The `optmatch`

package implements the optimal full matching algorithm for
bipartite matching problems. Given a matrix describing the distances between
two groups (where one group is represented by row entries, and the other by
column entries), the algorithm finds a matching between units that
minimizes the average within grouped distances. This algorithm is a popular
choice for covariate balancing applications (e.g. propensity score matching),
but it also can be useful for design stage applications such as blocking. For
more on the application and its implementation, see:

```
Hansen, B.B. and Klopfer, S.O. (2006) Optimal full matching and
related designs via network flows, JCGS 15 609-627.
```

`optmatch`

is available on CRAN:

```
> install.packages("optmatch")
> library("optmatch")
```

In addition to the optimal full matching algorithm, the package contains useful functions for generating distance specifications, combining and editing distance specifications, and summarizing and displaying matches. This walk through shows how to use these tools in your matching workflow.

Before we start, let's generate some simulated data. We will have two groups,
the "treated" and "control" groups. Without our knowledge, nature assigned
units from a pool into one of these two groups. The probability of being a
treated unit depends on some covariates. In the vector `Z`

, let a 1 denote
treated units and 0 denote control units

```
set.seed(20120111) # set this to get the exact same answers as I do
n <- 26 # chosen so we can divide the alphabet in half
W <- data.frame(w1 = rbeta(n, 4, 2), w2 = rbinom(n, 1, p = .33))
# nature assigns to treatment
tmp <- numeric(n)
tmp[sample(1:n, prob = W$w1^(1 + W$w2), size = n/2)] <- 1
W$z <- tmp
# for convenience, let's give the treated units capital letter names
tmp <- character(n)
tmp[W$z == 1] <- LETTERS[1:(n/2)]
tmp[W$z == 0] <- letters[(26 - n/2 + 1):26]
rownames(W) <- tmp
```

As we can see with a simple table and plot, these groups are not balanced on the covariates, as they would be (in expectation) with a randomly assigned treatment.

```
table(W$w2, W$z)
library(lattice) ; densityplot(W$w1, groups = W$z)
```

The next steps use the covariates to pair up similar treated and control
units. For more on assessing the amount and severity of imbalance between
groups on observed covariates, see the
RItools `R`

package.

These two groups are different, but how different are individual treated units
from individual control units? In answering this question, we will produce
several distance specifications: matrices of treated units (rows) by control
units (columns) with entries denoting distances. `optmatch`

provides several
ways of generating these matrices so that you don't have to do it by hand.

Let's begin with a simple Euclidean distance on the space defined by `W`

:

```
distances <- list()
distances$euclid <- match_on(z ~ w1 + w2, data = W, method = "euclidean")
```

The `method`

argument tells the `match_on`

function how to compute the
distances over the space defined by the formula. The default method extends the
simple Euclidean distance by rescaling the distances by the covariance of the
variables, the Mahalanobis
distance:

```
distances$mahal <- match_on(z ~ w1 + w2, data = W)
```

You can write additional distance computation functions. See the documentation
for `match_on`

for more details on how to create these functions.

To create distances, we could also try regressing the treatment indicator on
the covariates and computing the difference distance for each treated and
control pair. To make this process easier, `match_on`

has methods for `glm`

objects (and for big data problems, `bigglm`

objects):

```
propensity.model <- glm(z ~ w1 + w2, data = W, family =
binomial())
distances$propensity <- match_on(propensity.model)
```

The `glm`

method is a wrapper around the `numeric`

method for `match_on`

. The
`numeric`

method takes a vector of scores (for example, the linear prediction
for each unit from the model) and a vector indicating treatment status (`z`

)
for each unit. This method returns the absolute difference between each
treated and control pair on their scores (additionally,
the `glm`

method rescales the data before invoking the `numeric`

method). If
you wish to fit a "caliper" to your distance matrix, a hard limit on allowed
distances between treated and control units, you can pass a `caliper`

argument, a scalar numeric value. Any treated and control pair that is larger
than the caliper value will be replaced by `Inf`

, an unmatchable value. The
`caliper`

argument also applies to `glm`

method. Calipers are covered in more
detail in the next section.

