Causality in a Social World - Guanglei Hong - E-Book

Causality in a Social World E-Book

Guanglei Hong

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Causality in a Social World introduces innovative new statistical research and strategies for investigating moderated intervention effects, mediated intervention effects, and spill-over effects using experimental or quasi-experimental data. The book uses potential outcomes to define causal effects, explains and evaluates identification assumptions using application examples, and compares innovative statistical strategies with conventional analysis methods. Whilst highlighting the crucial role of good research design and the evaluation of assumptions required for identifying causal effects in the context of each application, the author demonstrates that improved statistical procedures will greatly enhance the empirical study of causal relationship theory. Applications focus on interventions designed to improve outcomes for participants who are embedded in social settings, including families, classrooms, schools, neighbourhoods, and workplaces.

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CONTENTS

Cover

Title page

Preface

Part I: OVERVIEW

1 Introduction

1.1 Concepts of moderation, mediation, and spill-over

1.2 Weighting methods for causal inference

1.3 Objectives and organization of the book

1.4 How is this book situated among other publications on related topics?

References

2 Review of causal inference concepts and methods

2.1 Causal inference theory

2.2 Applications to Lord’s paradox and Simpson’s paradox

2.3 Identification and estimation

Appendix 2.1: Potential bias in a prima facie effect

Appendix 2.2: Application of the causal inference theory to Lord’s paradox

References

3 Review of causal inference designs and analytic methods

3.1 Experimental designs

3.2 Quasiexperimental designs

3.3 Statistical adjustment methods

3.4 Propensity score

Appendix 3.A: Potential bias due to the omission of treatment-by-covariate interaction

Appendix 3.B: Variable selection for the propensity score model

References

4 Adjustment for selection bias through weighting

4.1 Weighted estimation of population parameters in survey sampling

4.2 Weighting adjustment for selection bias in causal inference

4.3 MMWS

Appendix 4.A: Proof of MMWS-adjusted mean observed outcome being unbiased for the population average potential outcome

Appendix 4.B: Derivation of MMWS for estimating the treatment effect on the treated

Appendix 4.C: Theoretical equivalence of MMWS and IPTW

Appendix 4.D: Simulations comparing MMWS and IPTW under misspecifications of the functional form of a propensity score model

References

5 Evaluations of multivalued treatments

5.1 Defining the causal effects of multivalued treatments

5.2 Existing designs and analytic methods for evaluating multivalued treatments

5.3 MMWS for evaluating multivalued treatments

5.4 Summary

Appendix 5.A: Multiple IV for evaluating multivalued treatments

References

Part II: MODERATION

6 Moderated treatment effects: concepts and existing analytic methods

6.1 What is moderation?

6.2 Experimental designs and analytic methods for investigating explicit moderators

6.3 Existing research designs and analytic methods for investigating implicit moderators

Appendix 6.A: Derivation of bias in the fixed-effects estimator when the treatment effect is heterogeneous in multisite randomized trials

Appendix 6.B: Derivation of bias in the mixed-effects estimator when the probability of treatment assignment varies across sites

Appendix 6.C: Derivation and proof of the population weight applied to mixed-effects models for eliminating bias in multisite randomized trials

References

7 Marginal mean weighting through stratification for investigating moderated treatment effects

7.1 Existing methods for moderation analyses with quasiexperimental data

7.2 MMWS estimation of treatment effects moderated by individual or contextual characteristics

7.3 MMWS estimation of the joint effects of concurrent treatments

References

8 Cumulative effects of time-varying treatments

8.1 Causal effects of treatment sequences

8.2 Existing strategies for evaluating time-varying treatments

8.3 MMWS for evaluating 2-year treatment sequences

8.4 MMWS for evaluating multiyear sequences of multivalued treatments

8.5 Conclusion

Appendix 8.A: A saturated model for evaluating multivalued treatments over multiple time periods

References

Part III: MEDIATION

9 Concepts of mediated treatment effects and experimental designs for investigating causal mechanisms

9.1 Introduction

9.2 Path coefficients

9.3 Potential outcomes and potential mediators

9.4 Causal effects with counterfactual mediators

9.5 Population causal parameters

9.6 Experimental designs for studying causal mediation

References

10 Existing analytic methods for investigating causal mediation mechanisms

10.1 Path analysis and SEM

10.2 Modified regression approaches

10.3 Marginal structural models

10.4 Conditional structural models

10.5 Alternative weighting methods

10.6 Resampling approach

10.7 IV method

10.8 Principal stratification

10.9 Sensitivity analysis

10.10 Conclusion

Appendix 10.A: Bias in path analysis estimation due to the omission of treatment-by-mediator interaction

References

11 Investigations of a simple mediation mechanism

11.1 Application example: national evaluation of welfare-to-work strategies

11.2 RMPW rationale

11.3 Parametric RMPW procedure

11.4 Nonparametric RMPW procedure

11.5 Simulation results

11.6 Discussion

Appendix 11.A: Causal effect estimation through the RMPW procedure

Appendix 11.B: Proof of the consistency of RMPW estimation

References

12 RMPW extensions to alternative designs and measurement

12.1 RMPW extensions to mediators and outcomes of alternative distributions

12.2 RMPW extensions to alternative research designs

12.3 Alternative decomposition of the treatment effect

References

13 RMPW extensions to studies of complex mediation mechanisms

13.1 RMPW extensions to moderated mediation

13.2 RMPW extensions to concurrent mediators

13.3 RMPW extensions to consecutive mediators

13.4 Discussion

Appendix 13.A: Derivation of RMPW for estimating population average counterfactual outcomes of two concurrent mediators

Appendix 13.B: Derivation of RMPW for estimating population average counterfactual outcomes of consecutive mediators

