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Keywords:

  • Body-mass index;
  • Causality;
  • Directed acyclic graphs;
  • Dyad;
  • Genes;
  • Homophily;
  • Instrumental variable;
  • Longitudinal;
  • Mendelian randomization;
  • Peer effect;
  • Social network;
  • Two-stage least squares

Summary

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 Directed Acyclic Graphs (DAGs)
  5. 3 Graphical IV Criteria
  6. 4 Causal Models for Peer Effects in Dyads
  7. 5 Potential Outcomes Representation
  8. 6 Dyadic Instrumental Variables Analysis
  9. 7 Friend and Spouse Peer Effect Analysis of the FHS Network
  10. 8 Conclusion
  11. 9 Supplementary Web Appendix
  12. 10 Acknowledgements
  13. References
  14. Supporting Information

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.


1 Introduction

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 Directed Acyclic Graphs (DAGs)
  5. 3 Graphical IV Criteria
  6. 4 Causal Models for Peer Effects in Dyads
  7. 5 Potential Outcomes Representation
  8. 6 Dyadic Instrumental Variables Analysis
  9. 7 Friend and Spouse Peer Effect Analysis of the FHS Network
  10. 8 Conclusion
  11. 9 Supplementary Web Appendix
  12. 10 Acknowledgements
  13. References
  14. Supporting Information

We develop instrumental variable (IV) methods for the estimation of causal peer effects using longitudinal dyadic data from a social network. A peer effect (social contagion, induction) occurs when a behavior, trait, or characteristic of an individual's peers (those to whom she is connected, or alters) affects her own (the ego's) health behavior. While evidence exists of associations of observed traits (phenotypes and behaviors) among groups of individuals (such as obesity (Christakis and Fowler, 2007), smoking (Christakis and Fowler, 2008), and alcohol use (Rosenquist et al., 2010)), experiments to prove that such associations are causal are often difficult or impossible due to practical or ethical limitations on randomization, albeit with a few exceptions (Wing and Jeffery, 1999; Centola, 2010; Fowler and Christakis, 2010).

Observational analyses may suffer from selection bias due to non-random assignment of treatment. The challenges are magnified in network contexts as confounding takes several structurally different forms. In addition to the spread of health traits because of peer influence, clusters of similar individuals may form due to both homophily (“birds of a feather flock together”) and unmeasured common causes affecting socially connected individuals (confounding). Because each of these phenomena may lead to correlations between the phenotypes of connected individuals (Christakis and Fowler, 2007; Shalizi and Thomas, 2011), methods to parse these associations apart are required.

One approach to causal inference with observational data emulates randomized trials by using an instrumental variable (IV), a variable that influences exposure but, conditional on the exposure, has no influence on the outcome (Angrist, Imbens, and Rubin, 1996). However, the literature on the use of IVs to estimate peer effects is limited. Randomized dorm-room assignments have been used to estimate peer effects among college students (Sacerdote, 2001) and military recruits (Carrell, Fullerton, and West, 2009). In other settings, covariates averaged over neighboring observations (contextual variables) have been used as IVs for peer effects (Fletcher, 2008).

Directed acyclic graphs (DAGs) can clarify the identification problems of IV analysis for peer effects by focusing attention on the causal relationships among variables to better align the identification strategy with scientific judgments (Pearl, 2009). We use DAGs to (1) identify subtle dependencies that complicate estimation of peer effects, (2) succinctly notate causal data generating models, and (3) prove theorems about identifiability conditions for causal peer effects. We illustrate our methods using networks with a simple structure consisting of disjoint pairs of individuals (dyads), with no influence (interference) between dyads.

Our motivating application concerns peer effects in the Framingham Heart Study (FHS) (Christakis and Fowler, 2007), specifically the utility of using recently sequenced genetic data to develop IVs for peer effects on body mass index among friends and spouses. The appeal of genes as IVs is that they are inherently randomized by a naturally occurring process, are assigned at conception, and are not directly visible and hence, unlikely to directly influence other individuals. Several recent methodological papers discuss Mendelian randomization as IVs (Didelez, Meng, and Sheehan, 2010; Vansteelandt et al., 2011; Palmer et al., 2012) but none consider peer effects. Our paper explores promises as well as pitfalls facing the use of Mendelian randomization as IVs in the study of peer effects.

In Sections 'Directed Acyclic Graphs (DAGs)''Causal Models for Peer Effects in Dyads', we introduce DAGs to develop several increasingly general causal models for peer effects involving IVs to account for latent homophily and unmeasured confounding. Our models accommodate several other features often considered obstacles to identifying peer effects, including pleiotropy (genes affecting multiple individual characteristics), population stratification, and gene-based homophily. Section 'Potential Outcomes Representation' outlines the potential outcomes representation of our preferred causal model. Estimation of these models of peer effects using longitudinal dyadic network data is described in Section 'Dyadic Instrumental Variables Analysis'. Section 'Friend and Spouse Peer Effect Analysis of the FHS Network' describes the FHS network of friend and spouse ties and evaluates the linked genetic alleles as potential IVs for peer effects. Section 'Conclusion' concludes with a discussion.

