knitr::opts_chunk$set(collapse = TRUE, prompt=TRUE)

All the software used in this worksheet is freely available. The *R*
statistical package is installed for you in the lab, but you may
download and install *R* for Windows, Mac, and Linux systems from:
https://www.r-project.org.

The following document walks through a common propensity score
matching work-flow in *R*. Example *R* code will appear with a `>`

indicating the command prompt. You may type this code yourself ---
each line is a command to *R*. Output will follow prefaced by
`##`

. (In *R*, `#`

represents a comment; any command preceded by any
number of `#`

's will not be executed.) For example:

2 + 2

(Note that when entering the code yourself, do not include the `>`

in
your command. Also, for longer lines of code in this document, the
text may wrap onto a second line, with the second line preceded by a
`+`

sign. When entering the code yourself, you do not have to wrap the
lines, and do not include the `+`

.)

*R* stores data in named variables using the arrow operator:

my.variable <- 2 + 2 my.variable * 3

*R* add-on packages are available to install directly from *R*:

install.packages("optmatch")

Thi command will ask you to select a CRAN server. Any server will do. You may
also be asked whether you'd like to set up a "personal library to install
packages into"; if so, answer yes. (The default personal library location that
*R* will suggest should be OK.) You'll only need to run these commands the first
time you want to use **optmatch** on a particular computer, or when you install
a new version of *R*.

Attach extension packages that we'll be using for matching and associated diagnostics:

```
library(optmatch)
```

```
library(optmatch)
```

You'll do this each time you start a new *R* session and want to run
matching commands.

To load the nuclear plants data, enter

```
data(nuclearplants)
```

```
data(nuclearplants)
```

To see the first six rows:

```
head(nuclearplants)
```

For more on the variables here, enter

help("nuclearplants")

You can directly access a variable within this data frame as follows. (Try typing in the commands to see what they do.)

nuclearplants$pt table(nuclearplants$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.

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.

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) if (requireNamespace("pander", quietly = TRUE)) { pander::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`.') } else { show(p) }

For basic summary information about this match, try

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))

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)

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.

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) summary(cap.noadj)

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
`RItools::balanceTest()`

function zeroes in on balance, and facilitates checking
balance on multiple variables at the same time. Here are some examples:

install.packages("RItools") library(RItools) balanceTest(pr ~ cap + t2, data = nuke.nopt)

if (requireNamespace("RItools", quietly = TRUE)) { library(RItools) } else { cat("RItools package not installed properly") }

if (requireNamespace("RItools", quietly = TRUE)) { RItools::balanceTest(pr ~ cap + t2, data = nuke.nopt) }

balanceTest(pr ~ cap + t2 + strata(pm) - 1, data = nuke.nopt) # The `- 1` suppresses the unmatched output to make the output cleaner

if (requireNamespace("RItools", quietly = TRUE)) { RItools::balanceTest(pr ~ cap + t2 + strata(pm) - 1, data = nuke.nopt) }

**Exercises.**

- Compare
`pm`

,`tm`

and the unmatched samples in terms of balance on`t2`

. - Compare
`pm`

,`tm`

and the unmatched samples in terms of balance on`date`

. - Compare
`pm`

to Mahalanobis pair matching on`t1`

in terms of balance on`date`

. - 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.

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, let's 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!

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

balanceTest(pr ~ date + t1 + t2 + cap + ne + ct + bw + cum.n, data = nuclearplants) balanceTest(pr ~ date + t1 + t2 + cap + ne + ct + bw + cum.n + pt + strata(ps.pm2) - 1, 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 <- balanceTest(pr ~ date + t1 + t2 + cap + ne + ct + bw + cum.n + strata(ps.pm2), data = nuclearplants) plot(myb) print(myb, digits=1)

if (requireNamespace("RItools", quietly = TRUE)) { myb <- RItools::balanceTest(pr ~ date + t1 + t2 + cap + ne + ct + bw + cum.n + strata(ps.pm2), data = nuclearplants) print(myb, digits=1) plot(myb) } else { cat("RItools package not installed properly") }

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 `balanceTest()`

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:

- the "best balance overall," as measured by Chi-square statistics;
- the "best balance overall," as indicated by the largest standardized difference among the covariates;
- the best balance on
`date`

and`cap`

, as measured by the larger of the standardized differences for these two variables; and - 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.

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))

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)

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.

Below are some suggestions of other data sets you could try this out on.

For another interesting (toy) data set, you can use the "tli" data set in the
**xtable** package.

install.packages("xtable") # if not already installed data(tli, package = "xtable")

You might compare test scores for kids with `disadvg=="YES"`

to those of kids
with `disadvg=="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 collected many of the data sets discussed in his *Design of
Observational Studies* (2010) into the
**DOS** and
**DOS2** packages.

install.packages("DOS") # if not already installed install.packages("DOS2") # if not already installed data(package = "DOS") data(package = "DOS2")

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") # 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") # if not already installed Hmisc:::getHdata(rhc, what = "all")

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