References

Part IV: SPILL-OVER

14 Spill-over of treatment effects: concepts and methods

14.1 Spill-over: A nuisance, a trifle, or a focus?

14.2 Stable versus unstable potential outcome values: An example from agriculture

14.3 Consequences for causal inference when spill-over is overlooked

14.4 Modified framework of causal inference

14.5 Identification: Challenges and solutions

14.6 Analytic strategies for experimental and quasiexperimental data

14.7 Summary

References

15 Mediation through spill-over

15.1 Definition of mediated effects through spill-over in a cluster randomized trial

15.2 Identification and estimation of the spill-over effect in a cluster randomized design

15.3 Definition of mediated effects through spill-over in a multisite trial

15.4 Identification and estimation of spill-over effects in a multisite trial

15.5 Consequences of omitting spill-over effects in causal mediation analyses

15.6 Quasiexperimental application

15.7 Summary

Appendix 15.1: Derivation of the weight for estimating the population average counterfactual outcome

E

[

Y

(1,

p

,0,M

(

p

))]

Appendix 15.2: Derivation of bias in the ITT effect due to the omission of spill-over effects

References

Index

End User License Agreement

List of Tables

Chapter 02

Table 2.1 Glossary of basic concepts in causal inference

Table 2.2 Potential outcomes and population average causal effect

Chapter 03

Table 3.1 Propensity score and its balancing property

Table 3.2 Within-stratum mean difference in the reading outcome between retained and promoted students

Chapter 04

Table 4.1 Propensity score stratification and MMWS computation

Chapter 05

Table 5.1 Computation of marginal mean weight through stratification for four treatment groups

Table 5.2 Between treatment group differences in logit propensity scores before and after weighting

Table 5.3 MMWS for a multidosage treatment

Chapter 06

Table 6.1 Potential outcomes and joint effects of two concurrent treatments

Chapter 07

Table 7.1 Marginal mean weight through stratification for four treatment groups: Hispanic

Table 7.2 Marginal mean weight through stratification for four treatment groups: Non-Hispanic

Table 7.3 Computed MMWS for evaluating the joint effects of instructional time and grouping

Chapter 08

Table 8.1 Glossary for the causal effects of 2-year treatment sequences

Table 8.2 Repeated observations at level 1 for a two-level model for evaluating 2-year treatment sequences

Chapter 09

Table 9.1 Glossary of causal effects in mediation analysis

Table 9.2 Potential mediators and potential outcomes

Table 9.3 A comparison of experimental designs for studying causal mediation

Chapter 10

Table 10.1 Identification assumptions for estimating the natural direct and indirect effects

Table 10.2 Comparisons of identification assumptions across the alternative analytic methods

Chapter 11

Table 11.1 RMPW applied to data from a sequentially randomized design

Table 11.2 RMPW applied to data from a sequentially randomized block design

Table 11.3 Parametric RMPW applied to data from a standard randomized experiment

Table 11.4 Nonparametric RMPW applied to data from a standard randomized experiment

Table 11.5 Parametric and nonparametric RMPW estimation of the causal effects

Chapter 12

Table 12.1 Parametric RMPW for analyzing data from multisite randomized trials

Table 12.2 Parametric RMPW for the second set of treatment effect decomposition

Chapter 13

Table 13.1 Glossary of potential outcomes associated with two concurrent mediators

Table 13.2 Decomposition of the population average total effect with two noninteracting concurrent mediators

Table 13.3 Decomposition of the population average total effect with two interacting concurrent mediators

Table 13.4 Sequential ignorability required by the RMPW strategy for two concurrent mediators

Table 13.5 Parametric RMPW for treatment effect decomposition with two noninteracting concurrent mediators

Table 13.6 Parametric RMPW for treatment effect decomposition with two interacting concurrent mediators

Table 13.7 Decomposition of the population average total effect with two interacting consecutive mediators

Table 13.8 Decomposition of the population average total effect with two interacting consecutive mediators and treatment-by-mediator interactions

Table 13.9 Sequential ignorability required by the RMPW strategy for two consecutive mediators

Table 13.10 Parametric RMPW for treatment effect decomposition with two consecutive mediators

Table 13.11 Parametric RMPW for treatment effect decomposition with two interacting consecutive mediators

Table 13.12 Parametric RMPW for treatment effect decomposition with two interacting consecutive mediators and treatment-by-mediator interactions

Chapter 15

Table 15.1 Decomposition of the total treatment effect in a cluster randomized trial

Table 15.2 Decomposition of the total treatment effect in a multisite randomized trial

List of Illustrations

Chapter 02

Figure 2.1

Lord’s paradox: statistician 1’s gain score analysis

Figure 2.2

Lord’s paradox: statistician 2’s analysis of covariance

Chapter 03

Figure 3.1

Treatment effect estimation through an analysis of covariance:

Figure 3.2

Biased estimation of the treatment effect due to the omission of a treatment- by-covariate interaction

Figure 3.3

Identify common support by comparing the distribution of the estimated logit of propensity score between the treated group and the control group

Chapter 04

Figure 4.1

Proportionate sample, disproportionate sample, and weighted sample

Figure 4.2

Propensity score-based weighting for removing selection bias

Chapter 05

Figure 5.1

Analysis of covariance for evaluating multivalued treatments:

Figure 5.2

Common support for evaluating four-category treatments. The use of this information does not imply endorsement by the publisher.

Chapter 06

Figure 6.1

A hypothetical example of aptitude–treatment interaction

Chapter 07

Figure 7.1

Mean of learning approaches for Hispanic and non-Hispanic language minority students by ELL services: (a) unweighted and (b) weighted

Figure 7.2

Weighted estimates of kindergartners’ yearly growth rate in literacy by time and grouping

Chapter 08

Figure 8.1

Potential 2-year growth trajectories of a hypothetical student associated with alternative treatment sequences

Figure 8.2

Endogeneity of time-varying treatments to time-varying covariates

Figure 8.3

Sequential randomization of 2-year treatment sequences

Figure 8.4

Sequential stratification of 2 years of treatment data

Figure 8.5

Causal effects of 2-year treatment sequences

Chapter 09

Figure 9.1

Path diagram representing the effect of

Z

on

Y

indirectly through

M

Chapter 11

Figure 11.1

Hypothesized causal relationships between the treatment (Z), mediator (M), and outcome (Y)

Figure 11.2

Average potential outcomes under a hypothetical sequentially randomized experiment

Chapter 13

Figure 13.1

Two concurrent mediators

Figure 13.2

Two consecutive mediators

Guide

Cover

Table of Contents

Begin Reading

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Causality in a Social World

Moderation, Meditation and Spill-over

 

Guanglei Hong

Department of Comparative Human Development, University of Chicago, USA

 

 

 

 

 

 

This edition first published 2015© 2015 John Wiley & Sons, Ltd

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Preface

A scientific mind in training seeks comprehensive descriptions of every interesting phenomenon. In such a mind, data are to be pieced together for comparisons, contrasts, and eventually inferences that are required for understanding causes and effects that may have led to the phenomenon of interest. Yet without disciplined reasoning, a causal inference would slip into the rut of a “casual inference” as easily as making a typographical error.

