Cited by 702Advanced Causal Inference TechniquesOptimal Experimental Design MethodsStatistical Methods in Clinical Trials

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Comparison of Multivariate Matching Methods: Structures, Distances, and Algorithms

Gu, Xing Sam, Rosenbaum, Paul R.·1993·Journal of Computational and Graphical Statistics

DOI 10.1080/10618600.1993.10474623

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Abstract

A comparison and evaluation is made of recent proposals for multivariate matched sampling in observational studies, where the following three questions are answered: (1) Algorithms: In current statistical practice, matched samples are formed using “nearest available” matching, a greedy algorithm. Greedy matching does not minimize the total distance within matched pairs, though good algorithms exist for optimal matching that do minimize the total distance. How much better is optimal matching than greedy matching? We find that optimal matching is sometimes noticeably better than greedy matching in the sense of producing closely matched pairs, sometimes only marginally better, but it is no better than greedy matching in the sense of producing balanced matched samples. (2) Structures: In common practice, treated units are matched to one control, called pair matching or 1–1 matching, or treated units are matched to two controls, called 1–2 matching, and so on. It is known, however, that the optimal structure is a full matching in which a treated unit may have one or more controls or a control may have one or more treated units. Optimal 1 — k matching is compared to optimal full matching, finding that optimal full matching is often much better. (3) Distances: Matching involves defining a distance between covariate vectors, and several such distances exist. Three recent proposals are compared. Practical advice is summarized in a final section.

