Preface xv Part I Overview 1 1 Introduction 3 1.1 Concepts of moderation, mediation, and spill-over 3 1.1.1 Moderated treatment effects 5 1.1.2 Mediated treatment effects 7 1.1.3 Spill-over effects of a treatment 8 1.2 Weighting methods for causal inference 10 1.3 Objectives and organization of the book 11 1.4 How is this book situated among other publications on related topics? 12 2 Review of causal inference concepts and methods 18 2.1 Causal inference theory 18 2.1.1 Attributes versus causes 18 2.1.2 Potential outcomes and individual-specific causal effects 19 2.1.3 Inference about population average causal effects 22 2.1.3.1 Prima facie effect 24 2.1.3.2 Ignorability assumption 25 2.2 Applications to Lord s paradox and Simpson s paradox 27 2.2.1 Lord s paradox 27 2.2.2 Simpson s paradox 31 2.3 Identification and estimation 34 2.3.1 Selection bias 35 2.3.2 Sampling bias 35 2.3.3 Estimation efficiency 36 Appendix 2.1: Potential bias in a prima facie effect 36 Appendix 2.2: Application of the causal inference theory to Lord s paradox 37 3 Review of causal inference designs and analytic methods 40 3.1 Experimental designs 40 3.1.1 Completely randomized designs 40 3.1.2 Randomized block designs 41 3.1.3 Covariance adjustment for improving efficiency 43 3.1.4 Multilevel experimental designs 43 3.2 Quasiexperimental designs 44 3.2.1 Nonequivalent comparison group designs 44 3.2.2 Other quasiexperimental designs 45 3.3 Statistical adjustment methods 46 3.3.1 ANCOVA and multiple regression 46 3.3.1.1 ANCOVA for removing selection bias 46 3.3.1.2 Potential pitfalls of ANCOVA with a vast between-group difference 47 3.3.1.3 Bias due to model misspecification 48 3.3.2 Matching and stratification 50 3.3.3 Other statistical adjustment methods 51 3.3.3.1 The IV method 51 3.3.3.2 DID analysis 54 3.4 Propensity score 55 3.4.1 What is a propensity score? 56 3.4.2 Balancing property of the propensity score 57 3.4.3 Pooling conditional treatment effect estimate: Matching, stratification, and covariance adjustment 60 3.4.3.1 Propensity score matching 61 3.4.3.2 Propensity score stratification 62 3.4.3.3 Covariance adjustment for the propensity score 66 3.4.3.4 Sensitivity analysis 66 Appendix 3.A: Potential bias due to the omission of treatment-by-covariate interaction 70 Appendix 3.B: Variable selection for the propensity score model 71 4 Adjustment for selection bias through weighting 76 4.1 Weighted estimation of population parameters in survey sampling 77 4.1.1 Simple random sample 77 4.1.2 Proportionate sample 78 4.1.3 Disproportionate sample 79 4.2 Weighting adjustment for selection bias in causal inference 80 4.2.1 Experimental result 81 4.2.2 Quasiexperimental result 81 4.2.3 Sample weight for bias removal 82 4.2.4 IPTW for bias removal 84 4.3 MMWS 86 4.3.1 Theoretical rationale 86 4.3.1.1 MMWS for a discrete propensity score 87 4.3.1.2 MMWS for a continuous propensity score 88 4.3.1.3 MMWS for estimating the treatment effect on the treated 89 4.3.2 MMWS analytic procedure 91 4.3.3 Inherent connection and major distinctions between MMWS and IPTW 93 Appendix 4.A: Proof of MMWS-adjusted mean observed outcome being unbiased for the population average potential outcome 95 Appendix 4.B: Derivation of MMWS for estimating the treatment effect on the treated 96 Appendix 4.C: Theoretical equivalence of MMWS and IPTW 97 Appendix 4.D: Simulations comparing MMWS and IPTW under misspecifications of the functional form of a propensity score model 97 5 Evaluations of multivalued treatments 100 5.1 Defining the causal effects of multivalued treatments 100 5.2 Existing designs and analytic methods for evaluating multivalued treatments 102 5.2.1 Experimental designs and analysis 102 5.2.1.1 Randomized experiments