As a doctoral student thirsting for rigorous methodological training at the University of Michigan, I was very fortunate to be among the first group of students on the campus attending a causal inference seminar in 2000 jointly taught by Yu Xie from the Department of Sociology, Susan Murphy from the Department of Statistics, and Stephen Raudenbush from the School of Education. The course was unique in both content and pedagogy. The instructors put together a bibliography drawn from the past and current causal inference literature originated from multiple disciplines. In the spirit of “reciprocal teaching,” the students were deliberately organized into small groups, each representing a mix of disciplinary backgrounds, and took turns to teach the weekly readings under the guidance of a faculty instructor. This invaluable experience revealed to me that the pursuit of causal inference is a multidisciplinary endeavor. In the rest of my doctoral study, I continued to be immersed in an extraordinary interdisciplinary community in the form of the Quantitative Methodology Program (QMP) seminars led by the above three instructors who were then joined by Ben Hansen (Statistics), Richard Gonzalez (Psychology), Roderick Little (Biostatistics), Jake Bowers (Political Science), and many others. I have tried, whenever possible, to carry the same spirit into my own teaching of causal inference courses at the University of Toronto and now at the University of Chicago. In these courses, communicating understandings of causal problems across disciplinary boundaries has been particularly challenging but also stimulating and gratifying for myself and for students from various departments and schools.

Under the supervision of Stephen Raudenbush, I took a first stab at the conceptual framework of causal inference in my doctoral dissertation by considering peer spill-over in school settings. From that point on, I have organized my methodological research to tackle some of the major obstacles to causal inferences in policy and program evaluations. My work addresses issues including (i) how to conceptualize and evaluate the causal effects of educational treatments when students’ responses to alternative treatments depend on various features of the organizational settings including peer composition, (ii) how to adjust for selection bias in evaluating the effects of concurrent and consecutive multivalued treatments, and (iii) how to conceptualize and analyze causal mediation mechanisms. My methodological research has been driven by the substantive interest in understanding the role of social institutions—schools in particular—in shaping human development especially during the ages of fast growth (e.g., childhood, adolescence, and early adulthood). I have shared methodological advances with a broad audience in social sciences through applying the causal inference methods to prominent substantive issues such as grade retention, within-class grouping, services for English language learners, and welfare-to-work strategies. These illustrations are used extensively in this book.

Among social scientists, awareness of methodological challenges in causal investigations is higher than ever before. In response to this rising demand, causal inference is becoming one of the most productive scholarly fields in the past decade. I have had the benefit of following the work and exchanging ideas with some fellow methodologists who are constantly contributing new thoughts and making breakthroughs. These include Daniel Almirall, Howard Bloom, Tom Cook, Michael Elliot, Ken Frank, Ben Hansen, James Heckman, Jennifer Hill, Martin Huber, Kosuke Imai, Booil Jo, John R. Lockwood, David MacKinnon, Daniel McCaffrey, Luke Miratrix, Richard Murnane, Derek Neal, Lindsey Page, Judea Pearl, Stephen Raudenbush, Sean Reardon, Michael Seltzer, Youngyun Shin, Betsy Sinclair, Michael Sobel, Peter Steiner, Elizabeth Stuart, Eric Thetchen Tchetchen, Tyler VanderWeele, and Kazuo Yamaguchi. I am grateful to Larry Hedges who offered me the opportunity of guest editing the Journal of Research in Educational Effectiveness special issue on the statistical approaches to studying mediator effects in education research in 2012. Many of the colleagues whom I mentioned above contributed to the open debate in the special issue by either proposing or critically examining a number of innovative methods for studying causal mechanisms. I anticipate that many of them are or will soon be writing their own research monographs on causal inference if they have not already done so. Therefore, readers of this book are strongly recommended to browse past and future titles by these and other authors for alternative perspectives and approaches. My own understanding, of course, will continue to evolve as well in the coming years.

One has to be ambitious to stay upfront in substantive areas as well as in methodology. The best strategy apparently is to learn from substantive experts. During different phases of my training and professional career, I have had the opportunities to work with David K. Cohen, Brian Rowan, Deborah Ball, Carl Corter, Janette Pelletier, Takako Nomi, Esther Geva, David Francis, Stephanie Jones, Joshua Brown, and Heather D. Hill. Many of my substantive insights were derived from conversations with these mentors and colleagues. I am especially grateful to Bob Granger, former president of the William T. Grant Foundation, who reassured me that “By staying a bit broad you will learn a lot and many fields will benefit from your work.”

Like many other single-authored books, this one is built on the generous contributions of students and colleagues who deserve special acknowledgement. Excellent research assistance from Jonah Deutsch, Joshua Gagne, Rachel Garrett, Yihua Hong, Xu Qin, Cheng Yang, and Bing Yu was instrumental in the development and implementation of the new methods presented in this book. Richard Congdon, a renowned statistical software programmer, brought revolutionary ideas to interface designs in software development with the aim of facilitating users’ decision-making in a series of causal analyses. Richard Murnane, Stephen Raudenbush, and Michael Sobel carefully reviewed and critiqued multiple chapters of the manuscript draft. Additional comments and suggestions on earlier drafts came from Howard Bloom, Ken Frank, Ben Hansen, Jennifer Hill, Booil Jo, Ben Kelcey, David MacKinnon, and Fan Yang. Terese Schwartzman provided valuable assistance in manuscript preparation. The writing of this book was supported by a Scholars Award from the William T. Grant Foundation and an Institute of Education Sciences (IES) Statistical and Research Methodology in Education Grant from the U.S. Department of Education. All the errors in the published form are of course the sole responsibility of the author.

