Identification and estimation of treatment and interference effects in observational studies on networks

1.1 Motivation

• Interference makes each unit’s potential outcomes depend on other units’ treatments, creating statistical challenges; researchers also seek to measure spillovers to leverage beneficial effects or address harmful ones.The introduction illustrates interference through infectious-disease prevention, peer learning, and social-media sharing.Source: §1.1 · pp. 2–3

1.2 Related work

• Prior strategies include partial-interference designs and methods for network data, while observational network causal inference remains an early-stage literature with differing interference and dependence assumptions.Some network methods allow general interference but establish results under partial interference or IID assumptions; another allows dependence from contagion and homophily.Source: §1.2 · pp. 4–5

1.3 Contributions

• The framework limits interference to immediate neighbors and summarizes their treatment, defining treatment and spillover estimands that are marginalized over the observed neighborhood-treatment distribution.The authors describe these estimands as more descriptive than approaches based on hypothetical treatment-vector interventions.Source: §1.3 · p. 5
• The authors extend unconfoundedness to network settings, prove identification results, and discuss the assumption’s plausibility for dependent data.Their assumption is described as weaker than related assumptions because it requires independence of treatments and each unit’s potential outcomes.Source: §1.3 · pp. 5–6
• Bias from wrongly assuming SUTVA depends on both the extent of interference and residual association between individual and neighborhood treatment after covariate adjustment.The paper uses simulations across varying interference levels and treatment correlations to validate the analytical bias results.Source: §1.3 · p. 6
• The joint propensity score balances covariates across individual- and neighborhood-treatment arms, supporting adjustment by individual-treatment subclassification and parametric neighborhood-score adjustment.The proposed score can include individual, neighbor, and network characteristics.Source: §1.3 · p. 6

2.1 Notation

• The network defines each unit’s neighborhood; the framework assumes ties are fixed and correctly measured, and distinguishes individual covariates from neighborhood structural and aggregated covariates.Neighborhoods are defined by links, and the methods can be extended to directed networks.Source: §2.1 · pp. 7–8

2.2 Potential outcomes and neighborhood interference

• Neighborhood interference allows outcomes to depend on neighbors’ treatments through a specified summary function, but not on treatments outside the neighborhood.The summary can represent treated-neighbor counts, proportions, or weighted treatment sums.Source: §2.2 · p. 9

2.3 Individual and neighborhood treatments

• The joint treatment pairs an individual’s treatment with neighborhood exposure, which is mapped from neighbors’ assignments rather than assigned directly.Individual treatment may be assigned or self-selected; neighborhood exposure follows from the treatment mapping and network.Source: §2.3 · pp. 10–11

2.4 Causal estimands: Main effects and spillover effects

• Main effects compare individual treatment at fixed neighborhood exposure, while spillover effects compare exposure g with 0 at fixed individual treatment; overall effects average over observed exposure, and total effect is their sum.This definition fixes a unit’s treatment while leaving neighbors’ treatments at their observed values.Source: §2.4 · pp. 11–13

2.5 Unconfoundedness of the joint treatment

• Unconfoundedness of both individual and neighborhood treatment given covariates identifies the average dose-response function and, consequently, main and spillover effects.The covariates must account for potential confounders of the joint treatment and outcomes.Source: §2.5 · pp. 13–14

2.6 Conditional main and spillover effects

• Conditional effects can focus on units with a specified individual or neighborhood treatment, such as zero neighborhood exposure, and require weaker identifying conditions than marginal effects.The paper reports definitions, identification results, and possible estimation methods for these effects in Appendix A.Source: §2.6 · p. 15

3.1 Naive Estimator

• Under interference, potential outcomes indexed only by individual treatment are not well-defined, and estimators comparing Zi arms regardless of Gi generally do not estimate the main effect.The naive adjustment considers individual covariates but may omit neighborhood covariates.Source: §3.1 · p. 16

3.2 Outline of Bias Results

• The bias analysis contrasts τ with τobs_X⋆ and derives expressions for cases where unconfoundedness holds or fails, highlighting conditional dependence between Zi and Gi as a bias source.When unconfoundedness fails, the expression combines bias from interference with bias from unmeasured confounders.Source: §3.2 · pp. 16–17