with multiple treatment arms 102 5.2.1.2 Identification under ignorability 102 5.2.1.3 ANOVA 103 5.2.2 Quasiexperimental designs and analysis 105 5.2.2.1 ANCOVA and multiple regression 105 5.2.2.2 Propensity score-based adjustment 108 5.2.2.3 Other adjustment methods 112 5.3 MMWS for evaluating multivalued treatments 112 5.3.1 Basic rationale 113 5.3.2 Analytic procedure 114 5.3.2.1 MMWS for a multinomial treatment measure 115 5.3.2.2 MMWS for an ordinal treatment measure 120 5.3.3 Identification assumptions 121 5.4 Summary 123 Appendix 5.A: Multiple IV for evaluating multivalued treatments 124 Part II Moderation 127 6 Moderated treatment effects: concepts and existing analytic methods 129 6.1 What is moderation? 129 6.1.1 Past discussions of moderation 130 6.1.1.1 Purpose of moderation research 130 6.1.1.2 What qualifies a variable as a moderator? 132 6.1.2 Definition of moderated treatment effects 133 6.1.2.1 Treatment effects moderated by individual or contextual characteristics 133 6.1.2.2 Joint effects of concurrent treatments 134 6.2 Experimental designs and analytic methods for investigating explicit moderators 136 6.2.1 Randomized block designs 137 6.2.1.1 Identification assumptions 137 6.2.1.2 Two-way ANOVA 138 6.2.1.3 Multiple regression 139 6.2.2 Factorial designs 140 6.2.2.1 Identification assumptions 140 6.2.2.2 Analytic strategies 142 6.3 Existing research designs and analytic methods for investigating implicit moderators 142 6.3.1 Multisite randomized trials 143 6.3.1.1 Causal parameters and identification assumptions 144 6.3.1.2 Analytic strategies 145 6.3.2 Principal stratification 149 Appendix 6.A: Derivation of bias in the fixed-effects estimator when the treatment effect is heterogeneous in multisite randomized trials 151 Appendix 6.B: Derivation of bias in the mixed-effects estimator when the probability of treatment assignment varies across sites 153 Appendix 6.C: Derivation and proof of the population weight applied to mixed-effects models for eliminating bias in multisite randomized trials 153 7 Marginal mean weighting through stratification for investigating moderated treatment effects 159 7.1 Existing methods for moderation analyses with quasiexperimental data 159 7.1.1 Analysis of covariance and regression-based adjustment 161 7.1.1.1 Treatment effects moderated by subpopulation membership 161 7.1.1.2 Treatment effects moderated by a concurrent treatment 164 7.1.2 Propensity score-based adjustment 165 7.1.2.1 Propensity score matching and stratification 166 7.1.2.2 Inverse-probability-of-treatment weighting 167 7.2 MMWS estimation of treatment effects moderated by individual or contextual characteristics 168 7.2.1 Application example 170 7.2.2 Analytic procedure 170 7.3 MMWS estimation of the joint effects of concurrent treatments 174 7.3.1 Application example 174 7.3.2 Analytic procedure 175 7.3.3 Joint treatment effects moderated by individual or contextual characteristics 179 8 Cumulative effects of time-varying treatments 185 8.1 Causal effects of treatment sequences 186 8.1.1 Application example 186 8.1.2 Causal parameters 187 8.1.2.1 Time-varying treatments 187 8.1.2.2 Time-varying potential outcomes 187 8.1.2.3 Causal effects of 2-year treatment sequences 187 8.1.2.4 Causal effects of multiyear treatment sequences 190 8.2 Existing strategies for evaluating time-varying treatments 190 8.2.1 The endogeneity problem in nonexperimental data 190 8.2.2 SEM 191 8.2.3 Fixed-effects econometric models 192 8.2.4 Sequential randomization 192 8.2.5 Dynamic treatment regimes 193 8.2.6 Marginal structural models and structural nested models 194 8.3 MMWS for evaluating 2-year treatment sequences 195 8.3.1 Sequential ignorability 195 8.3.2 Propensity scores 196 8.3.3 