In `matchit()`

, setting `method = "optimal"`

performs optimal pair
matching. The matching is optimal in the sense that that sum of the absolute
pairwise distances in the matched sample is as small as possible. The method
functionally relies on `optmatch::fullmatch()`

.

Advantages of optimal pair matching include that the matching order is not required to be specified and it is less likely that extreme within-pair distances will be large, unlike with nearest neighbor matching. Generally, however, as a subset selection method, optimal pair matching tends to perform similarly to nearest neighbor matching in that similar subsets of units will be selected to be matched.

This page details the allowable arguments with `method = "optmatch"`

.
See `matchit()`

for an explanation of what each argument means in a general
context and how it can be specified.

Below is how `matchit()`

is used for optimal pair matching:

```
matchit(formula,
data = NULL,
method = "optimal",
distance = "glm",
link = "logit",
distance.options = list(),
estimand = "ATT",
exact = NULL,
mahvars = NULL,
antiexact = NULL,
discard = "none",
reestimate = FALSE,
s.weights = NULL,
ratio = 1,
min.controls = NULL,
max.controls = NULL,
verbose = FALSE,
...)
```

## Arguments

- formula
a two-sided formula object containing the treatment and covariates to be used in creating the distance measure used in the matching. This formula will be supplied to the functions that estimate the distance measure.

- data
a data frame containing the variables named in

`formula`

. If not found in`data`

, the variables will be sought in the environment.- method
set here to

`"optimal"`

.- distance
the distance measure to be used. See

`distance`

for allowable options. Can be supplied as a distance matrix.- link
when

`distance`

is specified as a method of estimating propensity scores, an additional argument controlling the link function used in estimating the distance measure. See`distance`

for allowable options with each option.- distance.options
a named list containing additional arguments supplied to the function that estimates the distance measure as determined by the argument to

`distance`

.- estimand
a string containing the desired estimand. Allowable options include

`"ATT"`

and`"ATC"`

. See Details.- exact
for which variables exact matching should take place.

- mahvars
for which variables Mahalanobis distance matching should take place when

`distance`

corresponds to a propensity score (e.g., for caliper matching or to discard units for common support). If specified, the distance measure will not be used in matching.- antiexact
for which variables ant-exact matching should take place. Anti-exact matching is processed using

`optmatch::antiExactMatch()`

.- discard
a string containing a method for discarding units outside a region of common support. Only allowed when

`distance`

is not`"mahalanobis"`

and not a matrix.- reestimate
if

`discard`

is not`"none"`

, whether to re-estimate the propensity score in the remaining sample prior to matching.- s.weights
the variable containing sampling weights to be incorporated into propensity score models and balance statistics.

- ratio
how many control units should be matched to each treated unit for k:1 matching. For variable ratio matching, see section "Variable Ratio Matching" in Details below.

- min.controls, max.controls
for variable ratio matching, the minimum and maximum number of controls units to be matched to each treated unit. See section "Variable Ratio Matching" in Details below.

- verbose
`logical`

; whether information about the matching process should be printed to the console. What is printed depends on the matching method. Default is`FALSE`

for no printing other than warnings.- ...
additional arguments passed to

`optmatch::fullmatch()`

. Allowed arguments include`tol`

and`solver`

. See the`optmatch::fullmatch()`

documentation for details. In general,`tol`

should be set to a low number (e.g.,`1e-7`

) to get a more precise solution.The arguments

`replace`

,`caliper`

, and`m.order`

are ignored with a warning.

## Details

### Mahalanobis Distance Matching

Mahalanobis distance matching can be done one of two ways:

If no propensity score needs to be estimated,

`distance`

should be set to`"mahalanobis"`

, and Mahalanobis distance matching will occur using all the variables in`formula`

. Arguments to`discard`

and`mahvars`

will be ignored. For example, to perform simple Mahalanobis distance matching, the following could be run:With this code, the Mahalanobis distance is computed using`matchit(treat ~ X1 + X2, method = "nearest", distance = "mahalanobis")`

`X1`

and`X2`

, and matching occurs on this distance. The`distance`

component of the`matchit()`

output will be empty.If a propensity score needs to be estimated for common support with

`discard`

,`distance`

should be whatever method is used to estimate the propensity score or a vector of distance measures, i.e., it should not be`"mahalanobis"`

. Use`mahvars`

to specify the variables used to create the Mahalanobis distance. For example, to perform Mahalanobis after discarding units outside the common support of the propensity score in both groups, the following could be run:With this code,`matchit(treat ~ X1 + X2 + X3, method = "nearest", distance = "glm", discard = "both", mahvars = ~ X1 + X2)`

`X1`

,`X2`

, and`X3`

are used to estimate the propensity score (using the`"glm"`

method, which by default is logistic regression), which is used to identify the common support. The actual matching occurs on the Mahalanobis distance computed only using`X1`

and`X2`

, which are supplied to`mahvars`

. The estimated propensity scores will be included in the`distance`

component of the`matchit()`

output.

