A Weighting Framework for Clusters as Confounders in Observational Studies

TL;DR

The paper tackles cluster-confounded observational causal inference by disentangling global and local balance, and introduces a cohesive weighting framework that unifies cluster-membership and cluster-level covariate approaches. It analyzes model-based IPW with random intercepts, hierarchical balancing weights, and Mundlak balancing weights, highlighting that RI-IPW often fails to address within-cluster imbalance, while Hierarchical and Mundlak methods address both balance types under different assumptions. Mundlak balancing weights leverage cluster-level sufficient statistics and an exponential-family structure to reduce local-balance constraints, enabling effective adjustment in settings with small clusters, at the cost of stronger assumptions. Through simulations and two applied cases in education and health services, the paper provides practical guidance on when to use each method and how to balance global versus local covariate differences, with implications for more robust causal inference in clustered observational studies.

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.

Figures (9)

• Figure 1: Bias and Root Mean Square Error of balancing weights, IPW based on a random intercept model, hierarchical balancing weights, and Mundlak weights, varying the degree of cluster confounding.
• Figure 2: Balance metrics for three cluster adjustment methods in the education application.
• Figure 3: Estimates for the effect of not attending a pre-school program on test scores for four different covariate adjustment strategies.
• Figure 4: Balance metrics for three cluster adjustment methods in the surgery application.
• Figure 5: Estimates for the effect of surgery on adverse events for five different covariate adjustment strategies.