# Artless Automatic Matching, Version 2 Implements a simple version of multivariate matching using a propensity score, near-exact matching, near-fine balance, and robust Mahalanobis distance matching. You specify the variables, and the program does everything else. Should you be artful, not artless? See the notes. ```r artlessV2(dat, z, x = NULL, pr = NULL, xm = NULL, near = NULL, fine = NULL, ncontrols = 1, rnd = 2, solver = "rlemon") ``` ## Arguments - `dat`: A dataframe containing the data set that will be matched. Let N be the number of rows of dat. - `z`: A binary vector with N coordinates where z[i]=1 if the ith row of dat describes a treated individual and z[i]=0 if the ith row of dat describes a control. - `x`: x is a numeric matrix with N rows. If pr is NULL, then the covariates in x are used to estimate a propensity score using a linear logit model that predicts z from x. An error will stop the program if pr and x are both NULL. If neither pr nor x is NULL, then a harmless warning message will remind you that your propensity score, pr, was used in matching and x was not used to estimate the propensity score. The balance table describes the covariates in x; so, those covariates should be continuous variables or binary variables that can be described by a mean or a proportion, not nominal categories. - `pr`: A vector with N coordinates containing an estimated propensity or similar quantity. If pr is NULL, then the program estimates the propensity score; see the discussion of x above. An error will stop the program if both pr and x are NULL. - `xm`: xm is a numeric matrix with N rows. The covariates in xm are used to define a robust Mahalanobis distance between treated and control individuals. Use of a matrix xm is optional. - `near`: A numeric vector of length N or a numeric matrix with N rows. Each column of near should represent levels of a nominal covariate with two or a few levels. The variables in near are used in near-exact matching. Use of a matrix near is optional. - `fine`: A numeric vector of length N or a numeric matrix with N rows. Each column of fine should represent levels of a nominal covariate with two or a few levels. The variables in fine are used in near-fine balancing. Use of a matrix fine is optional. - `ncontrols`: A positive integer. ncontrols is the number of controls to be matched to each treated individual. The default is matched pairs, i.e., one control. - `rnd`: A nonnegative integer. The balance table is rounded for display to rnd digits. - `solver`: Either "rlemon" or "rrelaxiv". The rlemon solver is automatically available without special installation. The rrelaxiv requires a special installation. See the note. ## Details This function builds a matched treated-control sample from an unmatched data set. It asks you to designate roles for specific covariates, and it does the rest. It is described as ``artless automatic matching'' because it makes decisions by default. Perhaps you could make better decisions; if so, perhaps try alittleArt() in this package, which give you much more control over decisions. For even more control over matching decisions, try the iTOS package. artlessV2() will often create a reasonable matched sample with little effort; however, it also could be used as a first step in learning the art of constructing a matched sample. Wittgenstein spoke of a the ``ladder you throw away after you have climbed it,'' and artlessV2() can also serve that function. ## Returns - **match**: A dataframe containing the matched data set. match contains the rows of dat in a different order. match adds two columns to dat, called mset and matched, which identify matched pairs or matched sets. Specifically, matched is TRUE if a row is in the matched sample and is FALSE otherwise. Rows of dat that are in the same matched set have the same value of mset. The rows of match are sorted by mset with the treated individual before the matched controls. The unmatched controls with matched=FALSE appear as the last rows of match. When you analyze the matched data to estimate treatment effects, please be careful to remove rows of match with matched==FALSE. The rows with matched==FALSE are only useful in understanding what matching has accomplished (or failed to accomplish); see, for instance, Figure 4.2 in Rosenbaum (2025). - **balance**: A matrix called the balance table. The matrix has one row for each covariate in x. It also has a first row for the propensity score. There are five columns. Column 1 is the mean of the covariate in the treated group. Column 2 is the mean of the covariate in the matched control group. Column 