2 Directed Acyclic Graphs (DAGs)

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 Directed Acyclic Graphs (DAGs)
  5. 3 Graphical IV Criteria
  6. 4 Causal Models for Peer Effects in Dyads
  7. 5 Potential Outcomes Representation
  8. 6 Dyadic Instrumental Variables Analysis
  9. 7 Friend and Spouse Peer Effect Analysis of the FHS Network
  10. 8 Conclusion
  11. 9 Supplementary Web Appendix
  12. 10 Acknowledgements
  13. References
  14. Supporting Information

We use DAGs to encode the structural (i.e., causal) assumptions of our causal models and prove their identifiability. DAGs represent variables as nodes and the direct causal effects between them as edges. Missing edges denote sharp null hypotheses of no direct causal effect. All DAGs considered in this paper are so-called causal DAGs (Pearl, 2009), which are assumed to contain all observed and unobserved common causes in the process. Paths are non-intersecting sequences of adjacent edges, regardless of the direction of the arrows. Causal paths between a treatment and an outcome contain only edges that point away from treatment and toward the outcome. All other paths are noncausal, or spurious, paths. Variable M is a collider on a path if the path contains the formation inline image (i.e., both edges point to M). All variables directly or indirectly caused by a given variable are called its descendants. Brackets around a variable indicate that the variable has been conditioned on; for example, inline image.

The d-separation rule (Pearl, 1988) translates between the causal assumptions encoded in the DAG and the associations observable in data. A path is said to be d-separated or blocked if (1) it contains a non-collider variable that has been conditioned on, such as M in inline image (where M is a mediator) or inline image (where M is a common cause or confounder), or if (2) it contains a collider variable, inline image, and neither the collider nor any of its descendants has been conditioned on. Paths that are not d-separated are said to be d-connected, unblocked, or open. In causal DAGs, variables that are d-separated along all paths are statistically independent; and variables that are d-connected along at least one path may be associated (Verma and Pearl, 1988). The crucial point is that conditioning on a non-collider blocks the flow of association along a path, whereas conditioning on a collider or one of its descendants may induce an association.

Under conventional axioms (Pearl, 2009; Richardson and Robins, 2013), causal DAGs and potential outcomes are equivalent notational systems for predicting statistical associations and identifying the causal effects of an intervention. Since IV is principally an identification strategy for linear models, we henceforth assume that the DAG represents a linear model, making no assumptions about the distribution of the variables (e.g., joint normality).

3 Graphical IV Criteria

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 Directed Acyclic Graphs (DAGs)
  5. 3 Graphical IV Criteria
  6. 4 Causal Models for Peer Effects in Dyads
  7. 5 Potential Outcomes Representation
  8. 6 Dyadic Instrumental Variables Analysis
  9. 7 Friend and Spouse Peer Effect Analysis of the FHS Network
  10. 8 Conclusion
  11. 9 Supplementary Web Appendix
  12. 10 Acknowledgements
  13. References
  14. Supporting Information

We apply versions of the graphical criteria for detecting IVs for the total causal effect of treatment (variable) T on outcome (variable) Y in linear models developed by Brito and Pearl (2002).

Single-IV Criterion: Let inline image denote the DAG that represents the assumed causal model, and let inline image be inline image after removing all edges emanating from T (inline image represents the null hypothesis of no treatment effect). Then G is an IV for the total causal effect of T on Y conditional on a set of variables Z (the so-called conditioning set, which may be empty) if:

  1. Z contains no descendant of T in inline image.
  2. There is an unblocked path between G and T in inline image after conditioning on Z.
  3. There is no unblocked path between G and Y in inline image after conditioning on Z.

The first and third conditions give the exclusion restriction: except for the causal effect of T on Y, the IV G must be independent of Y given Z. (However, these conditions do not imply that G is independent of Y conditional on inline image—in the presence of an unmeasured cause of T and Y, conditioning on T opens a path from G to Y (Hernán and Robins, 2006).) The IV criterion generalizes to multiple treatments and multiple IVs (IV-sets).

IV-Set Criterion: For multivariate inline image, let inline image be inline image after removing all edges emanating from T. Then a multivariate inline image is an IV-set for the joint causal effect of T on Y conditional on a set of variables Z if:

  1. Z contains no descendant of T in inline image.
  2. For every inline image there exists, for some k, an unblocked path, called inline image, between inline image and inline image in inline image after conditioning on Z, such that inline image have no nodes in common.
  3. For inline image there are no unblocked paths between inline image and Y in inline image after conditioning on Z.

It follows from condition 2 that inline image for an IV-set G. Importantly, an IV-set G may exist for T even if no variable inline image individually is a valid IV for any single variable inline image (Brito, 2010). Note that IV sets identify not only the joint effect of T on Y but also the direct effect of each inline image on Y not mediated by inline image, which may coincide with the total causal effect of inline image on Y.

4 Causal Models for Peer Effects in Dyads

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 Directed Acyclic Graphs (DAGs)
  5. 3 Graphical IV Criteria
  6. 4 Causal Models for Peer Effects in Dyads
  7. 5 Potential Outcomes Representation
  8. 6 Dyadic Instrumental Variables Analysis
  9. 7 Friend and Spouse Peer Effect Analysis of the FHS Network
  10. 8 Conclusion
  11. 9 Supplementary Web Appendix
  12. 10 Acknowledgements
  13. References
  14. Supporting Information

We first present the common core of our causal models for peer effects (on BMI for illustration) to explicate the two central identification challenges: common cause confounding and homophily bias. We then discuss a series of more realistic models for peer effects and evaluate conditions under which each model can be identified via IV analysis.

