Interpretation of the Individual Effect Under Treatment Spillover

Abstract

Some interventions are intended to benefit both individuals and the groups to which they belong. When a treatment given to one person exerts a causal effect on others, the treatment is said to exhibit spillover, dissemination, or interference. However, defining meaningful causal effects under spillover can be challenging. In this commentary, we discuss the meaning of the “individual effect,” a quantity proposed to summarize the effect of treatment on the person who receives it, when spillover may be present.

Some interventions may include important spillover or dissemination effects between study participants (1, 2). For example, vaccines, cash transfers, and education programs may exert a causal effect on study participants beyond those to whom individual treatment is assigned. In a recent paper, Buchanan et al. (3) provided a causal definition of the “individual effect” of an intervention in networks of people who inject drugs. Their work builds on a definition of the “direct effect,” randomization design, and framework for causal inference under interference introduced by Hudgens and Halloran (4). Some researchers have suggested that the “direct effect” may not always have a causal interpretation (5–7). In this commentary, we discuss interpretation of the individual effect when a spillover or dissemination effect exists.

POTENTIAL OUTCOMES AND CAUSAL EFFECTS

In their paper, Buchanan et al. (3) introduce potential outcome notation for the effect of an intervention on an undesirable outcome (e.g., risk behavior, fatal overdose, human immunodeficiency virus (HIV) infection) among people who inject drugs. Let $Y_{ki}$ be an indicator for that outcome, where $k=\mathrm{1,}\dots ,K$ is the cluster and $i=\mathrm{1,}\dots ,n_k$ is the individual within that cluster. A cluster is a group of people for which one individual’s potential outcome may be a function of treatments assigned to other persons in that group. When this is the case, the treatment is said to exhibit “spillover” or “dissemination,” a type of interference between experimental units. Potential outcomes for individuals in one cluster are assumed to be a function of treatments given to other persons within their own cluster, and not of treatments given to persons in other clusters. In this commentary, we use the terms “spillover” and “dissemination”—introduced by Buchanan et al. (3)—interchangeably.

Let $X_k$ be an indicator that cluster k is treated, meaning that a single cluster member is the “index” participant who directly receives the intervention. We use the terms “treatment” and “intervention” interchangeably. Let $R_{ki}$ be the indicator that individual i is the index subject, with exactly 1 index individual per cluster. Define the vector of index indicators as ${\mathbf{R}}_k=(R_{k1},\dots ,R_{k{n}_k})$. Define the individual potential outcome as $Y_{ki}(\mathbf{r},x)$, where ${\mathbf{R}}_k=\mathbf{r}$ is the vector of index subject indicators and $X_k=x$ is the group-level treatment indicator. Because ${\sum }_i{R}_{ki}=1$ for all clusters k, the potential outcome notation $Y_{ki}(\mathbf{r},x)$ reduces unambiguously to $Y_{ki}(r,x)$, where $r=R_{ki}$ indicates that individual i is the index subject; whenever $R_{ki}=1$, it is implicit that $R_{kj}=0$ for $j\ne i$. Buchanan et al. ensure that $Y_{ki}\mathrm{(0,\; 1)}$ is well defined by assuming that it does not matter which fellow cluster member of i is treated when i is untreated: “Conditional on baseline covariates, we assume that there is exchangeability between the $2\times n_k$ possible configurations when there is 1 index participant in network k” (3, p. 2451). The $Y_{ki}(r,x)$ notation introduced by Buchanan et al. (3) does not accommodate the case of more than 1 treated individual per cluster. Buchanan et al. define the risk difference (RD) for the individual effect as

$${\mathrm{RD}}^I=\mathbb{E}[Y_{ki}\mathrm{(1,1)}-Y_{ki}\mathrm{(0,1)],}$$

that for the disseminated effect as

$${\mathrm{RD}}^D=\mathbb{E}[Y_{ki}\mathrm{(0,1)}-Y_{ki}\mathrm{(0,0)],}$$

and that for the composite effect as

$${\mathrm{RD}}^{\mathrm{Comp}}=\mathbb{E}[Y_{ki}\mathrm{(1,1)}-Y_{ki}\mathrm{(0,0)],}$$

where expectation is with respect to the potential outcomes for individuals i across clusters k in the study (3). By definition, the composite effect can be written as the sum of the individual and disseminated effects, ${\mathrm{RD}}^{\mathrm{Comp}}={\mathrm{RD}}^I+{\mathrm{RD}}^D$.

MEANING OF THE INDIVIDUAL EFFECT

The potential outcome notation of Buchanan et al. (3) implicitly encodes 2 distinct types of exposure to the intervention for subject i in a treated cluster k. First, if subject i is the treated index subject ($R_{ki}=1$, $X_k=1$), then i receives exposure to the intervention via their own treatment and no disseminated exposure from another cluster member, because no other cluster members can be treated. Second, if subject i is a nonindex subject in a treated cluster ($R_{ki}=0$, $X_k=1$), then i receives no direct exposure to the intervention but receives disseminated exposure from 1 treated index subject in their cluster. Figure 1 shows how the individual effect RD^I contrasts potential outcomes by changing both of these types of exposures simultaneously.

Figure 1.

