Page 1 of 44
Chapter 16
Heterogeneous Agents, Social Interactions, and Causal Inference
Guanglei Hong and Stephen W. Raudenbush
University of Chicago
Abstract
Most causal analysesin the social sciences depend on the assumption that eachparticipant possesses a single potential outcome under each possible treatment assignment. Rubin (1986) labeled this the “Stable Unit Treatment Value Assumption” (SUTVA).Under SUTVA, theindividual-specific impact of a treatment depends neither on the mechanism by which the treatment is assigned nor on thetreatment assignments of other individuals. However, in the social world, heterogeneous agents enact most interventions of interest: teachers implement curricula, psychologists enact family therapy, precinct captains supervise community policing. Moreover, the potential outcomes of one participant will often depend the treatment assignment of other participants (classmates, family members, neighbors). This chapter presents a model that relaxes the conventionalSUTVA by incorporating agents and social interactions. We define a treatment setting for an individual participant as a local environment constituted by a set of agents and participants along with their treatment assignments. Our model assigns a single potential outcome to each participant in each of such treatment settings. In a cluster-randomized trial, if no interference exists between clusters and if cluster composition remains intact,the treatment setting is fixed for all participants in a cluster and SUTVA becomes reasonable. However, when participants are assigned to treatments within clusters, we need a model for within-cluster interference among participants. When clusters are spatially contiguous, social interactions generate interference between clusters. We alsoincorporate new models for interference as a part of the meditation mechanism. In general, when SUTVA is relaxed, new causal questions come to light. We illustrate these ideas using studies of grade retention in elementary school, community policing in cities, school-wide interventions for behavioral improvement, and system-wide curricular changes for promoting math learning.
1. Introduction
In this chapter we focus on two pervasive features of social interventions designed to increasehuman health, skills, or productivity. First, the interventions are usually delivered byhuman agents – physicians, teachers, case workers, therapists, or workplace managers who tend to be heterogeneous in beliefs, training, and experience. Second, the interventions typically target the behaviors of participants clustered in organizational settings. Social interactions among these participants may influence how participants respond to the interventions. Heterogeneous agents and social interactions among participants thus contribute to participants’ potential outcomes.
The counterfactual account of causalitycan provide a conceptually powerful framework for studying such interventions, with profound implications for the design of causal-comparative studies and the framing of research questions. However, the vast majority of evaluation studies to date have relied, explicitly or implicitly, on the assumption that heterogeneity among agents and social interactions among participants are irrelevant in generating participant-specific potential outcomes. If this assumption is correct, each participant possesses a single potential outcome under each treatment condition, and causal effects are comparisons among these potential outcomes. However, if heterogeneous agents and socially interacting participants influence treatment enactment, a richer class of causal models is required, one that generates for each participanta setof potential outcomes for each possible treatment assignment,suggesting novel causal questionsthat are often theoretically interesting.
Building on key contributions of Hong (2004), Verbitsky and Raudenbush (2004, Forthcoming), Hong and Raudenbush (2006), Sobel (2006), Rosenbaum (2007), Hudgens and Halloran (2008), Tchetgen Tchetgen & VanderWeele (2012), and Manski (forthcoming), our aim in this chapter is to scrutinize the “Stable Unit Treatment ValueAssumption” (SUTVA)for applicationsin the social sciences and particularly in social policy analyses.We discuss how to relax the assumption to accommodate heterogeneous agents and socially interacting participants.
Section 2 illustrates the importance of heterogeneous agents and social interactions for understanding causality with a series of representative examples. We consider the cluster randomized trial, interference within clusters, interference between clusters, and spilloversin mediational models. Section 3 formalizes our general causal framework that incorporates heterogeneous agents and social interactions. Sections 4 to 8 applythis framework to the representative examples. Section 9 concludes.
2. The Role of Agents and Social Interactions: Illustrative Examples
2.1. The Conventional Paradigm: Agents and Social Interactions Ignored
The conventional paradigm for modeling potential outcomes often applies well to medicine. The aim is to compare a new “experimental” medication E to a conventional or “control” medication C. Each patient possesses a potential outcome under each of these two conditions, and the difference between the two potential outcomes is a patient-specific causal effect of E versus C. The average of these effects in the population is the population-average causal effect. A key implicit assumption, originally identified by Rubin (1986) as the “Stable Unit Treatment-Value Assumption” (or “SUTVA”), is that a) how the treatment is enacted is irrelevant to each patient’s potential outcome; and b) the treatment assignment of one patient has no influence on the potential outcome of another patient. Entailment (a) would fail if, for example, physicians vary in their skill in encouraging patients to comply with their prescribed medication. Entailment (b) would fail if, as a result of receiving the new drug, a patient interacts differently with a spouse who is also in the experiment. While possible failures of (a) and (b) might arguably be negligible in the case of a drug trial, these entailments may be highly implausible in a study of two alternative reading curricula, where (a) would imply that teachers do not vary in their skill in enacting the curricula in their instruction and (b) would imply that the classroom peers have no effect on a child’s potential response.