The final convenience method of `match_on`

is using an arbitrary function. This
function is probably most useful for advanced users of `optmatch`

. See the
documentation of the `match_on`

function for more details on how to write your
own arbitrary computation functions.

We have created several representations of the matching problem, using Euclidean distance, Mahalanobis distance, the estimated propensity score, and an arbitrary function. We can combine these distances into single metric using standard arithmetic functions:

```
distances$all <- with(distances, euclid + mahal + propensity)
```

You may find it convenient to work in smaller pieces at first and then stitch
the results together into a bigger distance. The `rbind`

and `cbind`

functions let us
add additional treated and control entries to a distance specification for
each of the existing control and treated units, respectively. For example, we
might want to combine a Mahalanobis score for units `n`

through `s`

with a
propensity score for units `t`

through `z`

:

```
W.n.to.s <- W[c(LETTERS[1:13], letters[14:19]),]
W.t.to.z <- W[c(LETTERS[1:13], letters[20:26]),]
mahal.n.to.s <- match_on(z ~ w1 + w2, data = W.n.to.s)
ps.t.to.z <- match_on(glm(z ~ w1 + w2, data = W.t.to.z, family = binomial()))
distances$combined <- cbind(mahal.n.to.s, ps.t.to.z)
```

The `exactMatch`

function creates "stratified" matching problems, in which
there are subgroups that are completely separate. Such matching problems are
often much easier to solve than problems where a treated unit could be
connected to any control unit.

There is another method for creating reduced matching problems. The `caliper`

function compares each entry in an existing distance specification and
disallows any that are larger than a specified value. For example, we can trim
our previous combined distance to anything smaller than the median value:

```
distances$median.caliper <- caliper(distances$all, median(distances$all))
distances$all.trimmed <- with(distances, all + median.caliper)
```

Like the `exactMatch`

function, the results of `caliper`

used the sparse
matrix representation mentioned above, so can be very efficient for large,
sparse problems. As noted previously, if using the `glm`

or `numeric`

methods
of `match_on`

, passing the caliper's width in the `caliper`

argument can be more
efficient.

In addition to the space advantages of only storing the finite entries in a
sparse matrix, the results of `exactMatch`

and `caliper`

can be used to speed
up computation of *new* distances. The `match_on`

function that we saw earlier
has an argument called `within`

that helps filter the resulting
computation to only the finite entries in the `within`

matrix. Since `exactMatch`

and `caliper`

use finite entries denote valid pairs, they make excellent sources of
the `within`

argument.

Instead of creating the entire Euclidean distance matrix and *then* filtering
out cross-strata matches, we use the results of `exactMatch`

to compute only
the interesting cases:

```
tmp <- exactMatch(z ~ w2, data = W)
distances$exact <- match_on(z ~ w1, data = W, within = tmp)
```

Users of previous versions of `optmatch`

may notice that the `within`

argument is similar to the old `structure.formula`

argument. Like
`within`

, `structure.formula`

focused distance on within strata pairs.
Unlike `structure.formula`

, the `within`

argument allows using *any*
distance specification as an argument, including those created with `caliper`

. For
example, here is the Mahalanobis distance computed only for units that differ
by less than one on the propensity score.

```
distances$mahal.trimmed <- match_on(z ~ w1 + w2, data = W,
within = match_on(propensity.model, caliper = 1))
```

Now that we have generated several distances specifications, let's put them to use. Here is the simplest way to evaluate all distances specifications:

```
matches <- lapply(distances, function(x) { fullmatch(x, data = W) })
```

The result of the matching process is a named factor, where the names
correspond to the units (both treated and control) and the levels of the
factors are the matched groups. Including the `data`

argument is highly
recommended. This argument will make sure that the result of `fullmatch`

will
be in the same order as the original `data.frame`

that was used to build the
distance specification. This will make appending the results of `fullmatch`

on to the original `data.frame`

much more convenient.

The `fullmatch`

function as several arguments for fine tuning the allowed
ratio of treatment to control units in a match, and how much of the pool to
throw away as unmatchable. One common pattern for these arguments are pairs:
one treated to one control unit. Not every distance specification is amendable
to this pattern (e.g. when there are more treated units than control units in
`exactMatch`

created stratum). However, it can be done with the Mahalanobis
distance matrix we created earlier:

```
mahal.match <- pairmatch(distances$mahal, data = W)
```

Like `fullmatch`

, `pairmatch`

also allows fine tuning the ratio of matches to
allow larger groupings. It is can be helpful as it computes what percentage of
the group to throw away, giving better odds of successfully finding a matching
solution.

