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")
The estimated average causal effect.
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.
The estimated average causal effect without any
BiasAdjust. If BiasAdjust is not requested, this is the
same as est.
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.
The conditional standard error. The practitioner should not generally use this.
A list which contains the matched datasets produced by
Match. Three datasets are included in this list: Y,
Tr and X.
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.
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.
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.
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.
The original number of observations in the dataset.
The original number of weighted observations in the dataset.
The original number of treated observations (unweighted).
The number of observations in the matched dataset.
The number of weighted observations in the matched dataset.
The caliper which was used.
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.
The value of the exact function argument.
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.
The number of matches which were dropped either because of caliper or exact matching.
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.
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.
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.
A matrix containing the covariates for which we wish to make bias adjustments.
A matrix containing the covariates for which the variance
of the causal effect may vary. Also see the Var.calc option,
which takes precedence.
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.
A scalar for the number of matches which should be
found. The default is one-to-one matching. Also see the ties
option.
A logical scalar for whether regression adjustment
should be used. See the Z matrix.
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.
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.
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.
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
ties=FALSE often results in a large speedup. Whether two
potential matches are close enough to be considered tied, is
controlled by the distance.tolerance
option.
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.
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.
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.
A vector the same length as Y which
provides observation specific weights.
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.
A logical flag for whether the population or sample variance is returned.
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
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.
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))
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.
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, https://www.jsekhon.com.
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. https://www.jsekhon.com
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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