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Optimal estimation of generalized causal effects in cluster-randomized trials with multiple outcomes

Xinyuan Chen, Fan Li

TL;DR

This work develops a unified potential outcomes framework for generalized causal effects in cluster-randomized trials with multiple outcomes. It defines cluster-pair and individual-pair estimands (cp-GCE and ip-GCE) via a flexible contrast function $w$ to accommodate non-prioritized and prioritized outcomes, accounting for informative cluster sizes. The authors derive nonparametric moment estimators, efficient influence functions, and covariate-adjusted model-robust and debiased machine learning (DML) estimators, proving consistency, asymptotic normality, and semiparametric efficiency under mild regularity; they also propose subsampling for computational efficiency with preserved asymptotic properties. Through extensive simulations and a chronic-pain CRT data example, the DML estimators demonstrate superior precision and robustness, offering practical guidance for covariate adjustment and outcome prioritization in CRT analyses. Overall, the framework enables robust, efficient global summaries of treatment effects across multiple outcomes in CRTs, with clear interpretability under both non-prioritized and prioritized settings and potential extensions to covariate-adaptive designs.

Abstract

Cluster-randomized trials (CRTs) are widely used to evaluate group-level interventions and increasingly collect multiple outcomes capturing complementary dimensions of benefit and risk. Investigators often seek a single global summary of treatment effect, yet existing methods largely focus on single-outcome estimands or rely on model-based procedures with unclear causal interpretation or limited robustness. We develop a unified potential outcomes framework for generalized treatment effects with multiple outcomes in CRTs, accommodating both non-prioritized and prioritized outcome settings. The proposed cluster-pair and individual-pair causal estimands are defined through flexible pairwise contrast functions and explicitly account for potentially informative cluster sizes. We establish nonparametric estimation via weighted clustered U-statistics and derive efficient influence functions to construct covariate-adjusted estimators that integrate debiased machine learning with U-statistics. The resulting estimators are consistent and asymptotically normal, attain the semiparametric efficiency bounds under mild regularity conditions, and have analytically tractable variance estimators that are proven to be consistent under cross-fitting. Simulations and an application to a CRT for chronic pain management illustrate the practical utility of the proposed methods.

Optimal estimation of generalized causal effects in cluster-randomized trials with multiple outcomes

TL;DR

This work develops a unified potential outcomes framework for generalized causal effects in cluster-randomized trials with multiple outcomes. It defines cluster-pair and individual-pair estimands (cp-GCE and ip-GCE) via a flexible contrast function to accommodate non-prioritized and prioritized outcomes, accounting for informative cluster sizes. The authors derive nonparametric moment estimators, efficient influence functions, and covariate-adjusted model-robust and debiased machine learning (DML) estimators, proving consistency, asymptotic normality, and semiparametric efficiency under mild regularity; they also propose subsampling for computational efficiency with preserved asymptotic properties. Through extensive simulations and a chronic-pain CRT data example, the DML estimators demonstrate superior precision and robustness, offering practical guidance for covariate adjustment and outcome prioritization in CRT analyses. Overall, the framework enables robust, efficient global summaries of treatment effects across multiple outcomes in CRTs, with clear interpretability under both non-prioritized and prioritized settings and potential extensions to covariate-adaptive designs.

Abstract

Cluster-randomized trials (CRTs) are widely used to evaluate group-level interventions and increasingly collect multiple outcomes capturing complementary dimensions of benefit and risk. Investigators often seek a single global summary of treatment effect, yet existing methods largely focus on single-outcome estimands or rely on model-based procedures with unclear causal interpretation or limited robustness. We develop a unified potential outcomes framework for generalized treatment effects with multiple outcomes in CRTs, accommodating both non-prioritized and prioritized outcome settings. The proposed cluster-pair and individual-pair causal estimands are defined through flexible pairwise contrast functions and explicitly account for potentially informative cluster sizes. We establish nonparametric estimation via weighted clustered U-statistics and derive efficient influence functions to construct covariate-adjusted estimators that integrate debiased machine learning with U-statistics. The resulting estimators are consistent and asymptotically normal, attain the semiparametric efficiency bounds under mild regularity conditions, and have analytically tractable variance estimators that are proven to be consistent under cross-fitting. Simulations and an application to a CRT for chronic pain management illustrate the practical utility of the proposed methods.
Paper Structure (16 sections, 5 theorems, 19 equations, 2 figures, 5 tables)

This paper contains 16 sections, 5 theorems, 19 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Notation, estimands and assumptions
  3. Estimating the generalized causal effects
  4. Nonparametric moment estimators
  5. The efficient influence functions
  6. Model-robust estimators
  7. The debiased machine learning estimators
  8. Computationally efficient estimation using subsamples
  9. Simulation studies
  10. Simulation designs
  11. Simulation results
  12. Additional simulations with finite-sample adjustments
  13. A data example
  14. Analysis of prioritized outcomes
  15. Analysis of non-prioritized outcomes
  16. ...and 1 more sections

Key Result

Theorem 1

Under Assumptions asp:sutva-asp:cl-ran, for $v=C,I$, if regularity conditions (P1)-(P4) in Section S1 of the Supplementary Materials hold, then $\widehat{\bm{\lambda}}_v^{\mathrm{np}}\overset{p}{\to}\bm{\lambda}_v$; furthermore, if regularity conditions (P5)-(P7) also hold, then $m^{1/2}(\widehat{\b

Figures (2)

  • Figure 1: An illustrative schematic of (a) cp-GCE and (b) ip-GCE definitions. In each panel, $(i,j)$ represents individual unit $j$ in cluster $i$; the two yellow (blue) clusters denote two independent random draws from the population of clusters by setting $A_i=1$ ($A_i=0$).
  • Figure 2: Violin plots of estimators from simulation studies I and II. NP: nonparametric; MR: model-robust; DML: debiased machine learning; SJZ: Smith2025.

Theorems & Definitions (15)

  • Example 1: Non-prioritized, dimension-wise comparison
  • Example 2: Prioritized, joint comparison
  • Example 3: Non-prioritized, joint comparison
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Theorem 2
  • Theorem 3
  • ...and 5 more