MMWS computation 197 8.3.3.1 MMWS adjustment for year 1 treatment selection 197 8.3.3.2 MMWS adjustment for two-year treatment sequence selection 198 8.3.4 Two-year growth model specification 199 8.3.4.1 Growth model in the absence of treatment 200 8.3.4.2 Growth model in the absence of confounding 200 8.3.4.3 Weighted 2-year growth model 202 8.4 MMWS for evaluating multiyear sequences of multivalued treatments 204 8.4.1 Sequential ignorability of multiyear treatment sequences 204 8.4.2 Propensity scores for multiyear treatment sequences 204 8.4.3 MMWS computation 205 8.4.4 Weighted multiyear growth model 205 8.4.5 Issues of sample size 206 8.5 Conclusion 207 Appendix 8.A: A saturated model for evaluating multivalued treatments over multiple time periods 207 Part III Mediation 211 9 Concepts of mediated treatment effects and experimental designs for investigating causal mechanisms 213 9.1 Introduction 214 9.2 Path coefficients 215 9.3 Potential outcomes and potential mediators 216 9.3.1 Controlled direct effects 217 9.3.2 Controlled treatment-by-mediator interaction effect 217 9.4 Causal effects with counterfactual mediators 219 9.4.1 Natural direct effect 219 9.4.2 Natural indirect effect 220 9.4.3 Natural treatment-by-mediator interaction effect 220 9.4.4 Unstable unit treatment value 221 9.5 Population causal parameters 222 9.5.1 Population average natural direct effect 224 9.5.2 Population average natural indirect effect 225 9.6 Experimental designs for studying causal mediation 225 9.6.1 Sequentially randomized designs 228 9.6.2 Two-phase experimental designs 228 9.6.3 Three- and four-treatment arm designs 230 9.6.4 Experimental causal-chain designs 231 9.6.5 Moderation-of-process designs 231 9.6.6 Augmented encouragement designs 232 9.6.7 Parallel experimental designs and parallel encouragement designs 232 9.6.8 Crossover experimental designs and crossover encouragement designs 233 9.6.9 Summary 234 10 Existing analytic methods for investigating causal mediation mechanisms 238 10.1 Path analysis and SEM 239 10.1.1 Analytic procedure for continuous outcomes 239 10.1.2 Identification assumptions 242 10.1.3 Analytic procedure for discrete outcomes 245 10.2 Modified regression approach 246 10.2.1 Analytic procedure for continuous outcomes 246 10.2.2 Identification assumptions 247 10.2.3 Analytic procedure for binary outcomes 248 10.3 Marginal structural models 250 10.3.1 Analytic procedure 250 10.3.2 Identification assumptions 252 10.4 Conditional structural models 252 10.4.1 Analytic procedure 252 10.4.2 Identification assumptions 253 10.5 Alternative weighting methods 254 10.5.1 Analytic procedure 254 10.5.2 Identification assumptions 256 10.6 Resampling approach 256 10.6.1 Analytic procedure 256 10.6.2 Identification assumptions 257 10.7 IV method 257 10.7.1 Rationale and analytic procedure 257 10.7.2 Identification assumptions 258 10.8 Principal stratification 259 10.8.1 Rationale and analytic procedure 259 10.8.2 Identification assumptions 260 10.9 Sensitivity analysis 261 10.9.1 Unadjusted confounding as a product of hypothetical regression coefficients 261 10.9.2 Unadjusted confounding reflected in a hypothetical correlation coefficient 262 10.9.3 Limitations when the selection mechanism differs by treatment 264 10.9.4 Other sensitivity analyses 265 10.10 Conclusion 265 10.10.1 The essentiality of sequential ignorability 265 10.10.2 Treatment-by-mediator interactions 266 10.10.3 Homogeneous versus heterogeneous causal effects 266 10.10.4 Model-based assumptions 266 Appendix 10.A: Bias in path analysis estimation due to the omission of treatment-by-mediator interaction 267 11 Investigations of a simple mediation mechanism 273 11.1 Application example: national evaluation of