# Matching in *R* using the optmatch and RItools packages In optmatch: Functions for Optimal Matching

table(nuclearplants$pt) with(nuclearplants, table(pt))  The variable you will have just viewed and tabulated, pt, is a dummy for whether the plant was built with "partial turnkey guarantees." These plants were not comparable to the others in terms of construction costs. Let's exclude them for the time being, for simplicity. To do this we'll create a data table (in R jargon, a "data frame") of just those observations for which pt is 0: nuke.nopt <- subset(nuclearplants, pt == 0)  To inspect its first six or last six entries, do head(nuke.nopt) tail(nuke.nopt)  To view this as presenting a matching problem, we'll think of plants built on the site of a previously existing plant (pr == 1) as the treatment group and plants on new sites (pr == 0) as comparisons. # Optimal pair matching and 1:k matching ## Pair matching To check the number of treated and control plants: table(nuke.nopt$pr)


To get the pair match minimizing the mean paired distance on cap, among all collections of 7 non-overlapping pairs, do

pairmatch(pr ~ cap, data = nuke.nopt)


For a more readable report of who gets matched to whom, type

print(pairmatch(pr ~ cap, data = nuke.nopt), grouped = TRUE)


For matching on both date and cap, you'd type pairmatch(pr ~ cap + date, ...) instead of pairmatch(pr ~ cap, ...). We'll talk later about how this combines discrepancies on the two variables. For now, note the form of the output this command generates: a variable of the same length as the variables making up nuke.nopt, assigning a distinct name to each matched set. To fix your intuition, you might try connecting up below the units that pairmatch() has placed in the same matched sets.

library(pander)
a <- with(nuke.nopt, data.frame(
Plant=row.names(nuke.nopt),
Date=round(date-65, 1),
Capacity=round(x=(cap-400),digits=-1))[as.logical(pr),])

b <- with(nuke.nopt, data.frame(
Plant=row.names(nuke.nopt),
Date=round(date-65, 1),
Capacity=round(x=(cap-400),digits=-1))[!as.logical(pr),])

rownames(a) <- NULL
rownames(b) <- NULL

c <- cbind(data.frame(rbind(as.matrix(a), matrix(nrow=nrow(b)-nrow(a), ncol=3))), b)
pandoc.table(c, style="multiline", missing="",
caption='New-site (left columns) versus existing-site (right columns) plants. "date" is date-65; "capacity" is cap-400.')


summary(pairmatch(pr ~ cap, data = nuke.nopt))


If you've already typed in the pairmatch(...) part, you can use the up-arrow, Home and End keys to avoid having to re-type. Alternatively, to assign the name "pm" to the matching result, do

pm <- pairmatch(pr ~ cap, data = nuke.nopt)


Now, you can just type print(pm, grouped = TRUE) or summary(pm).

The following would give a basic matched analysis of the effect of new or existing site on construction costs is given with the help of R's linear modeling function. In effect, the existing site effect is estimated as one "way" in a two-way \textsc{anova}, the other "way" being the factor variable that represents the matching result, i.e. pm.

summary(lm(cost ~ pr + pm, data = nuke.nopt))


## Matching with multiple controls

There are other types of matches you might want to try. Here's how to create matched triples (each treatment group unit is matched to two control group units):

tm <- pairmatch(pr ~ cap, controls = 2, data = nuke.nopt)


There will be further variations suggested on the slides.

## Did matching work?

It's possible to give the software an impossible list of requirements for a match. For instance, try running the following:

pairmatch(pr ~ cap, controls = 3, data=nuke.nopt)


The problem here is that the data don't have 3 comparison units to go with each treatment unit, since we have 7 treatment units but only 19 comparison units.

Matching can also fail because the distance matrix embodies matching constraints that are impossible to meet. In these cases the matching function will generally run without complaint, although it won't create any matches. Here is an example, where the caliper is so narrow as to forbid all possible matches:

pairmatch(pr ~ cap + cost, caliper=.001, data = nuke.nopt)


Behind the scenes, the caliper argument restricts how the maximum distance between matched objects. For example, consider Table 1 above. Plants A and H are 1.3 units apart in date. If we assigned caliper=1, they could never be matched because they exceed the caliper limit.

