Package 'Matching' reference manual

Title:	Multivariate and Propensity Score Matching with Balance Optimization
Description:	Provides functions for multivariate and propensity score matching and for finding optimal balance based on a genetic search algorithm. A variety of univariate and multivariate metrics to determine if balance has been obtained are also provided. For details, see the paper by Jasjeet Sekhon (2007, <doi:10.18637/jss.v042.i07>).
Authors:	Jasjeet Singh Sekhon [aut, cre], Theo Saarinen [aut]
Maintainer:	Jasjeet Singh Sekhon <[email protected]>
License:	GPL-3
Version:	4.10-15
Built:	2024-11-13 05:59:24 UTC
Source:	https://github.com/jasjeetsekhon/matching

Univariate Balance Tests

Description

This function provides a number of univariate balance metrics. Generally, users should call MatchBalance and not this function directly.

Usage

balanceUV(Tr, Co, weights = rep(1, length(Co)), exact = FALSE, ks=FALSE,
          nboots = 1000, paired=TRUE, match=FALSE,
          weights.Tr=rep(1,length(Tr)), weights.Co=rep(1,length(Co)),
          estimand="ATT")
balanceUV(Tr, Co, weights = rep(1, length(Co)), exact = FALSE, ks=FALSE,
          nboots = 1000, paired=TRUE, match=FALSE,
          weights.Tr=rep(1,length(Tr)), weights.Co=rep(1,length(Co)),
          estimand="ATT")

Arguments

`Tr`	A vector containing the treatment observations.
`Co`	A vector containing the control observations.
`weights`	A vector containing the observation specific weights. Only use this option when the treatment and control observations are paired (as they are after matching).
`exact`	A logical flag indicating if the exact Wilcoxon test should be used instead of the test with a correction. See `wilcox.test` for details.
`ks`	A logical flag for if the univariate bootstrap Kolmogorov-Smirnov (KS) test should be calculated. If the ks option is set to true, the univariate KS test is calculated for all non-dichotomous variables. The bootstrap KS test is consistent even for non-continuous variables. See `ks.boot` for more details.
`nboots`	The number of bootstrap samples to be run for the `ks` test. If zero, no bootstraps are done. Bootstrapping is highly recommended because the bootstrapped Kolmogorov-Smirnov test only provides correct coverage even for non-continuous covariates. At least 500 `nboots` (preferably 1000) are recommended for publication quality p-values.
`paired`	A flag for if the paired `t.test` should be used.
`match`	A flag for if the `Tr` and `Co` objects are the result of a call to `Match`.
`weights.Tr`	A vector of weights for the treated observations.
`weights.Co`	A vector of weights for the control observations.
`estimand`	This determines if the standardized mean difference returned by the `sdiff` object is standardized by the variance of the treatment observations (which is done if the estimand is either "ATE" or "ATT") or by the variance of the control observations (which is done if the estimand is "ATC").

Value

`sdiff`	This is the standardized difference between the treated and control units multiplied by 100. That is, 100 times the mean difference between treatment and control units divided by the standard deviation of the treatment observations alone if the estimand is either `ATT` or `ATE`. The variance of the control observations are used if the estimand is `ATC`.
`sdiff.pooled`	This is the standardized difference between the treated and control units multiplied by 100 using the pooled variance. That is, 100 times the mean difference between treatment and control units divided by the pooled standard deviation as in Rosenbaum and Rubin (1985).
`mean.Tr`	The mean of the treatment group.
`mean.Co`	The mean of the control group.
`var.Tr`	The variance of the treatment group.
`var.Co`	The variance of the control group.
`p.value`	The p-value from the two-sided weighted `t.test`.
`var.ratio`	var.Tr/var.Co.
`ks`	The object returned by `ks.boot`.
`tt`	The object returned by two-sided weighted `t.test`.
`qqsummary`	The return object from a call to `qqstats` with standardization—i.e., balance test based on the empirical CDF.
`qqsummary.raw`	The return object from a call to `qqstats` without standardization–i.e., balance tests based on the empirical QQ-plot which retain the scale of the variable.

Author(s)

Jasjeet S. Sekhon, UC Berkeley, [email protected], https://www.jsekhon.com.

References

Sekhon, Jasjeet S. 2011. "Multivariate and Propensity Score Matching Software with Automated Balance Optimization.” Journal of Statistical Software 42(7): 1-52. 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

Rosenbaum, Paul R. and Donald B. Rubin. 1985. “Constructing a Control Group Using Multivariate Matched Sampling Methods That Incorporate the Propensity Score.” The American Statistician 39:1 33-38.

Hollander, Myles and Douglas A. Wolfe. 1973. Nonparametric statistical inference. New York: John Wiley & Sons.

Examples

data(lalonde)
attach(lalonde)

foo  <- balanceUV(re75[treat==1],re75[treat!=1])
summary(foo)

data(lalonde)
attach(lalonde)

foo  <- balanceUV(re75[treat==1],re75[treat!=1])
summary(foo)

Genetic Matching

Description

This function finds optimal balance using multivariate matching where a genetic search algorithm determines the weight each covariate is given. Balance is determined by examining cumulative probability distribution functions of a variety of standardized statistics. By default, these statistics include t-tests and Kolmogorov-Smirnov tests. A variety of descriptive statistics based on empirical-QQ (eQQ) plots can also be used or any user provided measure of balance. The statistics are not used to conduct formal hypothesis tests, because no measure of balance is a monotonic function of bias and because balance should be maximized without limit. The object returned by GenMatch can be supplied to the Match function (via the Weight.matrix option) to obtain causal estimates. GenMatch uses genoud to perform the genetic search. Using the cluster option, one may use multiple computers, CPUs or cores to perform parallel computations.

