Feedback-robust negative-control identification of contagion effects under homophily

In observational dyadic and network data, contagion (peer) effects — the causal effect of one person's trait or behavior on an associated person's later outcome — are generically confounded with homophily, the tendency of connected individuals to share latent traits. Liu and Tchetgen Tchetgen (2022) proposed a regression-based negative-control adjustment: a variable such as the alter's follow-up outcome is used to detect and remove homophily bias when estimating the contagion effect of the alter's baseline trait on the ego's follow-up outcome. Two of the directions left open — the parametric model required for the nameship (dyad-formation) mechanism and inference under dependent, overlapping network observations — have been substantially advanced by later work on double negative controls and network-dependent estimation. The open question is identification of contagion effects by negative-control-type adjustments that remains valid when peer influence may operate at unobserved intermediate times between measurement waves, which invalidates candidate negative controls through intermediate-outcome confounding. This serves one objective: trustworthy negative-control inference about contagion in the presence of homophily.

Background

A peer effect or contagion effect is the causal effect of one person's exposure or behavior on the outcome of a person connected to them; in network studies the two members of a tie are called the alter (whose behavior is the exposure) and the ego (whose outcome is studied), and a dyad is a pair linked by a relationship such as friendship or named contact. Estimating contagion effects from observational network data is notoriously hard. Homophily — the tendency of people to form ties with others who share similar traits, including unobserved ones — makes connected people behave similarly even with no influence at all, so naive associations conflate social influence with latent similarity (Shalizi and Thomas, 2011). The problem is compounded by the reflection problem, the linear-algebraic impossibility of separating endogenous peer influence from correlated contextual effects in simultaneous-outcome models (Manski, 1993). These concerns were sharpened by prominent contagion claims, most famously the spread of obesity over the Framingham social network (Christakis and Fowler, 2007), whose causal interpretation remains contested. Negative controls are variables that cannot plausibly affect the outcome but are related to the unmeasured confounder: a negative control exposure should be associated with the confounding mechanism while having no direct causal effect on the outcome, so its observed association with the outcome signals residual bias. Liu and Tchetgen Tchetgen (2022) developed a regression-based negative-control adjustment for dyadic data. In their design, the dyad is formed by nameship (whether one member names the other as a contact, encoded in indicators $R_1,R_2$ and a nameship type $S$), the exposure $A$ is the alter's baseline trait (e.g. baseline obesity), the outcome $Y$ is the ego's follow-up trait, and the negative control exposure $Z$ is the alter's follow-up trait (e.g. follow-up BMI), which cannot directly cause the ego's contemporaneous outcome but is correlated with the latent homophily variables $U_1,U_2$ through tie formation. Identification requires untestable conditions: $Z$ must be a valid negative control (associated with nameship, no direct effect on $Y$, no residual confounding of $Z$ and $Y$), the homophily variable must be mean-independent of $(A,Z)$ given covariates $C$, and a conditional location-shift restriction (their equations (3)–(4)) must hold; because the observed sample conditions on $S\ge1$, a parametric model $\Pr(S\mid A,C,Z;\theta)$ for the nameship mechanism is fitted (in a first stage of a two-step algorithm in the arXiv preprint; by joint maximum likelihood in the published Biometrics version); conditional independence of the naming indicators, $R_1\perp R_2\mid A,C,Z$, is assumed for convenience in the preprint (the published version instead models nameship dependence directly through an odds-ratio interaction $\exp(\delta R_1 R_2)$ in the joint likelihood); and asymptotics assume independent, non-overlapping dyads. The paper's own discussion, amplified by Ogburn and VanderWeele (2014) on causal diagrams for interference (spillover between units), identifies the sharpest threat: if contagion can operate at unobserved intermediate times between the two measurement waves, then ego and alter intermediate outcomes confound the negative control $Z$ with $Y$, invalidating the identifying assumptions (valid only under the sharp null of no contagion, i.e. the assumption that no individual's outcome is affected by contagion). Later literature has built substantially on this program: a general framework of double negative controls (a negative control outcome paired with a negative control exposure) now gives nonparametric identification of peer effects with dependence-robust inference (Egami and Tchetgen Tchetgen, 2024); placebo outcomes (pre-treatment outcomes that cannot be affected by the treatment and therefore act as negative-control outcomes) under a structural stationarity assumption address diffusion bias in spatial panel settings (Egami, 2024); dual instrumental variables yield triply robust dyadic estimators (Luo et al., 2025); Bayesian latent-position models adjust for latent homophily through the network itself (Um et al., 2026); and machine-learning constructions reduce the variance of proxy-based contagion estimators (Fatemi and Zheleva, 2023). None of these resolves the question below.

Problem Statement

Open problem (Liu and Tchetgen Tchetgen, 2022). Setting: observational dyadic or network data with (at least) two measurement occasions per person, an exposure $A$ equal to the alter's baseline trait, an outcome $Y$ equal to the ego's follow-up trait, and latent homophily variables that confound the association between $A$ and $Y$ through tie formation. Target: credible causal inference about the contagion effect of $A$ on $Y$ using negative-control-type adjustments, where a candidate negative control exposure $Z$ (e.g. the alter's follow-up trait) is associated with the homophily/tie-formation mechanism but is assumed to have no direct causal effect on $Y$. The following question remains open: Feedback-robust identification. Determine whether any negative-control-type adjustment can identify the contagion effect when peer influence and interaction may operate at unobserved intermediate times between the measurement waves, under which the standard candidate negative controls are confounded with the outcome through intermediate outcomes of ego and alter (the intermediate-time confounding structure of the source's Figure 3); if such an adjustment exists, construct the identifying assumptions and an estimator, and if not, characterize the additional data or assumptions required. Answers may address network (not only dyadic) observation patterns, since the dependence complication has been treated in related frameworks, but must confront the intermediate-time feedback regime head-on rather than assuming it away.

References

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