If before matching you want to remove just the subjects lacking a counterpart within caliper distance, you can do pairmatch(..., remove.unmatchables = TRUE). That won't help with the minuscule caliper above, but with less extreme calipers it helps you salvage a few matches.

## How closely did I match?

Getting back to a matching that succeeded, note that summary() reports information about how close the matches are.

summary(pm)


## Did matching balance the covariate?

Comparing overt biases before and after matching. An assessment of the unmatched difference between the groups on cap can be had via:

cap.noadj <- lm(cap ~ pr, data = nuke.nopt)


The output is suppressed, as most of it is not relevant to balance. This variation hones in on the part that is:

summary(lm(cap ~ pr, data = nuke.nopt))$coeff["pr",]  (Note again the use of square brackets, [ and ], for specifying subsets of a matrix. With R one has to carefully distinguish square brackets, curly brackets and parentheses.) Here is a parallel calculation that takes the match pm into account. summary(lm(cap ~ pr + pm, data = nuke.nopt))$coeff["pr",]


The RItools package's xBalance function zeroes in on balance, and facilitates checking balance on multiple variables at the same time. Here are some examples:

xBalance(pr ~ cap + t2, report="all", data=nuke.nopt)
xBalance(pr ~ cap + t2 + strata(pm),
data=nuke.nopt,


Exercises.

1. Compare pm, tm and the unmatched samples in terms of balance on t2.
2. Compare pm, tm and the unmatched samples in terms of balance on date.
3. Compare pm to Mahalanobis pair matching on t1 in terms of balance on date.
4. Compare Mahalanobis pair matching on cap and date to Mahalanobis pair matching on cap, date and each of t1,t2. Add the last two variables in one at a time, so that you're comparing a total of three matches. Compare on balance in cap and t2.

Section "Checking balance in general", below, presents convenient ways to do balance assessment for many variables at once. Before getting to that let's discuss try matching with propensity scores.

# Propensity Score Matching

## Propensity score fitting in R

Logistic regression models are fit in R using the function glm(), with "family" argument set to "binomial." Example:

psm <- glm(pr ~ date + t1 + t2 + cap + ne + ct + bw + cum.n + pt,
family = binomial, data = nuclearplants)


The fitted logistic regression is then stored in the object "psm." The propensity scores can be accessed with psm\$fitted.values (estimated probabilities) or scores(psm) (estimated logits of probabilities).

It's often a good idea to compare the groups' distributions on the propensity score.

boxplot(psm)


The groups do overlap, if not greatly. It may be wise to restrict the sample to the region of overlap, at least roughly. A propensity caliper would help with this.

First, lets match directly on the propensity score without restricting the sample to the region of overlap.

ps.pm <- pairmatch(psm, data = nuclearplants)
summary(ps.pm)


To restrict to the overlapping region, we want to imply a caliper to the distances generated by the propensity score model. To do this requires a more explicit generation of the match, involving separate steps for generation of the distances followed by matching upon those distances.

First, we create a distance matrix based upon psm:

psm.dist <- match_on(psm, data=nuclearplants)


psm.dist is a matrix with an entry corresponding to the distance between each potential pair of treatment and control units. We can caliper directly on this distance matrix,

caliper(psm.dist, 2)


Entries which are Inf will never be matched. Adding the caliper to psm.dist will disallow matching between units which differ by more than 2 standard deviations.

(Note that this differs from the previous use of caliper directly in pairmatch, where the caliper is applied directly to the distances instead of smartly upon the standard deviations between the propensity scores.)