Usage

GenMatch(Tr, X, BalanceMatrix=X, estimand="ATT", M=1, weights=NULL,
         pop.size = 100, max.generations=100,
         wait.generations=4, hard.generation.limit=FALSE,
         starting.values=rep(1,ncol(X)),
         fit.func="pvals",
         MemoryMatrix=TRUE,
         exact=NULL, caliper=NULL, replace=TRUE, ties=TRUE,
         CommonSupport=FALSE, nboots=0, ks=TRUE, verbose=FALSE,
         distance.tolerance=1e-05,
         tolerance=sqrt(.Machine$double.eps),
         min.weight=0, max.weight=1000,
         Domains=NULL, print.level=2,
         project.path=NULL,
         paired=TRUE, loss=1,
         data.type.integer=FALSE,
         restrict=NULL,
         cluster=FALSE, balance=TRUE, ...)
GenMatch(Tr, X, BalanceMatrix=X, estimand="ATT", M=1, weights=NULL,
         pop.size = 100, max.generations=100,
         wait.generations=4, hard.generation.limit=FALSE,
         starting.values=rep(1,ncol(X)),
         fit.func="pvals",
         MemoryMatrix=TRUE,
         exact=NULL, caliper=NULL, replace=TRUE, ties=TRUE,
         CommonSupport=FALSE, nboots=0, ks=TRUE, verbose=FALSE,
         distance.tolerance=1e-05,
         tolerance=sqrt(.Machine$double.eps),
         min.weight=0, max.weight=1000,
         Domains=NULL, print.level=2,
         project.path=NULL,
         paired=TRUE, loss=1,
         data.type.integer=FALSE,
         restrict=NULL,
         cluster=FALSE, balance=TRUE, ...)

