Sensitivity analysis for contamination in egocentric-network randomized trials with interference

Technical Summary: Sensitivity Analysis for Contamination in Egocentric-Network Randomized Trials with Interference

Problem Statement
Egocentric-Network Randomized Trials (ENRTs) are increasingly utilized to estimate causal effects under interference when collecting complete sociocentric network data is logistically infeasible. In an ENRT, a set of "egos" (indexes) is sampled, and each ego recruits a subset of their neighbors ("alters"). Treatments are randomized across egos. Standard analytical approaches assume that the observed ego-networks are disjoint in the underlying population network; that is, alters are connected to only one ego, and egos are not connected to other egos.

However, the underlying population network often contains edges connecting these disjoint observed components (e.g., an alter connected to multiple egos, or egos connected to each other). This phenomenon, termed contamination, leads to missing edges in the observed network. Consequently, the exposure status (whether a unit has a treated neighbor) calculated from the observed network is misspecified. The authors demonstrate that under a design-based inference framework, the standard Horvitz-Thompson (HT) estimators for both direct and indirect effects are biased whenever such contamination exists. Specifically, naive estimators tend to underestimate indirect effects and overestimate direct effects.

Methodology
The paper proposes a design-based sensitivity analysis framework to address contamination without assuming a stochastic superpopulation model for the network. The underlying population network is treated as fixed but partially unobserved.

1. Bias-Corrected Estimators:

   • Indirect Effect: The authors derive a bias-corrected estimator for the indirect effect on alters ($\widehat{IE}_{adj}$). This estimator adjusts the naive HT estimator using sensitivity parameters representing the probability ($\pi^a_i$) or expected number of missing alter-ego edges.
   • Direct Effect: A bias-corrected estimator for the direct effect on egos ($\widehat{DE}_{adj}$) is derived. This requires two sensitivity parameters: the probability of missing ego-ego edges ($\pi^e_i$) and a ratio parameter ($\kappa$) representing the ratio of the average direct effect among egos exposed to at least one treated neighbor to the effect among unexposed egos.
   • Variance Estimation: Conservative variance estimators are developed to account for the dependence induced by latent edges between ego-networks.

2. Sensitivity Parameters and Specification:
   The framework relies on specifying the probability or expected number of missing edges ($\rho^e_{ij}$ and $\rho^a_{ij}$). The authors provide four practical guidelines for specifying these parameters:

   • Homogeneous probabilities (constant probability for all potential missing edges).
   • Homogeneous number of edges (specifying the total expected count of missing edges).
   • Heterogeneous probabilities (using covariates and homophily measures, e.g., logistic link functions based on distance).
   • Heterogeneous number of edges (combining expected counts with heterogeneous probabilities).

3. Augmentation and Cross-Fitting:
   To improve asymptotic precision without compromising design-based validity, the authors augment the bias-corrected estimators with working outcome regression models. They implement a conditional two-fold cross-fitting procedure tailored to the ENRT design (splitting ego-networks rather than individual units) to avoid finite-sample bias while preserving independence assumptions.

4. Analysis Approaches:
   The framework is implemented via two methods:

   • Grid Sensitivity Analysis (GSA): Evaluating estimates across a grid of plausible sensitivity parameter values.
   • Probabilistic Bias Analysis (PBA): Sampling sensitivity parameters from specified distributions (e.g., Beta, Poisson, Uniform) to generate a distribution of bias-corrected estimates, accounting for both statistical uncertainty and bias uncertainty.

5. Internal Validation:
   The paper outlines how sensitivity parameters can be calibrated using internal validation data, such as recall surveys where participants report exposure to intervention terms. This allows for the estimation of exposure probabilities directly, reducing reliance on pure sensitivity analysis.

Key Results

• Theoretical Findings: The authors prove that naive HT estimators are biased under contamination. The direction of the bias for the indirect effect is generally towards the null (underestimation). The bias for the direct effect depends on the ratio $\kappa$; if direct effects are larger in the presence of treated neighbors, the naive estimator is biased away from the null (overestimation).
• Simulation Study: Simulations based on the HPTN 037 design show that naive estimators exhibit substantial bias and poor coverage as contamination increases. The proposed bias-corrected estimators (both standard and augmented) exhibit negligible bias across scenarios. The augmented estimators provide valid variance estimates and nominal coverage rates, whereas non-augmented estimators can be conservative. The study also finds that misspecifying the direction of $\kappa$ can lead to severe bias, while misspecifying its magnitude (but correct direction) still outperforms the naive estimator.
• Application to HPTN 037: The framework was applied to re-analyze the HIV Prevention Trials Network 037 study.
  • Using internal validation data (recall of intervention terms), the authors estimated the number of missing edges.
  • Grid and Probabilistic Analysis: Results indicated that ignoring contamination leads to a significant underestimation of the indirect effect (by at least 18% in some scenarios) and a slight overestimation of the direct effect.
  • The bias-corrected estimates suggested a substantially larger indirect effect and a smaller direct effect compared to previous analyses that assumed disjoint networks.

Significance and Claims
The paper claims to provide a flexible, design-based tool for assessing the robustness of causal estimates in ENRTs to network contamination. Its primary contributions are:

1. Bias Correction: Providing explicit bias-corrected estimators for direct and indirect effects that account for missing edges, moving beyond the restrictive assumption of disjoint ego-networks.
2. Sensitivity Framework: Introducing a novel sensitivity analysis framework that quantifies epistemic uncertainty regarding latent edges without requiring a generative network model.
3. Practical Implementation: Offering concrete guidelines for specifying sensitivity parameters and demonstrating the utility of internal validation data to calibrate these parameters.
4. Empirical Impact: Demonstrating through the HPTN 037 re-analysis that standard assumptions of disjoint networks can lead to misleading policy conclusions, specifically underestimating the spillover (indirect) benefits of peer-driven interventions.

The authors emphasize that their approach prioritizes robust internal validity within the sample, avoiding strong unverifiable assumptions about the recruitment process or the population network generation, which are often required by alternative model-based approaches.