4.1 The Two Identification Problems: Confounding and Homophily Bias

Figure 1 gives the core of our causal models for a longitudinally observed population of independent dyads including individuals 1 and 2. Let inline image denote BMI, the phenotype of interest for individual inline image at time t and let q denote the number of periods before the present that the tie was formed (Figure 1 depicts the case when inline image). Current BMI may affect the same individual's subsequent BMI: inline image, inline image, inline image. Additionally, each individual's present BMI may affect the other's subsequent BMI (peer effect); inline image, and inline image, inline image. We assume there were no effects of 1 and 2 on each other prior to tie-formation.

image

Figure 1. 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, inline image. Latent homophily bias arises from implicit conditioning on the social tie inline image, which opens the noncausal path inline image, inline image. Confounding bias arises from unobserved common causes, inline image, satisfying inline image. Although presented for the case when inline image, other cases are represented by dropping (when inline image) or adding (when inline image) inline image and the analogous edges to those involving inline image, inline image. Variables U and C are unobserved, all others are observed.

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BMI is affected by two more types of variables, each assumed to be at least partially unobserved. The first is a vector of individual-specific unobserved variables inline image, inline image, (inline image, inline image) such as metabolic functioning, food preferences, etc. Second, each individual's BMI is potentially affected by shared environmental exposures, inline image, such as local food sources, restaurant commercials, food fads, etc. Thus, inline image for some or all of inline image; Figure 1 depicts a case where inline image corresponds to an event at inline image. Finally, inline image represents the existence of a social tie between individuals 1 and 2.

Taking the perspective of individual 1, the goal is to identify the total causal effect of inline image on inline image; that is, the effect of 2's BMI at time inline image on 1's subsequent BMI at time inline image. Without loss of generality, we focus on the peer effect from inline image to inline image. In the causal model of Figure 1, presented with inline image, treatment inline image and outcome inline image share three sources of association—one causal and two spurious. First, treatment may affect the outcome along the causal path inline image, the causal effect we aim to identify. Second, they may be associated due to unobserved shared environmental confounding by inline image along the unblocked non-causal paths inline image and inline image. Third, and centrally for this investigation, treatment and outcome may be associated due to the preferential (nonrandom) formation of social ties. The status of inline image may be affected by inline image, because, for example, people bond preferentially with others holding similar tastes in food (homophily—“birds of a feather flock together”) or with opposite tastes (heterophily—“opposites attract”). This preferential formation turns inline image into a collider variable. Investigating peer effects among individuals linked by a social tie necessarily implies conditioning on the social tie. Since inline image is a collider, conditioning on it opens the noncausal path inline image, and hence induces a noncausal association between treatment and outcome. Bias due to falsely considering this association as causal is generically known as homophily bias (Shalizi and Thomas, 2011) and constitutes a type of selection bias (Elwert and Christakis, 2008; Elwert, 2013). This spurious association cannot be eliminated by conditioning on any set of observed variables if the sources of tie formation are at least partially unobserved, and it will exist even if the causal effect of inline image on inline image is zero. In fact, using Pearl (1995), it can be shown that common cause confounding in inline image and homophily in inline image prevent non-parametric identification of the causal effect of inline image on inline image under the causal model of Figure 1.

4.2 IV Identification for Various Causal Models of Peer Effects

We now investigate the identification of peer effects despite confounding and homophily bias in several more realistic causal models. Figures 2 and 3 elaborate on the model in Figure 1 in two ways: first, by explicitly adding the observed exogeneous covariates inline image (such as gender, age, education, and the geographic distance between ego's and alter's residences) and, second, by adding inline image (such as genes or other isomorphic variables) affecting BMI but not tie-formation for inline image. We do not index inline image by t but note that these variables may contain time-varying elements.

image

Figure 2. DAG involving time-invariant IV inline image for causal estimation of inline image when inline image. The variables inline image and inline image (inline image) are observed and unobserved individual predictors of inline image, respectively, that may also affect tie-formation. While inline image can be conditioned on inline image cannot, necessitating the use of IV-methods. When inline image (one follow-up period), inline image instruments inline image; when inline image (the case presented here), inline image instruments both inline image and inline image; and so on until inline image instruments inline image. IV identification is reliant on inline image being observed so that they can be instrumented (if diminline image) and inline image not being causes of inline image (i.e., they cannot contribute to homophily).

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image

Figure 3. DAG involving time-varying instrumental variable (IV), inline image, assumed to be a cause of inline image through the interaction of inline image with a time-varying variable (e.g., age) in inline image, inline image (presented when inline image). The variables inline image and inline image (inline image) are observed and unobserved individual predictors of inline image, respectively, that may also affect tie-formation. While inline image can be conditioned on inline image cannot, necessitating the use of IV-methods. By conditioning on inline image and inline image, the noncausal pathways from inline image to inline image (e.g., inline image), inline image, are blocked making inline image a valid IV. If inline image is added to the DAG, it is necessary to condition on inline image.

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Figures 2 and 3 differ in only one, albeit crucial, respect. The model in Figure 2 provides for a scenario where the time-invariant (assigned at conception) gene G alone is the instrument, whereas Figure 3 supposes that gene expression varies over time due to an interaction with a time-varying covariate in X, GX. We shall refer to these as gene-alone and gene-interaction identification, respectively.