Arguments

`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.
`BalanceMatrix`	A matrix containing the variables we wish to achieve balance on. This is by default equal to `X`, but it can in principle be a matrix which contains more or less variables than `X` or variables which are transformed in various ways. See the examples.
`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.
`weights`	A vector the same length as `Y` which provides observation specific weights.
`pop.size`	Population Size. This is the number of individuals `genoud` uses to solve the optimization problem. The theorems proving that genetic algorithms find good solutions are asymptotic in population size. Therefore, it is important that this value not be small. See `genoud` for more details.
`max.generations`	Maximum Generations. This is the maximum number of generations that `genoud` will run when optimizing. This is a soft limit. The maximum generation limit will be binding only if `hard.generation.limit` has been set equal to TRUE. Otherwise, `wait.generations` controls when optimization stops. See `genoud` for more details.
`wait.generations`	If there is no improvement in the objective function in this number of generations, optimization will stop. The other options controlling termination are `max.generations` and `hard.generation.limit`.
`hard.generation.limit`	This logical variable determines if the `max.generations` variable is a binding constraint. If `hard.generation.limit` is FALSE, then the algorithm may exceed the `max.generations` count if the objective function has improved within a given number of generations (determined by `wait.generations`).
`starting.values`	This vector's length is equal to the number of variables in `X`. This vector contains the starting weights each of the variables is given. The `starting.values` vector is a way for the user to insert one individual into the starting population. `genoud` will randomly create the other individuals. These values correspond to the diagonal of the `Weight.matrix` as described in detail in the `Match` function.
`fit.func`	The balance metric `GenMatch` should optimize. The user may choose from the following or provide a function: `pvals`: maximize the p.values from (paired) t-tests and Kolmogorov-Smirnov tests conducted for each column in `BalanceMatrix`. Lexical optimization is conducted—see the `loss` option for details. `qqmean.mean`: calculate the mean standardized difference in the eQQ plot for each variable. Minimize the mean of these differences across variables. `qqmean.max`: calculate the mean standardized difference in the eQQ plot for each variable. Minimize the maximum of these differences across variables. Lexical optimization is conducted. `qqmedian.mean`: calculate the median standardized difference in the eQQ plot for each variable. Minimize the median of these differences across variables. `qqmedian.max`: calculate the median standardized difference in the eQQ plot for each variable. Minimize the maximum of these differences across variables. Lexical optimization is conducted. `qqmax.mean`: calculate the maximum standardized difference in the eQQ plot for each variable. Minimize the mean of these differences across variables. `qqmax.max`: calculate the maximum standardized difference in the eQQ plot for each variable. Minimize the maximum of these differences across variables. Lexical optimization is conducted. Users may provide their own `fit.func`. The name of the user provided function should not be backquoted or quoted. This function needs to return a fit value that will be minimized, by lexical optimization if more than one fit value is returned. The function should expect two arguments. The first being the `matches` object returned by `GenMatch`—see below. And the second being a matrix which contains the variables to be balanced—i.e., the `BalanceMatrix` the user provided to `GenMatch`. For an example see https://www.jsekhon.com.
`MemoryMatrix`	This variable controls if `genoud` sets up a memory matrix. Such a matrix ensures that `genoud` will request the fitness evaluation of a given set of parameters only once. The variable may be TRUE or FALSE. If it is FALSE, `genoud` will be aggressive in conserving memory. The most significant negative implication of this variable being set to FALSE is that `genoud` will no longer maintain a memory matrix of all evaluated individuals. Therefore, `genoud` may request evaluations which it has previously requested. When the number variables in `X` is large, the memory matrix consumes a large amount of RAM. `genoud`'s memory matrix will require significantly less memory if the user sets `hard.generation.limit` equal to TRUE. Doing this is a good way of conserving memory while still making use of the memory matrix structure.
`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. Obviously, if `exact` matching is done using all of the covariates, one should not be using `GenMatch` unless the `distance.tolerance` has been set unusually high.
`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. The `ecaliper` object which is returned by `GenMatch` shows the enforced caliper on the scale of the `X` variables. 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 `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.
`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 object `matches` to reconstruct the matched dataset. Seriously, don't use this option; use the `caliper` option instead.
`nboots`	The number of bootstrap samples to be run for the `ks` test. By default this option is set to zero so no bootstraps are done. See `ks.boot` for additional details.
`ks`	A logical flag for if the univariate bootstrap Kolmogorov-Smirnov (KS) test should be calculated. If the ks option is set to true, the univariate KS test is calculated for all non-dichotomous variables. The bootstrap KS test is consistent even for non-continuous variables. By default, the bootstrap KS test is not used. To change this see the `nboots` option. If a given variable is dichotomous, a t-test is used even if the KS test is requested. See `ks.boot` for additional details.
`verbose`	A logical flag for whether details of each fitness evaluation should be printed. Verbose is set to FALSE if the `cluster` option is used.
`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.
`min.weight`	This is the minimum weight any variable may be given.
`max.weight`	This is the maximum weight any variable may be given.
`Domains`	This is a `ncol(X)` $\times 2$ matrix. The first column is the lower bound, and the second column is the upper bound for each variable over which `genoud` will search for weights. If the user does not provide this matrix, the bounds for each variable will be determined by the `min.weight` and `max.weight` options.
`print.level`	This option controls the level of printing. There are four possible levels: 0 (minimal printing), 1 (normal), 2 (detailed), and 3 (debug). If level 2 is selected, `GenMatch` will print details about the population at each generation, including the best individual found so far. If debug level printing is requested, details of the `genoud` population are printed in the "genoud.pro" file which is located in the temporary `R` directory returned by the `tempdir` function. See the `project.path` option for more details. Because `GenMatch` runs may take a long time, it is important for the user to receive feedback. Hence, print level 2 has been set as the default.
`project.path`	This is the path of the `genoud` project file. By default no file is produced unless `print.level=3`. In that case, `genoud` places its output in a file called "genoud.pro" located in the temporary directory provided by `tempdir`. If a file path is provided to the `project.path` option, a file will be created regardless of the `print.level`. The behavior of the project file, however, will depend on the `print.level` chosen. If the `print.level` variable is set to 1, then the project file is rewritten after each generation. Therefore, only the currently fully completed generation is included in the file. If the `print.level` variable is set to 2 or higher, then each new generation is simply appended to the project file. No project file is generated for `print.level=0`.
`paired`	A flag for whether the paired `t.test` should be used when determining balance.
`loss`	The loss function to be optimized. The default value, `1`, implies "lexical" optimization: all of the balance statistics will be sorted from the most discrepant to the least and weights will be picked which minimize the maximum discrepancy. If multiple sets of weights result in the same maximum discrepancy, then the second largest discrepancy is examined to choose the best weights. The processes continues iteratively until ties are broken. If the value of `2` is used, then only the maximum discrepancy is examined. This was the default behavior prior to version 1.0. The user may also pass in any function she desires. Note that the option 1 corresponds to the `sort` function and option 2 to the `min` function. Any user specified function should expect a vector of balance statistics ("p-values") and it should return either a vector of values (in which case "lexical" optimization will be done) or a scalar value (which will be maximized). Some possible alternative functions are `mean` or `median`.
`data.type.integer`	By default, floating-point weights are considered. If this option is set to `TRUE`, search will be done over integer weights. Note that before version 4.1, the default was to use integer weights.
`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 restriction 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))`
`cluster`	This can either be an object of the 'cluster' class returned by one of the `makeCluster` commands in the `parallel` package or a vector of machine names so that `GenMatch` can setup the cluster automatically. If it is the latter, the vector should look like: `c("localhost","musil","musil","deckard")`. This vector would create a cluster with four nodes: one on the localhost another on "deckard" and two on the machine named "musil". Two nodes on a given machine make sense if the machine has two or more chips/cores. `GenMatch` will setup a SOCK cluster by a call to `makePSOCKcluster`. This will require the user to type in her password for each node as the cluster is by default created via `ssh`. One can add on usernames to the machine name if it differs from the current shell: "username@musil". Other cluster types, such as PVM and MPI, which do not require passwords, can be created by directly calling `makeCluster`, and then passing the returned cluster object to `GenMatch`. For an example of how to manually setup up a cluster with a direct call to `makeCluster` see https://www.jsekhon.com. For an example of how to get around a firewall by ssh tunneling see: https://www.jsekhon.com.
`balance`	This logical flag controls if load balancing is done across the cluster. Load balancing can result in better cluster utilization; however, increased communication can reduce performance. This option is best used if each individual call to `Match` takes at least several minutes to calculate or if the nodes in the cluster vary significantly in their performance. If cluster==FALSE, this option has no effect.
`...`	Other options which are passed on to `genoud`.

