Completing instrumental-variable identification of peer effects in longitudinal dyadic data: non-linear models and unremedied homophily and confounding structures

Whether traits and behaviors spread causally through social ties (peer effects, or social contagion) or merely co-occur because similar people form ties with each other (homophily) is a central open problem of observational network science. O'Malley, Elwert, Rosenquist, Zaslavsky and Christakis (2014) showed that in longitudinal dyadic data, genetic variants and time-varying gene expression can serve as instrumental variables that identify peer effects even under homophily and unobserved confounding, but only in linear homogeneous-effect models, while several realistic data-generating structures were left without remedies in one or the other of the paper's two causal models — homophily on contemporaneous measured phenotype in both; pleiotropy on unobservables and shared-context confounding of the genotype under the gene-alone model (Figure 2); homophily on, and shared-context confounding of, instrument-period gene expression under the gene-interaction model (Figure 3) — and the Framingham Heart Study application suffered weak instruments. Subsequent work has advanced neighboring subproblems — alternative identification devices (negative controls, latent-space and Bayesian nonparametric models) and a 2025 dual-IV nonparametric preprint — but completion of IV identification for peer effects in longitudinal dyadic data, beyond linear models and robust to the confounding structures the source leaves unremedied, remains open.

Background

Whether an individual's traits and behaviors are influenced by those of their friends and partners — causal peer effects, also called social contagion or social induction — is a long-standing question in social science and epidemiology. Observational estimation is hard because social contagion is entangled with homophily, the tendency of people to form social ties with others who are similar to them, and with unobserved confounding by shared environment. Shalizi and Thomas (2011) proved that in observational social-network studies these mechanisms are generically confounded: distinguishing contagion from homophily requires strong parametric or covariate assumptions. The prominent finding of Christakis and Fowler (2007) that obesity spreads through the Framingham Heart Study (FHS) social network remains contested precisely for this reason; a general survey of the area is given by VanderWeele and An (2013).

Instrumental variables (IV) offer a potential way out: an instrument is a variable that affects the peer's (the alter's) treatment but has no pathway to the focal individual's (the ego's) outcome except through the alter's treatment, so that it isolates exogenous variation in the treatment. In econometrics, Bramoullé, Djebbari and Fortin (2009) showed that peer effects can be identified through intransitive network structure (using outcomes of 'friends of friends' as instruments). O'Malley, Elwert, Rosenquist, Zaslavsky and Christakis (2014) studied genes as instruments for peer effects of traits such as body mass index (BMI): fixed genetic variants (they used variants in FTO and MC4R, the two loci with the strongest established associations with BMI) and, more promisingly, time-varying gene expression are invisible to one's peers and are transmitted as if randomly, making them structurally isomorphic to randomized encouragement. In epidemiology this genetic-IV idea is known as Mendelian randomization. The authors represented the assumed data-generating processes with directed acyclic graphs (DAGs) — graphs whose nodes are variables and whose arrows denote direct causal effects, constrained to contain no directed cycles — and proved graphical criteria, due to Brito and Pearl, for reading off from the DAG whether a single variable, or a set of variables acting jointly as instruments (an IV set), satisfies the instrumental conditions, applying them to establish identification of peer effects in a dyad observed over time under latent homophily and confounding. Their three main results: (i) a single fixed gene generally fails as an IV if it affects past values of the alter's treatment; (ii) multiple fixed genes, or time-varying gene expression, can identify peer effects if exclusion violations are instrumented along with the focal treatment; (iii) identification survives many complications often considered fatal, such as pleiotropy (a gene affecting multiple traits) on observables and, under the gene-interaction model, on unobservables, homophily on past phenotype, past and ongoing homophily on genotype, population stratification (ancestry-related genetic structure correlated with both genotype and environment), and gene expression that is endogenous to past phenotype and past gene expression.