4.2.1 Gene-alone identification

We now evaluate whether inline image can serve as an IV for inline image under various conditioning strategies, where Z denotes the variables conditioned on. Figure 2 includes several different cases based on q. We first suppose the number of periods since tie-formation is inline image and then inline image, and finally draw conclusions for general q. The case when inline image can be thought of as estimating a single peer effect over the entire follow-up period since tie-formation at inline image while other cases allow the peer effect to be incrementalized, which is useful if there are time-varying predictors. In this section, we again focus on the peer effect from inline image to inline image.

Theorem 1. Assume that inline image in the causal model represented by Figure 2. Then inline image is an IV for the total causal effect inline image conditional on inline image.

Proof 1. Condition (1) of the single-IV criterion is met because inline image is not a descendant of inline image. Condition (2) is met because the path inline image is a direct effect and hence is unblocked. Condition (3) is met because all paths from inline image to inline image in inline image pass through the colliders inline image and inline image; since neither inline image nor inline image is conditioned on, and inline image is not a descendant of either, all paths from inline image to inline image in inline image are blocked. □

The model in Figure 2 permits conditioning on certain additional variables.

Corollary 1. In Figure 2 with inline image, any subset of inline image can be conditioned on in addition to inline image without affecting the IV identifiability of inline image.

Proof 2. The single-IV criterion is met because (1) no variable in Z descends from inline image; (2) is trivially met; (3) all paths from inline image to inline image in inline image pass through the colliders inline image and inline image, which block these paths and are not opened by conditioning on Z since no variable in Z descends from inline image or inline image. □

Corollary 1 is useful because all variables in Z are associated with the outcome inline image—conditional on inline image and the other variables in Z—such that conditioning on them will reduce variance in inline image and lead to more precise estimates.

Gene-alone identification fails when inline image when inline image is univariate in Figure 2 because no amount of conditioning can remedy several exclusion violations. For example, the open path inline image can only be blocked by conditioning on inline image or inline image; but doing so would necessarily induce another exclusion violation by opening the path inline image as inline image is a collider on this path and inline image descends from this collider. However, the total causal effect of inline image can be identified via the IV-set criterion if inline image is multivariate (e.g., representing multiple genes, or multiple alleles of the same gene, that each affect inline image over inline image; diminline image).

Theorem 2. In the causal model represented by Figure 2 with inline image, if diminline image, then inline image is an IV set for the total causal effect of inline image on inline image after conditioning on inline image.

Proof 3. The IV-set criterion for the joint causal effect of inline image and inline image on inline image is met because (1) inline image does not descend from inline image or inline image; (2) inline image and inline image are open and share no nodes (since inline image is multivariate); (3) all paths from inline image to inline image must pass through inline image, inline image, or inline image, which are colliders in inline image; since neither inline image, inline image, inline image, nor any of their descendants are conditioned on, all paths from inline image to inline image are blocked. Finally, since the total causal effect of inline image on inline image is not-mediated by inline image, IV set identification of the joint causal effect of inline image and inline image on inline image implies identification of the total causal effect of inline image on inline image. □

Corollary 2. Theorem 2 generalizes to arbitrary inline image, diminline image, where inline image instruments inline image with any subset of inline image together with inline image as the conditioning set.

Proof 4. Directly extend the proof of Theorem 2 and Corollary 1. □

The solution to the identification problem in Figure 2 when inline image, inline image, inline image, and diminline image involves an unusual use of IV. Whereas typically IVs are used to identify treatment effects, here, inline image both identifies the treatment effect and remedies the exclusion violation that would occur if the paths inline image were not accounted for by instrumenting inline image for inline image.

Corollary 2 illustrates that inline image faces an increasing challenge with the duration of the social tie as all values of the alter phenotype over inline image must be instrumented. Because inline image has limited dimension this will eventually be impossible. The central limitation of gene-alone identification, however, is that it breaks down under homophily on phenotype.

Corollary 3. If inline image for any inline image is added to Figure 2 then inline image of any dimension is not a valid IV to identify the total causal effect of inline image on inline image, conditional on inline image.

Proof 5. Because inline image is a descendant of inline image, conditioning on inline image is equivalent to conditioning on inline image, which opens the unblockable noncausal path inline image, among others, inline image, representing an exclusion violation. □

Therefore, we next look beyond using genes alone as IVs.

4.2.2 Gene-interaction identification

Even though genes themselves are not time-varying, their expression often is. The causal model analogous to that of Figure 2 but with time-varying gene expression is shown in Figure 3. Let inline image denote a variable representing individual k's (inline image) gene-by-age expression at time t (here the notation GX reflects that age is an element of X). The edges inline image and inline image are included at all periods to represent varying gene expression due to age.

Theorem 3. In Figure 3 the effect inline image, inline image (the case inline image is presented), is identified by using inline image to instrument inline image conditional on inline image, inline image, and inline image.

Proof 6. Because inline image only affects inline image the single-IV criterion applies. Therefore, after conditioning on inline image, inline image, and inline image an analogous argument as for Theorem 1 completes the proof. □

Corollary 4. Under the DAG in Figure 3, inline image, inline image and inline image may be added for inline image, inline image without compromising IV-identification based on inline image.