Value

`value`	The fit values at the solution. By default, this is a vector of p-values sorted from the smallest to the largest. There will generally be twice as many p-values as there are variables in `BalanceMatrix`, unless there are dichotomous variables in this matrix. There is one p-value for each covariate in `BalanceMatrix` which is the result of a paired t-test and another p-value for each non-dichotomous variable in `BalanceMatrix` which is the result of a Kolmogorov-Smirnov test. Recall that these p-values cannot be interpreted as hypothesis tests. They are simply measures of balance.
`par`	A vector of the weights given to each variable in `X`.
`Weight.matrix`	A matrix whose diagonal corresponds to the weight given to each variable in `X`. This object corresponds to the `Weight.matrix` in the `Match` function.
`matches`	A matrix where the first column contains the row numbers of the treated observations in the matched dataset. The second column contains the row numbers of the control observations. And the third column contains the weight that each matched pair is given. These objects may not correspond respectively to the `index.treated`, `index.control` and `weights` objects which are returned by `Match` because they may be ordered in a different way. Therefore, end users should use the objects returned by `Match` because those are ordered in the way that users expect.
`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`.

Author(s)

Jasjeet S. Sekhon, UC Berkeley, [email protected], https://www.jsekhon.com.

References

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

Sekhon, Jasjeet Singh and Walter R. Mebane, Jr. 1998. "Genetic Optimization Using Derivatives: Theory and Application to Nonlinear Models.” Political Analysis, 7: 187-210. https://www.jsekhon.com

Examples

data(lalonde)
attach(lalonde)

#The covariates we want to match on
X = cbind(age, educ, black, hisp, married, nodegr, u74, u75, re75, re74)

#The covariates we want to obtain balance on
BalanceMat <- cbind(age, educ, black, hisp, married, nodegr, u74, u75, re75, re74,
                    I(re74*re75))

#
#Let's call GenMatch() to find the optimal weight to give each
#covariate in 'X' so as we have achieved balance on the covariates in
#'BalanceMat'. This is only an example so we want GenMatch to be quick
#so the population size has been set to be only 16 via the 'pop.size'
#option. This is *WAY* too small for actual problems.
#For details see https://www.jsekhon.com.
#
genout <- GenMatch(Tr=treat, X=X, BalanceMatrix=BalanceMat, estimand="ATE", M=1,
                   pop.size=16, max.generations=10, wait.generations=1)

#The outcome variable
Y=re78/1000

#
# Now that GenMatch() has found the optimal weights, let's estimate
# our causal effect of interest using those weights
#
mout <- Match(Y=Y, Tr=treat, X=X, estimand="ATE", Weight.matrix=genout)
summary(mout)

#                        
#Let's determine if balance has actually been obtained on the variables of interest
#                        
mb <- MatchBalance(treat~age +educ+black+ hisp+ married+ nodegr+ u74+ u75+
                   re75+ re74+ I(re74*re75),
                   match.out=mout, nboots=500)

# For more examples see: https://www.jsekhon.com.
data(lalonde)
attach(lalonde)

#The covariates we want to match on
X = cbind(age, educ, black, hisp, married, nodegr, u74, u75, re75, re74)

#The covariates we want to obtain balance on
BalanceMat <- cbind(age, educ, black, hisp, married, nodegr, u74, u75, re75, re74,
                    I(re74*re75))

#
#Let's call GenMatch() to find the optimal weight to give each
#covariate in 'X' so as we have achieved balance on the covariates in
#'BalanceMat'. This is only an example so we want GenMatch to be quick
#so the population size has been set to be only 16 via the 'pop.size'
#option. This is *WAY* too small for actual problems.
#For details see https://www.jsekhon.com.
#
genout <- GenMatch(Tr=treat, X=X, BalanceMatrix=BalanceMat, estimand="ATE", M=1,
                   pop.size=16, max.generations=10, wait.generations=1)

#The outcome variable
Y=re78/1000

#
# Now that GenMatch() has found the optimal weights, let's estimate
# our causal effect of interest using those weights
#
mout <- Match(Y=Y, Tr=treat, X=X, estimand="ATE", Weight.matrix=genout)
summary(mout)

#                        
#Let's determine if balance has actually been obtained on the variables of interest
#                        
mb <- MatchBalance(treat~age +educ+black+ hisp+ married+ nodegr+ u74+ u75+
                   re75+ re74+ I(re74*re75),
                   match.out=mout, nboots=500)

# For more examples see: https://www.jsekhon.com.

Gerber and Green Dataset used by Imai

Description

This is the dataset used by Imai (2005) to replicate and evaluate the field experiment done by Gerber and Green (2000). The accompanying demo replicates Imai's propensity score model which is then used to estimate the causal effect of get-out-the-vote telephone calls on turnout.

Usage

data(GerberGreenImai)data(GerberGreenImai)

Format

A data frame with 10829 observations on the following 26 variables.