Project Editors Richard Davis, Prachi Sinha Sahay, and Liz Wingett and Assistant Editor Heather Kay at Wiley, and project Managers P. Jayapriya and R. Jayavel at SPi Global have been wonderful to work with throughout the manuscript preparation and final production of the book. Debbie Jupe, Commissioning Editor at Wiley, was remarkably effective in initiating the contact with me and in organizing anonymous reviews of the book proposal and of the manuscript. These constructive reviews have shaped the book in important ways. I cannot agree more with one of the reviewers that “causality cannot be established on a pure statistical ground.” The book therefore highlights the importance of substantive knowledge and research designs and places a great emphasis on clarifying and evaluating assumptions required for identifying causal effects in the context of each application. Much caution is raised against possible misusage of statistical techniques for analyzing causal relationships especially when data are inadequate. Yet I maintain that improved statistical procedures along with improved research designs would greatly enhance our ability in attempt to empirically examine well-articulated causal theories.

Part IOVERVIEW

1Introduction

According to an ancient Chinese fable, a farmer who was eager to help his crops grow went into his field and pulled each seedling upward. After exhausting himself with the work, he announced to his family that they were going to have a good harvest, only to find the next morning that the plants had wilted and died. Readers with minimal agricultural knowledge may immediately point out the following: the farmer’s intervention theory was based on a correct observation that crops that grow taller tend to produce more yield. Yet his hypothesis reflects a false understanding of the cause and the effect—that seedlings pulled to be taller would yield as much as seedlings thriving on their own.

In their classic Design and Analysis of Experiments, Hinkelmann and Kempthorne (1994; updated version of Kempthorne, 1952) discussed two types of science: descriptive science and the development of theory. These two types of science are interrelated in the following sense: observations of an event and other related events, often selected and classified for description by scientists, naturally lead to one or more explanations that we call “theoretical hypotheses,” which are then screened and falsified by means of further observations, experimentation, and analyses (Popper, 1963). The experiment of pulling seedlings to be taller was costly, but did serve the purpose of advancing this farmer’s knowledge of “what does not work.” To develop a successful intervention, in this case, would require a series of empirical tests of explicit theories identifying potential contributors to crop growth. This iterative process gradually deepens our knowledge of the relationships between supposed causes and effects—that is, causality—and may eventually increase the success of agricultural, medical, and social interventions.

1.1 Concepts of moderation, mediation, and spill-over

Although the story of the ancient farmer is fictitious, numerous examples can be found in the real world in which well-intended interventions fail to produce the intended benefits or, in many cases, even lead to unintended consequences. “Interventions” and “treatments,” used interchangeably in this book, broadly refer to actions taken by agents or circumstances experienced by an individual or groups of individuals. Interventions are regularly seen in education, physical and mental health, social services, business, politics, and law enforcement. In an education intervention, for example, teachers are typically the agents who deliver a treatment to students, while the impact of the treatment on student outcomes is of ultimate causal interest. Some educational practices such as “teaching to the test” have been criticized to be nearly as counterproductive as the attempt of helping seedlings grow by pulling them upward. “Interventions” and “treatments” under consideration do not exclude undesired experiences such as exposure to poverty, abuse, crime, or bereavement. A treatment, planned or unplanned, becomes a focus of research if there are theoretical reasons to anticipate its impact, positive or negative, on the well-being of individuals who are embedded in social settings including families, classrooms, schools, neighborhoods, and workplaces.

In social science research in general and in policy and program evaluations in particular, questions concerning whether an intervention works and, if so, which version of the intervention works, for whom, under what conditions, and why are key to the advancement of scientific and practical knowledge. Although most empirical investigations in the social sciences concentrate on the average effect of a treatment for a specific population as opposed to the absence of such a treatment (i.e., the control condition), in-depth theoretical reasoning with regard to how the causal effect is generated, substantiated by compelling empirical evidence, is crucial for advancing scientific understanding.

First, when there are multiple versions or different dosages of the treatment or when there are multiple versions of the control condition, a binary divide between “the treatment” and “the control” may not be as informative as fine-grained comparisons across, for example, “treatment version A,” “treatment version B,” “control version A,” and “control version B.” For example, expanding the federally funded Head Start program to poor children is expected to generate a greater benefit when few early childhood education alternatives are available (call it “control version A”) than when there is an abundance of alternatives including state-sponsored preschool programs (call it “control version B”).

Second, the effect of an intervention will likely vary among individuals or across social settings. A famous example comes from medical research: the well-publicized cardiovascular benefits of initiating estrogen therapy during menopause were contradicted later by experimental findings that the same therapy increased postmenopausal women’s risk for heart attacks. The effect of an intervention may also depend on the provision of some other concurrent or subsequent interventions. Such heterogeneous effects are often characterized as moderated effects in the literature.

Third, alternative theories may provide competing explanations for the causal mechanisms, that is, the processes through which the intervention produces its effect. A theoretical construct characterizing the hypothesized intermediate process is called a mediator of the intervention effect. The fictitious farmer never developed an elaborate theory as to what caused some seedlings to surpass others in growth. Once scientists revealed the causal relationship between access to chemical nutrients in soil and plant growth, wide applications of chemically synthesized fertilizers finally led to a major increase in crop production.

Finally, it is well known in agricultural experiments that a new type of fertilizer applied to one plot may spill-over to the next plot. Because social connections among individuals are prevalent within organizations or through networks, an individual’s response to the treatment may similarly depend on the treatment for other individuals in the same social setting, which may lead to possible spill-overs of intervention effects among individual human beings.

Answering questions with regard to moderation, mediation, and spill-over poses major conceptual and analytic challenges. To date, psychological research often presents well-articulated theories of causal mechanisms relating stimuli to responses. Yet, researchers often lack rigorous analytic strategies for empirically screening competing theories explaining the observed effect. Sociologists have keen interest in the spill-over of treatment effects transmitted through social interactions yet have produced limited evidence quantifying such effects. As many have pointed out, in general, terminological ambiguity and conceptual confusion have been prevalent in the published applied research (Holmbeck, 1997; Kraemer et al., 2002, 2008).

