Several matching methods require or can involve the distance between treated and control units. Options include the Mahalanobis distance, propensity score distance, or distance between user-supplied values. Propensity scores are also used for common support via the discard options and for defining calipers. This page documents the options that can be supplied to the distance argument to matchit().

There are four ways to specify the distance argument: 1) as the string "mahalanobis", 2) as a string containing the name of a method for estimating propensity scores, 3) as a vector of values whose pairwise differences define the distance between units, or 4) as a distance matrix containing all pairwise differences.

When distance is specified as one of the allowed strings (described below) other than "mahalanobis", a propensity score is estimated using the variables in formula and the method corresponding to the given argument. This propensity score can be used to compute the distance between units as the absolute difference between the propensity scores of pairs of units. In this respect, the propensity score is more like a "position" measure than a distance measure, since it is the pairwise difference that form the distance rather than the propensity scores themselves. Still, this naming convention is used to reflect their primary purpose without committing to the status of the estimated values as propensity scores, since transformations of the scores are allowed and user-supplied values that are not propensity scores can also be supplied (detailed below). Propensity scores can also be used to create calipers and common support restrictions, whether or not they are used in the actual distance measure used in the matching, if any.

In addition to the distance argument, two other arguments can be specified that relate to the estimation and manipulation of the propensity scores. The link argument allows for different links to be used in models that require them such as generalized linear models, for which the logit and probit links are allowed, among others. In addition to specifying the link, the link argument can be used to specify whether the propensity score or the linearized version of the propensity score should be used; by specifying link = "linear.{link}", the linearized version will be used.

The distance.options argument can also be specified, which should be a list of values passed to the propensity score-estimating function, for example, to choose specific options or tuning parameters for the estimation method. If formula, data, or verbose are not supplied to distance.options, the corresponding arguments from matchit() will be automatically supplied. See the Examples for demonstrations of the uses of link and distance.options. When s.weights is supplied in the call to matchit(), it will automatically be passed to the propensity score-estimating function as the weights argument unless otherwise described below.

Allowable options

Below are the allowed options for distance:

"glm"

The propensity scores are estimated using a generalized linear model (e.g., logistic regression). The formula supplied to matchit() is passed directly to glm(), and predict.glm() is used to compute the propensity scores. The link argument can be specified as a link function supplied to binomial(), e.g., "logit", which is the default. When link is prepended by "linear.", the linear predictor is used instead of the predicted probabilities. distance = "glm" with link = "logit" (logistic regression) is the default in matchit().

"gam"

The propensity scores are estimated using a generalized additive model. The formula supplied to matchit() is passed directly to mgcv::gam(), and mgcv::predict.gam() is used to compute the propensity scores. The link argument can be specified as a link function supplied to binomial(), e.g., "logit", which is the default. When link is prepended by "linear.", the linear predictor is used instead of the predicted probabilities. Note that unless the smoothing functions s(), te(), ti(), or t2() are used in formula, a generalized additive model is identical to a generalized linear model and will estimate the same propensity scores as glm. See the documentation for mgcv::gam(), mgcv::formula.gam(), and mgcv::gam.models() for more information on how to specify these models. Also note that the formula returned in the matchit() output object will be a simplified version of the supplied formula with smoothing terms removed (but all named variables present).

"gbm"

The propensity scores are estimated using a generalized boosted model. The formula supplied to matchit() is passed directly to gbm::gbm(), and gbm::predict.gbm() is used to compute the propensity scores. The optimal tree is chosen using 5-fold cross-validation by default, and this can be changed by supplying an argument to method to distance.options; see gbm::gbm.perf() for details. The link argument can be specified as "linear" to use the linear predictor instead of the predicted probabilities. No other links are allowed. The tuning parameter defaults differ from gbm::gbm(); they are as follows: n.trees = 1e4, interaction.depth = 3, shrinkage = .01, bag.fraction = 1, cv.folds = 5, keep.data = FALSE. These are the same defaults as used in WeightIt and twang, except for cv.folds and keep.data. Note this is not the same use of generalized boosted modeling as in twang; here, the number of trees is chosen based on cross-validation or out-of-bag error, rather than based on optimizing balance. twang should not be cited when using this method to estimate propensity scores.

