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Estimating Average Causal Effects Under General Interference, with Application to a Social Network Experiment
Peter M. Aronow, Cyrus Samii
TL;DR
Interference between units complicates causal-effect estimation because outcomes can depend on peers’ treatment assignments, particularly in social-network experiments. The paper develops a randomization-based framework using treatment designs, exposure mappings, and causal estimands, and establishes estimation and inference results for arbitrary but known interference. It evaluates the framework with school-network simulations and a field experiment on anti-conflict norms and behavior.
Problem
Interference violates the traditional assumption that a unit’s potential outcomes do not depend on other units’ treatment assignments, limiting standard causal analyses of network experiments.
Method
The paper combines a known experimental design, an exposure mapping from assignments to exposures, and causal estimands with inverse-probability-based estimators.
Results
The framework provides estimators for average unit-level causal effects and randomization variance, establishes consistency and asymptotic normality under local dependence, and is evaluated in school-network simulations and a field experiment.
Takeaways & Limitations
The framework supports causal inference for arbitrary interference and treatment-assignment dependencies while allowing estimands beyond average unit-level effects.
Takeaways & Limitations
The inference results require local dependence conditions, and the framework assumes the exposure mapping fully characterizes interference; misspecification can occur otherwise.
Abstract
from arXiv · showhide
This paper presents a randomization-based framework for estimating causal effects under interference between units, motivated by challenges that arise in analyzing experiments on social networks. The framework integrates three components: (i) an experimental design that defines the probability distribution of treatment assignments, (ii) a mapping that relates experimental treatment assignments to exposures received by units in the experiment, and (iii) estimands that make use of the experiment to answer questions of substantive interest. We develop the case of estimating average unit-level causal effects from a randomized experiment with interference of arbitrary but known form. The resulting estimators are based on inverse probability weighting. We provide randomization-based variance estimators that account for the complex clustering that can occur when interference is present. We also establish consistency and asymptotic normality under local dependence assumptions. We discuss refinements including covariate-adjusted effect estimators and ratio estimation. We evaluate empirical performance in realistic settings with a naturalistic simulation using social network data from American schools. We then present results from a field experiment on the spread of anti-conflict norms and behavior among school students.
1. Introduction.
The paper develops a randomization-based framework for estimating causal effects when treatment effects can transmit between units, especially through social networks. It combines a known treatment-assignment design, an exposure mapping, and causal estimands, then applies the framework to simulations and a school field experiment.
- Interference occurs when treatment effects spill over or otherwise depend across units, making conventional no-interference assumptions inadequate.The paper motivates methods for estimating both direct and indirect exposure effects.
- The framework combines the treatment-assignment distribution, an exposure mapping from assignments to received exposures, and causal estimands.It is developed for randomized experiments with arbitrary but known interference.
- The proposed estimators target average unit-level causal effects and include randomization-variance estimators, consistency results, confidence intervals, ratio estimators, and covariate-adjusted refinements.The variance estimators account for clustering induced by interference and are conservative.
- A naturalistic simulation using school social-network data evaluates finite-sample performance, followed by a field experiment on anti-conflict norms and behavior among middle school students.The field experiment randomly assigned schools to host the program and students within host schools to participate directly, allowing peer transmission to be studied.
- The framework extends earlier hierarchical experiments with partial interference to arbitrary interference and treatment-assignment dependencies.It also supports studying how indirect effects vary with individual characteristics.
2. Related Literature.
The paper broadens causal estimation beyond hierarchical experiments with partial interference. It targets average unit-level effects under arbitrary interference and treatment-assignment dependencies, including effects that may vary with unit characteristics.
- The framework generalizes estimation and inference theory from hierarchical experiments with partial interference to arbitrary interference and treatment-assignment dependencies.
- It accommodates both group-average and average unit-level causal effects, with the latter often central when studying characteristics that moderate treatment effects.
- Treatment assignments and treatment-induced exposures are distinguished because interference can constrain exposure patterns even when assignments are manipulated by the design.The paper gives spatial spillover as an example in which assignment locations affect interference patterns.
3. Treatment Assignment and Exposure Mappings.
The framework separates randomized treatment assignments from the exposures those assignments induce through unit-specific traits. It defines exposure mappings and generalized exposure probabilities as the basis for estimating causal effects and their variances.
- An experimental design randomly selects a treatment-assignment vector from supported assignments with known probabilities.The analysis assumes Pr(Z = z) is known for every assignment z in the design’s support.