Corollary 4 (proof omitted) illustrates that exploiting time-varying gene expression is advantageous in three ways. First, it allows genetic homophily at (or before) inline image, inline image. Second, it allows homophily on the phenotype of interest up to but not including inline image. This restriction appears reasonable given prior work suggesting that changes in physical appearance (e.g., BMI) have minimal impact on tie-dissolution even if initial similar appearance led to tie-formation (O'Malley and Christakis, 2011). Third, the requirements for identification do not get more onerous with q. These flexibilities centrally motivate our adoption of Figure 3 as the primary causal model in our empirical analysis.

4.2.3 Relaxing further assumptions

In observational data settings, it is important to evaluate the extent to which a given identification strategy is consistent with multiple plausible causal models. Table 1 summarizes several substantively important elaborations of the causal models in Figures 2 and 3, all of which consist of adding edges; that is, relaxing assumptions (proofs omitted).

Table 1. Extensions to DAGs and their consequence when inline image and individual 1 is the ego
PhenomenonEffectChange to ZApplies to figure
  1. a

    Including unmeasured prior phenotype, inline image for inline image and inline image.

  2. b

    Shared ancestry of individuals 1 and 2.

  3. c

    Add indicator variables for each dyad to Z.

Homophily oninline imageNo implication3
measured phenotypeinline imageNo remedy2, 3
(inline image)inline imageNo remedy2, 3
Homophily oninline imageNo implication3
measured genotypeinline imageNo implication3
(inline image)inline imageNo remedy3
Pleiotropy oninline imageAdd inline image2
observablesinline imageNo implication3
Pleiotropy oninline imageNo remedy2
unobservablesainline imageNo implication3
Populationinline imageAdd dyad2, 3
stratificationbinline imagefixed effectsc 
Inter-phenotypeinline imageNo implication2, 3
Peer effectinline imageNo implication2, 3
 inline imageNo implication2, 3
Predictorinline imageNo implication2, 3
Associationsinline imageAdd inline image2, 3
 inline imageAdd inline image2, 3
Confounding oninline imageNo remedy2
genotype orinline imageNo remedy3
gene expressioninline imageAdd inline image3
Epigeneticinline imageAdd Y2(0)3
Effectsinline imageNo implication3
Serial dependentinline imageAdd inline image3
gene-expressioninline imageAdd inline image3
Relationshipinline imageNo implication2, 3
status (inline image)inline imageNo implication2, 3
 inline imageNo implication2, 3

First, as noted previously, homophily on the phenotype at any time is lethal for gene-alone identification with a single IV under the model of Figure 2, but homophily on phenotype prior to inline image is not lethal for identifying the peer effect from inline image to t under Figure 3.

Second, inline image may be pleiotropic; that is, affect not only BMI, but also other characteristics of the individual. In Figure 2, inline image may additionally affect observed covariates inline image (necessitating conditioning on inline image) but not unobserved features directly affecting social-tie formation; that is, inline image (because of the irreparable exclusion violation inline image). By contrast, in Figure 3, adding inline image, inline image and even inline image are unproblematic, as is inline image and inline image, inline image, (but not inline image or inline image). Importantly, pleiotropy on unobservables (inline image) includes effects of genes on latent pre-tie formation phenotype (which by virtue of being unobserved is an element of inline image). Pleiotropy on latent pre-tie formation phenotype thus ruins IV identification only in the case of Figure 2, but it does not ruin IV identification in Figure 3.

Third, population stratification describes an association between inline image and inline image based on sharing attributes due to common ancestry (Didelez and Sheehan, 2007). To protect the exclusion restriction, one should control for race and ethnicity and ensure (to the extent possible) that members of the dyad are not directly related (e.g., using the method in Price et al. (2006)). However, because ethnic origin (e.g., Irish, German, Greek) is seldom available within general racial groups, including dyad fixed-effects is a more rigorous strategy of accounting for population stratification.

Fourth, our results also accommodate inter-phenotype peer effects; if inline image affects inline image, inline image, the results above hold. Even if 2's unobserved characteristics, inline image, affect inline image, our results continue to hold. Fifth, effects of 2's observed characteristics on unobserved shared environmental exposures (e.g., via residential choice), inline image, or on 1's observed characteristics, inline image, have no implications. Sixth, epigenetic confounding on unobserved contextual factors, inline image, inline image, can be accounted for by conditioning on inline image under Figure 3. Even under epigenetic effects due to the phenotype, which imply the addition of inline image, inline image, to Figure 3, identifiability is not affected except if t < q then Y2(t – 1) must be added to Z.

Finally, if inline image, inline image, (serial dependence) is added to Figure 3 it is necessary to condition on inline image in addition to inline image and inline image for inline image (for inline image) to be an IV. Therefore, inline image must not be fully determined by inline image, inline image, and inline image. Likewise, if inline image is added to Figure 3 then inline image must be added to Z. In summary, the IV and IV-set criteria permit identification of peer effects in a surprisingly large class of causal models with latent homophily and confounding.

5 Potential Outcomes Representation

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 Directed Acyclic Graphs (DAGs)
  5. 3 Graphical IV Criteria
  6. 4 Causal Models for Peer Effects in Dyads
  7. 5 Potential Outcomes Representation
  8. 6 Dyadic Instrumental Variables Analysis
  9. 7 Friend and Spouse Peer Effect Analysis of the FHS Network
  10. 8 Conclusion
  11. 9 Supplementary Web Appendix
  12. 10 Acknowledgements
  13. References
  14. Supporting Information

From hereon, we assume the causal model of Figure 3 and its extensions, which gives IV point identification under linearity and homogeneity (Brito and Pearl, 2002). We now exhibit model form assumptions using the potential outcomes representation of the DAG in Figure 3. We explicitly allow for time-varying elements of inline image, inline image, and inline image by adding the subscript inline image, use bold-face font to denote vectors, and use lower-case letters to denote observed and counterfactual values of random variables.

