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Estimating Peer Effects in Longitudinal Dyadic Data Using Instrumental Variables

DOI:10.1111/biom.12172
Authors:
James O'Malley at Dartmouth College
James O'Malley
Felix Elwert at University of Wisconsin–Madison
Felix Elwert
(James) Niels Rosenquist at Massachusetts General Hospital
(James) Niels Rosenquist
Alan M Zaslavsky at Harvard Medical School
Alan M Zaslavsky
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Abstract and Figures

The identification of causal peer effects (also known as social contagion or induction) from observational data in social networks is challenged by two distinct sources of bias: latent homophily and unobserved confounding. In this paper, we investigate how causal peer effects of traits and behaviors can be identified using genes (or other structurally isomorphic variables) as instrumental variables (IV) in a large set of data generating models with homophily and confounding. We use directed acyclic graphs to represent these models and employ multiple IV strategies and report three main identification results. First, using a single fixed gene (or allele) as an IV will generally fail to identify peer effects if the gene affects past values of the treatment. Second, multiple fixed genes/alleles, or, more promisingly, time-varying gene expression, can identify peer effects if we instrument exclusion violations as well as the focal treatment. Third, we show that IV identification of peer effects remains possible even under multiple complications often regarded as lethal for IV identification of intra-individual effects, such as pleiotropy on observables and unobservables, homophily on past phenotype, past and ongoing homophily on genotype, inter-phenotype peer effects, population stratification, gene expression that is endogenous to past phenotype and past gene expression, and others. We apply our identification results to estimating peer effects of body mass index (BMI) among friends and spouses in the Framingham Heart Study. Results suggest a positive causal peer effect of BMI between friends.
Directed acyclic graph (DAG) representing the common core of causal models for peer effects with observational data. The target of interest is the total causal effect of individual 2's (the alter's) phenotype on individual 1's (the ego's) subsequent phenotype, . Latent homophily bias arises from implicit conditioning on the social tie , which opens the noncausal path , . Confounding bias arises from unobserved common causes, , satisfying . Although presented for the case when , other cases are represented by dropping (when ) or adding (when ) and the analogous edges to those involving , . Variables U and C are unobserved, all others are observed.
Directed acyclic graph (DAG) representing the common core of causal models for peer effects with observational data. The target of interest is the total causal effect of individual 2's (the alter's) phenotype on individual 1's (the ego's) subsequent phenotype, . Latent homophily bias arises from implicit conditioning on the social tie , which opens the noncausal path , . Confounding bias arises from unobserved common causes, , satisfying . Although presented for the case when , other cases are represented by dropping (when ) or adding (when ) and the analogous edges to those involving , . Variables U and C are unobserved, all others are observed.
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DAG involving time-invariant IV for causal estimation of when . The variables and () are observed and unobserved individual predictors of , respectively, that may also affect tie-formation. While can be conditioned on cannot, necessitating the use of IV-methods. When (one follow-up period), instruments ; when (the case presented here), instruments both and ; and so on until instruments . IV identification is reliant on being observed so that they can be instrumented (if dim) and not being causes of (i.e., they cannot contribute to homophily).
DAG involving time-invariant IV for causal estimation of when . The variables and () are observed and unobserved individual predictors of , respectively, that may also affect tie-formation. While can be conditioned on cannot, necessitating the use of IV-methods. When (one follow-up period), instruments ; when (the case presented here), instruments both and ; and so on until instruments . IV identification is reliant on being observed so that they can be instrumented (if dim) and not being causes of (i.e., they cannot contribute to homophily).
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DAG involving time-varying instrumental variable (IV), , assumed to be a cause of through the interaction of with a time-varying variable (e.g., age) in , (presented when ). The variables and () are observed and unobserved individual predictors of , respectively, that may also affect tie-formation. While can be conditioned on cannot, necessitating the use of IV-methods. By conditioning on and , the noncausal pathways from to (e.g., ), , are blocked making a valid IV. If is added to the DAG, it is necessary to condition on .
DAG involving time-varying instrumental variable (IV), , assumed to be a cause of through the interaction of with a time-varying variable (e.g., age) in , (presented when ). The variables and () are observed and unobserved individual predictors of , respectively, that may also affect tie-formation. While can be conditioned on cannot, necessitating the use of IV-methods. By conditioning on and , the noncausal pathways from to (e.g., ), , are blocked making a valid IV. If is added to the DAG, it is necessary to condition on .
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Fitted values of BMI, , across the individuals in the FHS sample are obtained from a regression of BMI on exam (categorical), gender, birth era (categorical), year born, marital status, number of siblings, and smoking status. The smooth curves are computed using a generalized additive spline regression model with smoothing factors judiciously chosen to capture local trends but not overfit the data.
Fitted values of BMI, , across the individuals in the FHS sample are obtained from a regression of BMI on exam (categorical), gender, birth era (categorical), year born, marital status, number of siblings, and smoking status. The smooth curves are computed using a generalized additive spline regression model with smoothing factors judiciously chosen to capture local trends but not overfit the data.
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Supplementary resources (2)

