A Weighting Framework for Clusters as Confounders in Observational Studies

TL;DR

This paper introduces a unified weighting framework for addressing confounding in observational studies with clustered data, proposing new methods that improve balance control at both global and local levels.

Contribution

It develops a comprehensive weighting framework and two novel estimators—hierarchical balancing weights and Mundlak balancing weights—that enhance confounder control in clustered observational data.

Findings

Hierarchical balancing weights effectively control global and local balance.

Mundlak balancing weights accommodate small clusters with all treated or untreated units.

Simulation and real data applications demonstrate improved confounding adjustment.

Abstract

When units in observational studies are clustered in groups, such as students in schools or patients in hospitals, researchers often address confounding by adjusting for cluster-level covariates or cluster membership. In this paper, we develop a unified weighting framework that clarifies how different estimation methods control two distinct sources of imbalance: global balance (differences between treated and control units across clusters) and local balance (differences within clusters). We show that inverse propensity score weighting (IPW) with a random effects propensity score model -- the current standard in the literature -- targets only global balance and constant level shifts across clusters, but imposes no constraints on local balance. We then present two approaches that target both forms of balance. First, hierarchical balancing weights directly control global and local balance through a constrained optimization problem. Second, building on the recently proposed Generalized Mundlak approach, we develop a novel Mundlak balancing weights estimator that adjusts for cluster-level sufficient statistics rather than cluster indicators; this approach can accommodate small clusters where all units are treated or untreated. Critically, these approaches rest on different assumptions: hierarchical balancing weights require only that treatment is ignorable given covariates and cluster membership, while Mundlak methods additionally require an exponential family structure. We then compare these methods in a simulation study and in two applications in education and health services research that exhibit very different cluster structures.