3 is the mean of the covariate among all controls prior to matching. Column 4 is the difference between columns 1 and 2 divided by a pooled estimate of the standard deviation of the covariate before matching. Column 5 is the difference between columns 1 and 3 divided by a pooled estimate of the standard deviation of the covariate before matching. Notice that columns 4 and 5 have the same denominator, but different numerators. Tom Love (2002) suggests a graphical display of this information. ## See Also `alittleArt` ## References Bertsekas, D. P., Tseng, P. (1988) The Relax codes for linear minimum cost network flow problems. Annals of Operations Research, 13, 125-190. Bertsekas, D. P. (1990) The auction algorithm for assignment and other network flow problems: A tutorial. Interfaces, 20(4), 133-149. Bertsekas, D. P., Tseng, P. (1994) RELAX-IV: A Faster Version of the RELAX Code for Solving Minimum Cost Flow Problems. Greifer, N. and Stuart, E.A., (2021). Matching methods for confounder adjustment: an addition to the epidemiologist’s toolbox. Epidemiologic Reviews, 43(1), pp.118-129. Hansen, B. B. and Klopfer, S. O. (2006) "Optimal full matching and related designs via network flows". Journal of computational and Graphical Statistics, 15(3), 609-627. ('optmatch' package) Hansen, B. B. (2007) Flexible, optimal matching for observational studies. R News, 7, 18-24. ('optmatch' package) Love, Thomas E. (2002) Displaying covariate balance after adjustment for selection bias. Joint Statistical Meetings. Vol. 11. [https://chrp.org/love/JSM_Aug11_TLove.pdf](https://chrp.org/love/JSM_Aug11_TLove.pdf) Niknam, B.A. and Zubizarreta, J.R. (2022). <10.1001/jama.2021.20555> Using cardinality matching to design balanced and representative samples for observational studies. JAMA, 327(2), pp.173-174. Pimentel, S. D., Yoon, F., & Keele, L. (2015) Variable‐ratio matching with fine balance in a study of the Peer Health Exchange. Statistics in Medicine, 34(30), 4070-4082. Pimentel, S. D., Kelz, R. R., Silber, J. H. and Rosenbaum, P. R. (2015) Large, sparse optimal matching with refined covariate balance in an observational study of the health outcomes produced by new surgeons. Journal of the American Statistical Association, 110, 515-527. Rosenbaum, P. R. and Rubin, D. B. (1985) Constructing a control group using multivariate matched sampling methods that incorporate the propensity score. The American Statistician, 39, 33-38. Rosenbaum, P. R. (1989) Optimal matching for observational studies. Journal of the American Statistical Association, 84(408), 1024-1032. Rosenbaum, P. R., Ross, R. N. and Silber, J. H. (2007) Minimum distance matched sampling with fine balance in an observational study of treatment for ovarian cancer. Journal of the American Statistical Association, 102, 75-83. Rosenbaum, P. R. (2020a) Design of Observational Studies (2nd Edition). New York: Springer. Rosenbaum, P. R. (2020b). Modern algorithms for matching in observational studies. Annual Review of Statistics and Its Application, 7(1), 143-176. Rosenbaum, P. R. and Zubizarreta, J. R. (2023). Optimization Techniques in Multivariate Matching. Handbook of Matching and Weighting Adjustments for Causal Inference, pp.63-86. Boca Raton: FL: Chapman and Hall/CRC Press. Rosenbaum, P. R. (2025) Introduction to the Theory of Observational Studies. New York: Springer. Rubin, D. B. (1980) Bias reduction using Mahalanobis-metric matching. Biometrics, 36, 293-298. Stuart, E.A., (2010). Matching methods for causal inference: A review and a look forward. Statistical Science, 25(1), 1-21. Yang, D., Small, D. S., Silber, J. H. and Rosenbaum, P. R. (2012) Optimal matching with minimal deviation from fine balance in a study of obesity and surgical outcomes. Biometrics, 68, 628-636. Yu, Ruoqi, and P. R. Rosenbaum. Directional penalties for optimal matching in observational studies. Biometrics 75, no. 4 (2019): 1380-1390. Yu, R., Silber, J. H., & Rosenbaum, P. R. (2020) Matching methods for observational studies derived from large administrative databases. Statistical Science, 35(3), 338-355. Yu, R. (2021) Evaluating and improving a matched comparison of antidepressants and bone density. Biometrics, 77(4), 1276-1288. Yu R. & Rosenbaum, P. R. (2022) Graded matching for large observational studies. Journal of Computational and Graphical Statistics, 31(4):1406-1415. Yu, R. (2023) How well can fine balance work for covariate balancing? Biometrics. 