Cited by 369Advanced Causal Inference TechniquesStatistical Methods and InferenceStatistical Methods and Bayesian Inference

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Stable Weights that Balance Covariates for Estimation With Incomplete Outcome Data

Zubizarreta, José R.·2015·Journal of the American Statistical Association

DOI 10.1080/01621459.2015.1023805

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Abstract

Weighting methods that adjust for observed covariates, such as inverse probability weighting, are widely used for causal inference and estimation with incomplete outcome data. Part of the appeal of such methods is that one set of weights can be used to estimate a range of treatment effects based on different outcomes, or a variety of population means for several variables. However, this appeal can be diminished in practice by the instability of the estimated weights and by the difficulty of adequately adjusting for observed covariates in some settings. To address these limitations, this article presents a new weighting method that finds the weights of minimum variance that adjust or balance the empirical distribution of the observed covariates up to levels prespecified by the researcher. This method allows the researcher to balance very precisely the means of the observed covariates and other features of their marginal and joint distributions, such as variances and correlations and also, for example, the quantiles of interactions of pairs and triples of observed covariates, thus, balancing entire two- and three-way marginals. Since the weighting method is based on a well-defined convex optimization problem, duality theory provides insight into the behavior of the variance of the optimal weights in relation to the level of covariate balance adjustment, answering the question, how much does tightening a balance constraint increases the variance of the weights? Also, the weighting method runs in polynomial time so relatively large datasets can be handled quickly. An implementation of the method is provided in the new package sbw for R. This article shows some theoretical properties of the resulting weights and illustrates their use by analyzing both a dataset from the 2010 Chilean earthquake and a simulated example.