### Estimand

The `estimand`

argument controls whether control units are selected to be matched with treated units
(`estimand = "ATT"`

) or treated units are selected to be matched with
control units (`estimand = "ATC"`

). The "focal" group (e.g., the
treated units for the ATT) is typically made to be the smaller treatment
group, and a warning will be thrown if it is not set that way unless
`replace = TRUE`

. Setting `estimand = "ATC"`

is equivalent to
swapping all treated and control labels for the treatment variable. When
`estimand = "ATC"`

, the `match.matrix`

component of the output
will have the names of the control units as the rownames and be filled with
the names of the matched treated units (opposite to when `estimand = "ATT"`

). Note that the argument supplied to `estimand`

doesn't
necessarily correspond to the estimand actually targeted; it is merely a
switch to trigger which treatment group is considered "focal".

### Variable Ratio Matching

`matchit()`

can perform variable
ratio matching, which involves matching a different number of control units
to each treated unit. When `ratio > 1`

, rather than requiring all
treated units to receive `ratio`

matches, the arguments to
`max.controls`

and `min.controls`

can be specified to control the
maximum and minimum number of matches each treated unit can have.
`ratio`

controls how many total control units will be matched: `n1 * ratio`

control units will be matched, where `n1`

is the number of
treated units, yielding the same total number of matched controls as fixed
ratio matching does.

Variable ratio matching can be used with any `distance`

specification.
`ratio`

does not have to be an integer but must be greater than 1 and
less than `n0/n1`

, where `n0`

and `n1`

are the number of
control and treated units, respectively. Setting `ratio = n0/n1`

performs a restricted form of full matching where all control units are
matched. If `min.controls`

is not specified, it is set to 1 by default.
`min.controls`

must be less than `ratio`

, and `max.controls`

must be greater than `ratio`

. See the Examples section of
`method_nearest()`

for an example of their use, which is the same
as it is with optimal matching.

## Note

Optimal pair matching is a restricted form of optimal full matching
where the number of treated units in each subclass is equal to 1, whereas in
unrestricted full matching, multiple treated units can be assigned to the
same subclass. `optmatch::pairmatch()`

is simply a wrapper for
`optmatch::fullmatch()`

, which performs optimal full matching and is the
workhorse for `method_full`

. In the same way, `matchit()`

uses `optmatch::fullmatch()`

under the hood, imposing the restrictions that
make optimal full matching function like optimal pair matching (which is
simply to set `min.controls >= 1`

and to pass `ratio`

to the
`mean.controls`

argument). This distinction is not important for
regular use but may be of interest to those examining the source code.

The option `"optmatch_max_problem_size"`

is automatically set to
`Inf`

during the matching process, different from its default in
*optmatch*. This enables matching problems of any size to be run, but
may also let huge, infeasible problems get through and potentially take a
long time or crash R. See `optmatch::setMaxProblemSize()`

for more details.

## Outputs

All outputs described in `matchit()`

are returned with
`method = "optimal"`

. When `include.obj = TRUE`

in the call to
`matchit()`

, the output of the call to `optmatch::fullmatch()`

will be
included in the output. When `exact`

is specified, this will be a list
of such objects, one for each stratum of the `exact`

variables.

## References

In a manuscript, be sure to cite the following paper if using
`matchit()`

with `method = "optimal"`

:

Hansen, B. B., & Klopfer, S. O. (2006). Optimal Full Matching and Related Designs via Network Flows. Journal of Computational and Graphical Statistics, 15(3), 609–627. doi:10.1198/106186006X137047

For example, a sentence might read:

*Optimal pair matching was performed using the MatchIt package (Ho,
Imai, King, & Stuart, 2011) in R, which calls functions from the optmatch
package (Hansen & Klopfer, 2006).*

## See also

`matchit()`

for a detailed explanation of the inputs and outputs of
a call to `matchit()`

.

`optmatch::fullmatch()`

, which is the workhorse.

`method_full`

for optimal full matching, of which optimal pair
matching is a special case, and which relies on similar machinery.