79(3), 2346-2356. Zhang, B., D. S. Small, K. B. Lasater, M. McHugh, J. H. Silber, and P. R. Rosenbaum (2023) Matching one sample according to two criteria in observational studies. Journal of the American Statistical Association, 118, 1140-1151. Zubizarreta, J.R., 2012. Using mixed integer programming for matching in an observational study of kidney failure after surgery. Journal of the American Statistical Association, 107(500), pp.1360-1371. Zubizarreta, J. R., Reinke, C. E., Kelz, R. R., Silber, J. H. and Rosenbaum, P. R. (2011) Matching for several sparse nominal variables in a case control study of readmission following surgery. The American Statistician, 65(4), 229-238. Zubizarreta, J.R., Stuart, E.A., Small, D.S. and Rosenbaum, P.R. eds. (2023). Handbook of Matching and Weighting Adjustments for Causal Inference. Boca Raton: FL: Chapman and Hall/CRC Press. ## Author(s) Paul R. Rosenbaum ## Note -- The following are some practical tips on how to use artlessV2. -- Most covariates that you want to balance should be included in the propensiy score, either in pr or in x. -- A small number of nominal covariates with a few levels can be placed in near or in fine. Both near and fine covariates are given overriding importance; so, if you place too many covariates in near or fine, or if they have too many levels, they will override everything else, and the match quality will be poor. You can alter the importance given to near or fine by switching from artlessV2() to alittleArt(), and then you can place more covariates in near or fine. The same covariate can appear, perhaps in different forms, in x, xm, near and fine. In the example, a five-level education variable is in x and xm, and a two-level education variable formed from the five-level education variable is in fine. Again, there are better ways to handle education in alittleArt(). -- An attempt is made to exactly match for covariates in near. In the example, near contains two binary covariates, namely female and dontSmoke. This means that the match will try whenever possible to match women to women and men to men, nonsmokers to nonsmokers, and smokers to smokers. Other considerations are subbordinated to this goal. -- An attempt is made to balance covariates in fine. In the example, fine includes a covariate expressing four broad age categories, one low education category (less than high school), and a binary covariate distinguishing daily-smokers from everyone else. This means that the match will work hard to have the same proportion of people with less-than-high-school education in treated and control groups, but it will not prioritize pairing two people with less-than-high-school education. Although subbordinate to near exact matching, fine balance is given more importance than other considerations. Whether or not fine is NULL, an attempt is made to finely balance quantiles of the propensity score. -- Three separate attempts are made to, first, balance the propensity score in the sense of fine balance, and second to pair closely for the propensity score, and third to avoid controls with propensity scores below all treated individuals. More emphasis is given to balancing the propensity score, much less to pairing for it. The match also tries in a limited way to avoid using many controls whose propensity scores are below the minimum propensity score in the treated group. Again, alittleArt() gives you control over priorities, including the ability to assign priority zero to a goal. -- An attempt is made to pair closely for covariates in xm; however, this task has the lowest priority of the several goals. A continuous covariate, like age or bmi, might be placed in x and in xm. Covariates in xm are given roughly equal importance; so, do not put unimportant covariates in xm. Binary covariates may be included in xm, but not nominal covariates with more than two values. -- The covariates in x could include, say: (i) a quadratic in age, (age-mean(age))^2, (ii) an interaction, (age-mean(age))*(bmi-mean(bmi)), or (iii) spline terms computed from age. Alternatively, you can build your own propensity score in pr or substitute a different kind of score, rather than automatically using a linear logit model fitted by maximum likelihood. -- Usually, the first match you construct is imperfect, and you see this in the balance table or in plots of the matched data. So, you make small adjustments to x, xm, near and fine to fix the imperfections. You would have much more control over these adjustment if you used alittleArt() instead of artlessV2(). The match should be finalized before any outcome information is examined. Taking the first match without looking at it, without improving it, is not artless; it is incompetent. -- There can exist treated and control groups that cannot