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Comparison of Multivariate Matching Methods: Structures, Distances, and Algorithms
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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“Previous research has shown that while matching approaches generally involve the selection of the same sets of controls for the overall matched samples, optimal matching is superior in terms of minimizing the distance within each pair [18].”
BackgroundBottigliengo, D., Baldi, I., Lanera, C., Lorenzoni, G., Bejko, J., Bottio, T., Tarzia, V., Carrozzini, M., Gerosa, G., Berchialla, P., et al. (2021) · BMC Medical Research Methodology · doi:10.1186/s12874-021-01454-z · Cited by 73
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“**We decided to fix the number of treated and control subjects, as in [14], instead of randomly assigning to treatment each unit by means of another Bernoulli random variable, as in [16], to have a fixed number of maximum possible pairs and be able to better compare the number of units matched across simulations.”
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Stone, C. A. (2020) · doi:10.7275/qkqa-9k50 · Cited by 51
“Gu and Rosenbaum (1993) found that “…optimal matching is often better than nearest neighbor matching when the goal is to minimize the average distance within pairs…” (p.413).”
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Critical Issues and Common Pitfalls in Designing and Conducting Impact Studies in Education
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“Disregarding born in south, balancing success is more or less equal.16 This finding is in line with Gu and Rosenbaum (1993) who observe that in terms of balance, optimal matching seems to have no advantage over greedy matching.”
BackgroundMethodResultInfluentialUnsupervised Discovery of Implicit Gender Bias
Field, A., & Tsvetkov, Y. (2020) · Conference on Empirical Methods in Natural Language Processing · doi:10.48550/arxiv.2004.08361 · Cited by 55
“Then, we use greedy matching to match each O TEXTi where the true value of W GENDERi is F with O TEXTj where the true value of W GENDERj is M and |ei(O TEXTi)− ej(O TEXTj)| is minimal (Gu and Rosenbaum, 1993).”
BackgroundMethodSalimi, B., Parikh, H., Kayali, M., Getoor, L. C., Roy, S., & Suciu, D. M. (2020) · ACM SIGMOD International Conference on Management of Data (SIGMOD) · doi:10.1145/3318464.3389759 · Cited by 45
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“6 standard deviations (SD) to match the pairs [22].”
BackgroundLeite, W. L., Jimenez, F., Kaya, Y., Stapleton, L. M., MacInnes, J. W., & Sandbach, R. (2015) · Multivariate Behavioral Research · doi:10.1080/00273171.2014.991018 · Cited by 62
“One-to-many matching is known to outperform one-to-one matching for estimating the ATT in general conditions (Cepeda, Boston, Farrar, & Strom, 2003; Gu & Rosenbaum, 1993).”
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Saha, K., Reddy, M. D., Mattingly, S. M., Moskal, E., Sirigiri, A., & de Choudhury, M. (2019) · Proc. ACM Hum. Comput. Interact. · doi:10.1145/3359239 · Cited by 54
“Therefore, with these characteristics as covariates (see Table 2), we draw on matching techniques from causal inference [71, 124] to match individuals using Mahalanobis Distance Matching [59].”
Fischer, J., Tuggle, C. T., Wes, A., & Kovach, S. (2014) · Journal of Plastic, Reconstructive & Aesthetic Surgery · doi:10.1016/j.bjps.2014.01.021 · Cited by 54
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“6 SD will remove approximately 90% of the bias in observed confounders.(11) Applicants who did not have an acceptable range of match were excluded.”
Michalek, J. (2012) · doi:10.2791/8228 · Cited by 54
“Moreover, matching on all the covariates using a distance measure, which effectively regards all interactions among the X covariates as equally important, does not work very well (Gu and Rosenbaum, 1993; Rubin and Thomas, 1996).”
BackgroundOptimizing matching and analysis combinations for estimating causal effects
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“The variable results of previous comparisons of matching methods attest that there is no single approach that best fits all scenarios (see for example 38,29,32 ).”
