vanderweele2015_appendix.org

Table of Contents for Appendix in VanderWeele 2015

A.1. EXPLANATION AND MECHANISM –459

A.2. MEDIATION: INTRODUCTION AND REGRESSION-BASED APPROACHES –461

A.2.1. Definitions and Identification –461

A.2.2. Regression Methods for Direct and Indirect Effects –465

A.2.3. Equivalence of the Product and Difference Methods for a Continuous Outcome and for a Rare Binary Outcome –476

A.2.4. The Product Method as a Valid Test of the Presence of Any Mediated Effect –477

A.3. SENSITIVITY ANALYSIS FOR MEDIATION –478

A.3.1. Sensitivity Analysis for Unmeasured Confounding for Total Effects on the Difference Scale –478

A.3.2. Sensitivity Analysis for Unmeasured Confounding for a Total Effect on a Ratio Scale –481

A.3.3. Sensitivity Analysis for Unmeasured Confounding for Controlled Direct Effects –484

A.4. MEDIATION ANALYSIS WITH SURVIVAL DATA –492

A.4.1. Definitions for Mediation in a Survival Context –492

A.4.2. Mediation with Accelerated Failure Time Models –494

A.4.3. Mediation with Proportional Hazards Models –496

A.4.4. Mediation with Additive Hazards Models –501

A.4.5. A Weighting Approach to Mediation with Survival Data –504

A.4.6. Sensitivity Analysis with Survival Data –505

A.5. MULTIPLE MEDIATORS –510

A.5.1. Notation –510

A.5.2. Regression-Based Approach –511

A.5.3. Weighting Approach –516

A.5.4. Sum of Individual Mediated Effects Versus Joint Mediated Effects –517

A.5.5. Marginal Structural Models for Controlled Direct Effects in the Presence of Exposure-Induced Confounding –518

A.5.6. Structural Mean Models for Controlled Direct Effects in the Presence of Exposure-Induced Confounding –519

A.5.7. Effect Decomposition and Exposure-Induced Confounding –520

A.5.8. Path-Specific Effects –522

A.5.9. Sensitivity Analysis for Exposure-Induced Confounding –524

A.6. MEDIATION ANALYSIS WITH TIME-VARYING EXPOSURES AND MEDIATORS –528

A.6.1. Notation and Definitions for Time-Varying Exposures and Mediators –528

A.6.2. Controlled Direct Effects with Time-Varying Exposures and Mediators –529

A.6.3. Natural Direct and Indirect Effect and their Randomized Interventional Analogues with Time-Varying Exposures and Mediators –530

A.6.4. Counterfactual Analysis of MacKinnon’s Three-Wave Mediation Model –534

A.7. SELECTED TOPICS IN MEDIATION ANALYSIS –536

A.7.1. Multiple Versions of the Mediator and Ill-Defined Mediators –536

A.7.2. Interpretation of Direct and Indirect Effect Estimates Without the Cross-World Independence Assumption –541

A.7.3. Direct and Indirect Effects in Health Disparities Research –542

A.8. OTHER TOPICS RELATED TO INTERMEDIATES –549

A.8.1. Principal Stratification –549

A.8.2. Surrogate Outcomes –552

A.8.3. Instrumental Variables and Mendelian Randomization –553

A.9. AN INTRODUCTION TO INTERACTION ANALYSIS –555

A.9.1. Standard Error for RERI –555

A.9.2. Interaction Versus Effect Modification –556

A.9.3. Attributing Effects to Interactions –557

A.10. MECHANISTIC INTERACTION –561

A.10.1. Sufficient Cause and Epistatic Interactions –561

A.10.2. Extension to n-Way Sufficient Cause Interaction –565

A.10.3. Other Extensions: Sufficient Cause Interactions with Dichotomized Continuous Exposures and Under Independence of Background Causes –575

A.10.4. Antagonism –578

A.11. BIAS ANALYSIS FOR INTERACTIONS –580

A.11.1. Sensitivity Analysis and Robustness for Additive Interaction –580

A.11.2. Sensitivity Analysis and Robustness for Multiplicative Interaction –584

A.11.3. Sensitivity Analysis for the Relative Excess Risk Due to Interaction –588

A.11.4. Measurement Error and Additive Interaction –590

A.11.5. Measurement Error and Multiplicative Interaction –593

A.12. INTERACTION IN GENETICS: INDEPENDENCE AND BOOSTING POWER –594

A.13. POWER AND SAMPLE SIZE CALCULATIONS FOR INTERACTION ANALYSIS –594

A.13.1. Power and Sample-Size Calculations for Interaction for Continuous Outcomes –594

A.13.2. Power and Sample-Size Calculations for Binary Outcomes: Multiplicative Interaction –596

A.13.3. Derivations for Case–Control Exposure Probabilities from the Probabilities in the Underlying Population –598

A.13.4. Multiplicative Interaction with Case-Only Data –600

A.13.5. Additive Interaction in Cohort Studies Using Linear Risk Model –600

A.13.6. Additive Interaction in Cohort Studies Using Logistic Regression and RERI –602

A.13.7. Multiplicative and Additive Interaction for the log-linear model –603

A.14. A UNIFICATION OF MEDIATION AND INTERACTION –606

A.14.1. A General Four-Way Decomposition –606

A.14.2. Continuous Outcomes and Linear Regression Models –611

A.14.3. Decomposition on a Ratio Scale and Logistic Regression Models –612

A.14.4. Decomposition in the Presence of an Exposure-Induced Mediator–Outcome Confounder –619

A.15. SOCIAL INTERACTIONS AND SPILLOVER EFFECTS –621

A.15.1. Notation and Definitions –621

A.15.2. Basic Spillover and Individual/Direct Effects –622

A.15.3. Infectiousness Effect –623

A.15.4. Contagion Versus Infectiousness Effects –626

A.15.5. Tests for Specific Forms of Interference Using Causal Interactions –630

A.15.6. Inferential Challenges with Many Individuals per Cluster –633

A.15.7. Spillover Effects and Observational Data –636