be matched. If all of the treated individuals are under age 20 and all of the controls are over age 50, then there is no way you can match for age. You could do regression or covariance adjustment for age, but of course it would be silly. Matching will often stop you from doing silly things, while regression will let you do silly things. ## Note Should you be artful rather than artless? Essentially, the artlessV2() function is setting priorities by default. This makes artlessV2() easy to use, but its default priorities might not be your priorities. Using artlessV2() is like buying clothing that is size medium, without regard to whether you are size medium. One alternative is to use the alittleArt() function in this package; it gives you greater control over priorities. An alternative is to use the matching methods in, say, the iTOS package. The artlessV2() function calls alittleArt(), which in turn calls the functions in the iTOS package. There are also many more options in the iTOS package. The connection artlessV2() -> alittleArt() -> iTOS is in the documentation for alittleArt(). What can artful use of alittleArt() or iTOS do that artlessV2() cannot? artlessV2() automatically sets priorities and penalties, but alittleArt() and iTOS let you adjust them. artlessV2() automatically gives an emphasis to the propensity score, and does this in a particular way, but alittleArt() or iTOS let you decide. The directional penalties of Yu and Rosenbaum (2019) need to be titrated to produce desired effects; they are in iTOS but not in artlessV2() and mostly not in alittleArt(). Near-exact and near-fine matching are implemented for nominal variables in artlessV2(), but alittleArt() and iTOS have other options for ordered categories. alittleArt() and iTOS let you give more emphasis to one covariate, less to another, but artlessV2() does this only indirectly through x, pr, xm, near and fine. In artlessV2() all variables in near are treated as equally important, and all variables in fine are treated as equally important, but alittleArt() and iTOS let you decide. Caliper matching is possible in iTOS but not in artlessV2() and not in alittleArt(). artlessV2() uses the control-control edge costs in Zhang et al. (2023) to avoid low propensity scores in the control group, but iTOS lets you use this feature in any way you prefer. alittleArt() offers a little control over this feature, but not much. The iTOS package is associated with Rosenbaum (2025), especially its Chapters 5 and 6. ## Note This note provides some references and detail about what the package is actually doing. You do not have to read this note to use the package. Matching using propensity scores and a Mahalanobis distance is discussed in Rosenbaum and Rubin (1985). The robust Mahalanobis distance is discussed in Section 9.3 of Rosenbaum (2020a) and more briefly in Section 4.1 of Rosenbaum (2020b). Near-exact matching (also known as almost-exact matching) is an attempt to match exactly for a few nominal covariates, while also matching for other things. It is described in Sections 10.3 and 10.4 of Rosenbaum (2020a) and more briefly in Section 4.3 of Rosenbaum (2020b). Near-exact matching is implemented by a large penalty added to a covariate distance: if two people are not exactly matched for a near-exact covariate, then the covariate distance between them is very large. Near-exact matching minimizes the number of individuals who are not exactly matched. Fine balance attempts to balance a covariate without pairing for it. For example, female is balanced if the treated and control groups have the same proportion of females, but female is exactly matched if females are always matched to females. Fine balance is discussed in Chapter 11 of Rosenbaum (2020a) and more briefly in Section 4.4 of Rosenbaum (2020b). Fine balance was introduced in Section 3.2 of Rosenbaum (1989), and is further developed in Rosenbaum, Ross and Silber (2007). If one seeks a match as close as possible to fine balance, then one is doing near-fine balance. Near-fine balance is often implemented using penalties for imbalances; see Yang et al. (2012), Pimentel et al. (2015) and Zhang et al. (2023). One can do near-exact matching and fine balancing of the same variable, perhaps leading the proportion of females to be exactly the same in treated and control groups, with pairs matched for female as often as is possible. See Zubizarreta et al. (2011) for discussion. To do that, you will need to use alittleArt(), reducing the penalty for near, increasing the penalty for fine. artlessV2() uses the control-control edge costs in Zhang et