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BackgroundBaşer, O. (2007) · Journal of Medical Economics · doi:10.3111/13696990701646577 · Cited by 44
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“Matching on propensity score is more biasreducing/robust than covariate matching on more than five covariates (Gu and Rosenbaum 1993; Rubin and Thomas 2000).”
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“Similar to previously published work, we used a matchedpropensity analysis, since this may be a superior technique and also provides an opportunity to incorporate significantly different covariates after matching into final risk-adjusted models [25,26].”
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Matching estimators and optimal bandwidth choice
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“For other pair-matching techniqes see e.g. Gu and Rosenbaum (1993) .”
BackgroundCausal inference with observational data: A tutorial on propensity score analysis
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“The propensity score was used to match cases with comparisons using a greedy matching method, which pairs each case patient with a unique comparison patient based on the closest propensity scores without replacement.(23) The mean of c-statistics was 0.”
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“For each patient of the PE group, a greedy-matching algorithm was used to select the control patients who most closely matched that patient in terms of the three matching factors [23].”
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“It is very similar to the often used “nearest neighbor” method but reduces the distance within each pair better [64].”
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“To do so, a logit model with an optimal matching method without replacement, where a one-to-one ratio of control units were matched to a treatment observation based on the smallest average absolute distance across matches, was estimated (Gu & Rosenbaum, 1993).”
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“Thus every individual from the comparison group whose probability (score) is close to that of a given treated individuals may be used as a counterfactual to the latter (Gu & Rosenbaum, 1993).”
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“The Mahalanobis distance is well known within the matching and observational study literature where it is used to find subsets of the treatment and control that are similar [Rubin (1976), Rosenbaum and Rubin (1985), Gu and Rosenbaum (1993), Rubin and Thomas (2000)].”
MethodOutcomes and revision rates in normal, overweight, and obese patients 5 years after lumbar fusion.
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“In this paper, we propose using propensity scores (Rosenbaum and Rubin, 1983) and matching algorithms (Gu and Rosenbaum, 1993) as a method for removing all biases associated with differences in the pre-treatment variables.”
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“Then, pairs of observations that had similar propensity scores, but differed in race, were matched using the nearest neighbor method, which matched the closest control for each treated unit one at a time (Gu and Rosenbaum, 1993).”
Jonsson, N., Nilsen, T., Gille-Johnson, P., Bell, M., Martling, C., Larsson, A., & Mårtensson, J. (2017) · Critical Care and Resuscitation · doi:10.1016/s1441-2772(23)00918-3 · Cited by 43
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“…(Barker 2nd, Chang and Gutin et al., 1998; Conners Jr., Speroff and Dawson et al., 1996; Rosenbaum and Rubin, 1984) on estimated propensity scores is used, often in combination with model-based adjustments (Lieberman, Cohen and Lang et al., 1996; Rich, 1998; Gu and Rosenbaum, 1993).”
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Leon, A. C., Solomon, D. A., Li, C., Fiedorowicz, J. G., Coryell, W. H., Endicott, J., & Keller, M. B. (2011) · American Journal of Psychiatry · doi:10.1176/appi.ajp.2011.11060948 · Cited by 54
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Ribas, L. G. D. S., Pressey, R. L., & Bini, L. M. (2021) · Biological reviews/Biological reviews of the Cambridge Philosophical Society · doi:10.1111/brv.12697 · Cited by 49
“…control j units, as follows: Dij= Xi−Xj ´ X−1 Xi−Xj ð2Þ A weakness of this measure is that the ability to represent differences between two units decreases as the number of covariates increases, due to the possible interactions in the calculation of the distance (Gu & Rosenbaum, 1993; Zhao, 2004).”
BackgroundMaloney, M. H., Schilz, S. R., Herrin, J., Sangaralingham, L., Shah, N., & Barkmeier, A. (2019) · Ophthalmology (Rochester, Minn.) · doi:10.1016/j.ophtha.2018.09.040 · Cited by 41
Kum, H., & Masterson, T. (2010) · Journal of Economic and Social Measurement · doi:10.3233/jem-2010-0332 · Cited by 41
“Gu and Rosenbaum [9] show that propensity score matching produces matched samples that are more balanced than the use of the Mahalanobis distance function or propensity score with a Mahalanobis caliper if there are many covariates and large imbalances in the covariates between data sets.”