## Examples

```
data("lalonde")
#1:1 optimal PS matching with exact matching on race
m.out1 <- matchit(treat ~ age + educ + race + nodegree +
married + re74 + re75, data = lalonde,
method = "optimal", exact = ~race)
#> Warning: Fewer control units than treated units in some `exact` strata; not all treated units will get a match.
m.out1
#> A matchit object
#> - method: 1:1 optimal pair matching
#> - distance: Propensity score
#> - estimated with logistic regression
#> - number of obs.: 614 (original), 232 (matched)
#> - target estimand: ATT
#> - covariates: age, educ, race, nodegree, married, re74, re75
summary(m.out1)
#>
#> Call:
#> matchit(formula = treat ~ age + educ + race + nodegree + married +
#> re74 + re75, data = lalonde, method = "optimal", exact = ~race)
#>
#> 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
#> nodegree 0.7081 0.5967 0.2450 . 0.1114
#> married 0.1892 0.5128 -0.8263 . 0.3236
#> 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
#> nodegree 0.1114
#> married 0.3236
#> re74 0.4470
#> re75 0.2876
#>
#> Summary of Balance for Matched Data:
#> Means Treated Means Control Std. Mean Diff. Var. Ratio eCDF Mean
#> distance 0.4972 0.4901 0.0325 1.0026 0.0056
#> age 25.7845 25.9569 -0.0241 0.4984 0.0711
#> educ 10.0948 10.3966 -0.1501 0.6260 0.0286
#> raceblack 0.7500 0.7500 0.0000 . 0.0000
#> racehispan 0.0948 0.0948 0.0000 . 0.0000
#> racewhite 0.1552 0.1552 0.0000 . 0.0000
#> nodegree 0.6810 0.6121 0.1517 . 0.0690
#> married 0.2414 0.2672 -0.0660 . 0.0259
#> re74 2862.5644 3046.0687 -0.0376 1.4317 0.0569
#> re75 1785.7677 1873.2021 -0.0272 1.6241 0.0506
#> eCDF Max Std. Pair Dist.
#> distance 0.0776 0.0436
#> age 0.2069 1.2024
#> educ 0.0776 1.1705
#> raceblack 0.0000 0.0000
#> racehispan 0.0000 0.0000
#> racewhite 0.0000 0.0000
#> nodegree 0.0690 0.9102
#> married 0.0259 0.5503
#> re74 0.2759 0.7299
#> re75 0.1379 0.8456
#>
#> Sample Sizes:
#> Control Treated
#> All 429 185
#> Matched 116 116
#> Unmatched 313 69
#> Discarded 0 0
#>
#2:1 optimal matching on the scaled Euclidean distance
m.out2 <- matchit(treat ~ age + educ + race + nodegree +
married + re74 + re75, data = lalonde,
method = "optimal", ratio = 2,
distance = "scaled_euclidean")
m.out2
#> A matchit object
#> - method: 2:1 optimal pair matching
#> - distance: Scaled Euclidean
#> - number of obs.: 614 (original), 555 (matched)
#> - target estimand: ATT
#> - covariates: age, educ, race, nodegree, married, re74, re75
summary(m.out2, un = FALSE)
#>
#> Call:
#> matchit(formula = treat ~ age + educ + race + nodegree + married +
#> re74 + re75, data = lalonde, method = "optimal", distance = "scaled_euclidean",
#> ratio = 2)
#>
#> Summary of Balance for Matched Data:
#> Means Treated Means Control Std. Mean Diff. Var. Ratio eCDF Mean
#> age 25.8162 26.4757 -0.0922 0.4947 0.0658
#> educ 10.3459 10.2730 0.0363 0.5810 0.0286
#> raceblack 0.8432 0.2351 1.6726 . 0.6081
#> racehispan 0.0595 0.1216 -0.2629 . 0.0622
#> racewhite 0.0973 0.6432 -1.8422 . 0.5459
#> nodegree 0.7081 0.6243 0.1843 . 0.0838
#> married 0.1892 0.4432 -0.6487 . 0.2541
#> re74 2095.5737 4120.1633 -0.4143 0.7972 0.1577
#> re75 1532.0553 2107.1613 -0.1786 1.0608 0.0935
#> eCDF Max Std. Pair Dist.
#> age 0.1405 0.5477
#> educ 0.0865 0.3750
#> raceblack 0.6081 1.6726
#> racehispan 0.0622 0.2629
#> racewhite 0.5459 1.8422
#> nodegree 0.0838 0.1843
#> married 0.2541 0.6487
#> re74 0.4081 0.4806
#> re75 0.2595 0.3195
#>
#> Sample Sizes:
#> Control Treated
#> All 429 185
#> Matched 370 185
#> Unmatched 59 0
#> Discarded 0 0
#>
```