PERSONS: Number persons in household
WARD: Ward of residence
QUESTION: Asked to commit to voting
MAILGRP: Sent mail
PHONEGRP: Phone batch #1
PERSNGRP: Personal contact attempted
APPEAL: Content of message
CONTACT: Personal contact occurred
MAILINGS: Number of mailings sent
AGE: Age of respondent
MAJORPTY: Democratic or Republican
VOTE96.0: Abstained in 1996
VOTE96.1: Voted in 1996
MAILCALL: Phone batch #2
VOTED98: Voted in 1998
PHNSCRPT: Script read to phone respondents
DIS.MC: Contacted by phone in batch #2
DIS.PHN: Contacted by phone in batch #1
PHN.C: Contacted by phone
PHNTRT1: Phone contact attempted (no blood or blood/civic)
PHNTRT2: Phone contact attempted (no blood)
PHN.C1: Contact occurred in phntrt1
PHN.C2: Contact occurred in phntrt2
NEW: New voter
phone: Contacted by phone
AGE2: Age squared

Details

The demo provided, entitled GerberGreenImai, uses Imai's propensity score model to estimate the causal effect of get-out-the-vote telephone calls on turnout. The propensity score model fails to balance age.

References

Gerber, Alan S. and Donald P. Green. 2000. “The Effects of Canvassing, Telephone Calls, and Direct Mail on Voter Turnout: A Field Experiment.” American Political Science Review 94: 653-663.

Gerber, Alan S. and Donald P. Green. 2005. “Correction to Gerber and Green (2000), replication of disputed findings, and reply to Imai (2005).” American Political Science Review 99: 301-313.

Imai, Kosuke. 2005. “Do Get-Out-The-Vote Calls Reduce Turnout? The Importance of Statistical Methods for Field Experiments.” American Political Science Review 99: 283-300.

Hansen, Ben B. Hansen and Jake Bowers. forthcoming. “Attributing Effects to a Cluster Randomized Get-Out-The-Vote Campaign.” Journal of the American Statistical Association.

Bootstrap Kolmogorov-Smirnov

Description

This function executes a bootstrap version of the univariate Kolmogorov-Smirnov test which provides correct coverage even when the distributions being compared are not entirely continuous. Ties are allowed with this test unlike the traditional Kolmogorov-Smirnov test.

Usage

ks.boot(Tr, Co, nboots=1000, alternative = c("two.sided","less","greater"),
        print.level=0)
ks.boot(Tr, Co, nboots=1000, alternative = c("two.sided","less","greater"),
        print.level=0)

Arguments

`Tr`	A vector containing the treatment observations.
`Co`	A vector containing the control observations.
`nboots`	The number of bootstraps to be performed. These are, in fact, really Monte Carlo simulations which are preformed in order to determine the proper p-value from the empiric.
`alternative`	indicates the alternative hypothesis and must be one of '"two.sided"' (default), '"less"', or '"greater"'. You can specify just the initial letter. See `ks.test` for details.
`print.level`	If this is greater than 1, then the simulation count is printed out while the simulations are being done.

Value

`ks.boot.pvalue`	The bootstrap p-value of the Kolmogorov-Smirnov test for the hypothesis that the probability densities for both the treated and control groups are the same.
`ks`	Return object from `ks.test`.
`nboots`	The number of bootstraps which were completed.

Author(s)

Jasjeet S. Sekhon, UC Berkeley, [email protected], https://www.jsekhon.com.

References

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

Abadie, Alberto. 2002. “Bootstrap Tests for Distributional Treatment Effects in Instrumental Variable Models.” Journal of the American Statistical Association, 97:457 (March) 284-292.

Examples

#
# 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 which defaults to 0).
#
rr  <- Match(Y=Y,Tr=Tr,X=X,M=1);
summary(rr)

#
# Do we have balance on 1975 income after matching?
#
ks  <- ks.boot(lalonde$re75[rr$index.treated], lalonde$re75[rr$index.control], nboots=500)
summary(ks)
#
# 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 which defaults to 0).
#
rr  <- Match(Y=Y,Tr=Tr,X=X,M=1);
summary(rr)

#
# Do we have balance on 1975 income after matching?
#
ks  <- ks.boot(lalonde$re75[rr$index.treated], lalonde$re75[rr$index.control], nboots=500)
summary(ks)

Lalonde Dataset

Description

Dataset used by Dehejia and Wahba (1999) to evaluate propensity score matching.

Usage

data(lalonde)data(lalonde)

Format

A data frame with 445 observations on the following 12 variables.

age: age in years.
educ: years of schooling.
black: indicator variable for blacks.
hisp: indicator variable for Hispanics.
married: indicator variable for martial status.
nodegr: indicator variable for high school diploma.
re74: real earnings in 1974.
re75: real earnings in 1975.
re78: real earnings in 1978.
u74: indicator variable for earnings in 1974 being zero.
u75: indicator variable for earnings in 1975 being zero.
treat: an indicator variable for treatment status.

Details

Two demos are provided which use this dataset. The first, DehejiaWahba, replicates one of the models from Dehejia and Wahba (1999). The second demo, AbadieImbens, replicates the models produced by Abadie and Imbens in their Matlab code. Many of these models are found to produce good balance for the Lalonde data.

References

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.

LaLonde, Robert. 1986. “Evaluating the Econometric Evaluations of Training Programs.” American Economic Review 76:604-620.

Multivariate and Propensity Score Matching Estimator for Causal Inference

Description

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.

Usage

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

Arguments

`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 `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.
`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.

Details

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).

Value

`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.

Author(s)

Jasjeet S. Sekhon, UC Berkeley, [email protected], https://www.jsekhon.com.

References

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

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.