Feature Selection as Causal Inference: Experiments with Text Classification
Paul, M. J. (2017) · Conference on Computational Natural Language Learning · doi:10.18653/v1/k17-1018 · Cited by 40
“In this work, we perform one-to-one matching without replacement using a greedy matching algorithm; Gu and Rosenbaum (1993) found no quality difference using greedy versus optimal matching.”
MethodHannan, E., Samadashvili, Z., Jordan, D., Sundt, T., Stamato, N., Lahey, S., Gold, J., Wechsler, A., Ashraf, M. H., Ruiz, C., et al. (2015) · Circulation. Cardiovascular Interventions · doi:10.1161/circinterventions.115.002744 · Cited by 40
Treatment patterns and associated costs with Parkinson's disease levodopa induced dyskinesia.
Suh, D., Pahwa, R., & Mallya, U. (2012) · Journal of Neurological Sciences · doi:10.1016/j.jns.2012.05.029 · Cited by 40
“, greedy matching method) to adjust for differences in characteristics between the two patients groups [31].”
MethodKim, J., & Seltzer, M. H. (2007) · doi:10.1037/e644002011-001 · Cited by 40
Learning Causal Effects From Observational Data in Healthcare: A Review and Summary
Shi, J.-P., & Norgeot, B. (2022) · Frontiers in Medicine · doi:10.3389/fmed.2022.864882 · Cited by 38
“These algorithms differ from each other in whether patients are chosen with or without replacement (24), whether matching is optimal, greedy (24), one-to-one, or one-to-many (25), and what metric is used to measure similarity between two patients (11, 23, 26, 27).”
BackgroundStatin use and survival following a diagnosis of ovarian cancer: A prospective observational study
Majidi, A., Na, R.-H., Jordan, S., Defazio, A., & Webb, P. (2020) · International Journal of Cancer · doi:10.1002/ijc.33333 · Cited by 38
Fischer, J., Wes, A., Tuggle, C. T., Nelson, J. A., Tchou, J., Serletti, J., Kovach, S., & Wu, L.-Z. C. (2014) · Journal of Plastic, Reconstructive & Aesthetic Surgery · doi:10.1016/j.bjps.2014.07.021 · Cited by 38
An Alternative Approach to Addressing Selection Into and Out of Social Settings
Sharkey, P. (2012) · Sociological Methods & Research · doi:10.1177/0049124112452391 · Cited by 38
A comparison of propensity score matching methods for reducing selection bias
Bai, H.-Y. (2011) · International Journal of Research & Method in Education · doi:10.1080/1743727x.2011.552338 · Cited by 38
“PSM intends to balance the treatment and comparison groups on multiple covariates simultaneously, which is potentially similar to the balance achieved in randomization trials (Gu and Rosenbaum 1993 ); therefore, PSM is regarded as an effective strategy from alternative designs (Glynn, Schneeweiss, and Sturmer 2006 ) a…”
MethodMatching and Weighting With Functions of Error-Prone Covariates for Causal Inference
Lockwood, J. R., & McCaffrey, D. F. (2015) · Journal of the American Statistical Association · doi:10.1080/01621459.2015.1122601 · Cited by 35
“Methods for choosing matched samples has been an active area of research for decades (Rosenbaum and Rubin 1985; Rosenbaum 1989; Gu and Rosenbaum 1993; Hansen and Klopfer 2006), and Stuart (2010) provided a review of that literature.”
BackgroundKwon, J., Park, E.-J., Jung, S., Sohn, H., Ryu, H., & Suh, H. S. (2015) · Journal of dentistry research · doi:10.1177/0022034515587862 · Cited by 37
Is Dexamethasone Associated with Recurrence of Ovarian Cancer?
