Balancing Covariates via Propensity Score Weighting

ABSTRACT

Covariate balance is crucial for unconfounded descriptive or causal comparisons. However, lack of balance is common in observational studies. This article considers weighting strategies for balancing covariates. We define a general class of weights—the balancing weights—that balance the weighted distributions of the covariates between treatment groups. These weights incorporate the propensity score to weight each group to an analyst-selected target population. This class unifies existing weighting methods, including commonly used weights such as inverse-probability weights as special cases. General large-sample results on nonparametric estimation based on these weights are derived. We further propose a new weighting scheme, the overlap weights, in which each unit’s weight is proportional to the probability of that unit being assigned to the opposite group. The overlap weights are bounded, and minimize the asymptotic variance of the weighted average treatment effect among the class of balancing weights. The overlap weights also possess a desirable small-sample exact balance property, based on which we propose a new method that achieves exact balance for means of any selected set of covariates. Two applications illustrate these methods and compare them with other approaches.

KEYWORDS:

• Balancing weights
• Causal inference
• Clinical equipoise
• Confounding
• Exact balance
• Overlap weights

Acknowledgments

The authors are grateful to the associate editor and two anonymous reviewers for comments that help improve the clarity and exposition of the article, to Peng Ding for insightful discussions, particularly the proof of Corollary 1, to Dylan Small for sharing the programming code for the RHC study, and to Maggie Nguyen for computational assistance.

Funding

Li and Morgan’s research is partially funded by NSF-SES grant 1424688.