A potential outcome inline image is the value of an outcome Y that would be observed if a variable V were set by intervention to inline image. An observed value of V is denoted v, distinguishing it from the counterfactual inline image. Therefore, inline image denotes the potential outcome that would result for individual 1 if individual 2's phenotype at inline image were set to inline image and her gene-expression were set to inline image.

Under the DAG in Figure 3, a causal model for the potential outcomes of inline image given the conditioning set inline image (which must include inline image and inline image) is

  • display math(1)

where inline image, inline image, inline image, and inline image are coefficients and inline image is a random error. We assume inline image has constant variance, which simplifies estimation, but note that the assumption can be relaxed without affecting identification. The involvement of inline image and inline image in (1) illustrates that causal models make no distinction between observed and unobserved covariates. Due to the exclusion restriction, inline image is absent from the right-hand-side of (1). Therefore, the left-hand-side of (1) may be denoted inline image. Then the peer effect we seek to estimate satisfies inline image for inline image.

6 Dyadic Instrumental Variables Analysis

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 Directed Acyclic Graphs (DAGs)
  5. 3 Graphical IV Criteria
  6. 4 Causal Models for Peer Effects in Dyads
  7. 5 Potential Outcomes Representation
  8. 6 Dyadic Instrumental Variables Analysis
  9. 7 Friend and Spouse Peer Effect Analysis of the FHS Network
  10. 8 Conclusion
  11. 9 Supplementary Web Appendix
  12. 10 Acknowledgements
  13. References
  14. Supporting Information

To implement IV analysis of (1), we use a two-stage least squares (2SLS) procedure. The “first-stage” of 2SLS regresses the endogeneous variable inline image, inline image, on the IV and the exogeneous variables in inline image (including inline image and inline image if conditioned on), yielding the regression

  • display math(2)

from which the fitted values, inline image, are computed. The second-stage applies OLS to

  • display math(3)

where inline image, estimating the peer effect inline image. Because inline image is an IV in (2), under OLS estimation inline image is orthogonal to inline image and inline image in (3), ensuring unbiased and statistically efficient IV-based estimates. The procedure generalizes to accommodate multiple heterogeneous effects such as two-period dependence (i.e., if inline image) and effect heterogeneity in observed effect modifiers (see Web Appendix).

6.1 Variance Estimation

Standard errors are computed using results from the general theory for 2SLS. Because the peer effects are of alter's lagged as opposed to contemporaneous phenotypes, the complications posed by the simultaneous involvement of the same observation as a predictor and an outcome (VanderWeele, Ogburn, and Tchetgen Tchetgen, 2012) are avoided. To account for repeated observations made on dyads over time, as outlined in the Web Appendix, we compute robust standard errors based on sandwich estimators (White, 1982).

7 Friend and Spouse Peer Effect Analysis of the FHS Network

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 Directed Acyclic Graphs (DAGs)
  5. 3 Graphical IV Criteria
  6. 4 Causal Models for Peer Effects in Dyads
  7. 5 Potential Outcomes Representation
  8. 6 Dyadic Instrumental Variables Analysis
  9. 7 Friend and Spouse Peer Effect Analysis of the FHS Network
  10. 8 Conclusion
  11. 9 Supplementary Web Appendix
  12. 10 Acknowledgements
  13. References
  14. Supporting Information

We illustrate our methods using a novel social network dataset constructed from the first seven health exams of the Offspring Cohort of the Framingham Heart Study (FHS), encompassing 32 years of follow-up. The Offspring Cohort includes 5124 individuals. Genetic data was available for 3462 distinct individuals, appearing in 22,361 exams (see Web Appendix).

The network ties considered here arise from participants naming friends and spouses at their health exams. Participants typically only named a single friend at each exam, which is likely to be the one with the most influence. Given the stability of the Framingham population from 1971 to 2003, approximately 50% of the nominated friend contacts were themselves also participants in the FHS and thus provided the same information, including BMI. Most spouses of FHS participants were also FHS participants. We estimate our model with a sample of 9270 unique dyads comprising spousal and nearly disjoint friendship dyads (ignoring occasional overlap of dyads when the same ego is named by multiple alters).

Because the fat mass and obesity gene (FTO) and the melanocortin-4 receptor gene (MC4R) have been confirmed through original and replication studies to be strongly associated with obesity (Speliotes et al., 2010), we consider them as IVs for peer effects of BMI. There is also evidence suggesting that genetic effects may be moderated by a person's age (Lasky-Su et al., 2008), justifying consideration of age-dependent gene expression as an IV.

Linearity is assumed for the data analysis and, moreover, we are interested in the linear peer effect of BMI itself. However, we note that in certain applications one might instead be interested in peer effects of obesity (BMI inline image), the effect of some other nonlinear transformation of BMI, or in the extent to which the peer effect of BMI is modified by age or some other individual characteristic of the alter (or the ego). While many interesting specifications could be considered, for illustration, we have chosen to focus on a linear specification.