Combining the above, we can now

ps.pm2 <- pairmatch(psm.dist, data = nuclearplants)
ps.pm3 <- pairmatch(psm.dist + caliper(psm.dist, 2), data = nuclearplants)
all.equal(ps.pm, ps.pm2, check.attributes=FALSE)
all.equal(ps.pm, ps.pm3, check.attributes=FALSE)
summary(ps.pm3)


Or you could match within calipers of the propensity score on some other distance, perhaps Mahalanobis distances based on selected covariates as recommended by Rubin and Thomas (2000, JASA) and others. For Mahalanobis matching on date, cap and the propensity score, for instance, combined with a propensity caliper of 1 pooled sd.

mhd1 <- match_on(pr ~ date + cap + scores(psm), data=nuclearplants)
mhpc.pm <- pairmatch(mhd1, caliper=1, data=nuclearplants)
summary(mhpc.pm) # oops
mhpc.pm <- pairmatch(mhd1, caliper=2, data=nuclearplants)
summary(mhpc.pm) # better!


## Checking balance in general

The RItools package has a convenient function for checking balance on many variables simultaneously. To get a sense of what it does, try this:

library(RItools)
xBalance(pr ~ date + t1 + t2 + cap + ne + ct + bw + cum.n, data = nuclearplants)
xBalance(pr ~ date + t1 + t2 + cap + ne + ct + bw + cum.n + pt +
strata(ps.pm2) -1, # the -1 just focuses the output a little
data = nuclearplants)


It can in the same display compare matching to no matching, on any of a number of axes. Here is a demonstration:

myb <- xBalance(pr ~ date + t1 + t2 + cap + ne + ct + bw + cum.n +
strata(ps.pm2),
data = nuclearplants,
"z.scores", "chisquare.test"))
plot(myb)
print(myb, digits=1)


For a very compact representation of the assessment, call summary() on the match, passing along the fitted propensity model as a second argument. If you've got RItools loaded, then this will call xBalance in the background, reporting the summary chi-square test results.

summary(ps.pm2, psm)


Exercise.

Try out {at least 3} different combinations of propensity score and Mahalanobis matching. Identify the matches that gives:

1. the "best balance overall," as measured by Chi-square statistics;
2. the "best balance overall," as indicated by the largest standardized difference among the covariates;
3. the best balance on date and cap, as measured by the larger of the standardized differences for these two variables; and
4. the best balance on date and cap, as measured by the larger of the standardized differences for these two variables, among those matches for which the overall imbalance p-value is no more than .1.

# Other topics in matching

## Full matching and matching with a varying number of controls

Try out for yourself and compare:

summary(fullmatch(pr ~ date + cap, data = nuke.nopt))
summary(fullmatch(pr ~ date + cap, data = nuke.nopt, min = 1))
summary(fullmatch(pr ~ date + cap, data = nuke.nopt, min = 2, max = 3))


## Subclassification before matching

Recall that the data set nuclearplants had 32 observations, 6 of which we excluded. These were plants built under "partial turnkey" guarantees (pt == 1), for which costs are difficult to compare with other plants. We might include the excluded plants by matching them only among themselves. Then we need to subclassify prior to matching.

This is a common and useful operation, and optmatch is designed to help you do it via a strata term in the matching formula.

pairmatch(pr ~ date + cap + scores(psm), data=nuclearplants)
pairmatch(pr ~ date + cap + scores(psm) + strata(pt), data=nuclearplants)


## Distances

You'll often want to do several variations on a match. It may save computation and typing time to store the distance you're using to match, if you're going to re-use that distance. To do so, you'll have to explicitly separate distance-making and matching, two steps that we've merged together thus far. We'll use the match_on() to create distances.

cap.dist <- match_on(pr ~ cap, data = nuke.nopt)
pm1 <- pairmatch(pr ~ cap, data=nuke.nopt)
pm2 <- pairmatch(cap.dist, data=nuke.nopt)
all.equal(pm1, pm2, check.attributes = FALSE)
summary(pm2)


What does a matching distance look like? Here's the upper-left corner of one of them:

round(cap.dist[1:3, 1:3], 1)


(Note the use of square brackets, [ and ], for specifying rows and columns of the distance matrix. If you find that this isn't working on a distance that you've produced, try as.matrix(my.dist)[1:3,1:3] or similar.)

