`Match`

implements a variety of algorithms for multivariate
matching including propensity score, Mahalanobis and inverse variance
matching. The function is intended to be used in conjunction with the
`MatchBalance`

function which determines the extent to which
`Match`

has been able to achieve covariate balance. In order to
do propensity score matching, one should estimate the propensity model
before calling `Match`

, and then send `Match`

the propensity
score to use. `Match`

enables a wide variety of matching
options including matching with or without replacement, bias
adjustment, different methods for handling ties, exact and caliper
matching, and a method for the user to fine tune the matches via a
general restriction matrix. Variance estimators include the usual
Neyman standard errors, Abadie-Imbens standard errors, and robust
variances which do not assume a homogeneous causal effect. The
`GenMatch`

function can be used to *automatically
find balance* via a genetic search algorithm which determines the
optimal weight to give each covariate.

```
Match(Y=NULL, Tr, X, Z = X, V = rep(1, length(Y)), estimand = "ATT", M = 1,
BiasAdjust = FALSE, exact = NULL, caliper = NULL, replace=TRUE, ties=TRUE,
CommonSupport=FALSE,Weight = 1, Weight.matrix = NULL, weights = NULL,
Var.calc = 0, sample = FALSE, restrict=NULL, match.out = NULL,
distance.tolerance = 1e-05, tolerance=sqrt(.Machine$double.eps),
version="standard")
```

- est
The estimated average causal effect.

- se
The Abadie-Imbens standard error. This standard error has correct coverage if

`X`

consists of either covariates or a known propensity score because it takes into account the uncertainty of the matching procedure. If an estimated propensity score is used, the uncertainty involved in its estimation is not accounted for although the uncertainty of the matching procedure itself still is.- est.noadj
The estimated average causal effect without any

`BiasAdjust`

. If`BiasAdjust`

is not requested, this is the same as`est`

.- se.standard
The usual standard error. This is the standard error calculated on the matched data using the usual method of calculating the difference of means (between treated and control) weighted by the observation weights provided by

`weights`

. Note that the standard error provided by`se`

takes into account the uncertainty of the matching procedure while`se.standard`

does not. Neither`se`

nor`se.standard`

take into account the uncertainty of estimating a propensity score.`se.standard`

does not take into account any`BiasAdjust`

. Summary of both types of standard error results can be requested by setting the`full=TRUE`

flag when using the`summary.Match`

function on the object returned by`Match`

.- se.cond
The conditional standard error. The practitioner should not generally use this.

- mdata
A list which contains the matched datasets produced by

`Match`

. Three datasets are included in this list:`Y`

,`Tr`

and`X`

.- index.treated
A vector containing the observation numbers from the original dataset for the treated observations in the matched dataset. This index in conjunction with

`index.control`

can be used to recover the matched dataset produced by`Match`

. For example, the`X`

matrix used by`Match`

can be recovered by`rbind(X[index.treated,],X[index.control,])`

. The user should generally just examine the output of`mdata`

.- index.control
A vector containing the observation numbers from the original data for the control observations in the matched data. This index in conjunction with

`index.treated`

can be used to recover the matched dataset produced by`Match`

. For example, the`X`

matrix used by`Match`

can be recovered by`rbind(X[index.treated,],X[index.control,])`

. The user should generally just examine the output of`mdata`

.- index.dropped
A vector containing the observation numbers from the original data which were dropped (if any) in the matched dataset because of various options such as

`caliper`

and`exact`

. If no observations were dropped, this index will be`NULL`

.- weights
A vector of weights. There is one weight for each matched-pair in the matched dataset. If all of the observations had a weight of 1 on input, then each matched-pair will have a weight of 1 on output if there are no ties.

- orig.nobs
The original number of observations in the dataset.

- orig.wnobs
The original number of weighted observations in the dataset.

- orig.treated.nobs
The original number of treated observations (unweighted).

- nobs
The number of observations in the matched dataset.

- wnobs
The number of weighted observations in the matched dataset.

- caliper
The

`caliper`

which was used.- ecaliper
The size of the enforced caliper on the scale of the

`X`

variables. This object has the same length as the number of covariates in`X`

.- exact
The value of the

`exact`

function argument.- ndrops
The number of weighted observations which were dropped either because of caliper or exact matching. This number, unlike

`ndrops.matches`

, takes into account observation specific weights which the user may have provided via the`weights`

argument.- ndrops.matches
The number of matches which were dropped either because of caliper or exact matching.

- Y
A vector containing the outcome of interest. Missing values are not allowed. An outcome vector is not required because the matches generated will be the same regardless of the outcomes. Of course, without any outcomes no causal effect estimates will be produced, only a matched dataset.

- Tr
A vector indicating the observations which are in the treatment regime and those which are not. This can either be a logical vector or a real vector where 0 denotes control and 1 denotes treatment.

- X
A matrix containing the variables we wish to match on. This matrix may contain the actual observed covariates or the propensity score or a combination of both. All columns of this matrix must have positive variance or

`Match`

will return an error.- Z
A matrix containing the covariates for which we wish to make bias adjustments.