We adjust for ego's gender, age, gender–age interaction, birth era, birth year, smoking status, number of siblings, geographic distance between residential locations of ego and alter at tie-formation, and gene–age interactions. Birth era accounts for whether an individual was born before 1942, between 1942 and 1948, or 1948 or later to capture possible cohort effects due to America's involvement in World War II. Because the offspring cohort is nearly 100% white, we do not adjust for race.

In addition, we adjust for wave number dummies to account for secular trends in BMI. Therefore, one can think of gene-age expression as random with respect to exam timing. Inclusion of alter's smoking status provides assurance against a possible pleiotropic effect between FTO and smoking and MC4R and smoking.

7.1 Representation of Genes

Genetic alleles are represented in inline image, inline image, by four dummy variables for two of the three possible states of each of FTO (states AA, AT, TT) and MC4R (states CC, CT, TT). The A and C alleles have been recognized by geneticists as the risk-alleles of FTO and MC4R, respectively. Having two copies of the risk-allele is the riskiest state followed by the one-copy heterozygous state. Therefore, we also include a fifth dummy variable corresponding to FTO = AA and MC4R = CC. While we could instrument 5 waves of phenotypes using gene-alone IV identification (Figure 2 and Corollary 2), we can relax more assumptions under gene–age interaction IV identification (Figure 3, Theorem 3, and Table 1). The age-dependent association of the FTO gene with BMI is clearly evident in Figure 4 (see Web Appendix for the same for MC4R).

image

Figure 4. Fitted values of BMI, inline image, across the inline image 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.

Download figure to PowerPoint

7.2 Dyadic Peer Effect Analyses

We estimate several statistical models, starting with one that is consistent with the causal model of Figure 3, as well as statistical models obtained by adding several of the Exclusions in Table 1. The four reported here condition on inline image, inline image, and inline image and are distinguished by whether inline image was excluded (as permitted in Figure 3) or conditioned on (to accommodate inline image) and by whether inline image was excluded or conditioned on (only allowed under Figure 3) to possibly improve precision. Because population stratification is a major concern in analyses involving genes and phenotypes, we include dyad fixed effects in all analyses. Thus, the five gene–age interaction variables of the alter (individual 2) are the IVs for inline image. We also performed analyses with the analogous five gene–ageinline image interaction variables as additional IVs; results remained essentially unchanged (not shown). We perform separate analyses for friends and spouses and use robust variance estimators to account for repeated observations over time (Section 'Variance Estimation').

7.3 Estimated Peer Effects

The IV estimates are consistent with positive BMI peer effects among friends and spouses (Table 2). Under the causal model of Figure 3 with inline image, the estimated BMI peer effect among friends (row 1) is positive and statistically significant (inline image, 95% CI inline image), whereas the BMI peer effect among spouses (row 5) is positive but not statistically significant (inline image, 95% CI inline image). In all other specifications (i.e., relaxations of Figure 3), the estimated BMI peer effects among friends and spouses are not statistically significant, although point estimates remain in the expected positive direction in most models. For many IV specifications, the corresponding OLS estimates differ appreciably, consistent with the presence of unobserved confounding and homophily bias in the OLS specifications.

Table 2. Dyadic peer effect analysis of lag alter BMI using time-varying gene–age expression as an instrument
Discretionary inline image termsIV Regression (2SLS)aRegression (OLS)
   
inline imageinline imageinline imagebEstimate95% CIEstimate95% CI
  1. a

    inline image are exogeneous covariates and inline image is an IV in all IV analyses. The elements of inline image, inline image, are: gender, age, gender–age interaction, birth era, birth year, smoking status, number of siblings, and (for inline image only) the geographic distance between residential locations of ego and alter at tie-formation. All models include dyad fixed effects. inline image and inline image are added to inline image as indicated in the two left-most columns.

  2. b

    The F-statistic is for the overall effect of the IV, inline image, in the first-stage equation. The critical value of the Cragg-Donald F-statistic, which quantifies the power of an IV, at the 20% level ranges from 6.71 to 6.77 across the models.

Nominated friend        
 
ExcludeExclude2.1500.8880.0631.713inline imageinline image0.100
ExcludeCovariate1.7310.874inline image1.7790.009inline image0.089
CovariateExclude1.1810.133inline image1.062inline imageinline image0.021
CovariateCovariate1.144inline imageinline image0.906inline imageinline image0.028
Spouse
 
ExcludeExclude4.0640.099inline image0.5220.0660.0390.094
ExcludeCovariate4.3510.101inline image0.4880.0320.0080.055
CovariateExclude0.268inline imageinline image1.6520.0500.0170.082
CovariateCovariate0.1810.906inline image3.6430.023inline image0.051

The imprecision (and resulting lack of significance) of many of our IV estimates is owed to relatively weak first stages. F-statistics indicate that only the causal models of Figure 3 (see inline image excluded rows of Table 2) have first stages at which IV strength is modest at best by conventional standards (e.g., under row 1, inline image for friends inline image for spouses) (Stock, Wright, and Yogo, 2002). Note, specifically that conditioning on inline image to account for possible serial dependence in gene expression (i.e., if inline image is added to Figure 3) results in a very weak first stage (e.g., inline image for spouses). This explains the noisy estimates of all rows with inline image as additional covariates in Table 2. Therefore, the absence of inline image is crucial to IV peer-effect estimation using FHS data. Other specifications (results not shown) yield first stages of similar strength. To improve precision, one might collect more data to increase sample size; or one might (we believe implausibly) assume the absence of unobserved population stratification, which would permit removal of the dyad fixed effects and result in a stronger first stage (results not shown).

