivation of Approximation for k......Page 275
4.4. Comments on the Approximations......Page 276
1. INTRODUCTION......Page 277
2.1. Overview of the Effects of Matching......Page 280
2.2. Analytic Approximations for the Matched Sample Moments......Page 281
2.3. Discussion of Practical Recommendations......Page 282
3. SIMULATION STUDY WITH ELLIPSOIDAL DATA......Page 284
3.1. Design......Page 285
3.2. Results When Close Matching Is Possible......Page 286
3.3. Trends for Difficult Matching Settings......Page 287
4. SIMULATION STUDY USING REAL DATA......Page 288
4.1. Simulation Design......Page 289
4.2. Results......Page 290
5. DISCUSSION OF BIAS REDUCTION AND PRECISION......Page 292
Derivation of the Approximations......Page 294
1.1. Combining Propensity Score Matching and Prognostic Covariate Adjustment......Page 296
1.2. Example......Page 298
1.3. The Analogy Between Propensity Score Matching and Randomization, and Between Special Matching and Experimental Blocking......Page 300
2.1. Covariate Distributions......Page 301
2.2. Matching Methods Using Linear Propensity Score Estimators......Page 302
2.3. Close Mean and Variance Matching on Special Covariates and the Estimated Propensity Score......Page 303
3.2. The Moments of X in Matched Samples with Close Matching......Page 304
3.3. Comments on the Results......Page 305
4. EXAMPLE......Page 307
5.1. Simulated Outcomes Nonlinear Function of the Covariates......Page 309
5.2. Results of Simulation......Page 310
5.3. The Effect of Matching on Different Covariate Components......Page 312
6. SUMMARY......Page 313
A.2. Decomposition Based on the Semi-Estimated Discriminant......Page 314
A.3. Distribution of W......Page 315
A.4. Derivation of the Results in Section A.3......Page 316
Results Involving Expectations of Matched Sample Standard Deviations......Page 317
PART VI: SOME APPLIED CONTRIBUTIONS......Page 319
19. Causal Inference in Retrospective Studies......Page 322
1.1. The Formal Model and the Definition of Unit-Level Causal Effects......Page 323
1.2. The Three Levels of Causal Inference......Page 325
1.4. The Role of Covariates in Prospective Studies......Page 327
2. Causal Inference in Retrospective Case-Control Studies......Page 328
2.1. The Standard Two-Way Table......Page 329
2.2. Relating Retrospective and Prospective Probabilities......Page 330
2.4. The Role of Covariates in Retrospective Studies......Page 332
3.1. The Data......Page 335
3.3. Simplifying the Analysis......Page 337
4.1. Risk Factors: Causal Agents Versus Attributes......Page 338
4.2. Randomization and Strong Ignorability......Page 339
4.3. The Role of Matching in Prospective and Retrospective Studies......Page 340
HISTORICAL NOTE......Page 341
1. INTRODUCTION......Page 342
2.1. Multichild Families......Page 344
2.3. Implementation......Page 345
3. GENERAL PROPENSITY SCORE METHODOLOGY......Page 347
3.1. Complete Data......Page 348
3.4. True Versus Estimated Propensity Scores......Page 350
3.5. Incomplete Data......Page 351
4.1. Single-Child Families......Page 352
5.2. Other Potential Advantages of the PMPD......Page 357
6. CONCLUSION......Page 358
Second-Year Lottery Winners......Page 359
ACKNOWLEDGMENTS......Page 360
1.1. Background on Propensity Scores......Page 361
1.2. The Database and Problem......Page 363
2.1. Estimation of Propensity Scores......Page 367
2.3. General Location Model With Complete Data......Page 368
2.4. Fitting the General Location Model With Missing Data......Page 369
2.5. Generalized Propensity Score Estimates: An Illustration......Page 370
3.1. Specific Models......Page 371