- V
A matrix containing the covariates for which the variance of the causal effect may vary. Also see the

`Var.calc`

option, which takes precedence.- estimand
A character string for the estimand. The default estimand is "ATT", the sample average treatment effect for the treated. "ATE" is the sample average treatment effect, and "ATC" is the sample average treatment effect for the controls.

- M
A scalar for the number of matches which should be found. The default is one-to-one matching. Also see the

`ties`

option.- BiasAdjust
A logical scalar for whether regression adjustment should be used. See the

`Z`

matrix.- exact
A logical scalar or vector for whether exact matching should be done. If a logical scalar is provided, that logical value is applied to all covariates in

`X`

. If a logical vector is provided, a logical value should be provided for each covariate in`X`

. Using a logical vector allows the user to specify exact matching for some but not other variables. When exact matches are not found, observations are dropped.`distance.tolerance`

determines what is considered to be an exact match. The`exact`

option takes precedence over the`caliper`

option.- caliper
A scalar or vector denoting the caliper(s) which should be used when matching. A caliper is the distance which is acceptable for any match. Observations which are outside of the caliper are dropped. If a scalar caliper is provided, this caliper is used for all covariates in

`X`

. If a vector of calipers is provided, a caliper value should be provided for each covariate in`X`

. The caliper is interpreted to be in standardized units. For example,`caliper=.25`

means that all matches not equal to or within .25 standard deviations of each covariate in`X`

are dropped. Note that dropping observations generally changes the quantity being estimated.- replace
A logical flag for whether matching should be done with replacement. Note that if

`FALSE`

, the order of matches generally matters. Matches will be found in the same order as the data are sorted. Thus, the match(es) for the first observation will be found first, the match(es) for the second observation will be found second, etc. Matching without replacement will generally increase bias. Ties are randomly broken when`replace==FALSE`

---see the`ties`

option for details.- ties
A logical flag for whether ties should be handled deterministically. By default

`ties==TRUE`

. If, for example, one treated observation matches more than one control observation, the matched dataset will include the multiple matched control observations and the matched data will be weighted to reflect the multiple matches. The sum of the weighted observations will still equal the original number of observations. If`ties==FALSE`

, ties will be randomly broken.*If the dataset is large and there are many ties, setting*Whether two potential matches are close enough to be considered tied, is controlled by the`ties=FALSE`

often results in a large speedup.`distance.tolerance`

option.- CommonSupport
This logical flag implements the usual procedure by which observations outside of the common support of a variable (usually the propensity score) across treatment and control groups are discarded. The

`caliper`

option is to be preferred to this option because`CommonSupport`

, consistent with the literature, only drops*outliers*and leaves*inliers*while the caliper option drops both. If`CommonSupport==TRUE`

, common support will be enforced on the first variable in the`X`

matrix. Note that dropping observations generally changes the quantity being estimated. Use of this option renders it impossible to use the returned objects`index.treated`

and`index.control`

to reconstruct the matched dataset. The returned object`mdata`

will, however, still contain the matched dataset. Seriously, don't use this option; use the`caliper`

option instead.- Weight
A scalar for the type of weighting scheme the matching algorithm should use when weighting each of the covariates in

`X`

. The default value of 1 denotes that weights are equal to the inverse of the variances. 2 denotes the Mahalanobis distance metric, and 3 denotes that the user will supply a weight matrix (`Weight.matrix`

). Note that if the user supplies a`Weight.matrix`

,`Weight`

will be automatically set to be equal to 3.- Weight.matrix
This matrix denotes the weights the matching algorithm uses when weighting each of the covariates in

`X`

---see the`Weight`

option. This square matrix should have as many columns as the number of columns of the`X`

matrix. This matrix is usually provided by a call to the`GenMatch`

function which finds the optimal weight each variable should be given so as to achieve balance on the covariates.For most uses, this matrix has zeros in the off-diagonal cells. This matrix can be used to weight some variables more than others. For example, if

`X`

contains three variables and we want to match as best as we can on the first, the following would work well:

`> Weight.matrix <- diag(3)`

`> Weight.matrix[1,1] <- 1000/var(X[,1])`

`> Weight.matrix[2,2] <- 1/var(X[,2])`

`> Weight.matrix[3,3] <- 1/var(X[,3])`

This code changes the weights implied by the inverse of the variances by multiplying the first variable by a 1000 so that it is highly weighted. In order to enforce exact matching see the`exact`

and`caliper`

options.- weights
A vector the same length as

`Y`

which provides observation specific weights.- Var.calc
A scalar for the variance estimate that should be used. By default

`Var.calc=0`

which means that homoscedasticity is assumed. For values of`Var.calc > 0`

, robust variances are calculated using`Var.calc`

matches.- sample
A logical flag for whether the population or sample variance is returned.

- distance.tolerance
This is a scalar which is used to determine if distances between two observations are different from zero. Values less than

`distance.tolerance`

are deemed to be equal to zero. This option can be used to perform a type of optimal matching- tolerance
This is a scalar which is used to determine numerical tolerances. This option is used by numerical routines such as those used to determine if a matrix is singular.