Supplementary Material 1
Data
April 2014
James O'Malley · Felix Elwert · (James) Niels Rosenquist · Alan M Zaslavsky · Nicholas Christakis
Supplementary Material 2
Data
April 2014
James O'Malley · Felix Elwert · (James) Niels Rosenquist · Alan M Zaslavsky · Nicholas Christakis
... We found the effect of an omitted confounder would need to have a combined relationship with the outcome (employees' purchases at transaction t) and predictor (coworkers' purchase with the employee at t − 1) larger than any measured for the observed covariates in our models before our statistically significant measures were rendered non-significant (Supplementary Table 10). Finally, we conducted an instrumental variables analysis using purchases by a second-degree coworker (the coworker's coworker) as a putatively exogenous source of variation in the coworker's purchasing, thus eliminating concerns of homophily to the extent that it manifests as unmeasured confounding ( Supplementary Fig. 9) 47 . For these analyses, we began with the same subset of data used above in Fig. 3 involving the strictest definition of a social tie (tie probability ≥0.6), but to justify the instrumental variable exclusion restriction, we excluded any second-degree alters of a given ego that were also first-degree alters of that same ego during the study period, as well second-degree alters that were egos in the analysis themselves. ...
... The exception to that caveat comes from our instrumental variables analysis that, if its assumptions hold, would remove all bias due to homophily, confounding or simultaneity. We reiterate our caution in accepting that our available instrument satisfies the exclusion restriction in the presence of homophily, confounding or simultaneity and the independence assumption required for valid instrumental variables estimation 47 . Future work could consider using different instrumental variables (including the special case of randomized assignment to peers/networks) in a completely identified, closed network, although these are challenging methods to execute in real-world scenarios. ...
... The data setup for each stage (that is, the timing of purchases included as dependent and independent variables) was identical to that used in the GEE analyses. In the first stage regression, purchases by second-degree alters (the primary alters' alters, Supplementary Fig. 9) were used as an instrument for primary alters' purchases 47 . To strengthen the plausibility of the exclusion restriction, only purchases by secondary alters who did not appear elsewhere in the data as either first-degree alters or egos were included as instruments. ...
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... In sociology and social sciences, the purpose of causal inference is to examine the association between social network and behaviors, also known as peer effects, social contagion or induction [173]- [175]. The peer effect means that the behavior, traits, or characteristics of an individual's peers (those he is connected to or alters) would affect his behavior [175]. ...
... In sociology and social sciences, the purpose of causal inference is to examine the association between social network and behaviors, also known as peer effects, social contagion or induction [173]- [175]. The peer effect means that the behavior, traits, or characteristics of an individual's peers (those he is connected to or alters) would affect his behavior [175]. Due to contextual confounding, peer selection, simultaneity bias and measurement error, [176] points that it is very difficult to estimate the peer effects from observational data but instrumental variables (IVs) can help to address these problems. ...
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... Past studies on inferring network effects have attempted to develop a better understanding of the role of homophily in network formation, and to test hypotheses about the relative prevalence of homophily and peer influence in social behavior using techniques such as instrumental variables [89], propensity score matching [90], SAOMs [17], [65]- [69], ERGMs [16], and latent space models [91], [92]. In this section, we will illustrate how our simulation strategy builds upon and extends [17], [93] to obtain estimates of homophily and influence based on network and behavioral characteristics across different sampling conditions. ...
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... It is plausible that there may be cross-trait social genetic effects given large phenotypic and genotypic correlations between BMI, drinking, and smoking (Liu et al. 2019) and the possibility that genes have 'pleiotropic effects' (having effects on multiple phenotypes; Lee et al. 2012;Visscher et al. 2017). A suggestion to study social influence effects using genes as instrumental variables forms another approach that may give insight here, but suffers from stricter assumptions (O'Malley et al. 2014). ...
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... Addressing collider bias in the causal modelling of social network data and learning-based hypotheses presents additional challenges (Lyons, 2011;Shalizi & Thomas, 2011). While some traditional approaches for addressing bias may assist with collider bias [i.e., counterfactual (Elwert & Christakis, 2008) or instrumental variable approaches (O'Malley et al., 2014)], several statistical models unique to network analysis have also been introduced as possible solutions. Shalizi and Thomas (2011) propose a model of tie formation and dissolution, aiming to identify instances where behaviors spread in response to tie formation and cease in response to tie dissolution, eliminating instances where an outcome is maintained by similar characteristics rather than social contagion (Krivitsky & Handcock, 2014). ...
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