al. (2013) to moderately penalize the use of a control whose propensity score is below the minimum propensity score in the treated group. In alittleArt(), you can exert control over this penalty or remove this feature by setting the penalty to zero. This package implements a very specific version of two-criteria matching from Zhang et al. (2023) using functions from the iTOS package. Two-criteria matching integrates a number of earlier techniques into a single network structure. artlessV2 picks several one-size-fits-all penalties for distances for two-criteria matching. An artful match might vary penalties in a thoughtful way to achieve a better, closer, more balanced match with a larger value of ncontrols. This package does not use asymmetric calipers and directional penalties from Yu and Rosenbaum (2019) because these are not easily automated, but the artful use of these techniques can produce a better match. They are available in limited form in the iTOS package. Directional penalties are very powerful tools, but they are easy to misuse; specifically, if you do not monitor and adjust what you are doing, you can reverse the direction of the bias without reducing its magnitude. The package uses optimal matching by minimum cost flow in a network. See Bertsekas (1990) for an introduction to this optimization technique, and see Rosenbaum (1989) for its application to matching in observational studies. The package uses by default the solver rlemon; it is available in R. The alternative, rrelaxiv, requires a special installation that will now be described. With solver="rrelaxiv", the package indirectly uses the callrelax() function in Samuel Pimentel's rcbalance package. This function was originally intended to call the excellent RELAXIV Fortan code of Bertsekas and Tseng (1988,1994). Unfortunately, that code has an academic license and is not available from CRAN; so, by default the package calls the rlemon function instead, which is available at CRAN. If you qualify as an academic, then you may be able to download the RELAXIV code from Github at and use it in artlessV2 by setting solver="rrelaxiv". artlessV2() uses a dense network, so it can match moderately large data sets, but not very large data sets. For very large data sets, see Yu et al. (2020) and Yu's bigmatch package in R. See also Yu and Rosenbaum (2022). Network optimization is only one of several optimization techniques that may be used in multivariate matching. See Niknam and Zubizarreta (2022), Zubizarreta (2012) and Rosenbaum and Zubizarreta (2023). ## Note The mathematical structure of artlessV2() is a very special implementation of the method in Zhang et al. (2023), and `alittleArt` is a somewhat more general implementation. The method is also described in Chapters 5 and 6 of Rosenbaum (2025). ## Examples ```r # The example below uses the binge data from the iTOS package. # See the documentation for binge in the iTOS package for more information. # library(iTOS) data(binge) b2<-binge[binge$AlcGroup!="P",] # Match binge drinkers to nondrinkers z<-1*(b2$AlcGroup=="B") # Treatment/control indicator b2<-cbind(b2,z) rm(z) rownames(b2)<-b2$SEQN attach(b2) # agec<-as.integer(ageC) # # x contains the variables in the propensity score # x<-data.frame(age,female,education,bmi,vigor,smokenow,smokeQuit,bpRX) # # Create nominal covariates to include in near or fine # smoke<-1*(smokenow==1) dontSmoke<-1*(smokenow==3) age50<-1*(age>=50) bmi30<-1*(bmi>=30) ed2<-1*(education<=2) smoke<-1*(smokenow==1) # # near contains covariates to be matched as exactly as possible # near<-cbind(female,dontSmoke) # # xm contains covariates in the robust Mahalanobis distance # Includes some continuous covariates. # xm<-cbind(age,bmi,vigor,smokenow,education) # # fine contains covariate that will be balanced, but not matched # fine<-cbind(ageC,ed2,smoke,dontSmoke) rm(agec,bmi30,smoke,ed2,age50) detach(b2) mc<-artlessV2(b2,b2$z,x,xm=xm,near=near,fine=fine,ncontrols=3) # # Here are the first two 1-to-3 matched sets. # mc$match[1:8,] # # You can check that every matched set is exactly matched for # female and nonsmoking. This is from near-exact matching. # In some other data set, the number of mismatches might be # minimized, not driven to zero. # # The balance table shows that large imbalances in covariates # existed before matching, but are much smaller after matching. # Look, for example, at the propensity score, female, and # the several versions of the smoking variable. # mc$balance m<-mc$match m<-m[m$matched,] # Remove the unmatched controls table(m$z) # 3 to 1 matching boxplot(m$age~m$z) ```

artlessV2 function