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()`

.

## 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).

## Allowable options

There are four ways to specify the `distance`

argument: 1) as a string containing the name of a method for
estimating propensity scores, 2) as a string containing the name of a method
for computing pairwise distances from the covariates, 3) as a vector of
values whose pairwise differences define the distance between units, or 4)
as a distance matrix containing all pairwise distances. The options are
detailed below.

### Propensity score estimation methods

When `distance`

is specified as the name of a method for estimating propensity scores
(described below), 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.
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.

The following methods for estimating propensity scores are allowed:

`"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()`

. (This used to be able to be requested as`distance = "ps"`

, which still works.)`"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`mgcv::s()`

,`mgcv::te()`

,`mgcv::ti()`

, or`mgcv::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.`"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.bart()`

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.

### Methods for computing distances from covariates

The following methods involve computing a distance matrix from the covariates themselves
without estimating a propensity score. Calipers on the distance measure 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 with these methods. See the individual matching methods pages for
whether these distances are allowed and how they are used. Each of these
distance measures can also be calculated outside `matchit()`

using its
corresponding function.

`"euclidean"`

The Euclidean distance is the raw distance between units, computed as $$d_{ij} = \sqrt{(x_i - x_j)(x_i - x_j)'}$$ It is sensitive to the scale of the covariates, so covariates with larger scales will take higher priority.

`"scaled_euclidean"`

The scaled Euclidean distance is the Euclidean distance computed on the scaled (i.e., standardized) covariates. This ensures the covariates are on the same scale. The covariates are standardized using the pooled within-group standard deviations, computed by treatment group-mean centering each covariate before computing the standard deviation in the full sample.

`"mahalanobis"`

The Mahalanobis distance is computed as $$d_{ij} = \sqrt{(x_i - x_j)\Sigma^{-1}(x_i - x_j)'}$$ where \(\Sigma\) is the pooled within-group covariance matrix of the covariates, computed by treatment group-mean centering each covariate before computing the covariance in the full sample. This ensures the variables are on the same scale and accounts for the correlation between covariates.

`"robust_mahalanobis"`

The robust rank-based Mahalanobis distance is the Mahalanobis distance computed on the ranks of the covariates with an adjustment for ties. It is described in Rosenbaum (2010, ch. 8) as an alternative to the Mahalanobis distance that handles outliers and rare categories better than the standard Mahalanobis distance but is not affinely invariant.

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`

.

### Distances supplied as a numeric vector or matrix

`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 previously 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 `mahalanobis_dist()`

or `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.

## 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")
# 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$model)
#>
#> Family: quasibinomial
#> Link function: logit
#>
#> Formula:
#> treat ~ s(age) + s(educ) + race + married + nodegree + re74 +
#> re75
#>
#> Parametric coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 5.436e-01 3.950e-01 1.376 0.16923
#> racehispan -2.447e+00 4.323e-01 -5.661 2.34e-08 ***
#> racewhite -2.995e+00 3.136e-01 -9.552 < 2e-16 ***
#> married -1.643e+00 3.437e-01 -4.781 2.20e-06 ***
#> nodegree 7.893e-01 4.800e-01 1.645 0.10060
#> re74 -9.838e-05 3.245e-05 -3.031 0.00254 **
#> re75 5.113e-05 5.001e-05 1.022 0.30706
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Approximate significance of smooth terms:
#> edf Ref.df F p-value
#> s(age) 7.488 8.143 6.782 <2e-16 ***
#> s(educ) 2.647 3.359 2.311 0.0628 .
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> R-sq.(adj) = 0.5 Deviance explained = 46.1%
#> GCV = 0.69813 Scale est. = 1.0287 n = 614
# CBPS for ATC matching w/replacement, using the just-
# identified version of CBPS (setting method = "exact"):
m.out3 <- matchit(treat ~ age + educ + race + married +
nodegree + re74 + re75, data = lalonde,
distance = "cbps", estimand = "ATC",
distance.options = list(method = "exact"),
replace = TRUE)
# Mahalanobis distance matching - no PS estimated
m.out4 <- matchit(treat ~ age + educ + race + married +
nodegree + re74 + re75, data = lalonde,
distance = "mahalanobis")
m.out4$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 0.4916 0.0817 1.0782 0.0225
#> age 25.9459 25.3874 0.0781 0.4201 0.0869
#> educ 10.4865 10.2883 0.0986 0.6448 0.0199
#> raceblack 0.7387 0.7207 0.0496 . 0.0180
#> racehispan 0.0991 0.1171 -0.0762 . 0.0180
#> racewhite 0.1622 0.1622 0.0000 . 0.0000
#> married 0.2072 0.2162 -0.0230 . 0.0090
#> nodegree 0.6486 0.6306 0.0396 . 0.0180
#> re74 2667.1135 2357.0686 0.0634 1.7471 0.0387
#> re75 1811.2988 1506.3709 0.0947 1.9333 0.0253
#> eCDF Max Std. Pair Dist.
#> distance 0.1441 0.0922
#> age 0.3063 0.9922
#> educ 0.0811 0.7707
#> raceblack 0.0180 0.0496
#> racehispan 0.0180 0.0762
#> racewhite 0.0000 0.0000
#> married 0.0090 0.4370
#> nodegree 0.0180 0.4756
#> re74 0.2432 0.5712
#> re75 0.0901 0.5442
#>
#> 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()
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)
```