3.3 Bias of Naive Estimator When Unconfoudedness Holds

• With sufficient X⋆ adjustment, naive SUTVA estimators are unbiased for τ despite interference when Zi and Gi are conditionally independent; otherwise bias reflects interference and residual treatment association.The bias depends on both the level of interference and the residual association between individual and neighborhood treatments after conditioning on X⋆.Source: §3.3 · pp. 18–19

3.4 Bias of Naive Estimator When Unconfoudedness Does Not hold

• When X⋆ leaves unmeasured confounding, naive-estimator bias combines interference and confounding; conditional independence of Zi and Gi or no interference leaves only confounding bias.Unmeasured confounders may include neighborhood covariates omitted under SUTVA-based adjustment.Source: §3.4 · p. 21

4.1 Joint Propensity Scores

• The joint propensity score is the probability of receiving individual treatment z and neighborhood treatment g given observed covariates x.It is defined as ψ(z; g; x) = P(Zi = z, Gi = g|Xi = x).Source: §4.1 · p. 22, §4 · p. 22
• At a fixed joint propensity score, covariates are balanced across treatment-exposure arms, and conditional unconfoundedness permits imputing potential outcomes across arms to adjust for main and spillover effects.This equivalence to conditioning on covariates requires Assumption 3 to hold given Xi.Source: §4.1 · pp. 22–23

4.2 Individual Propensity Score and Neighborhood Propensity Score

• Factoring the joint score yields separate neighborhood and individual propensity scores, on which unconfoundedness holds separately and which enable separate adjustment for the two treatments.The neighborhood score conditions on individual treatment and neighborhood-related covariates; the individual score conditions on individual-treatment covariates.Source: §4.2 · p. 24

5 Propensity Score-Based Estimator for Main Effects and Spillover Effects

• The estimator’s unbiasedness relies on sampling that preserves network connections, such as cluster sampling or egocentric sampling with nominated alters.Source: §5 · p. 24

5.1 Individual and Neighborhood Propensity Score Estimator

• Under Assumption 3, separate adjustment for individual and neighborhood propensity scores estimates the dose-response function and yields marginal main and spillover effects.The joint propensity score is factorized into individual and neighborhood scores because the bivariate treatment makes direct stratification or matching difficult.Source: §5.1 · p. 25
• The joint propensity score must include covariates sufficient for unconfoundedness and may differ from the unit-level assignment mechanism when neighbors’ treatments depend on non-neighboring covariates.Source: §5.1 · p. 25
• For spillover effects, neighborhood-score adjustment addresses covariate imbalance across compared exposure levels, while individual-score adjustment generalizes across individual-treatment arms; for main effects, this logic reverses.The individual treatment is binary and can use standard propensity-score adjustment, whereas neighborhood treatment uses a generalized propensity score and a model-based approach within individual-score subclasses.Source: §5.1 · p. 26

5.2 Estimating Procedure: Subclassification and GPS

• The procedure subclasses units by estimated individual propensity score, then models neighborhood propensity and outcomes within subclasses to estimate the dose-response function.Unlike Hirano and Imbens (2004), both models include individual treatment, the neighborhood treatment is discrete with a function-dependent domain, and the dose-response function may not be defined for every unit.Source: §5.2 · pp. 26–27

6 Realistic Simulation Study leveraging Add Health data

• Using friendship-network data from 29 Add Health schools (16,410 students), the study simulates insurance coverage and illness-related missed school days across four treatment–friend-exposure dependence scenarios.Friend exposure is the proportion insured among the five best friends, except in scenario 3, when it is the number of treated friends among all friends.Source: §6 · p. 28

6.1 Main Effect: Bias of na¨ive estimators and GPS-based estimator

• Ignoring interference can bias main-effect estimates in proportion to interference and treatment-neighborhood association, even after individual-covariate adjustment when that adjustment leaves treatment and neighborhood exposure dependent.When treatment and neighborhood exposure are conditionally independent given individual covariates, subclassification estimators are essentially unbiased; direct correlation between them can leave interference bias despite covariate adjustment.Source: §6.1 · pp. 30–31
• The proposed Subclassification & GPS estimator is unbiased across all scenarios and interference levels, while other adjustment methods retain interference or model-misspecification bias.It adjusts for both individual and neighborhood propensity scores.Source: §6.1 · pp. 31–32, Table 2 · p. 31