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Is It Who You Are or Where You Are? Accounting for Compositional Differences in Cross-Site Treatment Effect Variation
Lu, B., Ben‐Michael, E., Feller, A., & Miratrix, L. W. (2023) · Journal of Educational and Behavioral Statistics · doi:10.3102/10769986231155427 · Cited by 6
“Our proposed weighting approach draws on the recent literature on approximate balancing weights in observational causal inference (see, e.g., Athey et al., 2018; Hirshberg & Wager, 2021; Zubizarreta, 2015).”
Method
Covariate balancing functional propensity score for functional treatments in cross-sectional observational studies
Zhang, X., Wu, X., & Wang, Q. (2021) · Computational Statistics & Data Analysis · doi:10.1016/j.csda.2021.107303 · Cited by 6
Information theoretic approach to high‐dimensional multiplicative models: Stochastic discount factor and treatment effect
Qiu, C., & Otsu, T. (2022) · Quantitative Economics · doi:10.3982/qe1603 · Cited by 6
“In the recent literature of missing data analysis and causal inference, the covariate balancing approach explores the moment conditions in (7) to ﬁnd suitable weights used for estimation of θ (see, e.g., Zubizarreta (2015) and Chan, Yam, and Zhang (2016)).”
BackgroundMethod
Debiased inference on heterogeneous quantile treatment effects with regression rank scores
Giessing, A., & Wang, J. (2023) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1093/jrsssb/qkad075 · Cited by 10
Approximate balancing weights for clustered observational study designs
Ben‐Michael, E., Page, L. C., & Keele, L. J. (2024) · Statistics in Medicine · doi:10.1002/sim.10054 · Cited by 7
“In the non-clustered treatment assignment setting, many model classes have been considered, from sparse models 14 to models with expanding basis functions 26 to reproducing kernel Hilbert spaces.”
BackgroundMethod
Debiasing weighted multi-view k-means clustering based on causal regularization
Huang, X., Tao, H., Ni, H., & Hou, C. (2024) · Pattern Recognition · doi:10.1016/j.patcog.2024.111195 · Cited by 6
Semiparametric inference with missing data: Robustness to outliers and model misspecification
Cantoni, E., & de Luna, X. (2020) · Econometrics and Statistics · doi:10.1016/j.ecosta.2020.01.003 · Cited by 6
Examination of the Moderating Effect of Race on the Relationship between Vitamin D Status and COVID-19 Test Positivity Using Propensity Score Methods
Crandell, I., Rockwell, M. S., Whitehead, P. B., CARTER, K. F., & Hanlon, A. L. (2021) · Journal of the American Nutrition Association · doi:10.1080/07315724.2021.1948932 · Cited by 6
Robust Estimation of Propensity Score Weights via SubclassificationPreprint
Wang, L., Zhang, Y., Richardson, T. S., & Zhou, X. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1602.06366 · Cited by 6
“Our simulation setting is similar to that of Kang and Schafer (2007), which has become a standard setting for evaluating the performance of propensity score weighting methods (e.g., Imai and Ratkovic, 2014; Zubizarreta, 2015; Chan et al., 2015).”
BackgroundMethodResultInfluential
A Bayesian semiparametric framework for causal inference in high-dimensional dataPreprint
Antonelli, J., & Dominici, F. (2018) · arXiv (Cornell University) · Cited by 6
Bias-Aware Confidence Intervals for Empirical Bayes AnalysisPreprint
Ignatiadis, N., & Wager, S. (2019) · arXiv (Cornell University) · Cited by 6
Learning Decomposed Representation for Counterfactual InferencePreprint
Wu, A., Kuang, K., Yuan, J., Li, B., Wu, R., Zhu, Q., Zhuang, Y., & Wu, F. Q. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2006.07040 · Cited by 6
“t factor A, which only determine the outcome Y. Without the prior knowledge of confounding factors, previous methods, such as propensity score-based methods [4, 5, 21] and variables balancing methods [3, 28, 9], eliminated the confounding bias by treating all observed variables as confounders for balancing. However, b…”
BackgroundMethod
Covariate Distribution Balance via Propensity ScoresPreprint
Sant’Anna, P. H. C., Song, X., & Xu, Q. L. (2018) · SSRN Electronic Journal · doi:10.2139/ssrn.3258551 · Cited by 6
“It is worth mentioning that it is not clear whether such an attractive feature is shared by the calibration type estimators such as those in Hainmueller (2012), Zubizarreta (2015), and Wong and Chan (2018).”
BackgroundMethod
Causal Inference for Dummies: A Tutorial on Directed Acyclic Graphs and Balancing Weights
Primbs, M. A., Bijlstra, G., Holland, R. W., & Thoemmes, F. J. (2025) · Social Cognition · doi:10.1521/soco.2025.43.3.217 · Cited by 6
A Review of Causal Methods for High-Dimensional Data
Berkessa, Z. A., Läärä, E., & Waldmann, P. (2024) · IEEE Access · doi:10.1109/access.2024.3524261 · Cited by 5
Estimation of partially conditional average treatment effect by double kernel-covariate balancing
Wang, J., Wong, R. K. W., Yang, S., & Chan, K. C. G. (2022) · Electronic Journal of Statistics · doi:10.1214/22-ejs2000 · Cited by 5
“…makes achieving covariate balance signiﬁcantly more challenging than those for estimating the ATE, i.e. , when V is deterministic [e.g., 16, 20, 49, 9, 45, 46, 22, 43], for multiple reasons: (i) covariate balance is required for all v in a continuum, and (ii) the bandwidth h in kernel…”
Stratified analysis in comparative effectiveness studies that emulate randomized trials
Htoo, P. T., Glynn, R. J., Wang, S. V., Paik, J. M., Schneeweiß, S., Walker, A. M., & Patorno, E. (2023) · Pharmacoepidemiology and Drug Safety · doi:10.1002/pds.5716 · Cited by 5
“7 Total-cohort PS matching, on the other hand, could result in the potential imbalance in relevant prognostic factors 10,11,51 since it does not prioritize stronger versus weaker risk factors of the outcome and balances marginal distributions of all covariates on average.”
Background
Prioritizing Variables for Observational Study Design using the Joint Variable Importance Plot
Liao, L. D., Zhu, Y., Ngo, A. L., Chehab, R. F., & Pimentel, S. D. (2024) · The American Statistician · doi:10.1080/00031305.2024.2303419 · Cited by 5
“In a study using stable balancing weights inverse values of the outcome correlations plotted on the y -axis of the jointVIP can be used as balance tolerances (Zubizarreta 2015).”
So Many Choices: A Guide to Selecting Among Methods to Adjust for Observed Confounders
Keele, L. J., & Grieve, R. (2025) · Statistics in Medicine · doi:10.1002/sim.10336 · Cited by 9
“A newer class of weighting methods solve a convex optimization problem to find a set of balancing weights [46, 47].”
Average Treatment Effect Estimation in Observational Studies with Functional Covariates
Miao, R., Wu, X., & Zhang, X. (2022) · Statistics and Its Interface · doi:10.4310/20-sii632 · Cited by 5
More robust estimation of average treatment effects using kernel optimal matching in an observational study of spine surgical interventions
Kallus, N., Pennicooke, B. H., & Santacatterina, M. (2021) · Statistics in Medicine · doi:10.1002/sim.8904 · Cited by 5
Balance-Subsampled Stable Prediction Across Unknown Test Data
Kuang, K., Zhang, H., Wu, R., Wu, F. Q., Zhuang, Y., & Zhang, A. (2021) · ACM Transactions on Knowledge Discovery from Data · doi:10.1145/3477052 · Cited by 7
The Statistical Performance of Matching-Adjusted Indirect ComparisonsPreprint
Cheng, D. T.-Y., Ayyagari, R., & Signorovitch, J. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1910.06449 · Cited by 5
“MAIC is similar in spirit to a number of other re-weighting methods that seek to balance covariates between treatment groups to estimate causal effects (Zubizarreta, 2015; Li, Morgan and Zaslavsky, 2017) and identical to the covariate balancing propensity score used to estimate average treatment effects on the treated…”
BackgroundMethod
Varying impacts of letters of recommendation on college admissions: Approximate balancing weights for subgroup effects in observational studiesPreprint
Ben‐Michael, E., Feller, A., & Rothstein, J. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2008.04394 · Cited by 10
“, 1993), and have recently been extensively studied in the observational study context (e.g. Hainmueller, 2011; Zubizarreta, 2015; Athey et al., 2018; Hazlett, 2020; Hirshberg et al., 2019).”
BackgroundInfluential
Is it who you are or where you are? Accounting for compositional differences in cross-site treatment variationPreprint
Lu, B., Ben‐Michael, E., Feller, A., & Miratrix, L. W. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2103.14765 · Cited by 5
“See, for example, Zubizarreta (2015); Athey et al. (2018); Hirshberg and Wager (2020).”
BackgroundMethodInfluential
Automatic Double Machine Learning for Continuous Treatment EffectsPreprint
Klosin, S. (2021) · RePEc: Research Papers in Economics · doi:10.48550/arxiv.2104.10334 · Cited by 5
“This type of balancing condition for the binary treatment case is given in Chernozhukov et al. (2018b), Athey et al. (2016), Zubizarreta (2015).”
Background
Automatic Debiased Machine Learning via Neural Nets for Generalized Linear RegressionPreprint
Chernozhukov, V., Newey, W. K., Quintas-Martinez, V., et al. (2021) · arXiv (Cornell University) · Cited by 5
From Controlled to Undisciplined Data: Estimating Causal Effects in the Era of Data Science Using a Potential Outcome Framework
Dominici, F., Bargagli-Stoffi, F. J., & Mealli, F. (2021) · Harvard Data Science Review · doi:10.1162/99608f92.8102afed · Cited by 19
“…directly optimize weights or find optimal subsets of controls such that the mean or other characteristics of the covariates is the same in the treatment and control group (Diamond & Sekhon, 2013; F. Li et al., 2018; F. Li & Thomas, 2018; Zubizarreta, 2012, 2015; Zubizarreta, Paredes, et al., 2014).”
Background
Implications of pacemaker implantation after aortic valve surgery for endocarditis: a nationwide study
Bearpark, L., Dismorr, M., Franco‐Cereceda, A., et al. (2025) · European Journal of Cardio-Thoracic Surgery · doi:10.1093/ejcts/ezaf125 · Cited by 4
“Inverse probability of treatment weighting using optimization-based weights was used to obtain balanced groups, accounting for differences in baseline characteristics between the pacemaker and no pacemaker groups [21, 22].”
Modeling Versus Balancing Approaches to Addressing Instrumental Variables in Weighting: A Comparison of the Outcome‐Adaptive Lasso, Stable Balancing Weighting, and Stable Confounder Selection
Choi, B. Y., & Brookhart, M. A. (2025) · Pharmacoepidemiology and Drug Safety · doi:10.1002/pds.70173 · Cited by 4
Estimating Test-Retest Reliability in the Presence of Self-Selection Bias and Learning/Practice Effects
Belzak, W. C. M., & Lockwood, J. R., III (2024) · Applied Psychological Measurement · doi:10.1177/01466216241284585 · Cited by 4
“…How robust are inferences to alternative methods for computing weights, such as general calibration weighting from the survey sampling literature (Deville & S¨arndal, 1992), propensity score methods (Imai & Ratkovic, 2014; Rosenbaum & Rubin, 1983), and DEFF-minimizing weights (Zubizarreta, 2015)?”
Business Strategy and Cybersecurity Breaches
Li, T., & Walton, S. (2023) · Journal of Information Systems · doi:10.2308/isys-2022-033 · Cited by 4
Use of low titer O whole blood in infants and young children undergoing cardiac surgery with cardiopulmonary bypass
Griselli, M., Said, S. M., Spinella, P. C., et al. (2024) · Transfusion · doi:10.1111/trf.18014 · Cited by 4
“Optimization-based weighting 31,32 was selected based on having the best combination of covariate balance and efficiency; importantly, none of the procedures or comparisons included any of the outcome measures.”