Examples


# 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)
# 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)

Tests for Univariate and Multivariate Balance

Description

This function provides a variety of balance statistics useful for determining if balance exists in any unmatched dataset and in matched datasets produced by the Match function. Matching is performed by the Match function, and MatchBalance is used to determine if Match was successful in achieving balance on the observed covariates.

Usage

MatchBalance(formul, data = NULL, match.out = NULL, ks = TRUE,
             nboots=500, weights=NULL, digits=5, paired=TRUE, print.level=1)
MatchBalance(formul, data = NULL, match.out = NULL, ks = TRUE,
             nboots=500, weights=NULL, digits=5, paired=TRUE, print.level=1)

Arguments

`formul`	This formula does not estimate any model. The formula is simply an efficient way to use the R modeling language to list the variables we wish to obtain univariate balance statistics for. The dependent variable in the formula is usually the treatment indicator. One should include many functions of the observed covariates. Generally, one should request balance statistics on more higher-order terms and interactions than were used to conduct the matching itself.
`data`	A data frame which contains all of the variables in the formula. If a data frame is not provided, the variables are obtained via lexical scoping.
`match.out`	The output object from the `Match` function. If this output is included, `MatchBalance` will provide balance statistics for both before and after matching. Otherwise balance statistics will only be reported for the raw unmatched data.
`ks`	A logical flag for whether the univariate bootstrap Kolmogorov-Smirnov (KS) test should be calculated. If the ks option is set to true, the univariate KS test is calculated for all non-dichotomous variables. The bootstrap KS test is consistent even for non-continuous variables. See `ks.boot` for more details.
`weights`	An optional vector of observation specific weights.
`nboots`	The number of bootstrap samples to be run. If zero, no bootstraps are done. Bootstrapping is highly recommended because the bootstrapped Kolmogorov-Smirnov test provides correct coverage even when the distributions being compared are not continuous. At least 500 `nboots` (preferably 1000) are recommended for publication quality p-values.
`digits`	The number of significant digits that should be displayed.
`paired`	A flag for whether the paired `t.test` should be used after matching. Regardless of the value of this option, an unpaired `t.test` is done for the unmatched data because it is assumed that the unmatched data were not generated by a paired experiment.
`print.level`	The amount of printing to be done. If zero, there is no printing. If one, the results are summarized. If two, details of the computations are printed.

Details

This function can be used to determine if there is balance in the pre- and/or post-matching datasets. Difference of means between treatment and control groups are provided as well as a variety of summary statistics for the empirical CDF (eCDF) and empirical-QQ (eQQ) plot between the two groups. The eCDF results are the standardized mean, median and maximum differences in the empirical CDF. The eQQ results are summaries of the raw differences in the empirical-QQ plot.

Two univariate tests are also provided: the t-test and the bootstrap Kolmogorov-Smirnov (KS) test. These tests should not be treated as hypothesis tests in the usual fashion because we wish to maximize balance without limit. The bootstrap KS test is highly recommended (see the ks and nboots options) because the bootstrap KS is consistent even for non-continuous distributions. Before matching, the two sample t-test is used; after matching, the paired t-test is used.

Two multivariate tests are provided. The KS and Chi-Square null deviance tests. The KS test is to be preferred over the Chi-Square test because the Chi-Square test is not testing the relevant hypothesis. The null hypothesis for the KS test is equal balance in the estimated probabilities between treated and control. The null hypothesis for the Chi-Square test, however, is all of the parameters being insignificant; a comparison of residual versus null deviance. If the covariates being considered are discrete, this KS test is asymptotically nonparametric as long as the logit model does not produce zero parameter estimates.

NA's are handled by the na.action option. But it is highly recommended that NA's not simply be deleted, but one should check to make sure that missingness is balanced.

Value

`BeforeMatching`	A list containing the before matching univariate balance statistics. That is, a list containing the results of the `balanceUV` function applied to all of the covariates described in `formul`. Note that the univariate test results for all of the variables in `formul` are printed if `verbose > 0`.
`AfterMatching`	A list containing the after matching univariate balance statistics. That is, a list containing the results of the `balanceUV` function applied to all of the covariates described in `formul`. Note that the univariate test results for all of the variables in `formul` are printed if `verbose > 0`. This object is `NULL`, if no matched dataset was provided.
`BMsmallest.p.value`	The smallest p.value found across all of the before matching balance tests (including t-tests and KS-tests.
`BMsmallestVarName`	The name of the variable with the `BMsmallest.p.value` (a vector in case of ties).
`BMsmallestVarNumber`	The number of the variable with the `BMsmallest.p.value` (a vector in case of ties).
`AMsmallest.p.value`	The smallest p.value found across all of the after matching balance tests (including t-tests and KS-tests.
`AMsmallestVarName`	The name of the variable with the `AMsmallest.p.value` (a vector in case of ties).
`AMsmallestVarNumber`	The number of the variable with the `AMsmallest.p.value` (a vector in case of ties).

Author(s)

Jasjeet S. Sekhon, UC Berkeley, [email protected], https://www.jsekhon.com.

References

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

Abadie, Alberto. 2002. “Bootstrap Tests for Distributional Treatment Effects in Instrumental Variable Models.” Journal of the American Statistical Association, 97:457 (March) 284-292.

Hall, Peter. 1992. The Bootstrap and Edgeworth Expansion. New York: Springer-Verlag.

Wilcox, Rand R. 1997. Introduction to Robust Estimation. San Diego, CA: Academic Press.