- An exposure mapping is a function f: Ω × Θ → ∆ that maps assignments and unit-specific traits to treatment-induced exposure values.Traits may encode network ties, demographic differences, or first-, second-, and third-degree peer connections.
- Each unit’s exposure is D_i = f(Z, θ_i), and its generalized probability of exposure records the probability of every possible exposure condition.These probabilities are central to the estimation strategy.
- Indicator matrices across assignments encode which units receive each exposure, while probability matrices contain individual and joint exposure probabilities for variance estimation.Joint probabilities distinguish simultaneous exposure to the same or different conditions, with zero diagonal entries because one unit cannot have multiple exposures at once.
- When the assignment space is too large for exact construction, simulation can approximate marginal and joint exposure probabilities with arbitrary precision.The simulation approach uses R random assignment replicates and additive smoothing for marginal probabilities.
- The framework estimates average potential outcomes under exposure conditions, from which analysts can construct average unit-level causal effects and other causal quantities.
4. Average Potential Outcomes and Causal Effects.
The framework defines exposure-specific average potential outcomes and causal effects under interference, estimating them from observed outcomes using known exposure probabilities. Horvitz–Thompson estimators are unbiased under the design, while exact variance estimation can fail when relevant joint exposure probabilities or cross-potential-outcome terms are unavailable.
- Average Potential Outcomes and Causal Effects: Average unit-level causal effects compare mean potential outcomes under two exposure conditions, rather than relying only on group-level effects.The framework focuses on exposure-specific causal effects defined for arbitrary designs.
- Exposure-Based Potential Outcomes: A properly specified exposure mapping assigns each unit one potential outcome for each exposure condition, while consistency links observed outcomes to the realized exposure.These conditions restore a form of SUTVA at the exposure level despite interference in treatment assignments.
- Inverse-Probability Estimation: The Horvitz–Thompson estimator uses inverse exposure probabilities to estimate population totals and means from units observed under a given exposure.Units observed under an exposure form a known unequal-probability sample of the corresponding potential outcomes.
- Estimator Properties: Estimator 1 is unbiased, and its variance is characterized under the randomization distribution.The corresponding mean and difference-in-means estimators inherit the design-based construction.
- Variance Identification: Exact variance estimators require nonzero joint exposure probabilities, but some covariance and cross-potential-outcome components remain unidentified because each unit reveals only one potential outcome.When joint probabilities are zero, conservative variance approximations with nonnegative bias remain identifiable.
- Variance Identification: The framework nevertheless provides conservative variance estimators for exposure-specific means and causal-effect differences.These estimators are guaranteed to have nonnegative bias relative to the exact randomization variance.
5. Variance Estimators.
The paper develops randomization-based variance estimators for Horvitz–Thompson means and causal-effect contrasts under interference. Exact unbiasedness requires suitable joint exposure probabilities, while conservative estimators remain available when some probabilities are zero or covariance terms are unidentified.
- Variance Estimators: The proposed variance estimators are based on randomization-based Horvitz–Thompson variance expressions for exposure-specific means and causal-effect contrasts.The framework covers both individual exposure means and differences between exposure conditions.
- Exact Variance Estimation: When all joint exposure probabilities are positive, unbiased variance estimation is available for the Horvitz–Thompson estimator.The mean variance estimator scales the total variance estimator by 1/N^2.
- Zero Joint Probabilities: If some joint exposure probabilities are zero, the Horvitz–Thompson variance estimator is not unbiased, but its bias can be characterized.The paper also provides correction terms and conservative estimators for these cases.
- Covariance Estimation: The covariance between two Horvitz–Thompson totals is generally unidentified because only one potential outcome is observed per unit.A conservative approximation is available when cross-exposure joint probabilities are positive for distinct units.
- Causal-Effect Variance: The resulting conservative variance estimator for a causal-effect contrast combines variance terms and correction terms for the two exposure conditions.Its construction follows from the proposed conservative estimators and linearity of expectations.
- Asymptotic Properties: Consistency and asymptotically valid Wald-type confidence intervals follow under restrictions on how the design and exposure mapping grow across nested populations.The asymptotic argument considers a sequence of populations indexed by N.
6. Asymptotics and Intervals.
The paper establishes consistency, asymptotic normality, and valid Wald-type intervals under bounded outcomes, exposure probabilities, and suitably limited dependence. It also discusses efficiency refinements using covariates and ratio approximations.
- Consistency: Under Conditions 3 and 4, the Horvitz-Thompson causal-effect estimator converges to its target as the population grows.These conditions bound outcomes and exposure probabilities and restrict the growth of pairwise exposure dependence.