welfare-to-work strategies 274 11.1.1 Historical context 274 11.1.2 Research questions 275 11.1.3 Causal parameters 275 11.1.4 NEWWS Riverside data 277 11.2 RMPW rationale 277 11.2.1 RMPW in a sequentially randomized design 278 11.2.1.1 E[Y(0, M(0))] 279 11.2.1.2 E[Y(1, M(1))] 280 11.2.1.3 E[Y(1, M(0))] 280 11.2.1.4 E[Y(0, M(1))] 281 11.2.1.5 Nonparametric outcome model 282 11.2.2 RMPW in a sequentially randomized block design 283 11.2.3 RMPW in a standard randomized experiment 285 11.2.4 Identification assumptions 286 11.3 Parametric RMPW procedure 287 11.4 Nonparametric RMPW procedure 290 11.5 Simulation results 292 11.5.1 Correctly specified propensity score models 292 11.5.2 Misspecified propensity score models 294 11.5.3 Comparisons with path analysis and IV results 294 11.6 Discussion 295 11.6.1 Advantages of the RMPW strategy 295 11.6.2 Limitations of the RMPW strategy 295 Appendix 11.A: Causal effect estimation through the RMPW procedure 296 Appendix 11.B: Proof of the consistency of RMPW estimation 297 12 RMPW extensions to alternative designs and measurement 301 12.1 RMPW extensions to mediators and outcomes of alternative distributions 301 12.1.1 Extensions to a multicategory mediator 302 12.1.1.1 Parametric RMPW procedure 302 12.1.1.2 Nonparametric RMPW procedure 304 12.1.2 Extensions to a continuous mediator 304 12.1.3 Extensions to a binary outcome 306 12.2 RMPW extensions to alternative research designs 306 12.2.1 Extensions to quasiexperimental data 307 12.2.2 Extensions to data from cluster randomized trials 308 12.2.2.1 Application example and research questions 309 12.2.2.2 Estimation of the total effect 310 12.2.2.3 RMPW analysis of causal mediation mechanisms 310 12.2.2.4 Identification assumptions 312 12.2.2.5 Contrast with multilevel path analysis and SEM 312 12.2.2.6 Contrast with multilevel prediction models 313 12.2.3 Extensions to data from multisite randomized trials 313 12.2.3.1 Research questions and causal parameters 314 12.2.3.2 Estimation of the total effect and its between-site variation 315 12.2.3.3 RMPW analysis of causal mediation mechanisms 316 12.2.3.4 Identification assumptions 319 12.2.3.5 Contrast with multilevel path analysis and SEM 319 12.3 Alternative decomposition of the treatment effect 321 13 RMPW extensions to studies of complex mediation mechanisms 325 13.1 RMPW extensions to moderated mediation 325 13.1.1 RMPW analytic procedure for estimating and testing moderated mediation 326 13.1.2 Path analysis/SEM approach to analyzing moderated mediation 327 13.1.3 Principal stratification and moderated mediation 328 13.2 RMPW extensions to concurrent mediators 328 13.2.1 Treatment effect decomposition 329 13.2.1.1 Treatment effect decomposition without between-mediator interaction 329 13.2.1.2 Treatment effect decomposition with between-mediator interaction 331 13.2.2 Identification assumptions 333 13.2.3 RMPW procedure 333 13.2.3.1 Estimating E[Y(1,M1(1),M2(0))] 335 13.2.3.2 Estimating E[Y(1,M1(0),M2(0))] 335 13.2.3.3 Causal effect estimation with noninteracting concurrent mediators 336 13.2.3.4 Estimating E[Y(1,M1(0),M2(1))] 337 13.2.3.5 Causal effect estimation with interacting concurrent mediators 337 13.2.4 Contrast with the linear SEM approach 338 13.2.5 Contrast with the multivariate IV approach 339 13.3 RMPW extensions to consecutive mediators 340 13.3.1 Treatment effect decomposition 341 13.3.1.1 Natural direct effect of the treatment on the outcome 342 13.3.1.2 Natural indirect effect mediated by M1 only 342 13.3.1.3 Natural indirect effect mediated by M2 only 342 13.3.1.4 Natural indirect effect mediated by an M1-by-M2 interaction 343 13.3.1.5 Treatment-by-mediator interactions 343 13.3.2 Identification assumptions 345 13.3.3 RMPW