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

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.