Oliveira, G. D. D., McCarthy, R., Turan, A., Schink, J., Fitzgerald, P., & Sessler, D. (2014) · Anesthesia and Analgesia · doi:10.1213/ane.0b013e3182a5d656 · Cited by 37
Investigating Transit-Induced Displacement Using Eviction Data
Delmelle, E. C., Nilsson, I. M., & Bryant, A. (2020) · Housing Policy Debate · doi:10.1080/10511482.2020.1815071 · Cited by 47
“Although the two approaches generally produce the same set of matched samples, optimal matching is sometimes marginally better in producing closely matched pairs (Gu & Rosenbaum, 1993; Ho, Imai, King, & Stuart, 2011).”
ResultRandom Recursive Partitioning: a matching method for the estimation of the average treatment effect
Porro, G., & Iacus, S. M. (2008) · Journal of Applied Econometrics · doi:10.1002/jae.1026 · Cited by 34
“We concisely report the results in what follows: first we replicated the Monte Carlo experiment of Gu and Rosenbaum (1993).”
Miller, J. A., Toesca, D. A. S., Baclay, J. R. M., Vitzthum, L. K., Dubrowski, P., Pollom, E. L., & Chang, D. T. (2021) · International Journal of Radiation Oncology*Biology*Physics · doi:10.1016/j.ijrobp.2021.07.1698 · Cited by 44
Pimperl, A. F., Schulte, T., Mühlbacher, A. C., Rosenmöller, M., Busse, R., Groene, O., Rodríguez, H. P., & Hildebrandt, H. (2016) · Population Health Management · doi:10.1089/pop.2016.0036 · Cited by 49
“30 However, because optimal matching does not produce better balanced matched samples than greedy matching, 33 and greedy matching can be realized in an easier and faster way, the team decided to use greedy matching.”
Informative Subspace Learning for Counterfactual Inference
Chang, Y., & Dy, J. (2017) · Proceedings of the AAAI Conference on Artificial Intelligence · doi:10.1609/aaai.v31i1.10800 · Cited by 22
“Mahalanobis distance matching (MDM) computes the Mahalanobis distance between the feature vectors of different samples and works well for low-dimensional data (Rubin 1979; Rosenbaum 2002), but it can fail when the dimensionality of features becomes large or when the distribution of features are not Gaussian (Gu and Ro…”
BackgroundMethodInfluentialGupta, P., Richardson, T., Hall, M., Bertoch, D., Hebbar, K. B., Fortenberry, J., & Wetzel, R. (2016) · Critical Care Medicine · doi:10.1097/ccm.0000000000001837 · Cited by 35
“In addition, matching by propensity scores achieves a better balance of covariates between groups than other matching strategies (22, 23).”
BackgroundEffectiveness of a “Grass Roots” Statewide Enrichment Program for Gifted Elementary School Children
Golle, J., Zettler, I., Rose, N., Trautwein, U., Hasselhorn, M., & Nagengast, B. (2017) · Journal of Research on Educational Effectiveness · doi:10.1080/19345747.2017.1402396 · Cited by 49
“In contrast to nearest neighbor matching, optimal matching was used to minimize the average absolute distance across all matched pairs (Gu & Rosenbaum, 1993; Ho et al., 2011).”
MethodPan, W., & Bai, H. (2015) · BMC Medical Research Methodology · doi:10.1186/s12874-015-0049-3 · Cited by 40
“Both matching algorithms usually produce similar matched data when the size of the comparison group is large; whereas optimal matching gives rise to smaller overall distances within matched units [11, 12].”
BackgroundSmith, V. A., Arterburn, D. E., Berkowitz, T. S. Z., Olsen, M. K., Livingston, E. H., Yancy, W. S., Weidenbacher, H. J., & Maciejewski, M. L. (2019) · JAMA Surgery · doi:10.1001/jamasurg.2019.3732 · Cited by 38
Park, K., Sceats, L., Dehghan, M., Trickey, A. W., Wren, A. A., Wong, J. J., Bensen, R., Limketkai, B., Keyashian, K., & Kin, C. (2018) · Alimentary Pharmacology and Therapeutics · doi:10.1111/apt.14842 · Cited by 33
Metadata
- 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
- 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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