8 Conclusion

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 Directed Acyclic Graphs (DAGs)
  5. 3 Graphical IV Criteria
  6. 4 Causal Models for Peer Effects in Dyads
  7. 5 Potential Outcomes Representation
  8. 6 Dyadic Instrumental Variables Analysis
  9. 7 Friend and Spouse Peer Effect Analysis of the FHS Network
  10. 8 Conclusion
  11. 9 Supplementary Web Appendix
  12. 10 Acknowledgements
  13. References
  14. Supporting Information

We derived IV methodology for the estimation of peer effects using longitudinal data. A key methodological distinction of our approach, compared to past observational approaches, is that we account for latent common causes and homophily. An important theoretical finding is that latent homophily places severe demands on IVs. Genes have appeal as IVs due to their inherent randomness, lack of visibility to peers, and ongoing influence on the phenotype. However, ongoing influence on phenotype is problematic to time-invariant IVs such as genetic alleles as all past values of the alter's phenotype post tie-formation must be instrumented (even if they only have an indirect effect on ego's BMI). However, if variation in gene expression across age is used as an IV, the dimension of the instrumented variable does not need to increase with the duration of the social tie.

Using two genes widely recognized as having the strongest effects on BMI or obesity, we explored BMI peer effects among pairs of friends or spouses. Our analyses, which attempted to account for all sources of confounding, estimated large peer effects but lacked significance in all but one case.

Continued research on the use of genes as IVs for peer effects is motivated by the fact that, if this approach is successful, many important medical, sociological, and economic questions might be more rigorously answered than they have been in the past without having to make strong assumptions about absence of unobserved homophily or unobserved confounding. Conclusive evidence of peer effects would confirm that treatment of traits such as obesity, smoking, alcoholism, and depression could be improved by treating an individual's peers in addition to himself, or by intervening on the composition of his peer group to remove undesirable peer influences.

9 Supplementary Web Appendix

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 Directed Acyclic Graphs (DAGs)
  5. 3 Graphical IV Criteria
  6. 4 Causal Models for Peer Effects in Dyads
  7. 5 Potential Outcomes Representation
  8. 6 Dyadic Instrumental Variables Analysis
  9. 7 Friend and Spouse Peer Effect Analysis of the FHS Network
  10. 8 Conclusion
  11. 9 Supplementary Web Appendix
  12. 10 Acknowledgements
  13. References
  14. Supporting Information

Web Appendices, Tables, and Figures referenced in Sections 6, 6.1, 7, and 7.1 and additional references are available with this paper at the Biometrics website on Wiley Online Library. Example code, example data, and associated instructions for running the code are also available as a web supplement (same website).

10 Acknowledgements

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 Directed Acyclic Graphs (DAGs)
  5. 3 Graphical IV Criteria
  6. 4 Causal Models for Peer Effects in Dyads
  7. 5 Potential Outcomes Representation
  8. 6 Dyadic Instrumental Variables Analysis
  9. 7 Friend and Spouse Peer Effect Analysis of the FHS Network
  10. 8 Conclusion
  11. 9 Supplementary Web Appendix
  12. 10 Acknowledgements
  13. References
  14. Supporting Information

Research for the paper was supported by NIH grants R01 AG024448 and P01 AG031093 and by a grant from the Pioneer Portfolio of the Robert Wood Johnson Foundation. The Framingham Heart Study is conducted and supported by the National Heart, Lung, and Blood Institute (NHLBI) in collaboration with Boston University (Contract No. N01-HC-25195). This manuscript was not prepared in collaboration with investigators of the Framingham Heart Study and does not necessarily reflect the opinions or views of the Framingham Heart Study, Boston University, or NHLBI. Funding for SHARe Affymetrix genotyping was provided by NHLBI Contract N02-HL-64278. Data was downloaded from NIH dbGap, project #780, with accession phs000153.SocialNetwork.v6.p5.c2.NPU.

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  2. Summary
  3. 1 Introduction
  4. 2 Directed Acyclic Graphs (DAGs)
  5. 3 Graphical IV Criteria
  6. 4 Causal Models for Peer Effects in Dyads
  7. 5 Potential Outcomes Representation
  8. 6 Dyadic Instrumental Variables Analysis
  9. 7 Friend and Spouse Peer Effect Analysis of the FHS Network
  10. 8 Conclusion
  11. 9 Supplementary Web Appendix
  12. 10 Acknowledgements
  13. References
  14. Supporting Information
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Supporting Information

  1. Top of page
  2. Summary
  3. 1 Introduction
  4. 2 Directed Acyclic Graphs (DAGs)
  5. 3 Graphical IV Criteria
  6. 4 Causal Models for Peer Effects in Dyads
  7. 5 Potential Outcomes Representation
  8. 6 Dyadic Instrumental Variables Analysis
  9. 7 Friend and Spouse Peer Effect Analysis of the FHS Network
  10. 8 Conclusion
  11. 9 Supplementary Web Appendix
  12. 10 Acknowledgements
  13. References
  14. Supporting Information

Additional Supporting Information may be found in the online version of this article.

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biom12172-sm-0001-SuppData.pdf94KSupplementary Materials.
biom12172-sm-0001-SuppData_Code.zip315KSupplementary Materials Code.

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