- restrict
A matrix which restricts the possible matches. This matrix has one row for each restriction and three columns. The first two columns contain the two observation numbers which are to be restricted (for example 4 and 20), and the third column is the restriction imposed on the observation-pair. Negative numbers in the third column imply that the two observations cannot be matched under any circumstances, and positive numbers are passed on as the distance between the two observations for the matching algorithm. The most commonly used positive restriction is

`0`

which implies that the two observations will always be matched.Exclusion restrictions are even more common. For example, if we want to exclude the observation pair 4 and 20 and the pair 6 and 55 from being matched, the restrict matrix would be:

`restrict=rbind(c(4,20,-1),c(6,55,-1))`

- match.out
The return object from a previous call to

`Match`

. If this object is provided, then`Match`

will use the matches found by the previous invocation of the function. Hence,`Match`

will run faster. This is useful when the treatment does not vary across calls to`Match`

and one wants to use the same set of matches as found before. This often occurs when one is trying to estimate the causal effect of the same treatment (`Tr`

) on different outcomes (`Y`

). When using this option, be careful to use the same arguments as used for the previous invocation of`Match`

unless you know exactly what you are doing.- version
The version of the code to be used. The "fast" C/C++ version of the code does not calculate Abadie-Imbens standard errors. Additional speed can be obtained by setting

`ties=FALSE`

or`replace=FALSE`

if the dataset is large and/or has many ties. The "legacy" version of the code does not make a call to an optimized C/C++ library and is included only for historical compatibility. The "fast" version of the code is significantly faster than the "standard" version for large datasets, and the "legacy" version is much slower than either of the other two.

Jasjeet S. Sekhon, UC Berkeley, sekhon@berkeley.edu, http://sekhon.berkeley.edu/.

This function is intended to be used in conjunction with the
`MatchBalance`

function which checks if the results of this
function have actually achieved balance. The results of this function
can be summarized by a call to the `summary.Match`

function. If one wants to do propensity score matching, one should estimate the
propensity model before calling `Match`

, and then place the
fitted values in the `X`

matrix---see the provided example.

The `GenMatch`

function can be used to *automatically
find balance* by the use of a genetic search algorithm which determines
the optimal weight to give each covariate. The object returned by
`GenMatch`

can be supplied to the `Weight.matrix`

option of `Match`

to obtain estimates.

`Match`

is often much faster with large datasets if
`ties=FALSE`

or `replace=FALSE`

---i.e., if matching is done
by randomly breaking ties or without replacement. Also see the
`Matchby`

function. It provides a wrapper for
`Match`

which is much faster for large datasets when it can be
used.

Three demos are included: `GerberGreenImai`

, `DehejiaWahba`

,
and `AbadieImbens`

. These can be run by calling the
`demo`

function such as by `demo(DehejiaWahba)`

.

Sekhon, Jasjeet S. 2011. "Multivariate and Propensity Score
Matching Software with Automated Balance Optimization.''
*Journal of Statistical Software* 42(7): 1-52.
tools:::Rd_expr_doi("10.18637/jss.v042.i07")

Diamond, Alexis and Jasjeet S. Sekhon. 2013. "Genetic
Matching for Estimating Causal Effects: A General Multivariate
Matching Method for Achieving Balance in Observational Studies.''
*Review of Economics and Statistics*. 95 (3): 932--945.
http://sekhon.berkeley.edu/papers/GenMatch.pdf

Abadie, Alberto and Guido Imbens. 2006.
``Large Sample Properties of Matching Estimators for Average
Treatment Effects.'' *Econometrica* 74(1): 235-267.

Imbens, Guido. 2004. Matching Software for Matlab and Stata.

Also see `summary.Match`

,
`GenMatch`

,
`MatchBalance`

,
`Matchby`

,
`balanceUV`

,
`qqstats`

, `ks.boot`

,
`GerberGreenImai`

, `lalonde`

```
# Replication of Dehejia and Wahba psid3 model
#
# Dehejia, Rajeev and Sadek Wahba. 1999.``Causal Effects in
# Non-Experimental Studies: Re-Evaluating the Evaluation of Training
# Programs.''Journal of the American Statistical Association 94 (448):
# 1053-1062.
data(lalonde)
#
# Estimate the propensity model
#
glm1 <- glm(treat~age + I(age^2) + educ + I(educ^2) + black +
hisp + married + nodegr + re74 + I(re74^2) + re75 + I(re75^2) +
u74 + u75, family=binomial, data=lalonde)
#
#save data objects
#
X <- glm1$fitted
Y <- lalonde$re78
Tr <- lalonde$treat
#
# one-to-one matching with replacement (the "M=1" option).
# Estimating the treatment effect on the treated (the "estimand" option defaults to ATT).
#
rr <- Match(Y=Y, Tr=Tr, X=X, M=1);
summary(rr)
# Let's check the covariate balance
# 'nboots' is set to small values in the interest of speed.
# Please increase to at least 500 each for publication quality p-values.
mb <- MatchBalance(treat~age + I(age^2) + educ + I(educ^2) + black +
hisp + married + nodegr + re74 + I(re74^2) + re75 + I(re75^2) +
u74 + u75, data=lalonde, match.out=rr, nboots=10)
```

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