procedure 347 13.3.3.1 Estimating E[Y(1,M1(0),M2(0,M1(0)))] 347 13.3.3.2 Estimating E[Y(1,M1(1),M2(0,M1(0)))] 349 13.3.3.3 Estimating E[Y(1,M1(0),M2(1,M1(1)))] 349 13.3.3.4 Estimating E[Y(0,M1(1),M2(0,M1(0)))] and E[Y(0,M1(0),M2(1,M1(1)))] 351 13.3.4 Contrast with the linear SEM approach 353 13.3.5 Contrast with the sensitivity-based estimation of bounds for causal effects 354 13.4 Discussion 355 Appendix 13.A: Derivation of RMPW for estimating population average counterfactual outcomes of two concurrent mediators 355 Appendix 13.B: Derivation of RMPW for estimating population average counterfactual outcomes of consecutive mediators 358 Part IV Spill-over 363 14 Spill-over of treatment effects: concepts and methods 365 14.1 Spill-over: A nuisance, a trifle, or a focus? 365 14.2 Stable versus unstable potential outcome values: An example from agriculture 367 14.3 Consequences for causal inference when spill-over is overlooked 369 14.4 Modified framework of causal inference 371 14.4.1 Treatment settings 371 14.4.2 Simplified characterization of treatment settings 373 14.4.3 Causal effects of individual treatment assignment and of peer treatment assignment 375 14.5 Identification: Challenges and solutions 376 14.5.1 Hypothetical experiments for identifying average treatment effects in the presence of social interactions 376 14.5.2 Hypothetical experiments for identifying the impact of social interactions 380 14.5.3 Application to an evaluation of kindergarten retention 382 14.6 Analytic strategies for experimental and quasiexperimental data 384 14.6.1 Estimation with experimental data 384 14.6.2 Propensity score stratification 385 14.6.3 MMWS 386 14.7 Summary 387 15 Mediation through spill-over 391 15.1 Definition of mediated effects through spill-over in a cluster randomized trial 393 15.1.1 Notation 393 15.1.2 Treatment effect mediated by a focal individual s compliance 394 15.1.3 Treatment effect mediated by peers compliance through spill-over 394 15.1.4 Decomposition of the total treatment effect 395 15.2 Identification and estimation of the spill-over effect in a cluster randomized design 395 15.2.1 Identification in an ideal experiment 395 15.2.2 Identification when the mediators are not randomized 398 15.2.3 Estimation of mediated effects through spill-over 400 15.3 Definition of mediated effects through spill-over in a multisite trial 402 15.3.1 Notation 402 15.3.2 Treatment effect mediated by a focal individual s compliance 404 15.3.3 Treatment effect mediated by peers compliance through spill-over 404 15.3.4 Direct effect of individual treatment assignment on the outcome 405 15.3.5 Direct effect of peer treatment assignment on the outcome 405 15.3.6 Decomposition of the total treatment effect 405 15.4 Identification and estimation of spill-over effects in a multisite trial 406 15.4.1 Identification in an ideal experiment 407 15.4.2 Identification when the mediators are not randomized 409 15.4.3 Estimation of mediated effects through spill-over 410 15.4.3.1 Estimating E[Y(1,p,M(p),M (p))] and E[Y(0,0,0,M (0))] 410 15.4.3.2 Estimating E[Y(1,p,0,M (p))] 411 15.4.3.3 Estimating E[Y(1,p,0,M (0))] 412 15.4.3.4 Estimating E[Y(0,p,0,M (0))] 412 15.5 Consequences of omitting spill-over effects in causal mediation analyses 412 15.5.1 Biased inference in a cluster randomized trial 413 15.5.2 Biased inference in a multisite randomized trial 413 15.5.3 Biased inference of the local average treatment effect 415 15.6 Quasiexperimental application 416 15.7 Summary 419 Appendix 15.1: Derivation of the weight for estimating the population average counterfactual outcome E[Y(1, p, 0,M ( p))] 419 Appendix 15.2: Derivation of bias in the ITT effect due to the omission of spill-over effects 420 Index 423