A new framework for conceptualizing moderated, mediated, and spill-over effects has emerged relatively recently in the statistics and econometrics literature on causal inference (e.g., Abbring and Heckman, 2007; Frangakis and Rubin, 2002; Heckman and Vytlacil, 2007a, b; Holland, 1988; Hong and Raudenbush, 2006; Hudgens and Halloran, 2008; Jo, 2008; Pearl, 2001; Robins and Greenland, 1992; Sobel, 2008). The potential for further conceptual and methodological development and for broad applications in the field of behavioral and social sciences promises to greatly advance the empirical basis of our knowledge about causality in the social world.

This book clarifies theoretical concepts and introduces innovative statistical strategies for investigating the average effects of multivalued treatments, moderated treatment effects, mediated treatment effects, and spill-over effects in experimental or quasiexperimental data. Defining individual-specific and population average treatment effects in terms of potential outcomes, the book relates the mathematical forms to the substantive meanings of moderated, mediated, and spill-over effects in the context of application examples. It also explicates and evaluates identification assumptions and contrasts innovative statistical strategies with conventional analytic methods.

1.1.1 Moderated treatment effects

It is hard to accept the assumption that a treatment would produce the same impact for every individual in every possible circumstance. Understanding the heterogeneity of treatment effects therefore is key to the development of causal theories. For example, some studies reported that estrogen therapy improved cardiovascular health among women who initiated its use during menopause. According to a series of other studies, however, the use of estrogen therapy increased postmenopausal women’s risk for heart attacks (Grodstein et al., 1996, 2000; Writing Group for the Women’s Health Initiative Investigators, 2002). The sharp contrast of findings from these studies led to the hypothesis that age of initiation moderates the effect of estrogen therapy on women’s health (Manson and Bassuk, 2007; Rossouw et al., 2007). Revelations of the moderated causal relationship greatly enrich theoretical understanding and, in this case, directly inform clinical practice.

In another example, dividing students by ability into small groups for reading instruction has been controversial for decades. Many believed that the practice benefits high-ability students at the expense of their low-ability peers and exacerbates educational inequality (Grant and Rothenberg, 1986; Rowan and Miracle, 1983; Trimble and Sinclair, 1987). Recent evidence has shown that, first of all, the effect of ability grouping depends on a number of conditions including whether enough time is allocated to reading instruction and how well the class can be managed. Hence, instructional time and class management are among the additional moderators to consider for determining the merits of ability grouping (Hong and Hong, 2009; Hong et al., 2012a, b). Moreover, further investigations have generated evidence that contradicts the long-held belief that grouping undermines the interests of low-ability students. Quite the contrary, researchers have found that grouping is beneficial for students with low or medium prior ability—if adequate time is provided for instruction and if the class is well managed. On the other hand, ability grouping appears to have a minimal impact on high-ability students’ literacy. These results were derived from kindergarten data and may not hold for math instruction and for higher grade levels. Replications with different subpopulations and in different contexts enable researchers to assess the generalizability of a theory.

In a third example, one may hypothesize that a student is unlikely to learn algebra well in ninth grade without a solid foundation in eighth-grade prealgebra. One may also argue that the progress a student made in learning eighth-grade prealgebra may not be sustained without the further enhancement of taking algebra in ninth grade. In other words, the earlier treatment (prealgebra in eighth grade) and the later treatment (algebra in ninth grade) may reinforce one other; each moderates the effect of the other treatment on the final outcome (math achievement at the end of ninth grade). As Hong and Raudenbush (2008) argued, the cumulative effect of two treatments in a well-aligned sequence may exceed the sum of the benefits of two single-year treatments. Similarly, experiencing two consecutive years of inferior treatments such as encountering an incompetent math teacher who dampens a student’s self-efficacy in math learning in eighth grade and then again in ninth grade could do more damage than the sum of the effect of encountering an incompetent teacher only in eighth grade and the effect of having such a teacher only in ninth grade.

There has been a great amount of conceptual confusion regarding how a moderator relates to the treatment and the outcome. On one hand, many researchers have used the terms “moderators” and “mediators” interchangeably without understanding the crucial distinction between the two. An overcorrective attempt, on the other hand, has led to the arbitrary recommendation that a moderator must occur prior to the treatment and be minimally associated with the treatment (James and Brett, 1984; Kraemer et al., 2001, 2008). In Chapter 6, we clarify that, in essence, the causal effect of the treatment on the outcome may depend on the moderator value. A moderator can be a subpopulation identifier, a contextual characteristic, a concurrent treatment, or a preceding or succeeding treatment. In the earlier examples, age of estrogen therapy initiation and a student’s prior ability are subpopulation identifiers, the manageability of a class which reflects both teacher skills and peer behaviors characterizes the context, literacy instruction time is a concurrent treatment, while prealgebra is a preceding treatment for algebra. A moderator does not have to occur prior to the treatment and does not have to be independent of the treatment.

Once a moderated causal relationship has been defined in terms of potential outcomes, the researcher then chooses an appropriate experimental design for testing the moderation theory. The assumptions required for identifying the moderated causal relationships differ across different designs and have implications for analyzing experimental and quasiexperimental data. Randomized block designs are suitable for examining individual or contextual characteristics as potential moderators; factorial designs enable one to determine the joint effects of two or more concurrent treatments; and sequential randomized designs are ideal for assessing the cumulative effects of consecutive treatments. Multisite randomized trials constitute an additional type of designs in which the experimental sites are often deliberately sampled to represent a population of geographical locations or social organizations. Site memberships can be viewed as implicit moderators that summarize a host of features of a local environment. Replications of a treatment over multiple sites allow one to quantify the heterogeneity of treatment effects across the sites. Chapters 7 and 8 are focused on statistical methods for moderation analyses with quasiexperimental data. In particular, these chapters demonstrate, through a number of application examples, how a nonparametric marginal mean weighting through stratification (MMWS) method can overcome some important limitations of other existing methods.