"lasso", "ridge", "elasticnet"

The propensity scores are estimated using a lasso, ridge, or elastic net model, respectively. The formula supplied to matchit() is processed with model.matrix() and passed to glmnet::cv.glmnet(), and glmnet::predict.cv.glmnet() is used to compute the propensity scores. The link argument can be specified as a link function supplied to binomial(), e.g., "logit", which is the default. When link is prepended by "linear.", the linear predictor is used instead of the predicted probabilities. When link = "log", a Poisson model is used. For distance = "elasticnet", the alpha argument, which controls how to prioritize the lasso and ridge penalties in the elastic net, is set to .5 by default and can be changed by supplying an argument to alpha in distance.options. For "lasso" and "ridge", alpha is set to 1 and 0, respectively, and cannot be changed. The cv.glmnet() defaults are used to select the tuning parameters and generate predictions and can be modified using distance.options. If the s argument is passed to distance.options, it will be passed to predict.cv.glmnet(). Note that because there is a random component to choosing the tuning parameter, results will vary across runs unless a seed is set.

"rpart"

The propensity scores are estimated using a classification tree. The formula supplied to matchit() is passed directly to rpart::rpart(), and rpart::predict.rpart() is used to compute the propensity scores. The link argument is ignored, and predicted probabilities are always returned as the distance measure.

"randomforest"

The propensity scores are estimated using a random forest. The formula supplied to matchit() is passed directly to randomForest::randomForest(), and randomForest::predict.randomForest() is used to compute the propensity scores. The link argument is ignored, and predicted probabilities are always returned as the distance measure. When s.weights is supplied to matchit(), it will not be passed to randomForest because randomForest does not accept weights.

"nnet"

The propensity scores are estimated using a single-hidden-layer neural network. The formula supplied to matchit() is passed directly to nnet::nnet(), and fitted() is used to compute the propensity scores. The link argument is ignored, and predicted probabilities are always returned as the distance measure. An argument to size must be supplied to distance.options when using method = "nnet".

"cbps"

The propensity scores are estimated using the covariate balancing propensity score (CBPS) algorithm, which is a form of logistic regression where balance constraints are incorporated to a generalized method of moments estimation of of the model coefficients. The formula supplied to matchit() is passed directly to CBPS::CBPS(), and fitted is used to compute the propensity scores. The link argument can be specified as "linear" to use the linear predictor instead of the predicted probabilities. No other links are allowed. The estimand argument supplied to matchit() will be used to select the appropriate estimand for use in defining the balance constraints, so no argument needs to be supplied to ATT in CBPS.

"bart"

The propensity scores are estimated using Bayesian additive regression trees (BART). The formula supplied to matchit() is passed directly to dbarts::bart2(), and dbarts::fitted() is used to compute the propensity scores. The link argument can be specified as "linear" to use the linear predictor instead of the predicted probabilities. When s.weights is supplied to matchit(), it will not be passed to bart2 because the weights argument in bart2 does not correspond to sampling weights.

"mahalanobis"

No propensity scores are estimated. Rather than using the propensity score difference as the distance between units, the Mahalanobis distance is used instead. See mahalanobis() for details on how it is computed. The Mahalanobis distance is always computed using all the variables in formula. With this specification, calipers and common support restrictions cannot be used and the distance component of the output object will be empty because no propensity scores are estimated. The link and distance.options arguments are ignored. See individual methods pages for whether the Mahalanobis distance is allowed and how it is used. Sometimes this setting is just a placeholder to indicate that no propensity score is to be estimated (e.g., with method = "genetic"). To perform Mahalanobis distance matching and estimate propensity scores to be used for a purpose other than matching, the mahvars argument should be used along with a different specification to distance. See the individual matching method pages for details on how to use mahvars.

distance can also be supplied as a numeric vector whose values will be taken to function like propensity scores; their pairwise difference will define the distance between units. This might be useful for supplying propensity scores computed outside matchit() or resupplying matchit() with propensity scores estimated before without having to recompute them. distance can also be supplied as a matrix whose values represent the pairwise distances between units. The matrix should either be a square, with a row and column for each unit (e.g., as the output of a call to as.matrix(dist(.))), or have as many rows as there are treated units and as many columns as there are control units (e.g., as the output of a call to optmatch::match_on()). Distance values of Inf will disallow the corresponding units to be matched. When distance is a supplied as a numeric vector or matrix, link and distance.options are ignored.