6.2 Spillover Effects

• For spillover effects, the estimator using both propensity scores performs well across scenarios; omitting either score yields significant bias, and neighborhood-score adjustment outperforms individual-score-only adjustment.The reported comparisons concern estimators of ∆(0) and ∆(1).Source: Table 3 · p. 32, §6.2 · pp. 33–34, Table 4 · p. 33
• Greater insurance coverage among friends reduces missed school days; spillover effects are larger for uninsured students, while the individual coverage effect declines as friends’ coverage rises.The paper derives spillover effects from µ(z, g) − µ(z, 0) and main effects from µ(1, g) − µ(0, g).Source: §6.2 · pp. 34–35, Figure 2 · p. 35, Figure 3 · p. 35

7 Concluding Remarks

• SUTNVA defines outcomes through individual treatment and neighborhood treatment, enabling estimands that disentangle treatment and spillover effects; ignoring interference can combine confounder and interference bias.Interference-related bias is proportional to interference and treatment association, but is canceled when adjustment covariates make the two treatments independent; this result does not depend on the neighborhood-treatment function.Source: §7 · p. 36
• The proposed semiparametric estimator combines subclassification on the individual-treatment propensity score with parametric adjustment for neighborhood treatment, using a joint score that balances individual and neighborhood covariates.It estimates spillover effects as a dose-response function when interference operates through friends’ treatment, and its imputation approach supports different causal estimands.Source: §7 · p. 37
• Simulations assessed the proposed estimator for main and spillover effects, and it was consistent with the generating model; adjustment for the joint propensity score was needed across scenarios.The studies also evaluated bias from neglecting interference and validated the nonparametric bias formulas.Source: §7 · pp. 37–38
• The estimator is consistent with the stated interference model, but real-study estimates may retain bias from residual subclass imbalance or neighborhood propensity-model misspecification.The paper also identifies inference under network dependence and uncertainty about the network as open challenges, and proposes bootstrap procedures for egocentric or cluster sampling while noting other sampling schemes need further study.Source: §7 · p. 38

A.1 Conditional Main Effects

• The conditional main effect compares individual-treatment outcomes among units with neighborhood treatment Gi = g; conditional unconfoundedness for individual treatment supports unbiased estimation.The required adjustment covariates capture determinants of individual treatment given that neighborhood-treatment level.Source: §A.1 · p. 42

A.1.1 Conditional Propensity Scores of the Individual Treatment

• When covariates are high-dimensional or continuous, the conditional individual propensity score provides a balancing score and supports conditional unconfoundedness for individual treatment.Source: §A.1.1 · p. 43

A.1.2 Estimator based on Propensity Scores

• A subclassification estimator for the conditional main effect stratifies units with Gi = g by similar conditional individual propensity-score values and averages stratum estimates.The described procedure uses J strata.Source: §A.1.2 · pp. 43–44

A.2 Conditional Spillover Effects

• For control units, the conditional spillover effect compares outcomes at neighborhood treatment g versus 0; estimation can reduce this contrast to a binary neighborhood-treatment comparison under conditional unconfoundedness.The procedure focuses on units with Zi = 0 and a binary indicator for the specified neighborhood-treatment value.Source: §A.2 · pp. 44–45

A.2.1 Conditional Propensity Scores of the Neighborhood Treatment

• The conditional neighborhood propensity score is a balancing score.Source: §A.2.1 · p. 45
• Under Assumption 5, individual treatment Zi is conditionally unconfounded given the conditional neighborhood propensity score.Source: §A.2.1 · p. 45
• Conditional effects can be estimated with common propensity-score methods for binary treatments on a subset of units, but estimator performance is not illustrated here.The authors note that such estimators have been extensively assessed in the literature (e.g., Imbens & Rubin, 2015).Source: §A.2.1 · p. 45

B Data Generating Model for Simulations

• The simulation study uses the Add Health friendship network and students’ race and grade as its network and covariate inputs.Race is coded as 0 or 1, and grade ranges from 6 to 12.Source: B Data Generating Model for Simulations · p. 46
• Treatment and outcome variables are generated through different processes across four simulation scenarios.Source: B Data Generating Model for Simulations · p. 46