2.2. Modified Causal Framework Incorporating Treatment Settings
To generalize counter-factual models to heterogeneous agents and social settings, Hong (2004) defined a “treatment setting” as a specificlocal environment composed of one or more agentsand a set of participants along with the treatment assignments of those agents and participants. Under a given treatment, a participant may have as many potential outcome values as the number of possible treatment settings. Changing the treatment assignment of agents or peers may modify a participant’s potential outcome value even if that participant’s treatment assignment remains constant. This generalization relaxes SUTVA and enables one to pursue a rich class of meaningful causal questions. Person-specific causal effects can be defined as comparisons between potential outcomes associated withalternative treatments under a given treatment setting, or between potential outcomes associated with alternative treatment settings when a focal participant’s treatment assignment is fixed. In addition, one may investigate whether the treatment effect depends on the treatment setting.
2.3. Accommodating Within-Site Interference in Cluster Randomized Trials
A challenge then arises: If a treatment setting has an impact on an individual’s potential outcome value, shifting one participant’s treatment assignment may potentially perturb the outcome of every other participant. Causal inference seemingly becomes intractable.Cluster randomized trials provide a solution to this problem. If the person-level randomized trial is the gold standard when SUTVA holds, the cluster-randomized trial will often be the gold standard when we expect agents and social interactions to modify potential outcome values. In a cluster randomized trial, discussed in Section 4,every agent and every participant in a cluster are assigned to the same treatment. So there is one treatment setting per treatment condition, and the simple version of SUTVa holds: each participant possesses one and only one potential outcome under each treatment condition. However, two additional assumptions are needed (Hong and Raudenbush, 2006): a) there is no interference between clusters; and b) clusters are intact, meaning that cluster membership does not change in response to treatment.
2.4. Interference within Clusters in a Multi-Site Trial
Cluster-based assignment may not be practical for some treatments. In Section 4, we consider the case of grade retention (Hong and Raudenbush, 2006). A child who displaysrelatively poor progress in cognitive skills or social behavior may be retained in grade or promoted to the next grade. The child’s potential outcome values will plausibly depend not only on whether the child is retained or promoted but also on how many of his or her low-achieving peers in the same school are retained or promoted. This isbecause the treatment assignment of the peers will determine the focal child’s peer composition in class. It makes no sense to assign an entire school to be retained, so the cluster-randomized trial is not an option. Nevertheless, we can proceed in this type of multi-site trialsif important features of treatment settings can besummarized in a low-dimensional function. In the current example, one may characterize peer treatment assignment as the fraction of children in the same grade who are retained. This fraction varies from school to school while the retention treatment is assigned to individual students within schools. Hong and Raudenbush (2006) thus attempted to identify the effect of grade retention in schools retaining a relatively high fraction of students at risk of being retained and the effect of grade retention in schools retaining a relatively low fraction of such students.The authors assumed intact schools (i.e., children did not transfer schools as a result of being retained or promoted) and no interference between schools.
2.5. Interference between Clusters
Interference between clusters – generated by social interactions that cross cluster lines – is inevitablein some cases. Thus, a cluster randomized trial may fail to overcome the complexity induced by social interactions. In Section 6, we discuss a study of “community policing” in Chicago (Verbitsky-Shavitz and Raudenbush, Forthcoming). In this study, police districts were assigned to receive either community policing or regular policing. Police work is organized and carried out, however, in much smaller police beats, with about 12 beats per district. We expect spillover effects across beats: if community policing is effective, its impact in one beat should depend on the treatment assignment of other beats. In particular, effective policing in one beat may encourage offenders to operate elsewhere. Indeed, the evidence suggests that a beat not assigned to community policing will suffer when nearby beats do receive community policing; and being assigned to community policing appears particularly beneficial when surrounding beatsalso receive the intervention. To approximate how this intervention would work at scale, one might design a study that randomly assigns whole cities to the intervention, and we show how our generalized potential outcomes framework easily adapts to clarify the assumptions needed for this approximation to be valid.
2.6. Mediational Models with Spillovers
Spillover through social interactions is sometimes theorized as an important mechanism for the intervention effect. Section 7 shows how such mediational models can be representedwithin our generalized framework. One illustration is by Hong and Nomi (2012) who evaluated the effect of a system-wide algebra-for-all policy on student math outcomes mediated by class peer ability change.The policy was intended to improve the math learning of lower-achieving students who would have taken remedial math had algebra not been required. However, many schools created mixed ability algebra classes in response to the new policy. A rise in class peer ability for lower-achieving students seemingly contributed to an unintended negative side-effect of the policy possibly due to unfavorable social comparisons in mixed-ability classes.