William J. Conover (1971), Practical nonparametric statistics. New York: John Wiley & Sons. Pages 295-301 (one-sample "Kolmogorov" test), 309-314 (two-sample "Smirnov" test).

Shao, Jun and Dongsheng Tu. 1995. The Jackknife and Bootstrap. New York: Springer-Verlag.

Examples

#
# 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 which defaults to 0).
#
rr  <- Match(Y=Y,Tr=Tr,X=X,M=1);

#Let's summarize the output
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)
#
# 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 which defaults to 0).
#
rr  <- Match(Y=Y,Tr=Tr,X=X,M=1);

#Let's summarize the output
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)

Grouped Multivariate and Propensity Score Matching

Description

This function is a wrapper for the Match function which separates the matching problem into subgroups defined by a factor. This is equivalent to conducting exact matching on each level of a factor. Matches within each level are found as determined by the usual matching options. This function is much faster for large datasets than the Match function itself. For additional speed, consider doing matching without replacement—see the replace option. This function is more limited than the Match function. For example, Matchby cannot be used if the user wishes to provide observation specific weights.

Usage

Matchby(Y, Tr, X, by, estimand = "ATT", M = 1, ties=FALSE, replace=TRUE,
        exact = NULL, caliper = NULL, AI=FALSE, Var.calc=0,
        Weight = 1, Weight.matrix = NULL, distance.tolerance = 1e-05,
        tolerance = sqrt(.Machine$double.eps), print.level=1, version="Matchby", ...)
Matchby(Y, Tr, X, by, estimand = "ATT", M = 1, ties=FALSE, replace=TRUE,
        exact = NULL, caliper = NULL, AI=FALSE, Var.calc=0,
        Weight = 1, Weight.matrix = NULL, distance.tolerance = 1e-05,
        tolerance = sqrt(.Machine$double.eps), print.level=1, version="Matchby", ...)

Arguments

`Y`	A vector containing the outcome of interest. Missing values are not allowed.
`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.
`by`	A "factor" in the sense that `as.factor(by)` defines the grouping, or a list of such factors in which case their interaction is used for the grouping.
`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 (for all), 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.
`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 `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.
`replace`	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 is sorted. Thus, the match(es) for the first observation will be found first and then for the second etc. Matching without replacement will generally increase bias so it is not recommended. But if the dataset is large and there are many potential matches, setting `replace=false` often results in a large speedup and negligible or no bias. Ties are randomly broken when `replace==FALSE`—see the `ties` option for details.
`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 of `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 provide 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.
`AI`	A logical flag for if the Abadie-Imbens standard error should be calculated. It is computationally expensive to calculate with large datasets. `Matchby` can only calculate AI SEs for ATT. To calculate AI errors with other estimands, please use the `Match` function. See the `Var.calc` option if one does not want to assume homoscedasticity.
`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.
`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.
`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.
`print.level`	The level of printing. Set to '0' to turn off printing.
`version`	The version of the code to be used. The "Matchby" C/C++ version of the code is the fastest, and the end-user should not change this option.
`...`	Additional arguments passed on to `Match`.

Details

Matchby is much faster for large datasets than Match. But Matchby only implements a subset of the functionality of Match. For example, the restrict option cannot be used, Abadie-Imbens standard errors are not provided and bias adjustment cannot be requested. Matchby is a wrapper for the Match function which separates the matching problem into subgroups defined by a factor. This is the equivalent to doing exact matching on each factor, and the way in which matches are found within each factor is determined by the usual matching options.

Note that by default ties=FALSE although the default for the Match in GenMatch functions is TRUE. This is done because randomly breaking ties in large datasets often results in a great speedup. For additional speed, consider doing matching without replacement which is often much faster when the dataset is large—see the replace option.

There will be slight differences in the matches produced by Matchby and Match because of how the covariates are weighted. When the data is broken up into separate groups (via the by option), Mahalanobis distance and inverse variance will imply different weights than when the data is taken as whole.

Value

`est`	The estimated average causal effect.
`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 so that ties are taken into account.
`se`	The Abadie-Imbens standard error. This is only calculated if the `AI` option is `TRUE`. 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.
`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 `Matchby`. For example, the `X` matrix used by `Matchby` can be recovered by `rbind(X[index.treated,],X[index.control,])`.
`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 `Matchby`. For example, the `Y` matrix for the matched dataset can be recovered by `c(Y[index.treated],Y[index.control])`.
`weights`	The weights for each observation in the matched dataset.
`orig.nobs`	The original number of observations in the dataset.
`nobs`	The number of observations in the matched dataset.
`wnobs`	The number of weighted observations in the matched dataset.
`orig.treated.nobs`	The original number of treated observations.
`ndrops`	The number of matches which were dropped because there were not enough observations in a given group and because of caliper and exact matching.
`estimand`	The estimand which was estimated.
`version`	The version of `Match` which was used.

Author(s)

Jasjeet S. Sekhon, UC Berkeley, [email protected], https://www.jsekhon.com.

References

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

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.

