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

“13,29 Using one of these measures during OFM might lead to other results than in the presented case study.”

“Within each district, we used an optimal Mahalanobis matching approach to pair schools based on the matching variables (Gu & Rosenbaum, 1993; Rubin & Thomas, 2000).”

“To further study the approaches, 29 we will record computational time and average distance for various combinations 23 of distances, including Euclidean, Mahala-nobis, and rank-based Mahalanobis, and matching algorithms, 34 including greedy, and optimal.”

“이처럼 성향점수분석 기법에서는 원인처치를 받 거나 받지 않음으로써 단 하나의 결과만을 갖는 사례에 대해 다른 사례들로부터 정보를 빌려오는 방식으로 실현되지 않은 잠재적 결과, 즉 ‘대안사실(counterfactual)’(Gu & Rosenbaum, 1993)을 예측하고, 이것에 기초하여 효과추정치를 계산한다.”

“Matching within an observational study attempts to find matched pairs or sets by applying an algorithm (Gu & Rosenbaum, 1993).”

“Gu and Rosenbaum [46] stated that the use of Mahalanobis distance (with or without calipers) is best done when there are few covariates being adjusted as we have in this study.”

“The combination of a propensity caliper and Mahalanobis distance strives to achieve a good compromise between overall covariate balance and closeness of the covariates of matched subjects (see, e.g. Gu and Rosenbaum, 1993).”

“In a special case, Gu and Rosenbaum (1993) reported that if a binary covariate that takes values 1 and 0 with probabilities p and 1 − p; whenever p → 0, Mahalanobis distance tries to match a rare treated unit with this covariate equal to 1.”

“As opposed to greedy matching, full matching allows treatment group members to share a control group member as long as it reduces the average distance between matches [26,27].”

“introduced in Rosenbaum’s original manuscript [25] and investigated in detail by Austin and Stuart [26], we have used an adjusted propensity score by weighting factors.”

“Gu and Rosenbaum (1993) concluded that optimal matching and greedy matching perform equally in terms of creating groups with good balance, but the optimal matching does reduce the distance within pairs.”

“However, an increase in the number of covariates matched on could distort the ability of the Mahalanobis metric-distance approach to find units with similar distribution of covariates (Gu and Rosenbaum 1993; Rubin and Thomas 2000).”

“One advantage of full matching is that, unlike other matching methods, it has a flexible structure that does not constrain each treated unit to have the same number of control units (Gu and Rosenbaum 1993).”

“Among them are exact matching (Rosenbaum & Rubin, 1983), the nearest neighbor (e.g., Rubin, 1973), optimal matching (e.g., Gu & Rosenbaum, 1993), and subclassification (e.g., Cochran, 1968).”

“This approach is better suited to identify comparison units that more closely resemble the covariance 16 matrix of treatment units and more adequately identify matches in relation to propensity score strategies (Gu & Rosenbaum, 1993).”

“Where D is the variance-covariance matrix of X. Gu and Rosenbaum (1993) and Rubin and Thomas (2000) compare the performance of matching methods based on Mahalanobis metric matching and propensity score matching, and they find that the two distance measures perform similarly when there are a…”

“The total network-proximity-penalized covariate dissimilarity across all matched treated–control pairs is given by This quantity serves as the objective function to be minimized and is similar to those used in standard optimal matching problems (Gu and Rosenbaum, 1993, Stuart and Green, 2008).”

“However, Gu and Rosenbaum [63] showed that NNM outperformed OM in terms of balancing covariates, and therefore, we employed the NNM algorithm without replacement to match treated individuals with control individuals based on their PSs. NNM selects patients one by one from the treated group.”

“Metrics such as L1 distance (Iacus et al. 2011) and Mahalanobis Balance (MB) (Gu and Rosenbaum 1993; Rosenbaum and Rubin 1984) have been proposed as alternatives that capture overall distributional similarity.”

“We conducted a propensity score-matched [28], new-user design [29], active comparator [30], population-based cohort study using data from IMRD from January 1, 2000, to December 31, 2021.”

“27,28 Diferences in our primary and secondary outcomes were compared between the two groups after matching.”

“A typical method is Mahalanobis distance matching (MDM), which defines closeness using the Mahalanobis distance matrix (Gu and Rosenbaum 1993).”

