# Optmatch: Optimal Fullmatching for R

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")``````

## Using 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.

### Simulated data

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.

### Setting up distances

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.

### Combining and editing distances

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.

### Speeding up computation

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))``````

### Generating the match

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`.

## Using a development version of Optmatch

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)[http://github.com/markmfredrickson/optmatch].

### Installing a development version

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

`optmatch` is built using devtools which makes installing the current development version very easy. Simply install the `devtools` package and then use it to install from this repository.

``````install.packages("devtools")
devtools:::install_github("markmfredrickson/optmatch")``````

You may pass `ref=<branchname>` as an argument to `install_github` to install a branch other than "master", which is the default.

Note that this will install the development version globally, such that the existing release version from CRAN is overwritten. To revert to the current release version from CRAN, remove and re-install via the following

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

## Developing for Optmatch

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.