Examples

#
# Match exactly by racial groups and then match using the propensity score within racial groups
#

data(lalonde)

#
# Estimate the Propensity Score
#
glm1  <- glm(treat~age + I(age^2) + educ + I(educ^2) +
             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) after exactly
# matching on race using the 'by' option.  Estimating the treatment
# effect on the treated (the "estimand" option defaults to ATT).
rr  <- Matchby(Y=Y, Tr=Tr, X=X, by=lalonde$black, 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)

#
# Match exactly by racial groups and then match using the propensity score within racial groups
#

data(lalonde)

#
# Estimate the Propensity Score
#
glm1  <- glm(treat~age + I(age^2) + educ + I(educ^2) +
             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) after exactly
# matching on race using the 'by' option.  Estimating the treatment
# effect on the treated (the "estimand" option defaults to ATT).
rr  <- Matchby(Y=Y, Tr=Tr, X=X, by=lalonde$black, 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)

QQ Summary Statistics

Description

This function calculates a set of summary statistics for the QQ plot of two samples of data. The summaries are useful for determining if the two samples are from the same distribution. If standardize==TRUE, the empirical CDF is used instead of the empirical-QQ plot. The later retains the scale of the variable.

Usage

qqstats(x, y, standardize=TRUE, summary.func)
qqstats(x, y, standardize=TRUE, summary.func)

Arguments

`x`	The first sample.
`y`	The second sample.
`standardize`	A logical flag for whether the statistics should be standardized by the empirical cumulative distribution functions of the two samples.
`summary.func`	A user provided function to summarize the difference between the two distributions. The function should expect a vector of the differences as an argument and return summary statistic. For example, the `quantile` function is a legal function to pass in.

Value

`meandiff`	The mean difference between the QQ plots of the two samples.
`mediandiff`	The median difference between the QQ plots of the two samples.
`maxdiff`	The maximum difference between the QQ plots of the two samples.
`summarydiff`	If the user provides a `summary.func`, the user requested summary difference is returned.
`summary.func`	If the user provides a `summary.func`, the function is returned.

Author(s)

Jasjeet S. Sekhon, UC Berkeley, [email protected], https://www.jsekhon.com.

References

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

Diamond, Alexis and Jasjeet S. Sekhon. Forthcoming. "Genetic Matching for Estimating Causal Effects: A General Multivariate Matching Method for Achieving Balance in Observational Studies.” Review of Economics and Statistics. https://www.jsekhon.com

Examples

#
# 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 which defaults to 0).
#
rr  <- Match(Y=Y,Tr=Tr,X=X,M=1);
summary(rr)

#
# Do we have balance on 1975 income after matching?
#
qqout  <- qqstats(lalonde$re75[rr$index.treated], lalonde$re75[rr$index.control])
print(qqout)
#
# 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 which defaults to 0).
#
rr  <- Match(Y=Y,Tr=Tr,X=X,M=1);
summary(rr)

#
# Do we have balance on 1975 income after matching?
#
qqout  <- qqstats(lalonde$re75[rr$index.treated], lalonde$re75[rr$index.control])
print(qqout)

Summarizing output from balanceUV

Description

summary method for class balanceUV

Usage

  ## S3 method for class 'balanceUV'
summary(object, ..., digits=5)
## S3 method for class 'balanceUV'
summary(object, ..., digits=5)

Arguments

`object`	An object of class "`balanceUV`", usually, a result of a call to `balanceUV`.
`digits`	The number of significant digits that should be displayed.
`...`	Other options for the generic summary function.

Author(s)

Jasjeet S. Sekhon, UC Berkeley, [email protected], https://www.jsekhon.com.

Summarizing output from ks.boot

Description

summary method for class ks.boot

Usage

  ## S3 method for class 'ks.boot'
summary(object, ..., digits=5)
## S3 method for class 'ks.boot'
summary(object, ..., digits=5)

Arguments

`object`	An object of class "`ks.boot`", usually, a result of a call to `ks.boot`.
`digits`	The number of significant digits that should be displayed.
`...`	Other options for the generic summary function.

Author(s)

Jasjeet S. Sekhon, UC Berkeley, [email protected], https://www.jsekhon.com.

Summarizing output from Match

Description

summary method for class Match

Usage

  ## S3 method for class 'Match'
summary(object, ... , full=FALSE, digits=5)
## S3 method for class 'Match'
summary(object, ... , full=FALSE, digits=5)

Arguments

`object`	An object of class "`Match`", usually, a result of a call to `Match`.
`full`	A flag for whether the unadjusted estimates and naive standard errors should also be summarized.
`digits`	The number of significant digits that should be displayed.
`...`	Other options for the generic summary function.

Author(s)

Jasjeet S. Sekhon, UC Berkeley, [email protected], https://www.jsekhon.com.

Summarizing output from Matchby

Description

summary method for class Matchby

Usage

  ## S3 method for class 'Matchby'
summary(object, ... , digits=5)
## S3 method for class 'Matchby'
summary(object, ... , digits=5)

Arguments

`object`	An object of class "`Matchby`", usually, a result of a call to `Matchby`.
`digits`	The number of significant digits that should be displayed.
`...`	Other options for the generic summary function.

Author(s)

Jasjeet S. Sekhon, UC Berkeley, [email protected], https://www.jsekhon.com.

Package 'Matching'

Help Index

Univariate Balance Tests

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Genetic Matching

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Gerber and Green Dataset used by Imai

Description

Usage

Format

Details

References

See Also

Bootstrap Kolmogorov-Smirnov

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Lalonde Dataset

Description

Usage

Format

Details

References

See Also

Multivariate and Propensity Score Matching Estimator for Causal Inference

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Tests for Univariate and Multivariate Balance

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Grouped Multivariate and Propensity Score Matching

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

QQ Summary Statistics

Description

Usage

Arguments

Value

Author(s)

References