Authors: Zubizarreta, J. R.
Affiliations: Columbia University (US)
Corresponding author: Zubizarreta, José R.
ORCID: Zubizarreta, José R.
Venue: Journal of the American Statistical Association
Publisher: Informa UK Limited
Published: 2015-07-03
Volume: 110
Issue: 511
Pages: 910-922
ISSN: 0162-1459 (ISSN-L)
1537-274X
DOI: 10.1080/01621459.2015.1023805
OpenAlex: W2120817734
Semantic Scholar: 2425738e5b67fc909c6c41d7609e9d57e6791eb7
Cited by: 369
References: 54
Citations by year: 2026: 21 · 2025: 53 · 2024: 31 · 2023: 47 · 2022: 41 · 2021: 52 · 2020: 44 · 2019: 31 · 2018: 23 · 2017: 18 · 2016: 8
FWCI: 14.99
Type: journal article
Language: en
Open access: closed
Locations: Journal of the American Statistical Association (publishedVersion)
RePEc: Research Papers in Economics (submittedVersion)
Topics: Physical Sciences › Mathematics › Statistics and Probability › Advanced Causal Inference Techniques
Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
Keywords: covariate balance, inverse probability weighting, convex optimization, duality theory, treatment effect estimation, causal inference

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Stable Weights that Balance Covariates for Estimation With Incomplete Outcome Data

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