Outputs

When specifying an argument to distance that estimates a propensity score, the output of the function called to estimate the propensity score (e.g., the glm object when distance = "glm") will be included in the matchit() output object in the model component. When distance is anything other than "mahalanobis" and not matrix, the estimated or supplied distance measures will be included in the matchit() output object in the distance component.

Note

In versions of MatchIt prior to 4.0.0, distance was specified in a slightly different way. When specifying arguments using the old syntax, they will automatically be converted to the corresponding method in the new syntax but a warning will be thrown. distance = "logit", the old default, will still work in the new syntax, though distance = "glm", link = "logit" is preferred (note that these are the default settings and don't need to be made explicit).

Examples

data("lalonde")
# Linearized probit regression PS:
m.out1 <- matchit(treat ~ age + educ + race + married +
                    nodegree + re74 + re75, data = lalonde,
                  distance = "glm", link = "linear.probit")
if (requireNamespace("mgcv", quietly = TRUE)) {
# GAM logistic PS with smoothing splines (s()):
m.out2 <- matchit(treat ~ s(age) + s(educ) + race + married +
                    nodegree + re74 + re75, data = lalonde,
                  distance = "gam")
summary(m.out2$model)
}; if (requireNamespace("CBPS", quietly = TRUE)) {
# CBPS for ATC matching w/replacement, using the just-
# identified version of CBPS (setting method = "exact"):
m.out3 <- matchit(treat ~ age + educ + race + married +
                    nodegree + re74 + re75, data = lalonde,
                  distance = "cbps", estimand = "ATC",
                  distance.options = list(method = "exact"),
                  replace = TRUE)
}
#> 
#> Family: quasibinomial 
#> Link function: logit 
#> 
#> Formula:
#> treat ~ s(age) + s(educ) + race + married + nodegree + re74 + 
#>     re75
#> 
#> Parametric coefficients:
#>               Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)  5.436e-01  3.950e-01   1.376  0.16923    
#> racehispan  -2.447e+00  4.323e-01  -5.661 2.34e-08 ***
#> racewhite   -2.995e+00  3.136e-01  -9.552  < 2e-16 ***
#> married     -1.643e+00  3.437e-01  -4.781 2.20e-06 ***
#> nodegree     7.893e-01  4.800e-01   1.645  0.10060    
#> re74        -9.838e-05  3.245e-05  -3.031  0.00254 ** 
#> re75         5.113e-05  5.001e-05   1.022  0.30706    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Approximate significance of smooth terms:
#>           edf Ref.df     F p-value    
#> s(age)  7.488  8.143 6.782  <2e-16 ***
#> s(educ) 2.647  3.359 2.311  0.0628 .  
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> R-sq.(adj) =    0.5   Deviance explained = 46.1%
#> GCV = 0.69813  Scale est. = 1.0287    n = 614
# Mahalanobis distance matching - no PS estimated
m.out4 <- matchit(treat ~ age + educ + race + married +
                    nodegree + re74 + re75, data = lalonde,
                  distance = "mahalanobis")

m.out4$distance #NULL
#> NULL

# Mahalanobis distance matching with PS estimated
# for use in a caliper; matching done on mahvars
m.out5 <- matchit(treat ~ age + educ + race + married +
                    nodegree + re74 + re75, data = lalonde,
                  distance = "glm", caliper = .1,
                  mahvars = ~ age + educ + race + married +
                                nodegree + re74 + re75)