The second study, by Vanderweele, Hong, Jones, and Brown (Working paper),evaluated a school-wide intervention designed to reduce aggression and depression amongelementary school students. One would expect the effectiveness of the school-wide program to be mediated by the improvement of class quality by design. However, the analysts also hypothesized a second source of mediation: students in a focal class may benefit from the program if other classes in the same school raise their quality. This is because children interact not simplywith their classmates but also with those from other classes in the hallways or onthe playground. The study emphasized that, even in a cluster randomized trial, spillovers may occur if the mediator of interest is measured at a lower level than the treatment. In each case, evidence of spillovers enriches theoretical understanding and has direct implications for policymaking. See Chapter 12 in this volume for a related discussion on causal mediation analysis.
3. The Conventional Paradigm and Its Modification
We adopt the counterfactual account of causality throughout this chapter. In its simplest and most widely applied form, participants are assigned to one of two treatments. Perhaps the canonical example is a randomized clinical trial.
3.1. Potential Outcomes and Causal Effects
Let the random variabledenote possible treatment assignment of patient i: if patient i is assigned to receive the new, experimental drug; if that patient is assigned to receive the conventional, “control” drug. If , we will at some later time observe that patient’s health outcome . If, instead, , we will observe for that patient. A patient-specific causal effect of the experimental drug relative to the control for outcome Y is usually defined as
.(1)
Causal effects are thus comparisons between unit-specific potential outcomes, where each potential outcome is associated with a specific treatment assignment. Following Neyman (1923), Rubin (1974, 1978)developed the logic of this framework. Holland (1986) provides an elegant synthesis. A parallel and largely independent development of these ideas is found in economics whereinterest focuses on potential choices and outcomes associated with alternative courses of action (Haavelmo, 1944; Roy, 1951; Heckman, 1979).
One of the two potential outcomes is sure to be missing for each individual unit.Holland (1986) described thisfact as “the fundamental problem of causal inference.” To address this problem, he wrote, social science must abandon the project of estimating unit-specific causal effects and instead focus on aggregate estimands, most commonly the population mean causal effect
(2)
The population mean causal effectis the difference between two population means:, the population-average response under assignment to the experimental drug; and, the population-average response under assignment to the control drug.
3.2.Identification
We cannot directly computethe two population means and because we cannot simultaneously observe the same population under these two treatment conditions.Nonetheless, if treatment assignment is statistically independent of the potential outcomes, that is, if treatment assignment is ignorable,
,(3)
we have that
(4)
Random assignment of patients to drugs ensures ignorable treatment assignment and thereby is the foundation for the success of the randomized clinical trial in medicine. In general, if treatment assignment is ignorable, then where z is any possible value of the treatment assignment, categorical or continuous.
Researchers have adopted alternative approaches to causal inference with non-experimental data, each requiring a set of identification assumptions. Some approaches rely on statistical adjustment of observed covariates; some resort to an instrumental variable; and some take advantage of “natural experiments” and invoke model-based assumptions.
3.3 Stable Unit Treatment Value Assumption
In his discussion of Holland’s paper, Rubin (1986) called attention to a key assumption that had rarely been stated explicitly: the assumption that each patient in our example possesses one and only one potential outcome under a given treatment condition. He called this the “Stable Unit Treatment Value Assumption” (“SUTVA”) because the potential outcome value remains stable regardless of
a)the mechanism by which the treatment is assigned and
b)the treatment assignment of other units.
Suppose that a clinical trial involvesN patients and J physicians.If SUTVA does not hold, in the most general case, patient i would have as many potential outcomes as the combination of treatment assignments for all the units in the population. The patient’s potential outcome would also depend on the physician to which the patient is assigned.Following Hong (2004), we describe the potential outcome of each patient as a function of the patient’s own treatment assignment, the treatment assignment of other patients, as well as the assignment of the focal patient to a physician. Let represent the treatment assignment for all patients. For patient i treated by physician j, the potential outcome is denoted by
.(5)
SUTVAstates that the treatment assignments of patients other than i and the physician to which that patient is assigned have no effect on that patient’s potential outcome. And hence the potential outcome is a function of the focal patient’s treatment assignment only:
.(6)
Reports of clinical trials routinely but implicitly rely on SUTVA;the same practice has been adopted nearly universally and largely uncritically in the social sciences.One may argue that SUTVA is quite reasonable in our hypothetical drug trial. Few patients in the trial may know each other or interact. Even if they do, the medication taken may have little influence on these social interactions and even then such social interactions may not affect the long-term health of patients. Physicians may vary little in their skill in motivating patients to comply with the directions of their medication.