Once one has generated a match, you may wish to view the results. The results
of calls to `fullmatch`

or `pairmatch`

produce `optmatch`

objects (specialized
factors). This object has a special option to the `print`

method which groups
the units by factor level:

```
print(mahal.match, grouped = T)
```

If you wish to join the match factor back to the original `data.frame`

:

```
W.matched <- cbind(W, matches = mahal.match)
```

Make sure to include the `data`

argument to `fullmatch`

or `pairmatch`

,
otherwise results are not guaranteed to be in the same order as your original
`data.frame`

or `matrix`

.

This section will help you get the latest development version of `optmatch`

and
start using the latest features. Before starting, you should know which branch
you wish to install. Currently, the "master" branch is the main code base.
Additional features are added in their own branches. A list of branches is
available at (the optmatch project
page)[https://github.com/markmfredrickson/optmatch].

You must have the Fortran extensions for package building included. These can be had from CRAN: OS X, Windows.

We recommend using `dev_mode`

from the `devtools`

package to install
in-development version of the package so that you can keep the current CRAN
version as the primary package. Activating `dev_mode`

creates a secondary
library of packages which can only be accessed while in `dev_mode`

. Packages
normally installed can still be used, but if different versions are installed
normally and in `dev_mode`

, the `dev_mode`

version takes precedent if in
`dev_mode`

.

Install and load the `devtools`

package:

```
> install.packages("devtools")
> library("devtools")
```

Activate `dev_mode`

:

```
> dev_mode()
d>
```

Note that the prompt changes from `>`

to `d>`

to let you know you're in
`dev_mode`

. Now choose the development branch you want to use. To install
`master`

:

```
d> install_github("markmfredrickson/optmatch")
```

Either way, the package is then loaded in the usual fashion, provided you're
still in `dev_mode`

:

```
d> library(optmatch)
```

Once you've done this you can disable `dev_mode`

as follows

```
d> dev_mode()
>
```

The development version of the package remains loaded.

Note that if you load the package -- ie, enter `library(optmatch)`

(when the
package hasn't already been loaded otherwise) -- while *not* in `dev_mode`

, then
you'll get whatever version of the package may be installed in your library
tree, not this development version.

If you want to switch between versions of `RItools`

, we suggest re-starting R.

You may use RStudio to develop for Optmatch, by opening the `optmatch.Rproj`

file.
We suggest you ensure all required dependencies are installed by running

```
devtools::install_deps(dependencies = TRUE)
```

We prefer changes that include unit tests demonstrating the problem or showing
how the new feature should be added. The test suite uses the
testthat package to write and run tests.
(Please ensure you have the latest version of testthat (or at least v0.11.0),
as older versions stored the tests in a different directory, and may not
test properly.) See the `tests/testthat`

directory for examples. You can run
the test suite via Build -> Test Package.

New features should include inline Roxygen documentation.
You can generate all `.Rd`

documents from the `Roxygen`

code using Build ->
Document.

Finally, you can use Build -> Build and Reload or Build -> Clean and Rebuild to
load an updated version of `optmatch`

in your current RStudio session.
Alternatively, to install the developed version permanently, use Build -> Build
Binary Version, followed by

```
install.packages("../optmatch_VERSION.tgz", repo=NULL)
```

You can revert back to the current CRAN version by

```
remove.packages("optmatch")
install.packages("optmatch")
```

If you prefer not to use RStudio, you can develop using Make.

`make test`

: Run the full test suite.`make document`

: Update all documentation from Roxygen inline comments.`make interactive`

: Start up an interactive session with`optmatch`

loaded.`make check`

: Run`R CMD check`

on the package`make build`

: Build a binary package.`make vignette`

: Builds any vignettes in`vignettes/`

directory`make clean`

: Removes files built by`make vignette`

,`make document`

or`make check`

. Should not be generally necessary, but can be useful for debugging.

When your change is ready, make a pull request on github.

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