“This method pairs similar individuals so that the distance in covariate space is minimized across all matches (Gu and Rosenbaum, 1993; D. E. Ho et al., 2007).”

“…were exact-matched on region as well as demographic, clinical, and healthcare utilization factors [ 15 ] After exact matching, up to three matches for each surgical patient were selected based on minimizing a distance function including continu - ous covariates to reduce further confounding [ 18].”

“21,22 As a comparator to the Bayesian time-to-event analysis, we ran a Cox proportional hazards regression analysis on this newly created propensity score matched cohort.”

“The literature on causal inference provides several methods for such manipulation of the data with regard to the variables in fl uencing the self-selection, such as matching (Gu and Rosenbaum 1993) and inverse propensity weighting (IPW; Imbens 2000; Rosenbaum and Rubin 1983).”

“28 The use of replacement increases matching quality by allowing each control individual to serve as a match for multiple treated individuals, when appropriate, thereby reducing variability in the estimates.”

“The first step in MTP was to perform matching of the partial con-trol group ( n = 49) with RW patients using propensity scores (PS) according to an optimal matching algorithm in a 1:1 ratio [25] (details of the PS approach in Appendix 1).”

“…the closest control match for each treated unit is chosen without taking into account how other units will be or have been paired, “optimal” matching finds the matched samples with the smallest average absolute distance across all the matched pairs (Hansen and Klopfer 2006; Gu and Rosenbaum 1993).”

“…was conducted with “optimal matching,” which pairs each “treated unit” (MSHS in this case) with a similar “untreated unit” (HS program), using an algorithm that attempts to optimize a global criterion defined by the sum of the absolute pair distances in the matched sample (Gu & Rosenbaum, 1993).”

“…in order to accurately estimate the causal effect or discover correct causal relationships [33], [34], [35], such as propensity score method [36], [37], [38], front-door criterion [39], instrumental variable estimator combined with structural causal models or potential outcome framework…”

“However, Gu and Rosenbaum (1993) show that Mahalanobis metric-based matching may be susceptible to bias when many covariates are used.”

“Second, matching works well when the dimensionality of the covariate is low, but it easily fails when the dimensionality is high due to the curse of dimensionality (Gu & Rosenbaum, 1993).”

“To estimate the treatment effect τ from observations or outcome estimations, various τ estimators were developed, such as PSM [4], IPTW [5], DRE [6], TMLE [7] and ARB [8], etc.”

“We implemented the propensity score matching using one-to-one matching (Gu & Rosenbaum, 1993; Austin & Mamdani 2006).”

“As a side e ff ect, the selection of well-matched subject pairs results in a good balance be-tween groups of the selected subjects overall (Gu and Rosenbaum, 1993).”

“This is in contrast to optimal matching, where matches are selected to optimize the distance between propensity scores for the entire sample, which has been shown to perform comparably to greedy matching (Gu & Rosenbaum, 1993).”

“The one-to-k or caliper matching limits how different the participants are allowed to be, as it defines a caliper and selects all matches within the caliper (Austin, 2008; Gu & Rosenbaum, 1993).”

“However, MDM and CEM are particularly vulnerable to the curse of dimensionality discussed in Abadie and Imbens (2006) and demonstrated in Gu and Rosenbaum (1993).”

“The combination of a propensity-score caliper and rank-based Mahalanobis distance matrix strives to achieve a compromise between overall covariate balance and closeness of covariates of matched subjects (Sam Gu and Rosenbaum, 1993).”

“In the simulation, 100 control units and 100 treated units were drawn uniformly from the squares [0, 5](2) and [1, 6](2), respectively.”

“…be made regarding the choice of covariates, models for creating propensity scores, matching distances and algorithms, the estimation of treatment effects, and diagnosing the quality of matches (Caliendo & Kopeinig, 2008; Gu & Rosenbaum, 1993; Ho et al., 2007; Stuart, 2010; Stuart & Rubin, 2008).”

“Overall, earlier studies support the use of PSM as a tool for robust analysis (Gu and Rosenbaum, 1993; Heckman et a., 1998; Rosenbaum, 2004).”

“Our simulation study design combines elements of the designs in Gu and Rosenbaum (1993) and Austin (2009).”

“The optimal matching is guaranteed to find the global optimum through finding the path in a minimum cost flow in the network, which was discussed in [21, 24, 42] and is implemented in R package optimatch [39].”