Individual, disseminated, and composite intervention effects defined by Buchanan et al. (3) in a cluster of size 3. Cluster k is shown as a rectangle and subjects are shown as circles, with subject i labeled. Dark gray circles indicate a treated index subject, and light gray circles indicate a subject exposed to treatment via dissemination from a treated cluster member. The individual-effect risk difference (RD^I) compares the potential outcome in subject i when treated with no dissemination from other group members with the potential outcome in subject i when untreated with dissemination from the treated index subject.

Buchanan et al. interpret RD^I as the “effect on persons directly receiving an intervention beyond being in an intervention network” (3, p. 2450, Table 1). When the outcome is undesirable and $\mathbb{E}[Y_{ik}\mathrm{(0,0)]}\ge \mathbb{E}[Y_{ik}\mathrm{(0,1)]}\ge \mathbb{E}[Y_{ik}\mathrm{(1,1)]}$, RD^I may indeed summarize the additional benefit of being an index subject, “beyond” that of experiencing a disseminated effect. However, this monotonicity relationship may not always hold: An intervention with a strong disseminated effect might benefit untreated cluster members more than those who personally received treatment. Then the magnitude of RD^D would be greater than that of ${\mathrm{RD}}^{\mathrm{Comp}}$, and this interpretation of RD^I might not be meaningful.

A HYPOTHETICAL INTERVENTION TRIAL WITH A STRONG SPILLOVER OR DISSEMINATION EFFECT

One of the most effective interventions for reversing a potentially fatal opiate-related overdose is intranasal or injection administration of naloxone during an overdose. Consider a hypothetical study of clusters of drug injectors who are at risk of fatal opiate overdose and an intervention that involves dispensing a naloxone kit to 1 member of each cluster and training that person in its administration. The mechanism of action of this intervention in groups induces asymmetry in its effects: Overdose involves unconsciousness or incapacitation, so naloxone is rarely self-administered; instead, someone who possesses the medication and has been trained in its use can avert another cluster member’s overdose.

Suppose that the outcome of interest is fatal overdose, and treatment involves receipt of a naloxone kit from investigators and training in its use. A subject whose fellow cluster member is treated enjoys some protection against death due to overdose, because the treated subject can use their kit to reverse their cluster member’s overdose, so RD^D < 0. In contrast, the treated subject may derive less benefit from their own treatment because they cannot use their own naloxone kit to reverse their own overdose, except in rare cases (8). It is possible, but perhaps less likely, that their fellow cluster member might administer the treated subject’s naloxone kit to the treated subject, should the subject experience an overdose, implying ${\mathrm{RD}}^D<{\mathrm{RD}}^{\mathrm{Comp}}\le 0$. Clearly treatment is beneficial to any individual whose fellow cluster member receives it, and it is either beneficial to or ineffective for persons who only receive it themselves. However, the individual effect is ${\mathrm{RD}}^I={\mathrm{RD}}^{\mathrm{Comp}}-{\mathrm{RD}}^D>0$, so treatment seems to be harmful to the subject who receives it. Of course, naloxone is not harmful to anyone in this scenario; rather, the “individual effect” contrasts the small (or nonexistent) beneficial effect of individual treatment $({\mathrm{RD}}^{\mathrm{Comp}})$ against the larger beneficial disseminated effect of treatment (RD^D).

DISCUSSION

The quantity RD^I introduced by Buchanan et al. (3) is a well-defined statistical estimand. However, RD^I may be misleading because it contrasts the potential outcome in a person who receives treatment but no disseminated exposure with that in a person who receives no treatment but receives disseminated exposure from another subject. When the disseminated effect is large in comparison with the composite effect, RD^I can be positive (suggesting harm) even when the intervention is beneficial to treated individuals and their fellow cluster members. Buchanan et al. found a similar pattern in their evaluation of the HIV Prevention Trials Network 037 Study intervention (3, p. 2455, Table 4): The individual effect on any risk behavior RD^I was estimated to be null (ineffective), even though the disseminated and composite effects were estimated to be negative (beneficial). Two additional examples of substantial public health importance could show a similar pattern. Early access to antiretroviral therapy among HIV-positive persons is known to improve health and helps prevent transmission to HIV-negative partners (9–12). Likewise, preexposure prophylaxis among HIV-negative persons helps prevent HIV infection in treated individuals (13–15). While these interventions are known to benefit treated individuals and reduce HIV transmission within groups (16), a trial that computed an “individual effect” by contrasting ${\mathrm{RD}}^{\mathrm{Comp}}$ with RD^D might underestimate the benefit conferred by these interventions to treated individuals.

When might the individual effect RD^I be of scientific interest? First, resource constraints might necessitate a policy in which a single subject (or a fixed number of subjects) is treated per cluster, so a trial design that enforces this constraint may naturally reveal the quantity of interest. Second, RD^I may be of interest to investigators and research subjects because it summarizes the ethical trade-off in benefit or harm experienced by treated persons versus untreated persons. In the case of preexposure prophylaxis, these trade-offs might involve the benefit of reduced HIV infection risk for all cluster members, versus potential medication side effects for persons treated with preexposure prophylaxis (14). That is, RD^I answers the question, “Am I better off being treated or untreated, when someone in my cluster is treated?” If investigators desire a measure of the effect of an intervention on an individual subject while holding disseminated exposure constant, the “composite” effect ${\mathrm{RD}}^{\mathrm{Comp}}$ may be a more readily interpretable causal estimand.

Abbreviations

HIV

human immunodeficiency virus

RD

risk difference