Moderation questions are not restricted to treatment–outcome relationships. Rather, they are prevalent in investigations of mediation and spill-over. This is because heterogeneity of treatment effects may be explained by the variation in causal mediation mechanisms across subpopulations and contexts. It is also because the treatment effect for a focal individual may depend on whether there is spill-over from other individuals through social interactions. Yet similar to the confusion around the concept of moderation among applied researchers, there has been a great amount of misunderstanding with regard to mediation.

1.1.2 Mediated treatment effects

Questions about mediation are at the core of nearly every scientific theory. In its simplest form, a theory explains why treatment Z causes outcome Y by hypothesizing an intermediate process involving at least one mediator M that could have been changed by the treatment and could subsequently have an impact on the outcome. A causal mediation analysis then decomposes the total effect of the treatment on the outcome into two parts: the effect transmitted through the hypothesized mediator, called the indirect effect, and the difference between the total effect and the indirect effect, called the direct effect. The latter represents the treatment effect channeled through other unspecified pathways (Alwin and Hauser, 1975; Baron and Kenny, 1986; Duncan, 1966; Shadish and Sweeney, 1991). Most researchers in social sciences have followed the convention of illustrating the direct effect and the indirect effect with path diagrams and then specifying linear regression models postulated to represent the structural relationships between the treatment, the mediator, and the outcome.

In Chapter 9, we point out that the approach to defining the indirect effect and the direct effect in terms of path coefficients is often misled by oversimplistic assumptions about the structural relationships. In particular, this approach typically overlooks the fact that a treatment may generate an impact on the outcome not only by changing the mediator value but also by changing the mediator–outcome relationship (Judd and Kenny, 1981). An example in Chapter 9 illustrates such a case. An experiment randomizes students to either an experimental condition which provides them with study materials and encourages them to study for a test or to a control condition which provides neither study materials nor encouragement (Holland, 1988; Powers and Swinton, 1984). One might hypothesize that encouraging experimental group members to study will increase the time they spend studying, a focal mediator in this example, which in turn will increase their average test scores. One might further hypothesize that even without a change in study time, providing study materials to experimental group members will enable them to study more effectively than they otherwise would. Consequently, the effect of time spent studying might be greater under the experimental condition than under the control condition. Omitting the treatment-by-mediator interaction effect will lead to bias in the estimation of the indirect effect and the direct effect.

Chapter 11 describes in great detail an evaluation study contrasting a new welfare-to-work program emphasizing active participation in the labor force with a traditional program that guaranteed cash assistance without a mandate to seek employment (Hong, Deutsch, and Hill, 2011). Focusing on the psychological well-being of welfare applicants who were single mothers with preschool-aged children, the researchers hypothesized that the treatment would increase employment rate among the welfare recipients and that the treatment-induced increase in employment would likely reduce depressive symptoms under the new program. They also hypothesized that, in contrast, the same increase in employment was unlikely to have a psychological impact under the traditional program. Hence, the treatment-by-mediator interaction effect on the outcome was an essential component of the intermediate process.

We will show that by defining the indirect effect and the direct effect in terms of potential outcomes, one can avoid invoking unwarranted assumptions about the unknown structural relationships (Holland, 1988; Pearl, 2001; Robins and Greenland, 1992). The potential outcomes framework has further clarified that in a causal mediation theory, the treatment and the mediator are both conceivably manipulable. In the aforementioned example, a welfare applicant could be assigned at random to either the new program or the traditional one. Once having been assigned to one of the programs, the individual might or might not be employed due to various structural constraints, market fluctuations, or other random events typically beyond her control.

Because the indirect effect and the direct effect are defined in terms of potential outcomes rather than on the basis of any specific regression models, it becomes possible to distinguish the definitions of these causal effects from their identification and estimation. The identification step relates a causal parameter to observable population data. For example, for individuals assigned at random to an experimental condition, their average counterfactual outcome associated with the control condition is expected to be equal to the average observable outcome of those assigned to the control condition. Therefore, the population average causal effect can be easily identified in a randomized experiment. The estimation step then relates the sample data to the observable population quantities involved in identification while taking into account the degree of sampling variability.

A major challenge in identification is that the mediator–outcome relationship tends to be confounded by selection even if the treatment is randomized. Chapter 9 reviews various experimental designs that have been proposed by past researchers for studying causal mediation mechanisms. Chapter 10 compares the identification assumptions and the analytic procedures across a wide range of analytic methods. These discussions are followed by an introduction of the ratio-of-mediator-probability weighting (RMPW) strategy in Chapter 11. In comparison with most existing strategies for causal mediation analysis, RMPW relies on relatively fewer identification assumptions and model-based assumptions. Chapters 12 and 13 will show that this new strategy can be applied broadly with extensions to multilevel experimental designs and to studies of complex mediation mechanisms involving multiple mediators.

1.1.3 Spill-over effects of a treatment

It is well known in vaccination research that an unvaccinated person can benefit when most other people in the local community are vaccinated. This is viewed as a spill-over of the treatment effect from the vaccinated people to an unvaccinated person. Similarly, due to social interactions among individuals or groups of individuals, an intervention received by some may generate a spill-over impact on others affiliated with the same organization or connected through the same network. For example, effective policing in one neighborhood may drive offenders to operate in other neighborhoods. In evaluating an innovative community policing program, researchers found that a neighborhood not assigned to community policing tended to suffer if the surrounding neighborhoods were assigned to community policing. This is an instance in which a well-intended treatment generates an unintended negative spill-over effect. However, being assigned to community policy was found particularly beneficial when the surrounding neighborhoods also received the intervention (Verbitsky-Savitz and Raudenbush, ). In another example, the impact of a school-based mentoring program targeted at students displaying delinquent behaviors may be enhanced for a focal student if his or her at-risk peers are assigned to the mentoring program at the same time.