- Dependence assumptions: Local dependence is more restrictive than pairwise dependence because it constrains higher-order joint inclusion probabilities.The condition permits bounded-degree dependency graphs but fails when one treatment change can affect all N units.
- Confidence intervals: Under Conditions 3, 5, and 6, Wald-type intervals achieve at least 100(1 − α)% coverage asymptotically.Condition 6 excludes degenerate cases by requiring a positive limiting variance.
- Refinements: Covariate adjustment can improve efficiency, while ratio approximations may reduce mean squared error with little bias cost.These refinements are motivated by the unequal-probability sampling structure of the estimators.
7. Refinements.
The paper develops covariate-adjusted estimators and Hajek ratio refinements to improve efficiency while tracking their effects on unbiasedness, bias, and variance estimation.
- 7.1. Covariance Adjustment: Difference estimators use auxiliary covariates to reduce randomization variance without compromising unbiasedness.They also address location non-invariance affecting Horvitz-Thompson estimators.
- 7.1. Covariance Adjustment: Unbiasedness of the difference estimator requires Cov[g(xi, ξi(dk)), Zi] = 0.The function g is intended to approximate yi(dk) using auxiliary covariates and parameters.
- 7.1. Covariance Adjustment: Regression adjustment automates parameter selection but generally sacrifices unbiasedness while typically retaining consistency.Weighted least squares can estimate the adjustment parameters from the observed data.
- 7.2. Hajek Ratio Estimation Via Weighted Least Squares: The Hajek ratio estimator normalizes Horvitz-Thompson weights, often improving efficiency at the cost of finite-population bias and harder variance estimation.Its varying denominator shrinks unusually large estimates and raises unusually small ones.
- 7.2. Hajek Ratio Estimation Via Weighted Least Squares: The Hajek estimator is a ratio of two unbiased estimators, so it is not itself generally unbiased.Its bias tends to be small relative to sampling variability, and weighted least squares with Taylor linearization supports computation and variance estimation.
8. Misspecification.
When the exposure mapping fails to fully characterize interference, Horvitz-Thompson estimates target individual average potential outcomes rather than the originally intended potential-outcome means.
- 8. Misspecification: Exposure-mapping misspecification occurs when one observed exposure condition corresponds to multiple potential outcomes for a unit.This violates the assumption that the mapping fully characterizes interference.
- 8. Misspecification: Under misspecification, the Horvitz-Thompson estimator remains unbiased for the population mean of individual average randomization potential outcomes conditional on exposure.The corresponding contrast compares means under different restrictions on treatment assignments.
- 8. Misspecification: The resulting causal-effect estimate is not necessarily the contrast between uniquely defined exposure-specific potential-outcome means.It instead reflects averages over randomization potential outcomes induced by the exposure restrictions.
- 8. Misspecification: Inference for this misspecified effect does not follow immediately from the main results, although partial-interference results may apply.The paper specifically points to Liu and Hudgens (2014) for the partial-interference case.
9. A naturalistic simulation with social network data.
A naturalistic simulation applies the framework to undirected friendship networks from American schools, comparing exposure conditions and estimators under realistic interference patterns.
- Exposure mapping: The simulation defines four exposure conditions: direct plus indirect, isolated direct, indirect, and no exposure.The contrasts d10−d00, d11−d00, and d01−d00 isolate direct, combined, and indirect effects respectively.
- Simulation design: The study repeats the experiment across 144 school classes with an average class size of 626 students.Networks are undirected, with ties formed when either student nominated the other as a friend.
- Simulation design: Figure 1 maps treatment assignments onto student exposures, revealing degree-related exposure propensities and irregular clustering.These features motivate variance estimators that account for dependence among exposure conditions.
- Simulation results: The Horvitz-Thompson, Hajek, and WLS estimators show no perceivable bias, whereas OLS and DSM are heavily biased relative to estimate variability.Horvitz-Thompson is more variable than Hajek and WLS, while OLS and DSM intervals badly undercover.
- Simulation results: Horvitz-Thompson, Hajek, and WLS standard-error estimates are informative but conservative, producing empirical coverage above nominal levels.OLS aggregation bias arises from heterogeneous unit-level causal effects and inadequate conditioning on network degree.
- Field experiment: The field experiment studies how an anti-conflict program’s attitudinal and behavioral effects transmit through students’ social networks.Schools were assigned to host the program before students within host schools were assigned to participate.
10. Analysis of a social network field experiment.
The field experiment uses network-based exposure conditions to estimate direct and indirect effects of an anti-conflict program. Estimates indicate substantial direct and indirect effects on wristband wearing, while simulation results favor weighted estimators over OLS and DSM.
