1. What the Book Is About

2. New in the Second Edition

3. Acknowledgments

1. Introduction

1.1 Observational Studies

1.2 History and Development

1.3 Randomized Experiments

1.3.1 Fisher's Randomized Experiment

1.3.2 Types of Randomized Experiments and Statistical Tests

1.3.3 Critiques of Social Experimentation

1.4 Why and When a Propensity Score Analysis Is Needed

1.5 Computing Software Packages

1.6 Plan of the Book

2. Counterfactual Framework and Assumptions

2.1 Causality, Internal Validity, and Threats

2.2 Counterfactuals and the Neyman-Rubin Counterfactual Framework

2.3 The Ignorable Treatment Assignment Assumption

2.4 The Stable Unit Treatment Value Assumption

2.5 Methods for Estimating Treatment Effects

2.5.1 Design of Observational Study

2.5.2 The Seven Models

2.5.3 Other Balancing Methods

2.5.4 Instrumental Variables Estimator

2.5.5 Regression Discontinuity Designs

2.6 The Underlying Logic of Statistical Inference

2.7 Types of Treatment Effects

2.8 Treatment Effect Heterogeneity

2.8.1 The Importance of Studying Treatment Effect Heterogeneity

2.8.2 Checking the Plausability of the Unconfoundedness Assumption

2.8.3 A Methodological Note About the Hausman Test of Endogeneity

2.8.4 Tests of Treatment Effect Heterogeneity

2.8.5 Example

2.9 Heckman's Econometric Model of Causality
2.10 Conclusion

3. Conventional Methods for Data Balancing

3.1 Why Is Data Balancing Necessary? A Heuristic Example

3.2 Three Methods for Data Balancing

3.2.1 The Ordinary Least Squares Regression

3.2.2 Matching

3.2.3 Stratification

3.3 Design of the Data Simulation

3.4 Results of the Data Simulation

3.5 Implications of the Data Simulation

3.6 Key Issues Regarding the Application of OLS Regression

3.7 Conclusion

4. Sample Selection and Related Models

4.1 The Sample Selection Model

4.1.1 Truncation, Censoring, and Incidental Truncation

4.1.2 Why Is It Important to Model Sample Selection?

4.1.3 Moments of an Incidentally Truncated Bivariate Normal Distribution

4.1.4 The Heckman Model and Its Two-Step Estimator

4.2 Treatment Effect Model

4.3 Overview of the Stata Programs and Main Features of

**treatreg**
4.4 Examples

4.4.1 Application of the Treatment Effect model to Analysis of Observational Data

4.4.2 Evaluation of Treatment Effects from a Program With a Group Randomization Design

4.4.3 Running the Treatment Effect Model After Multiple Imputations of Missing Data

4.5 Conclusion

5. Propensity Score Matching and Related Models

5.1 Overview

5.2 The Problem of Dimensionality and the Properties of Propensity Scores

5.3 Estimating Propensity Scores

5.3.1 Binary Logistic Regression

5.3.2 Strategies to Specify a Correct Model—Predicting Propensity Scores

5.3.3 Hirano and Imbens's Method for Specifying Predictors Relying on Predetermined Critical *t* Values

5.3.4 Generalized Boosted Modeling

5.4 Matching

5.4.1 Greedy Matching

5.4.2 Optimal Matching

5.4.3 Fine Balance

5.5 Postmatching Analysis

5.5.1 Multivariate Analysis After Greedy Matching

5.5.2 Computing Indices of Covariate Imbalance

5.5.3 Outcome Analysis Using the Hodges-Lehmann Aligned Rank Test After Optimal Matching

5.5.4 Regression Adjustment Based on Sample Created by Optimal Pair Matching

5.5.5 Regression Adjustment Using Hodges-Lehmann Aligned Rank Scores After Optimal Matching

5.6 Propensity Score Matching With Multilevel Data

5.6.1 Overview of Statistical Approaches to Multilevel Data

5.6.2 Perspectives Extending the Propensity Score Analysis to the Multilevel Modeling

5.6.3 Estimation of the Propensity Scores Under the Context of Multilevel Modeling

5.6.4 Multilevel Outcome Analysis

5.7 Overview of the Stata and R Programs

5.8 Examples

5.8.1 Greedy Matching and Subsequent Analysis of Hazard Rates

5.8.2 Optimal Matching

5.8.3 Post-Full Matching Analysis Using the Hodges-Lehmann Aligned Rank Test

5.8.4 Post-Pair Matching Analysis Using Regression of Difference Scores

5.8.5 Multilevel Propensity Score Analysis

5.8.6 Comparison of Rand-*gbm* and Stata's *boost* Algorithms

5.9 Conclusion

6. Propensity Score Subclassification