The previous examples challenge the “stable unit treatment value assumption” (SUTVA) that has been invoked in most causal inference studies. This assumption states that an individual’s potential response to a treatment depends neither on how the treatment is assigned nor on the treatment assignments of other individuals (Rubin, 1980, 1986). Rubin and others believe that, without resorting to SUTVA, the causal effect of a treatment becomes hard to define and that causal inference may become intractable. However, as Sobel (2006) has illustrated with the Moving to Opportunity (MTO) experiment that offered housing vouchers enabling low-income families to move to low-poverty neighborhoods, there are possible consequences of violating SUTVA in estimating the treatment effects on outcomes such as safety and self-sufficiency. Because social interactions among individuals may affect whether one volunteers to participate in the study, whether one moves to a low-poverty neighborhood after receiving the housing voucher, as well as housing project residents’ subjective perceptions of their neighborhoods, the program may have a nonzero impact on the potential outcome of the untreated. As a result, despite the randomization of treatment assignment, the mean difference in the observed outcome between the treated units and the untreated units may be biased for the average treatment effect. Rather, the observable quantity is the difference between the effect of treating some rather than treating none for the treated and the effect of treating some rather than treating none for the untreated, the latter being the pure spill-over effect for the untreated.

In making attempts to relax SUTVA, researchers have proposed several alternative frameworks that incorporate possible spill-over effects (see Hong and Raudenbush, 2013, for a review). Hong (2004) presented a model that involves treatments and treatment settings. A treatment setting for an individual is a local environment constituted by a set of agents and participants along with their treatment assignments. An individual’s potential outcome value under a given treatment is assumed stable when the treatment setting is fixed; the potential outcome may take different values when the treatment setting shifts. One may investigate whether the treatment effect depends on the treatment setting. Applying this framework, Hong and Raudenbush (2006) examined the effect on a child’s academic growth of retaining the child in kindergarten rather than promoting the child to first grade when a relatively small proportion of low-achieving peers in the same school are retained as opposed to when a relatively large proportion of the peers are retained. Hudgens and Halloran (2008) presented a related framework in which the effect on an individual of the treatment received by this individual is distinguished from the effect on the individual of the treatment received by others in the same local community. These effects can be identified if communities are assigned at random to different treatment assignment strategies (such as retaining a large proportion of low-achieving students as opposed to retaining a small proportion of such students) and, subsequently, individuals within a community are randomized for treatment. Chapter 14 reviews these frameworks and discusses identification and estimation strategies.

Social contagion may also serve as an important channel through which the effect of an intervention is transmitted. For example, a school-wide intervention may reduce aggressive behaviors and thereby improve students’ psychological well-being by improving the quality of interpersonal relationships in other classes as well as in one’s own class. This is because children interact not simply with their classmates but also with those from other classes in the hallways or on the playground (VanderWeele et al., 2013). In this case, the spill-over between classes becomes a part of the mediation mechanism. In a study of student mentoring, whether an individual actually participated in the mentoring program despite the initial treatment assignment is typically viewed as a primary mediator. The proportion of one’s peers that participated in the program may act as a second mediator. A treatment may also exert its impact partly through regrouping individuals and thereby changing peer composition (Hong and Nomi, 2012). Chapter 15 discusses analytic strategies for detecting spill-over as a part of the causal mediation mechanism.

1.2 Weighting methods for causal inference

In most behavioral and social science applications, major methodological challenges arise due to the selection of treatments in a quasiexperimental study and the selection of mediator values in experimental and quasiexperimental studies. Statistical methods most familiar to researchers in social sciences are often inadequate for causal inferences with regard to multivalued treatments, moderation, mediation, and spill-over. The book offers a major revision to understanding of causality in the social world by introducing two complementary weighting strategies, both featuring nonparametric approaches to estimating these causal effects. The new weighting methods greatly simplify model specifications while enhancing the robustness of results by minimizing the reliance on some key assumptions.

The propensity score-based MMWS method removes selection bias associated with a large number of covariates by equating the pretreatment composition between treatment groups (Hong, 2010a, 2012; Huang et al., 2005). Unlike propensity score matching and stratification that are mostly restricted to evaluations of binary treatments, the MMWS method is flexible for evaluating binary and multivalued treatments by approximating a completely randomized experiment. In evaluating whether the treatment effects differ across subpopulations defined by individual characteristics or treatment settings, researchers may assign weights within each subpopulation in order to approximate a randomized block design. To investigate whether one treatment moderates the effect of another concurrent treatment, researchers may assign weights to the data to approximate a factorial randomized design. The method can also be used to assess whether the effect of an initial treatment is amplified or weakened by a subsequent treatment or to identify an optimal treatment sequence through approximating a sequential randomized experiment. Even though such analyses can similarly be conducted through inverse-probability-of-treatment weighting (IPTW) that has been increasingly employed in epidemiological research (Hernán, Brumback, and Robins, 2000; Robins, Hernán, and Brumback, 2000), IPTW is known for bias and imprecision in estimation especially when the propensity score models are misspecified in their functional forms (Hong, 2010a; Kang and Schafer, 2007; Schafer and Kang, 2008; Waernbaum, 2012). In contrast, the nonparametric MMWS method displays a relatively high level of robustness despite such misspecifications and also gains efficiency, as indicated by simulation results (Hong, 2010a).

To study causal mediation mechanisms, the RMPW method decomposes the total effect of a treatment into an “indirect effect” transmitted through a specific mediator and a “direct effect” representing unspecified mechanisms. In contrast with most existing methods for mediation analysis, the RMPW-adjusted outcome model is extremely simple and is nonparametric in nature. It generates estimates of the causal effects along with their sampling errors while adjusting for pretreatment covariates that confound the mediator–outcome relationships through weighting. The method applies regardless of the distribution of the outcome, the distribution of the mediator, or the functional relationship between the outcome and the mediator. Moreover, the RMPW method can easily accommodate data in which the mediator effect on the outcome may depend on the treatment assignment (Hong, 2010b). This is the case when a treatment produces its effects not only through changing the mediator value but also in part by altering the mediational process that normally produces the outcome (Judd and Kenny, 1981). One may use RMPW to further investigate whether the mediation mechanism varies across subpopulations and to disentangle complex mediation mechanisms involving multiple concurrent or consecutive mediators. A combination of RMPW with MMWS enables researchers to conduct mediation analysis when the treatment is not randomized. The RMPW approach to mediation analysis will be shown to have broad applicability to single-level and multilevel data (Hong & Nomi, 2012; Hong, Deutsch, and Hill, in press) when compared with path analysis, structural equation modeling (SEM), the instrumental variable method, and their recent extensions.