The identification theory is, however, restricted to linear models with homogeneous peer effects, and its Table 1 catalogues which extensions of the causal model are harmless, which are remediable by conditioning on observables (for example, conditioning on earlier gene expression $GX_{2(t-2)}$ to allow serial dependence in gene expression, adding earlier phenotype terms to allow epigenetic feedback, or adding dyad fixed effects for population stratification), and which have 'no remedy'. The 'no remedy' statuses are model-indexed (Table 1's 'Applies to figure' column): homophily on measured phenotype at or after $t-1$ (an arrow from the phenotype $Y_{k(t)}$ to the tie variable $A_{12}$ for $t\geq1$) has no remedy under either causal model; homophily on instrument-period gene expression ($GX_{k(1)}\to A_{12}$) and shared-context confounding of instrument-period gene expression ($C_{12}\to GX_{2(1)}$) have no remedy under the gene-interaction model (Figure 3, which the paper adopts as its primary causal model); and pleiotropy on unobservables ($G_2\to U_2$) and shared-context confounding of genotype ($C_{12}\to G_2$) have no remedy only under the gene-alone model (Figure 2) — under Figure 3, $G_2\to U_2$ is marked 'no implication'. In the FHS application, the first-stage $F$-statistic (the standard diagnostic for instrument strength, computed from the regression of the endogenous treatment on the instruments) for the gene-age interaction instruments was modest to weak by conventional standards (approximately 2.2 for friends and 4.1 for spouses, and at most about 0.27 for specifications that condition on $GX_{2(t-2)}$ to accommodate serial dependence in gene expression), so the estimated peer effects, while large, were mostly insignificant. The paper notes these weak first stages as a limitation of the application rather than posing them as a research question; what it leaves open is whether the identification results extend beyond linear homogeneous effects, and whether the structures its Table 1 marks 'no remedy' can be remedied or are provably irremediable.

Problem Statement

Complete the instrumental-variable identification theory for causal peer effects in longitudinal dyadic data, as left open by O'Malley, Elwert, Rosenquist, Zaslavsky and Christakis (2014). The objective is a complete account of when IV strategies identify peer effects under homophily and unobserved confounding in the source's longitudinal dyadic setting — admissible instruments being variables (genetic variants, time-varying gene expression, or others) that satisfy the source's graphical single-IV and IV-set conditions in its dyadic DAGs (Figure 2, the 'gene-alone' model; Figure 3, the 'gene-interaction' model, the paper's primary causal model, in which the time-varying instrument $GX_{2(t-1)}$ identifies $Y_{2(t-1)}\to Y_{1(t)}$ conditional on $(A_{12}, G_2, X_2)$) under latent homophily and shared-context confounding — such that every structure the source's Table 1 leaves as 'no remedy', under the causal model to which that status applies, is either remedied or proven irremediable. Concretely, what remains open is (1) whether the graphical IV and IV-set identification results extend beyond linear models with homogeneous effects to non-linear and nonparametric peer-effect settings in the same longitudinal dyadic setting, including effects of non-linear transformations of the phenotype (for example, peer effects of obesity, BMI $\geq 30$); the source already claims a linear-in-parameters two-stage-least-squares generalization to effect heterogeneity in observed effect modifiers, so that linear case is not itself the open target; and (2) whether the edge additions the source's Table 1 leaves as 'no remedy' — the statuses being model-indexed ('Applies to figure', the catalogue being stated for $q=2$ periods since tie formation with individual 1 as the ego) — can be remedied (conditioning sets, instrument configurations, or partial identification) or are non-identifiable: under Figure 3, homophily on measured phenotype at or after $t-1$ ($Y_{k(1)}\to A_{12}$ and $Y_{k(2)}\to A_{12}$), homophily on instrument-period gene expression ($GX_{k(1)}\to A_{12}$), and shared-context confounding of instrument-period gene expression ($C_{12}\to GX_{2(1)}$); under Figure 2, homophily on measured phenotype ($Y_{k(t)}\to A_{12}$), pleiotropy on unobservables ($G_2\to U_2$), and shared-context confounding of genotype ($C_{12}\to G_2$). Pleiotropy on unobservables is not an open item under Figure 3, where Table 1 marks $G_2\to U_2$ 'no implication'; the remedy-or-impossibility question for that structure concerns the gene-alone model.

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