3.3. Resultant Distributions of Propensity Scores and Their Logits......Page 372
3.4. Resultant Covariate Balance After Matching......Page 375
4. CONCLUSIONS......Page 378
1. INTRODUCTION – THE IMPORTANCE OF DESIGNING AN OBSERVATIONAL STUDY......Page 379
2. PROPENSITY SCORES – NO OUTCOME VARIABLES IN SIGHT......Page 381
3. HISTORY OF PROPENSITY SCORES – DISCRIMINANT MATCHING IN 1973 AND BEYOND; OBSERVED AND UNOBSERVED COVARIATES......Page 382
4. MAJOR TECHNIQUES FOR INITIAL DESIGN – MATCHING, SUBCLASSIFICATION, AND WEIGHTING......Page 383
5. TRADITIONAL BENCHMARKS FOR THE USE OF REGRESSION ADJUSTMENT – WHEN IS IT RELIABLE?......Page 384
6. COMPARING SMOKERS AND NEVER SMOKERS IN NMES......Page 387
7. MAHALANOBIS METRIC MATCHING OF SMOKERS AND NEVER SMOKERS WITHIN PROPENSITY SCORE CALIPERS......Page 389
8. FURTHER SUBCLASSIFICATION OF MATCHED SAMPLES OF SMOKERS AND NEVER SMOKERS......Page 391
9. OTHER APPROACHES TO DESIGNING AN OBSERVATIONAL STUDY IN NMES......Page 394
10. DISCUSSION......Page 395
ACKNOWLEDGMENTS......Page 396
PART VII: SOME FOCUSED APPLICATIONS......Page 397
23. Criminality in XYY and XXY Men......Page 399
CASE-FINDING PROCEDURE......Page 402
DOCUMENTARY DATA......Page 403
FREQUENCY OF XYY’S AND XXY’S AND OF CRIMINALS AMONG THEM......Page 405
CRIMINAL RATES AFTER ADJUSTMENT FOR BACKGROUND VARIABLES......Page 409
DISCUSSION AND CONCLUSIONS......Page 412
ACKNOWLEDGMENTS......Page 415
1.1. Causal Inference......Page 416
1.2. Four Modes of Statistical Inference for Causal Effects and the Existence of a Common Framework......Page 417
2.2. Basic Data......Page 418
2.4. Results of the Matching......Page 419
3.2. SUTVA......Page 422
3.3. The Posited Assignment Mechanism......Page 423
3.4. A Simplified Assignment Mechanism for the 4,000 Units......Page 424
4.1. Randomization-Based Tests of Sharp Null Hypothesis......Page 425
4.2. Bayesian Predictive Inference......Page 426
4.3. Randomization-Based Repeated-Sampling Inference......Page 428
4.4. Model-Based Repeated-Sampling Inference......Page 429
5.1. The Matched Sample Analyzed as a Paired-Comparison Experiment......Page 430
5.2. Some Second Looks at the Data and Posited Assignment Mechanism – What Happens Post-Intervention, Pre-Switch?......Page 431
6.2. Results Adjusted for Post-Intervention Pre-Switch Outcomes......Page 432
6.3. A Problem with This Treatment Assignment Mechanism – Missing Treatment Assignments......Page 433
6.4. Diagnostic Analyses Focusing on Those Less Likely to Quit......Page 434
7.1. Return to Treatment Assignment at Intervention Date......Page 435
7.2. A Confounded, Nonignorable Treatment Assignment Mechanism......Page 436
7.4. A Final Comment on This Assignment Mechanism......Page 437
8. CONCLUSION......Page 438
ACKNOWLEDGMENT......Page 439
25. In Utero Exposure to Phenobarbital and Intelligence Deficits in Adult Men......Page 440
Subjects......Page 442
Selection of Controls......Page 445
Procedures......Page 448
Data Analyses......Page 449
Study 1......Page 450
Study 2......Page 452
COMMENT......Page 454
26. Estimating Causal Effects from Large Data Sets Using Propensity Scores......Page 458
SUBCLASSIFICATION ON ONE CONFOUNDING VARIABLE......Page 459
PROPENSITY SCORE METHODS......Page 462
MORE THAN TWO TREATMENT CONDITIONS......Page 466
LIMITATIONS OF PROPENSITY SCORES......Page 467
CONCLUSION......Page 468
ACKNOWLEDGMENTS......Page 469
27. On Estimating the Causal Effects of DNR Orders......Page 470
ACKNOWLEDGMENT......Page 474
Conclusion: Advice to the Investigator......Page 475
References......Page 478
Author Index......Page 498
Subject Index......Page 502