Technical Summary: A Sensitivity Analysis Framework for Causal Inference Under Interference

1. Problem Statement

In causal inference, the assumption of no interference (the Stable Unit Treatment Value Assumption, or SUTVA) posits that a unit's outcome depends solely on its own treatment. However, in many applications, such as infectious disease studies, treatment received by one unit can influence the outcome of another, a phenomenon known as interference or spillover effects. Ignoring interference leads to systematic bias that cannot be mitigated by increasing sample size.

Existing literature on interference often assumes that all relevant confounders are observed and that the study population is representative of the target population. However, practitioners frequently face three simultaneous challenges:

1. Ignored Interference: Failure to adjust for spillover effects.
2. Unmeasured Confounding: The presence of unobserved variables affecting both treatment and outcome.
3. Lack of Transportability: The inability to generalize causal findings from a reference population to a target population due to differences in covariate distributions or treatment mechanisms.

Current sensitivity analysis frameworks typically address these issues in isolation. For instance, while VanderWeele et al. (2015) developed a framework for unmeasured confounding under interference, it does not account for ignored interference or transportability. Similarly, Forastiere et al. (2021) discuss bias from these factors but lack a structured sensitivity analysis approach with interpretable parameters for practitioners. This paper addresses the gap by proposing a unified framework to assess the systematic bias arising from the combination of these three complexities.

2. Methodology

2.1 Framework and Assumptions

The authors adopt the potential outcomes framework under neighborhood interference. They introduce an Outcome Exposure Mapping (Assumption 1), where a scalar function $g_i(\mathbf{A}_{N_i})$ summarizes the treatments of a unit's neighbors, reducing the dimensionality of potential outcomes. The observed outcome $Y$ is linked to potential outcomes via a consistency assumption (Assumption 2).

To handle unmeasured confounding and transportability, the authors relax standard identification assumptions:

• Weak Conditional Exchangeability (Assumption 5): Allows for unobserved confounders ($U$) but assumes conditional independence given observed covariates ($X$) and unobserved confounders. Crucially, this requires conditioning on common causes of personal treatment ($A$) and neighborhood treatment ($G$), as well as common causes of $A$ and the population indicator ($S$).
• Conditional Ignorability of Sampling (Assumption 7): Allows for unobserved differences between reference ($S=1$) and target ($S=2$) populations, provided that conditional on $G$ and specific covariates, the sampling is ignorable.

2.2 Bias Decomposition via Weighting Errors

The core methodology involves comparing a "naive" estimator (which ignores interference, unmeasured confounding, and transportability issues) against the true identification formulas derived under the relaxed assumptions. The authors define two types of weighting errors:

1. Multiplicative Error in Weights (MEW, $\varepsilon_a$): The ratio of the true propensity score (conditioning on all relevant $X$ and $U$) to the pseudo-propensity score (conditioning only on observed $X_{AY}$). This captures bias from ignored interference and unmeasured confounding.
2. Difference Error in Weights (DEW, $\tilde{\varepsilon}_S$): A term representing the difference between the naive sampling weight (1) and the true sampling weight required for transportability. This captures bias from lack of transportability.

2.3 Sensitivity Parameters

The bias is decomposed into a sum of covariance terms, each expressed as the product of a correlation coefficient and two standard deviations. These components serve as sensitivity parameters that practitioners can specify based on domain knowledge:

• For Interference/Confounding (Terms T1 & T2):
  • $\sigma_{Y(a,0)}$: Standard deviation of baseline potential outcomes.
  • $\sigma_{\varepsilon_a}$: Variability of the MEW scores (reflecting the strength of ignored confounding/interference).
  • $\rho_{Y(a,0), \varepsilon_a}$: Correlation between baseline outcomes and weighting errors.
  • $\sigma_{\gamma(a)}$: Variability of natural individual spillover effects.
  • $\rho_{\gamma(a), \varepsilon_a}$: Correlation between spillover effects and weighting errors.
• For Transportability (Terms T3 & T4):
  • $\sigma_{\tau(0)}$: Variability of controlled individual main effects.
  • $\sigma_{\tilde{\varepsilon}_S}$: Variability of the DEW scores.
  • $\rho_{\tau(0), \tilde{\varepsilon}_S}$: Correlation between direct effects and sampling weights.
  • $\sigma_{\gamma(1)-\gamma(0)}$: Variability of the difference in spillover effects.
  • $\rho_{\gamma(1)-\gamma(0), \tilde{\varepsilon}_S}$: Correlation between spillover differences and sampling weights.

The authors provide guidelines for bounding these parameters (e.g., using Popoviciu's inequality for outcome bounds or Bhatia-Davis inequality for weight bounds) and suggest setting correlation signs to maximize bias for worst-case scenarios.

2.4 Extension to Undefined Potential Outcomes

The paper extends the framework to settings where potential outcomes are undefined for units with insufficient neighbors (e.g., a unit with 0 neighbors cannot experience a neighborhood treatment of 2). By introducing a variable $N$ for the number of neighbors and conditioning the analysis on $N$, the authors derive identification formulas and bias decompositions (Theorem 3) that remain valid even when the exposure mapping domain varies by unit.

3. Key Results

• Theorem 1 (Bias Decomposition for $\phi_1$): The bias of a naive estimator for the natural average treatment effect in the reference population is decomposed into two parts:
  • T1: Covariance between MEW scores and baseline potential outcomes (driven by unmeasured confounding and the baseline level of interference).
  • T2: Covariance between MEW scores and natural spillover effects. This term highlights that even if common causes of $A$ and $G$ are absent, unmeasured confounding can still induce bias if it correlates with the heterogeneity of spillover effects.
• Theorem 2 (Bias Decomposition for $\phi_2$): The bias for the transported natural average treatment effect is the sum of the bias for $\phi_1$ plus two additional terms:
  • T3: Covariance between DEW scores and controlled direct effects.
  • T4: Covariance between DEW scores and the difference in spillover effects between treatment arms.
• Corollaries: The paper demonstrates that if the marginal distribution of neighborhood treatment ($G$) is identical between populations, transportability bias vanishes. It also shows that in the absence of interference, the framework reduces to existing sensitivity analysis methods for transportability (e.g., Huang, 2024).
• Theorem 3: Extends the bias decomposition to scenarios with undefined potential outcomes, showing that the structure of the bias remains similar but requires conditioning on the number of neighbors.

4. Significance and Contributions

The paper claims to be the first to develop a comprehensive sensitivity analysis framework that simultaneously accounts for ignored interference, unmeasured confounding, and lack of transportability.

• Novelty: Unlike previous works that treat these issues separately or require strong assumptions about the outcome generating process or the distribution of unmeasured confounders, this framework relies on weighting-based bias decomposition that is robust to various data complexities.
• Practical Utility: By parametrizing bias through easily interpretable sensitivity parameters (correlations and standard deviations), the framework allows practitioners to evaluate how robust their findings are to violations of standard assumptions. It enables researchers to gauge the strength of interference or confounding necessary to alter their conclusions.
• Generalization: The methodology generalizes previous weighting-based results (e.g., Shen et al., 2011; Hong et al., 2021) and extends them to the complex setting of interference and transportability.
• Handling Undefined Outcomes: The extension to undefined potential outcomes addresses a critical theoretical gap in interference literature, acknowledging that not all potential outcomes are well-defined for every unit.

In summary, the paper provides a structured, flexible, and theoretically grounded approach for practitioners to assess the systematic bias in causal estimates when interference, unmeasured confounding, and transportability issues are present but unobserved or unadjusted for in the data.