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Identification and estimation of causal mechanisms in cluster-randomized trials with post-treatment confounding using Bayesian nonparametrics

Yuki Ohnishi, Michael J. Daniels, Lei Yang, Fan Li

Abstract

Causal mediation analysis in cluster-randomized trials (CRTs) is essential for explaining how cluster-level interventions affect individual outcomes, yet it is complicated by interference, post-treatment confounding, and hierarchical covariate adjustment. We develop a Bayesian nonparametric framework that simultaneously accommodates interference and a post-treatment confounder that precedes the mediator. Identification is achieved through a multivariate Gaussian copula that replaces cross-world independence with a single dependence parameter, yielding a built-in sensitivity analysis to residual post-treatment confounding. For estimation, we introduce a nested common atoms enriched Dirichlet process (CA-EDP) prior that integrates the Common Atoms Model (CAM) to share information across clusters while capturing between- and within-cluster heterogeneity, and an Enriched Dirichlet Process (EDP) structure delivering robust covariate adjustment without impacting the outcome model. We provide formal theoretical support for our prior by deriving the model's key distributional properties, including its partially exchangeable partition structure, and by establishing convergence guarantees for the practical truncation-based posterior inference strategy. We demonstrate the performance of the proposed methods in simulations and provide further illustration through a reanalysis of a completed CRT.

Identification and estimation of causal mechanisms in cluster-randomized trials with post-treatment confounding using Bayesian nonparametrics

Abstract

Causal mediation analysis in cluster-randomized trials (CRTs) is essential for explaining how cluster-level interventions affect individual outcomes, yet it is complicated by interference, post-treatment confounding, and hierarchical covariate adjustment. We develop a Bayesian nonparametric framework that simultaneously accommodates interference and a post-treatment confounder that precedes the mediator. Identification is achieved through a multivariate Gaussian copula that replaces cross-world independence with a single dependence parameter, yielding a built-in sensitivity analysis to residual post-treatment confounding. For estimation, we introduce a nested common atoms enriched Dirichlet process (CA-EDP) prior that integrates the Common Atoms Model (CAM) to share information across clusters while capturing between- and within-cluster heterogeneity, and an Enriched Dirichlet Process (EDP) structure delivering robust covariate adjustment without impacting the outcome model. We provide formal theoretical support for our prior by deriving the model's key distributional properties, including its partially exchangeable partition structure, and by establishing convergence guarantees for the practical truncation-based posterior inference strategy. We demonstrate the performance of the proposed methods in simulations and provide further illustration through a reanalysis of a completed CRT.
Paper Structure (49 sections, 7 theorems, 101 equations, 4 figures, 3 tables)

This paper contains 49 sections, 7 theorems, 101 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background and literature review
  3. Motivating application
  4. Contributions
  5. Structural Assumptions, Causal Estimands, and Nonparametric Identification
  6. Notation, data structure, and causal estimands
  7. Nonparametric identification
  8. Bayesian Nonparametric Causal Mediation Analysis in CRTs
  9. Nested common atoms enriched Dirichlet process
  10. Distributional properties
  11. Posterior inference
  12. Sensitivity analysis for cross-world dependence
  13. Simulation Studies
  14. Application of the Bayesian Nonparametric Causal Mediation Method to the RPS Cluster-Randomized Trial
  15. Sensitivity analysis on cross-world dependence
  16. ...and 34 more sections

Key Result

Theorem 1

Under Assumption asmp:sutva--asmp:si, for $a, a' \in \{0,1\}$, $\mathbb{E}\left[\frac{1}{N} \sum_{j=1}^{N} Y_{\cdot j}(a, \mathbf{M}(a')) \right]$ are partially identified up to the conditional distribution of $\mathbf{D}(a') \mid \mathbf{D}(a), A, \mathbf{C},N$ as follows: Additionally, under Assumption asmp:sutva--asmp:cond_crossworld_indep_mediators, $\mathbb{E} \left[ \frac{1}{N}\sum_{j=1}^{

Figures (4)

  • Figure 1: Mediation directed acyclic graph for a CRT with a post-treatment confounder. Here, $\mathbf{X}_i$ and $N_i$ are baseline covariates and cluster size, $A_i$ is cluster-level treatment assignment, $D_{ij}$, $M_{ij}$ and $Y_{ij}$ are post-treatment confounder, mediator and outcome of individual $j$ in cluster $i$, and $\mathbf{D}_{i(-j)}$ and $\mathbf{M}_{i(-j)}$ are the vectors of post-treatment confounders and mediators excluding individual $j$. The blue, orange, and green arrows represent the natural direct effect, individual mediator effect, and spillover mediator effect, respectively.
  • Figure 2: Boxplots of posterior samples for all estimands under the two BNP priors (CA-EDP, nDDP), with a fixed $\rho=0$ (no cross-world dependence) and a prior on $\rho$ in \ref{['eq:prior_rho']}.
  • Figure 3: Boxplots of posterior samples for all estimands under the CA-EDP, with various values of $\rho \in \{0.1,0.3,0.5,0.7,0.9\}$.
  • Figure 4: Boxplots of posterior samples for all estimands under two BNP priors (CA-EDP, nDDP), with and without adjustment for the post-treatment confounder.

Theorems & Definitions (13)

  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • proof
  • Lemma 1
  • proof
  • proof
  • Lemma 2
  • ...and 3 more