Matching with a caliper of 2 pooled standard deviations on the cap variable:

round(cap.dist + caliper(cap.dist, 2), 1)


Entries of Inf or NaN in a distance matrix are interpreted as forbidden matches. Thus

pairmatch(cap.dist + caliper(cap.dist, 2), data = nuke.nopt)


matches on cap, insisting that paired units not differ by more than 2 pooled SDs in cap. If you would prefer to set a requirement on how much paired units can differ in cap in its original units, rather than standard units, then you would have to create cap.dist in a somewhat different way. There's an example on the help page for caliper(). Enter help(caliper) at the R command line.

## Using R for matching and another program for matched analysis

If you prefer to do your main work in another statistical package, you can use R for matching and balance assessment, and then re-import the matched data back into your preferred package for the main analysis. A typical work flow might look something like this:

2. Preprocess data (handle missing values, combine variables into single measures, etc).
3. Export data in a format R can read.
5. Perform matching and balance testing.

You already know how to perform the matching and balance testing, so what remains are the import and export steps, along with appending your matches to your original data.

When importing and exporting data, you must select a data format. In a broad sense, you have two options: proprietary data formats (for example .dta for Stata) or open standards (for example .csv, Comma Separated Values). The advantage of proprietary formats is that they may include additional information for R to use, such as the labels on a categorical variable. The potential pitfall is that R might not know how to read your particular file type. Open formats like CSV are easy to read and write, but you may lose variable names, labels, or other special forms of data. For reading and writing proprietary formats, consider the use of the foreign library in R. See for example read.dta and write.dta:

library(foreign)
?write.dta


For much more detail on this topic, see the "R Data Import/Export" manual.

If you use CSV files, consult your statistical package's manuals for details on how to export your data as a CSV. Once you have exported your data into a file, open the file (using a text editor, not Microsoft Word) to see if there is a header row of variable names. Then, to import your data use the following:

my.plants <- read.csv("nuclearplants.csv", header = TRUE)


If you do not have a header on your data change to header = FALSE.

At this point, you may proceed to conduct your matching as usual (for example using an example from earlier in this document). When you are done, you will need to append your match to your original data. The safest way to do this is using the following code snippet:

plant.match <- pairmatch(pr ~ cap, data = my.plants)
my.plants.extended <- data.frame(my.plants, matches = plant.match, check.rows=TRUE)


Finally, export your data as a .csv file (or a proprietary format such as with write.dta if that is what you are using):

write.csv(my.plants.extended, file = "nuclearplants-with-matches.csv")


As one final note, many of the balance tests included in RItools are available to Stata users directly from Stata. Information is available at the RItools webpage at {http://www.jakebowers.org/RItools.html}.

## Trying it out on your own

For another interesting (toy) data set, do

data(tli, package="xtable")


You might compare test scores for kids with disadvg=="YES" to those of kids withdisadvg=="NO" using propensity matching, in some combination with Mahalanobis matching and caliper matching. A check of propensity overlap may inform your decision as to whether to include a propensity caliper. Be sure to check for balance, and do check the structure of the matched sets.

Three sources of \textit{real} data can be gotten as follows. First, Paul Rosenbaum has posted many of the data sets discussed in his Design of Observational Studies (2010) to his web site. If you have an active internet connection then you can get them by (as of this writing):

download.file("http://www-stat.wharton.upenn.edu/~rosenbap/DOSdata.RData",
destfile="./DOSdata.RData")


Second, the "lalonde" data set, discussed by Lalonde (1986, Am. Econom. Rev.), Dehejia and Wahba (1999, JASA) and Smith and Todd (2005, J. Econom.), is bundled with several R packages, including "{arm}" and "Matching." To get it:

install.packages("arm", dep=T) # if not already installed
data(lalonde, package="arm")
help("lalonde", package="arm")


Third, the data used by Connors et al (1996, J. Am. Med. Assoc.) to examine costs, benefits and risks associated with right heart catheterization is bundled with Frank Harrell and collaborators' "Hmisc" package.

install.packages("Hmisc", dep=T) # if not already installed
Hmisc:::getHdata(rhc, what = "all")


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optmatch documentation built on May 29, 2017, 8:49 p.m.