1.3 Objectives and organization of the book

The book consists of four major units: an overview followed by three units focusing on moderation, mediation, and spill-over, respectively. The first unit provides an overview of the concepts of causal effects, moderation, mediation, and spill-over. After reviewing the existing research designs and analytic methods for causal effect estimation, it introduces the basic rationale of weighted estimation of population parameters in survey sampling and explains the extension of the weighting approach to causal inference. Each subsequent unit addresses research questions of increasing complexity around one of the three focal topics. Part II considers treatment effects moderated by individual or contextual characteristics, by a concurrent treatment, or by a prior or subsequent treatment. Part III shows how to investigate the case of a single mediator, of multiple mediators, and of moderated mediation effects. Part IV discusses the spill-over of treatment effects and the spill-over of mediated effects.

Throughout the book, the mathematics is kept to a minimum to ease reading. Derivations and proofs are left to appendices for readers with technical interest. Assuming that some readers may have had little prior training in basic probability theory, a topic not always covered in a systematic way in an introductory-level applied statistics course, Chapter 2 provides a glossary of the basic notation employed in causal inference. Readers who have already taken a course in causal inference or its equivalent may easily skip Chapters 2 and 3. Chapters 4–11 are suitable for use in an undergraduate- or graduate-level applied statistics course dealing with moderation and mediation analyses. Those who are working on cutting-edge problems related to causal mediation and spill-over may find the last four chapters of the book particularly engaging.

This book aims to make the new analytic methods readily accessible to a wide audience in the social and behavioral sciences. Readers with sufficient understanding of multiple regression and analysis of variance (ANOVA) will quickly grasp the logic of the weighting methods and will find them easy to implement with existing statistical software. For example, after applying MMWS to a sample, researchers can estimate the effects of multivalued treatments or a moderated treatment effect simply within the ANOVA framework. For mediational studies, the RMPW method generates parameter estimates corresponding to the direct effect and the indirect effect along with their estimated sampling variability and therefore greatly simplifies hypothesis testing. In addition to the templates, along with data examples, for implementing MMWS and RMPW in SAS, Stata, and R, the book is accompanied by stand-alone MMWS and RMPW software programs. The program interfaces are designed not only to greatly ease computation but also to assist the applied user with analytic decision-making. All these materials are available online free of charge at the publisher’s website: http://www.wiley.com/go/social_world.

Numerous examples from education, human development, psychology, sociology, and public policy motivate the development of innovative methods and help illustrate the concepts in the book. Analytic procedures are demonstrated through a series of case studies such as those discussed earlier. The examples are chosen to represent the causal questions often raised in behavioral and social science research and hence serve as prototypes for applying the analytic methods. In each case, statistical solutions are offered in sufficient detail to allow for replications.

1.4 How is this book situated among other publications on related topics?

Many other authors have contributed to research on moderation and mediation analyses and on causal inference in general. Such work has been systematically presented in four categories, in my view. The first category is textbooks on path analysis and SEM. The most popular ones include Kenneth A. Bollen’s (1989) Structural Equations with Latent Variables, David P. MacKinnon’s (2008) Introduction to Statistical Mediation Analysis, and Andrew F. Hayes’ (2013) Introduction to Mediation, Moderation, and Conditional Process Analysis: A Regression-Based Approach. Bollen’s book provides a lucid and precise introduction to SEM focusing on combining structural models with measurement models. MacKinnon’s book summarizes later developments extending SEM to multilevel data, longitudinal data, and categorical data and relaxing distributional assumptions by employing computer-intensive estimation methods. Hayes’s book offers a gentle introduction to the most basic mediation and moderation analyses with the aim of integrating the two within the regression framework. These books, however, do not place a major emphasis on clarifying the identification assumptions crucial for distinguishing between association and causation.

Books in the second category provide a healthy antidote to the standard methods that have been employed by social scientists in addressing causal questions. Among the most well known are Steven D. Levitt and Stephen J. Dubner’s (2005) Freakonomics: A Rogue Economist Explores the Hidden Side of Everything and Charles Manski’s (1995) Identification Problems in the Social Sciences. These books were aimed at raising critical awareness among social scientists and the general public with regard to the potentials and limitations of the existing methods. They were excellent in revealing analytic difficulties and uncertainties in social science research.

The third category includes books offering a general overview of research designs and analytic methods for causal inference, represented by William R. Shadish, Thomas D. Cook, and Donald T. Campbell’s (2002) Experimental and Quasi-Experimental Designs for Generalized Causal Inference, Stephen L. Morgan and Christopher Winship’s (2015) Counterfactuals and Causal Inference: Methods and Principles for Social Research, and Richard J. Murnane and John B. Willett’s (2011) Methods Matter: Improving Causal Inference in Educational and Social Science Research. Each book gives a comprehensive survey of a range of design options and analytic options for students in social sciences, yet none is focused on methods for investigating moderation, mediation, and spill-over.

The fourth category is research monographs each presenting a distinct cutting-edge approach to causal analysis. In this category, one may find Paul R. Rosenbaum’s (2002) Observational Studies and his (2009) Design of Observational Studies, Donald B. Rubin’s (2006) Matched Sampling for Causal Effects, and Judea Pearl’s (2009) Causality: Models, Reasoning, and Inference. The books by Rosenbaum or Rubin provide detailed discussions of how to use propensity score matching to overcome overt biases associated with the observables and how to use sensitivity analysis to account for hidden biases associated with the unobservables. The authors restricted the discussions primarily to evaluating the average effect of a binary treatment. Pearl’s book is unique in defining direct and indirect effects as mathematical functions of counterfactual outcomes. The definitions reflect an integrated understanding of probabilistic relationships and structural relationships. The book is also unique in its extensive use of graphical models and causal diagrams for determining identification. Yet the book may not satisfy practitioners searching for step-by-step analytic solutions directly applicable to their data.