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Statistics Commentary Series: Commentary No. 30: Propensity Scores.
Streiner, D. (2019) · Journal of Clinical Psychopharmacology · doi:10.1097/jcp.0000000000000991 · Cited by 3
“Even though optimal matching would seem optimal, the results tend to be similar in terms of balancing the groups.(9) The third decision (or rather set of decisions) involves yet another vague term that I have been using: how do we define a “close” match? Yet again, there are two options.”
Background
Evaluating selection out of health plans for medicaid beneficiaries with substance abuse
Normand, S., Belanger, A., & Frank, R. (2005) · Journal of Behavioral Health Services & Research · doi:10.1007/bf02287814 · Cited by 3
“8 331 (13) 33l (13) 0 1,215 (19) 855 (26) 17.”
Evaluation Strategies in Labor Economics -- An Application to Post-Secondary Education
Augurzky, B. (2000) · doi:10.11588/heidok.00001581 · Cited by 3
The role of Science Parks: a puzzle of growth, innovation and R&D investments
Lamperti, F., Mavilia, R., & Castellini, S. (2015) · The Journal of Technology Transfer · doi:10.1007/s10961-015-9455-2 · Cited by 116
“1), meaning that alternative distance functions, as for the Mahalanobis distance, turn out to not to be appropriated for our context (Gu and Rosenbaum 1993 [23]).”
Background
A Survey on Causality with Federated Learning: Challenges, Techniques, and Applications
Deng, J., Chen, H., Jiang, Z.-H., Wong, R. C.-W., & Ngai, E. C. H. (2026) · ACM Transactions on Knowledge Discovery from Data · doi:10.1145/3799419 · Cited by 2
“There are several distance metrics to measure the similarity among units [32], such as Mahalanobis distance [89], Euclidean distance [88], propensity score [23], and so on.”
Dynamic Ex-Post Baselines for Enhancing the Credibility of Improved Forest Management Carbon Offsets.
Yu, R.-X., Zhu, Z.-H., Qin, S.-Z., & Zhang, D. (2026) · Environmental Science and Technology · doi:10.1021/acs.est.5c11387 · Cited by 2
The relation of colchicine to total joint arthroplasty among people with gout in a population-based cohort study
Wang, Z.-Q., Tilley, S., Peloquin, C., Petrow, E., Clancy, M., & Neogi, T. (2025) · Arthritis & Rheumatology · doi:10.1002/art.43441 · Cited by 2
“Additionally, colchicine inhibits NLRP3 in ﬂ ammasome activation, thereby preventing the maturation and release of interleukin-1 β (IL-1 β ) and IL-18, further mitigating in ﬂ ammation.”
Optimal Pair Matching Combined with Machine Learning Predicts a Significant Reduction in Myocardial Infarction Risk in African Americans Following Omega-3 Fatty Acid Supplementation
Sun, S., Hara, A., Johnstone, L. M., Hallmark, B., Lu, E., Watkins, J. C., Thomson, C. A., Sergeant, S., Sergeant, S., Umans, J., et al. (2024) · Nutrients · doi:10.3390/nu16172933 · Cited by 2
Utility of real-world evidence in biosimilar development
Arani, R., Wang, J., Pang, D., Sinha, S. B., Uttenreuther-Fischer, M., & Chow, S.-C. (2024) · Journal of Biopharmaceutical Statistics · doi:10.1080/10543406.2024.2330217 · Cited by 2
“The Mahalanobis distance can work quite well when there are relatively few covariates (Amusa et al. 2022; Rubin 1979; Zhao 2004) but it does not perform as well when the covariates are not normally distributed or there are many covariates (Gu and Rosenbaum 1993).”
Local causal effects with continuous exposures: A matching estimator for the average causal derivative effect
Bong, S., & Lee, K. (2023) · doi:10.48550/arxiv.2311.18532 · Cited by 2
“Matching methods have been proposed to address the limitations of previous methods in estimating causal effects, and these methods have been well-established in the literature (Rubin, 1973; Rosenbaum, 1989; Gu and Rosenbaum, 1993; Heckman et al., 1998; Abadie and Imbens, 2002).”
Method
A mixed framework for causal impact analysis under confounding and selection biases: a focus on Egra dataset
Ayem, G. T., Ajibesin, A. A., Iorliam, A., & Nsang, A. S. (2023) · International Journal of Information Technology · doi:10.1007/s41870-023-01490-6 · Cited by 3
“The PSM algorithm employs the Greedy one-to-one matching technique for each similar (23) instance of the treated to the controlled [63].”
Multilevel Modeling Resolves Ambiguities in Analyses of Discipline Disproportionality: A Demonstration Comparing Title 1 Montessori and Non-Montessori Schools
LeBoeuf, L., Goldstein-Greenwood, J., & Lillard, A. S. (2023) · Journal of Research on Educational Effectiveness · doi:10.1080/19345747.2023.2186991 · Cited by 2
“Propensity score matching (PSM) can be an effective way to reduce bias in treatment-effect estimates when treatment is self-selected or otherwise non-randomly assigned (Gu & Rosenbaum, 1993; Harris & Horst, 2016; Steiner et al., 2015; Stuart, 2010).”
Background
An Improved Matching Practice for Augmenting a Randomized Clinical Trial with External Control
Li, J., Du, Y., Liu, H., & Yi, Y.-Y. (2023) · Therapeutic Innovation and Regulatory Science · doi:10.1007/s43441-023-00497-2 · Cited by 2
Propensity Score Matching with Cross-Classified Data Structures: A Comparison of Methods
Lee, Y., Leite, W. L., & Leroux, A. J. (2023) · The Journal of Experimental Education · doi:10.1080/00220973.2023.2164843 · Cited by 3
“Additionally, even though one-to-one matching is more common in applied research (Thoemmes & Kim, 2011), one-to-many matching has been found to outperform one-to-one matching (Cepeda et al., 2003; Gu & Rosenbaum, 1993).”
Background
The Association between Bundled Payment Participation and Changes in Medical Episode Outcomes among High-Risk Patients
Liao, J., Wang, E.-K., Isidro, U., Zhu, J., Cousins, D. S., & Navathe, A. (2022) · Healthcare · doi:10.3390/healthcare10122510 · Cited by 2
“Reference [23] is cited in the supplementary materials.”
Background
Causal inference for data centric engineering
Graham, D. (2022) · doi:10.48550/arxiv.2211.13618 · Cited by 2
“The matching estimator has been used extensively in the literature (see for example Rosenbaum 1989, 1995, 2002, Gu and Rosenbaum 1993, Rubin 1973a,b, 1979, Rubin and Thomas 1992a,b, 1996, 2000, Dehejia and Wahba 1999, Becker and Ichino 2002, Abadie and Imbens 2002).”