B.1 Correlation Structure of the Joint Treatment and Covariate Balance

• In Scenario 1, individual treatment depends on race and grade, while the small observed treatment association is attributed perhaps to homophily and disappears after conditioning on covariates.The passage attributes individual-covariate imbalance to the propensity score and small neighborhood-covariate differences presumably to homophily.Source: §B.1 · pp. 46–47
• In Scenario 2, individual and neighborhood treatments are dependent given individual covariates but independent conditional on neighborhood covariates.Neighborhood treatment depends on friends’ characteristics, with its dependence on friends’ grade attributed presumably to homophily.Source: §B.1 · pp. 47–48
• Scenario 3 generates treatment dependence through degree: neighborhood treatment counts treated friends, and the average count differs between individual treatment arms.The scenario’s treatment assignment uses individual covariates and student degree.Source: §B.1 · p. 52
• Scenario 4 retains residual dependence between individual and neighborhood treatments even after conditioning on individual, neighborhood, and other covariates.The assignment procedure directly correlates the treatments; covariate balance is reported in Tables 14–16.Source: §B.1 · pp. 52–53

B.2.1 Outcome Models for Main Effects Simulations

• For main-effect simulations, spillover effects are specified as δ(g; z) = δg, yielding Δ(z) = δE[Gi] for either individual treatment level.The interference parameter δ takes low, medium, and high levels; Scenario 3 uses values −0.3, −0.5, and −0.8, while other scenarios use −5, −8, and −10.Source: §B.2.1 · p. 54

B.2.2 Outcome Models for Spillover Effects Simulations

• For spillover-effect simulations, the main effect varies with neighborhood treatment as τ(g) = −10 + 3g, giving τ = −10 + 3E[Gi].Interference levels use δ values −5, −8, and −10, or −0.3, −0.5, and −0.8 in Scenario 3; Scenario 3 scales the last two equation terms by 1/10.Source: §B.2.2 · p. 55

B.3 True Main and Spillover Effects

• For the overall main effect τ, bias is not directly proportional to the association measure; it depends on P(Gi = g|Zi = 1, Xind_i = x) for each g.Source: §B.3 · p. 55
• Table 17 organizes scenarios by Zi–Gi correlation, true overall main effect τ, spillover effects Δ(0) and Δ(1), and partial correlation conditional on individual covariates.The true effects are computed as means of estimates across simulation data.Source: §B.3 · p. 55, Table 17 · p. 56

C Subclassification and Generalized Propensity Score Estimator

• The estimator subclasses units by individual propensity score, seeking similar scores and sufficient covariate balance between individual-treatment groups.Source: C Subclassification and Generalized… · p. 56
• Within each subclass, it models neighborhood propensity and outcome, predicts potential outcomes at joint treatment levels, and averages them to estimate dose-response functions.Source: C Subclassification and Generalized… · pp. 56–57

C.1 Individual Treatment, Neighborhood Treatment and Outcome Models

• In the simulations, individual treatment is modeled as Bernoulli, its propensity score is estimated by logistic regression, and five quintile subclasses are chosen with covariate balance.Source: §C.1 · p. 57
• The neighborhood-treatment model depends on Gi’s definition: proportion-treated-neighbor scenarios use binomial logistic regression, while the treated-neighbor-count scenario uses an alternative model.Source: §C.1 · p. 58
• For the simulations, the outcome model is linear in individual and neighborhood treatment, their interaction, and the estimated neighborhood propensity score; a cubic polynomial is recommended for more realistic settings.Source: §C.1 · p. 58
• At each joint treatment level, the procedure predicts the neighborhood propensity score and potential outcome for each unit, then averages predicted outcomes to estimate the dose-response function.Source: §C.1 · pp. 58–59

C.2 Statistical Inference and Interval Estimation

• Inference uses independent unit-level bootstrap resampling with replacement after fixing each unit’s neighborhood treatment and covariates as node attributes.The reported standard errors use 1,000 bootstrap replications, with 95% confidence intervals based on a normal approximation.Source: §C.2 · p. 59
• The estimator’s standard errors are consistent with the Monte Carlo root mean square errors reported in Tables 2, 3, and 4.Source: §C.2 · p. 59
• The variance estimator is appropriate when the observed sample arose through egocentric sampling, with sampled units carrying their neighborhood treatment and covariates regardless of neighbor resampling.The text says its properties should then be similar to those discussed in Imbens & Rubin (2015).Source: §C.2 · p. 59
• For dependent sampling, the proposed solution is not ideal; clustered bootstrap or network-community-based clusters are suggested for investigation.The paper notes that further investigation is needed when sampling is not independent, including when individuals within clusters form networks.Source: §C.2 · pp. 59–60