summary(m.out5)
#> 
#> Call:
#> matchit(formula = treat ~ age + educ + race + married + nodegree + 
#>     re74 + re75, data = lalonde, distance = "glm", mahvars = ~age + 
#>     educ + race + married + nodegree + re74 + re75, caliper = 0.1)
#> 
#> Summary of Balance for All Data:
#>            Means Treated Means Control Std. Mean Diff. Var. Ratio eCDF Mean
#> distance          0.5774        0.1822          1.7941     0.9211    0.3774
#> age              25.8162       28.0303         -0.3094     0.4400    0.0813
#> educ             10.3459       10.2354          0.0550     0.4959    0.0347
#> raceblack         0.8432        0.2028          1.7615          .    0.6404
#> racehispan        0.0595        0.1422         -0.3498          .    0.0827
#> racewhite         0.0973        0.6550         -1.8819          .    0.5577
#> married           0.1892        0.5128         -0.8263          .    0.3236
#> nodegree          0.7081        0.5967          0.2450          .    0.1114
#> re74           2095.5737     5619.2365         -0.7211     0.5181    0.2248
#> re75           1532.0553     2466.4844         -0.2903     0.9563    0.1342
#>            eCDF Max
#> distance     0.6444
#> age          0.1577
#> educ         0.1114
#> raceblack    0.6404
#> racehispan   0.0827
#> racewhite    0.5577
#> married      0.3236
#> nodegree     0.1114
#> re74         0.4470
#> re75         0.2876
#> 
#> 
#> Summary of Balance for Matched Data:
#>            Means Treated Means Control Std. Mean Diff. Var. Ratio eCDF Mean
#> distance          0.5096        0.4905          0.0865     1.0661    0.0244
#> age              25.9459       25.0450          0.1259     0.4271    0.0878
#> educ             10.4865       10.2793          0.1031     0.6640    0.0175
#> raceblack         0.7387        0.7207          0.0496          .    0.0180
#> racehispan        0.0991        0.0991          0.0000          .    0.0000
#> racewhite         0.1622        0.1802         -0.0608          .    0.0180
#> married           0.2072        0.2342         -0.0690          .    0.0270
#> nodegree          0.6486        0.6577         -0.0198          .    0.0090
#> re74           2667.1135     2215.3307          0.0925     1.8804    0.0429
#> re75           1811.2988     1529.5967          0.0875     1.8724    0.0244
#>            eCDF Max Std. Pair Dist.
#> distance     0.1441          0.0933
#> age          0.3243          0.9292
#> educ         0.0811          0.7841
#> raceblack    0.0180          0.0496
#> racehispan   0.0000          0.0541
#> racewhite    0.0180          0.1216
#> married      0.0270          0.5751
#> nodegree     0.0090          0.6143
#> re74         0.2432          0.5596
#> re75         0.0991          0.5166
#> 
#> Percent Balance Improvement:
#>            Std. Mean Diff. Var. Ratio eCDF Mean eCDF Max
#> distance              95.2       22.0      93.5     77.6
#> age                   59.3       -3.6      -8.0   -105.6
#> educ                 -87.5       41.6      49.5     27.2
#> raceblack             97.2          .      97.2     97.2
#> racehispan           100.0          .     100.0    100.0
#> racewhite             96.8          .      96.8     96.8
#> married               91.6          .      91.6     91.6
#> nodegree              91.9          .      91.9     91.9
#> re74                  87.2        4.0      80.9     45.6
#> re75                  69.9    -1303.5      81.8     65.5
#> 
#> Sample Sizes:
#>           Control Treated
#> All           429     185
#> Matched       111     111
#> Unmatched     318      74
#> Discarded       0       0
#> 

# User-supplied propensity scores
p.score <- fitted(glm(treat ~ age + educ + race + married +
                        nodegree + re74 + re75, data = lalonde,
                      family = binomial))

m.out6 <- matchit(treat ~ age + educ + race + married +
                    nodegree + re74 + re75, data = lalonde,
                  distance = p.score)

# User-supplied distance matrix using optmatch::match_on()
if (requireNamespace("optmatch", quietly = TRUE)) {
dist_mat <- optmatch::match_on(
              treat ~ age + educ + race + nodegree +
                married + re74 + re75, data = lalonde,
              method = "rank_mahalanobis")

m.out7 <- matchit(treat ~ age + educ + race + nodegree +
                    married + re74 + re75, data = lalonde,
                  distance = dist_mat)
}