Method
Fast-Track Management of Concurrent Percutaneous Coronary Intervention in Patients Scheduled for Endovascular Abdominal Aortic Aneurysm Repair
Weng, C., Wang, J.-R., Zhao, J.-C., Ma, Y.-K., Huang, B., Yang, Y., Yuan, D., Wang, T.-H., & Chen, X.-Y. (2022) · Journal of Endovascular Therapy · doi:10.1177/15266028221105179 · Cited by 2
“Matching ratios of 1:2 and an optimal approach 19 with minimal average distance were used to create a matched cohort.”
Method
Percutaneous Coronary Intervention in Patients With a History of Gastrointestinal Bleeding (From the Blue Cross Blue Shield of Michigan Cardiovascular Consortium).
Meloche, C., Seth, M., Madder, R., Kurlander, J. E., Yaser, J., Chattahi, J., Collins, J., Lingam, N., Arora, D. S., Gurm, H., et al. (2021) · American Journal of Cardiology · doi:10.1016/j.amjcard.2021.06.013 · Cited by 2
“We sought to evaluate the association between a history of gastrointestinal bleeding (GIB) and outcomes after PCI.”
Method
Comparison of statistical methodologies used to estimate the treatment effect on time-to-event outcomes in observational studies
Jreich, R., & Sebastien, B. (2021) · Journal of Biopharmaceutical Statistics · doi:10.1080/10543406.2021.1918140 · Cited by 2
Effect of Tourniquet Use During Surgical Treatment of Open Fractures
Gitajn, I., Werth, P., Sprague, S., Bzovsky, S., Petrisor, B., Jeray, K., O'Hara, N., Bhandari, M., & Slobogean, G. (2021) · Journal of Bone and Joint Surgery. American volume · doi:10.2106/jbjs.20.01458 · Cited by 2
Better experimental design by hybridizing binary matching with imbalance optimization
Krieger, A., Azriel, D., & Kapelner, A. (2020) · Canadian journal of statistics · doi:10.1002/cjs.11685 · Cited by 2
Discovering context specific causal relationships
Ma, S., Li, J., Liu, L., & Le, T. D. (2018) · Intelligent Data Analysis · doi:10.3233/ida-184010 · Cited by 2
“Various distance measure, e.g. Minkowski distance and Mahalanobis distance, can be used for the stratiﬁcation, but most of them does not perform well when there are many covariates under study [9].”
Value of Propensity Score Matching for Equalizing Comparator Groups in Observational Database Studies: A Case Study in Anti-infectives.
Mullins, C., Ernst, F. R., Krukas, M. R., Solomkin, J., Eckmann, C., Shelbaya, A., Quintana, A., & Reisman, A. (2016) · Clinical Therapeutics · doi:10.1016/j.clinthera.2016.10.013 · Cited by 2
Improving Covariate Balance in 2^K Factorial Designs via Rerandomization
Branson, Z., Dasgupta, T., & Rubin, D. (2015) · doi:10.48550/arxiv.1511.01973 · Cited by 2
“va = 2 p γ(p2 + 1, a2) γ(p2 , a2) , (9)”
Background
The impact of grants in the agriculture sector in Albania: a counterfactual approach
Preka, O. (2014) · AMS Dottorato Institutional Doctoral Theses Repository (University of Bologna) · doi:10.6092/unibo/amsdottorato/6716 · Cited by 2
No Base Left Behind
Nguyen-Hoang, P., Yeung, R., & Bogin, A. (2014) · doi:10.1177/1091142113482570 · Cited by 2
Data-Driven Uplift Modeling
Zahid, A., Kamiran, F., Abbas, S., Qureshi, B., & Karim, A. (2025) · IEEE Access · doi:10.1109/access.2025.3557468 · Cited by 2
“This observation is consistent with previous studies [84, e.g.].”
Design of Retrospective and Case-Control Studies in Oncology
Panageas, K., Goldman, D. A., & Kingham, T. (2018) · Springer eBooks · doi:10.1007/978-3-319-71324-3_9 · Cited by 2
Methods and Biostatistics in Oncology
Jo, R. A., & Riechelmann, R. (2018) · Cambridge International Law Journal · doi:10.1007/978-3-319-71324-3 · Cited by 2
Cadrelites? Career continuity, discontinuity, or disruption of former socialist elites in the early 1990s : an event history analysis on the basis of statistically matched data
Gebauer, R. (2012) · Social Science Open Access Repository (GESIS – Leibniz Institute for the Social Sciences) · doi:10.12759/hsr.37.2012.2.139-152 · Cited by 2
Do inclusionary zoning policies equitably disperse affordable housing? A comparative spatial analysis
Kontokosta, C. E. (2015) · Journal of Housing and the Built Environment · doi:10.1007/s10901-014-9430-5 · Cited by 1
Psychiatric readmissions following Soteria versus traditional inpatient care: a propensity score-matched cohort study.
Friedlander, A., Oren, S., Sinai, D., Bitan, D., Yoffe, R., Eitan, R., & Lichtenberg, P. (2026) · Psychiatry Research · doi:10.1016/j.psychres.2026.117263 · Cited by 1
Does Inflation Targeting Enhance Economic Performance? Evidence From Asian Economies
Sethi, C., & Mishra, B. R. (2026) · International Journal of Finance & Economics · doi:10.1002/ijfe.70211 · Cited by 1
The association between intraoperative ketamine administration and reduced postoperative opioid consumption among chronic opioid users after multilevel lumbar fusion.
Diltz, Z. R., Nguyen, A. Q., Wang, C., Liu, J. W., Sullivan, T. C., Lambert, B., & Saifi, C. (2026) · Journal of Neurosurgery : Spine · doi:10.3171/2025.12.spine25700 · Cited by 1
Cohort event monitoring of booster COVID-19 vaccine safety using patient-reported outcomes in pregnant women
Maisonneuve, E., Luxi, N., Bellitto, C., Ciccimarra, F., Raethke, M., van Hunsel, F., Lieber, T., Mulder, E., Riefolo, F., Villalobos, F., et al. (2025) · Frontiers in Drug Safety and Regulation · doi:10.3389/fdsfr.2025.1689349 · Cited by 1
“Women vaccinated during pregnancy were then 1:4 matched to non-pregnant women on the basis of their PS value, using the nearest neighbor matching procedure (Gu and Rosenbaum, 1993).”
Reduced mortality but elevated venous thromboembolism risk following knee and hip arthroplasty in patients with rheumatoid arthritis: A general population-based cohort study
Deng, X., Lu, N., Xie, D., Li, H., & Wang, H.-C. (2025) · PLoS ONE · doi:10.1371/journal.pone.0335453 · Cited by 1
Impact of genetically modified Brinjal (Bt brinjal) on farmers’ income and production in Pabna District, Bangladesh
Hossain, M. A., Hossain, K. K., & Joshi, N. P. (2025) · GM crops & food · doi:10.1080/21645698.2025.2560698 · Cited by 3
“34 This research employed PSM to reduce selection bias in observational data by simulating random treatment assignment through the pairing of treated and control groups.”