D.1 Proof of Theorem 1

• Theorem 1’s proof identifies the adjusted observed mean with a conditional potential-outcome mean using no multiple versions, neighborhood interference, and unconfoundedness assumptions.Source: §D.1 · pp. 60–61

D.2 Proof of Theorem 2.A

• Theorem 2.A’s proof uses iterated equations, the definition of Vg, and the unconfoundedness assumption.Source: §D.2 · p. 61

D.3 Proof of Corollary 2.A.1

• The Corollary 2.A.1 proof invokes unconfoundedness, consistency, and independence between Zi and Gi.Source: §D.3 · p. 62

D.4 Proof of Corollary 2.A.2

• D.4 continues the previous proof without assuming that Zi and Gi are conditionally independent given X⋆.The proof explicitly drops that independence assumption.Source: §D.4 · p. 63
• When the conditional outcome expectations do not depend on g, the no-interaction case yields a straightforward simplification of the bias formula.The passage states this condition and simplification for the individual and neighborhood treatments Zi and Gi.Source: §D.4 · p. 63

D.5 Proof of Theorem 2.B

• D.5 begins the proof by applying iterated equations and the definition of Vg.These are the stated steps used to obtain the proof's subsequent expressions.Source: §D.5 · p. 64
• The derivation uses conditional probabilities for Ui given Zi = 1 and Gi = g, alongside the corresponding conditional outcome expression.The supplied passage gives the treatment-1 probability term and the associated outcome-expression fragment.Source: §D.5 · p. 64
• It likewise uses conditional probabilities for Ui given Zi = 0 and Gi = g, alongside the corresponding conditional outcome expression.The supplied passage gives the treatment-0 probability term and the associated outcome-expression fragment.Source: §D.5 · p. 64
• The proof invokes the relationship between Zi and Ui and consistency, then adds and subtracts quantities that do not depend on g and u.The added-and-subtracted terms are conditional expectations under individual and neighborhood treatment values.Source: §D.5 · p. 64
• If Zi and Gi do not interact, the D.5 bias formula reduces to a simpler expression.The supplied proof passage states that the bias formula reduces under this condition.Source: §D.5 · p. 65

• The derivation uses conditional unconfoundedness, consistency, and independence of Zi and Gi given X⋆, then marginalizes over Gi.Source: §D.6 · pp. 65–66
• To derive the bias as with unmeasured confounding, the proof adds and subtracts conditional outcome expectations for Zi = 1 and Zi = 0.Source: §D.6 · p. 66
• Without interaction between individual treatment Zi and Ui, the bias formula simplifies.Source: §D.6 · p. 66

D.7 Proof of Corollary 2.B.2

• Without interaction between individual treatment and neighborhood treatment, the bias formula has a simplified expression.The supplied passage does not include the simplified formula itself.Source: §D.7 · p. 68

D.8 Proof of Proposition 3

• Conditioning on the joint propensity score makes the joint treatment-and-exposure probability equal that score, with or without conditioning on covariates.The proof establishes the covariate-conditioned equality directly and obtains the score-conditioned equality by iterated expectations.Source: §D.8 · p. 68

D.9 Proof of Proposition 4

• Under Assumption 3, conditioning additionally on the potential outcome leaves the joint treatment-and-exposure probability equal to the joint propensity score.The proof uses the law of iterated expectations and the fact that the score is a function of covariates.Source: §D.9 · pp. 68–69

D.10 Proof of Proposition 5

• Under Assumption 3, the Proposition 5 proof shows the conditional joint treatment-and-exposure probability given the potential outcome and propensity score equals ψ(z; g; Xi).The proof applies iterated expectations, using that λ(z; g; Xg_i) is a function of Xi.Source: §D.10 · pp. 69–70
• The Proposition 5 proof uses the factorization ψ(z; g; Xi) = λ(z; g; Xg_i) and refers to the binary-treatment propensity score φ(z; Xz_i).The supplied passages do not show the complete factorization expression.Source: §D.10 · p. 69