- Experimental design: The experiment constructs student networks from nomination data and defines exposure using school assignment, participation, and participant peers.Exposure conditions include direct, indirect, isolated direct, school, and no exposure categories.
- Experimental design: The analysis is restricted to 2,050 eligible students with nonzero probabilities of receiving all exposure conditions.Only eligible students can have a nonzero possibility of being in every exposure condition.
- Field-experiment results: Indirect exposure increases wristband-wearing probability by about 15 to 16 percentage points, with a 95% confidence interval of about 8 to 23 percentage points.The HT, Hajek, and WLS results mostly agree.
- Field-experiment results: Direct exposure increases wristband-wearing probability by about 30 percentage points, with confidence intervals ranging from about 5 to 50 or 21 to 40 percentage points.The intervals correspond to isolated direct exposure and combined direct-plus-indirect exposure, respectively.
- Interpretation: The program therefore shows substantial direct and indirect effects, and accounting for indirect effects is important for estimating its overall impact and cost-effectiveness.A naive participant-versus-nonparticipant comparison would drastically underestimate the program’s effect.
11. Conclusion.
The conclusion presents a general framework for causal inference under interference and emphasizes its design-based applicability. The approach uses randomization-based estimators while allowing minimal restrictions on potential outcomes.
- Scope and assumptions: Exposure probabilities and dependence between units may be unequal and non-obvious because of the experiment and exposure mapping.This dependence is an important feature of inference under interference.
- Estimation: The estimators use known treatment-assignment probabilities to support unbiased effect estimation and conservative variance estimation.Wald-type normal-approximation intervals are described as reasonable when clustering of exposure indicators is limited.
- Framework: The framework integrates experimental design, exposure mapping, and causal estimands for analyzing causal effects under interference.It is presented as applicable beyond social-network experiments when interference matters.
- Scope and assumptions: The framework is characterized as design-consistent and useful for evaluating alternative experimental designs in the presence of interference.Its consistency argument invokes the Strong Law of Large Numbers, while variance expressions account for correlated terms.
A.5. Key results for Propositions 5.1, 5.2, 5.3, and 5.4.
This section characterizes Horvitz–Thompson variance estimation under measurable and non-measurable sampling designs. When some joint inclusion probabilities are zero, the usual estimator is biased upward by an unobserved variance component, while modified estimators can remain conservative under stated conditions.
- Variance decomposition: The Horvitz–Thompson total estimator’s variance can be decomposed using inclusion indicators and pairwise covariances, including Cov(Ik, Il) = πkl − πkπl for distinct units.The diagonal covariance is Var(Ik) = πk(1 − πk).
- Sampling designs: Measurable designs require known positive πk and πkl for every unit and pair; otherwise the design is non-measurable.The standard variance estimator is unbiased because E(IkIl) = πkl.
- Non-measurable designs: When πkl = 0, the design never observes yk and yl together, so the corresponding variance component A cannot be directly estimated.Applying the variance estimator to such samples yields an unbiased estimator of Var(ˆt) + A rather than Var(ˆt).
- Conservative estimation: Var(ˆt) + A + A∗ ≥ Var(ˆt), so the proposed variance construction is conservative under the proposition’s conditions.The proof invokes Young’s inequality, and the associated Horvitz–Thompson estimator targets A∗.
- Conservative estimation: Choosing akl = bkl = 2 satisfies 1/akl + 1/bkl = 1 and provides an intuitive special case of the general construction.The paper notes that optimal values may be difficult to assign for every pair with πkl = 0.
- Asymptotic results: The Horvitz–Thompson estimators for µ(dk) and τ(dk, dl) are consistent under the stated boundedness and dependence conditions.The variance-estimator convergence argument uses a nonnegative bias relation and Chebyshev’s inequality.
A.7. Proof of Proposition 6.2.
The proof establishes convergence of the estimated variance for the causal-effect estimator under local-dependence conditions, then derives asymptotic normality and asymptotically valid Wald intervals.
- The variance of the scaled variance estimator is O(N^-1) under the boundedness and local-dependence conditions.
- The estimated variance converges to a value at least as large as the nonzero asymptotic variance of the causal-effect estimator.
- The proof bounds nonzero covariance contributions by counting only locally dependent observation quadruples.
- Studentized causal-effect estimates are asymptotically normal by the convergence of the variance estimator and Slutsky’s theorem.
- The resulting Wald-type intervals cover the target effect at least 100(1 − α)% of the time as N → ∞.