Authors: Gu, X. S., & Rosenbaum, P. R.
Affiliations: C.S. First Boston Hong Kong, Ltd.
University of Pennsylvania (US)
ORCID: Rosenbaum, Paul R.
Venue: Journal of Computational and Graphical Statistics
Publisher: Informa UK Limited
Published: 1993-12
Volume: 2
Issue: 4
Pages: 405-420
ISSN: 1061-8600 (ISSN-L)
1537-2715
DOI: 10.1080/10618600.1993.10474623
OpenAlex: W2030614017
Cited by: 702
References: 0
Citations by year: 2026: 20 · 2025: 29 · 2024: 27 · 2023: 32 · 2022: 35 · 2021: 41 · 2020: 46 · 2019: 37 · 2018: 51 · 2017: 36 · 2016: 36 · 2015: 40 · 2014: 36 · 2013: 35 · 2012: 27
FWCI: 1.01
Type: journal article
Language: en
Open access: closed
Locations: Journal of Computational and Graphical Statistics (publishedVersion)
Topics: Physical Sciences › Mathematics › Statistics and Probability › Advanced Causal Inference Techniques
Social Sciences › Decision Sciences › Management Science and Operations Research › Optimal Experimental Design Methods
Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods in Clinical Trials
Keywords: optimal matching, greedy matching, pair matching, covariate balance, distance measures, observational studies

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Comparison of Multivariate Matching Methods: Structures, Distances, and Algorithms

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