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)[https://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.

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.

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.