6.1 Overview

6.2 The Overlap Assumption and Methods to Address Its Violation

6.3 Structural Equation Modeling With Propensity Score Subclassification

6.3.1 The Need for Integrating SEM and Propensity Score Modeling Into One Analysis

6.3.2 Kaplan's (1999) Work to Integrate Propensity Score Subclassification With SEM

6.3.3 Conduct SEM With Propensity Score Subclassification

6.4 The Stratification-Multilevel Method

6.5 Examples

6.5.1 Stratification After Greedy Matching

6.5.2 Subclassification Followed by a Cox Proportional Hazards Model

6.5.3 Propensity Score Subclassification in Conjunction with SEM

6.6 Conclusion

7. Propensity Score Weighting

7.1 Overview

7.2 Weighting Estimators

7.2.1 Formulas for Creating Weights to Estimate ATE and ATT

7.2.2 A Corrected Version of Weights Estimating ATE

7.2.3 Steps in Propensity Score Weighting

7.3 Examples

7.3.1 Propensity Score Weighting With a Multiple Regression Outcome Analysis

7.3.2 Propensity Score Weighting With a Cox Proportional Hazards Model

7.3.3 Propensity Score Weighting With an SEM

7.3.4 Comparison of Models and Conclusions of the Study of the Impact of Poverty on Child Academic Achievement

7.4 Conclusion

8. Matching Estimators

8.1 Overview

8.2 Methods of Matching Estimators

8.2.1 Simple Matching Estimators

8.2.2 Bias-Corrected Matching Estimator

8.2.3 Variance Estimator Assuming Homoscedasticity

8.2.4 Variance Estimator Allowing for Heteroscedasticity

8.2.5 Large Sample Properties and Correction

8.3 Overview of the Stata Program

**nnmatch**
8.4 Examples

8.4.1 Matching With Bias-Corrected and Robust Variance Estimators

8.4.2 Efficacy Subset Analysis With Matching Estimators

8.5 Conclusion

9. Propensity Score Analysis With Nonparametric Regression

9.1 Overview

9.2 Methods of Propensity Score Analysis With Nonparametric Regression

9.2.1 The Kernel-Based Matching Estimators

9.2.2 Review of the Basic Concepts of Local Linear Regression **(lowess)**

9.2.3 Asymptotic and Finite-Sample Properties of Kernel and Local Linear Matching

9.3 Overview of the Stata Programs

**psmatch2** and

**bootstrap**
9.4 Examples

9.4.1 Analysis of Difference-in-Differences

9.4.2 Application of Kernel-Based Matching to One-Point Data

9.5 Conclusion

10. Propensity Score Analysis of Categorical or Continuous Treatments: Dosage Analyses

10.1 Overview

10.2 Modeling Doses With a Single Scalar Balancing Score Estimated by an Ordered Logistic Regression

10.3 Modeling Doses With Multiple Balancing Scores Estimated by a Multinomial Logit Model

10.4 The Generalized Propensity Score Estimator

10.5 Overview of the Stata

**gpscore** Program

10.6 Examples

10.6.1 Modeling Doses of Treatment With Multiple Balancing Scores Estimated by a Multinomial Logit Model

10.6.2 Modeling Doses of Treatment With the Generalized Propensity Score Estimator

10.7 Conclusion

11. Selection Bias and Sensitivity Analysis

11.1 Selection Bias: An Overview

11.1.1 Sources of Selection Bias

11.1.2 Overt Bias Versus Hidden Bias

11.1.3 Consequences of Selection Bias

11.1.4 Strategies to Correct for Selection Bias

11.2 A Monte Carlo Study Comparing Corrective Models

11.2.1 Design of the Monte Carlo Study

11.2.2 Results of the Monte Carlo Study

11.2.3 Implications

11.3 Rosenbaum's Sensitivity Analysis

11.3.1 The Basic Idea

11.3.2 Illustration of Wilcoxon's Signed Rank Test for Sensitivity Analysis of a Matched Pair Study

11.4 Overview of the Stata Program

**rbounds**
11.5 Examples

11.5.1 Sensitivity Analysis of the Effects of Lead Exposure

11.5.2 Sensitivity Analysis for the Study Using Pair Matching

11.6 Conclusion

12. Concluding Remarks

12.1 Common Pitfalls in Observational Studies: A Checklist for Critical Review

12.2 Approximating Experiments With Propensity Score Approaches

12.2.1 Criticism of Propensity Score Methods

12.2.2 Regression and Propensity Score Approaches: Do They Provide Similar Results?

12.2.3 Criticism of Sensitivity Analysis (Γ)

12.2.4 Group Randomized Trials

12.3 Other Advances in Modeling Causality

12.4 Directions for Future Development