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Matching (statistics)

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Matching is a statistical technique that evaluates the effect of a treatment by comparing the treated and the non-treated units in an observational study or quasi-experiment (i.e. when the treatment is not randomly assigned). The goal of matching is to reduce bias for the estimated treatment effect in an observational-data study, by finding, for every treated unit, one (or more) non-treated unit(s) with similar observable characteristics against which the covariates are balanced out. By matching treated units to similar non-treated units, matching enables a comparison of outcomes among treated and non-treated units to estimate the effect of the treatment reducing bias due to confounding.[1][2][3] Propensity score matching, an early matching technique, was developed as part of the Rubin causal model,[4] but has been shown to increase model dependence, bias, inefficiency, and power and is no longer recommended compared to other matching methods.[5] A simple, easy-to-understand, and statistically powerful method of matching known as Coarsened Exact Matching or CEM.[6]

Matching has been promoted by Donald Rubin.[4] It was prominently criticized in economics by Robert LaLonde (1986),[7] who compared estimates of treatment effects from an experiment to comparable estimates produced with matching methods and showed that matching methods are biased. Rajeev Dehejia and Sadek Wahba (1999) reevaluated LaLonde's critique and showed that matching is a good solution.[8] Similar critiques have been raised in political science[9] and sociology[10] journals.


When the outcome of interest is binary, the most general tool for the analysis of matched data is conditional logistic regression as it handles strata of arbitrary size and continuous or binary treatments (predictors) and can control for covariates. In particular cases, simpler tests like paired difference test, McNemar test and Cochran–Mantel–Haenszel test are available.

When the outcome of interest is continuous, estimation of the average treatment effect is performed.

Matching can also be used to "pre-process" a sample before analysis via another technique, such as regression analysis.[11]


Overmatching, or post-treatment bias, is matching for an apparent mediator that actually is a result of the exposure.[12] If the mediator itself is stratified, an obscured relation of the exposure to the disease would highly be likely to be induced.[13] Overmatching thus causes statistical bias.[13]

For example, matching the control group by gestation length and/or the number of multiple births when estimating perinatal mortality and birthweight after in vitro fertilization (IVF) is overmatching, since IVF itself increases the risk of premature birth and multiple birth.[14]

It may be regarded as a sampling bias in decreasing the external validity of a study, because the controls become more similar to the cases in regard to exposure than the general population.

See also[edit]


  1. ^ Rubin, Donald B. (1973). "Matching to Remove Bias in Observational Studies". Biometrics. 29 (1): 159–183. doi:10.2307/2529684. JSTOR 2529684.
  2. ^ Anderson, Dallas W.; Kish, Leslie; Cornell, Richard G. (1980). "On Stratification, Grouping and Matching". Scandinavian Journal of Statistics. 7 (2): 61–66. JSTOR 4615774.
  3. ^ Kupper, Lawrence L.; Karon, John M.; Kleinbaum, David G.; Morgenstern, Hal; Lewis, Donald K. (1981). "Matching in Epidemiologic Studies: Validity and Efficiency Considerations". Biometrics. 37 (2): 271–291. CiteSeerX doi:10.2307/2530417. JSTOR 2530417. PMID 7272415.
  4. ^ a b Rosenbaum, Paul R.; Rubin, Donald B. (1983). "The Central Role of the Propensity Score in Observational Studies for Causal Effects". Biometrika. 70 (1): 41–55. doi:10.1093/biomet/70.1.41.
  5. ^ King, Gary; Nielsen, Richard (October 2019). "Why Propensity Scores Should Not Be Used for Matching". Political Analysis. 27 (4): 435–454. doi:10.1017/pan.2019.11. hdl:1721.1/128459. ISSN 1047-1987.
  6. ^ Iacus, Stefano M.; King, Gary; Porro, Giuseppe (2011). "Multivariate Matching Methods That Are Monotonic Imbalance Bounding". Journal of the American Statistical Association. 106 (493): 345–361. doi:10.1198/jasa.2011.tm09599. hdl:2434/151476. ISSN 0162-1459. S2CID 14790456.
  7. ^ LaLonde, Robert J. (1986). "Evaluating the Econometric Evaluations of Training Programs with Experimental Data". American Economic Review. 76 (4): 604–620. JSTOR 1806062.
  8. ^ Dehejia, R. H.; Wahba, S. (1999). "Causal Effects in Nonexperimental Studies: Reevaluating the Evaluation of Training Programs" (PDF). Journal of the American Statistical Association. 94 (448): 1053–1062. doi:10.1080/01621459.1999.10473858.
  9. ^ Arceneaux, Kevin; Gerber, Alan S.; Green, Donald P. (2006). "Comparing Experimental and Matching Methods Using a Large-Scale Field Experiment on Voter Mobilization". Political Analysis. 14 (1): 37–62. doi:10.1093/pan/mpj001.
  10. ^ Arceneaux, Kevin; Gerber, Alan S.; Green, Donald P. (2010). "A Cautionary Note on the Use of Matching to Estimate Causal Effects: An Empirical Example Comparing Matching Estimates to an Experimental Benchmark". Sociological Methods & Research. 39 (2): 256–282. doi:10.1177/0049124110378098. S2CID 37012563.
  11. ^ Ho, Daniel E.; Imai, Kosuke; King, Gary; Stuart, Elizabeth A. (2007). "Matching as Nonparametric Preprocessing for Reducing Model Dependence in Parametric Causal Inference". Political Analysis. 15 (3): 199–236. doi:10.1093/pan/mpl013.
  12. ^ King, Gary; Zeng, Langche (2007). "Detecting Model Dependence in Statistical Inference: A Response". International Studies Quarterly. 51 (1): 231–241. doi:10.1111/j.1468-2478.2007.00449.x. ISSN 0020-8833. JSTOR 4621711. S2CID 12669035.
  13. ^ a b Marsh, J. L.; Hutton, J. L.; Binks, K. (2002). "Removal of radiation dose response effects: an example of over-matching". British Medical Journal. 325 (7359): 327–330. doi:10.1136/bmj.325.7359.327. PMC 1123834. PMID 12169512.
  14. ^ Gissler, M.; Hemminki, E. (1996). "The danger of overmatching in studies of the perinatal mortality and birthweight of infants born after assisted conception". Eur J Obstet Gynecol Reprod Biol. 69 (2): 73–75. doi:10.1016/0301-2115(95)02517-0. PMID 8902436.

Further reading[edit]

  • Angrist, Joshua D.; Pischke, Jörn-Steffen (2009). "Regression Meets Matching". Mostly Harmless Econometrics: An Empiricist's Companion. Princeton University Press. pp. 69–80. ISBN 978-0-691-12034-8.