Matching Algorithms for Causal Inference with Clustered Data

Description

Provides functions to perform matching algorithms for causal inference with clustered data, as described in B. Arpino and M. Cannas (2016) <doi:10.1002/sim.6880>. Pure within-cluster and preferential within-cluster matching are implemented. Both algorithms provide causal estimates with cluster-adjusted estimates of standard errors.

Details

Several strategies have been suggested for adapting propensity score matching to clustered data. Depending on researcher's belief about the strength of unobserved cluster level covariates it is possible to take into account clustering either in the estimation of the propensity score model (through the inclusion of fixed or random effects, e.g. Arpino and Mealli (2011)) and/or in the implementation of the matching algorithm (see, e.g. Rickles and Seltzer (2014); Arpino and Cannas (2016)). This package contains main function CMatch to adapt classic matching algorithms for causal inference to clustered data and a customized summary function to analyze the output. Depending on the type argument function CMatch calls either MatchW implementing a pure within-cluster matching or MatchPW implementing an approach which can be called "preferential" within-cluster matching. This approach first looks for matchable units within the same cluster and - if no match is found - continues the search in the remaining clusters. The functions also provide causal estimands with cluster-adjusted standard errors from fitting a multilevel model on matched data. CMatch returns an object of class ”CMatch” which can be be summarized and used as input of the CMatchBalance function to examine how much the procedure resulted in improved covariate balance. Although CMatch has been designed for dealing with clustered data, these algorithms can be used to force a perfect balance or to improve the balance of categorical variables, respectively. In this case, the "clusters" correspond to the levels of the categorical variable(s). When used for this purpouse the user should ignore the standard error (if provided). Note that Matchby from package Matching can be used for the same purpouse.

Author(s)

Massimo Cannas [aut, cre], Bruno Arpino [ctb], Elena Colicino [ctb]. A special thanks to Thomas W. Yee for his help in updating to version 2.1.

Maintainer: Massimo Cannas <massimo.cannas@unica.it>

References

Sekhon, Jasjeet S. (2011). Multivariate and Propensity Score Matching Software with Automated Balance Optimization. Journal of Statistical Software, 42(7): 1-52. http://www.jstatsoft.org/v42/i07/

Arpino, B., and Cannas, M. (2016). Propensity score matching with clustered data. An application to the estimation of the impact of caesarean section on the Apgar score. Statistics in Medicine, 35: 2074-2091. doi: 10.1002/sim.6880.

Rickles, J. H., and Seltzer, M. (2014). A Two-Stage Propensity Score Matching Strategy for Treatment Effect Estimation in a Multisite Observational Study. Journal of Educational and Behavioral Statistics, 39(6), 612-636. doi: 10.3102/1076998614559748

Arpino, B. and Mealli, F. (2011). The specification of the propensity score in multilevel observational studies. Computational Statistics & Data Analysis, 55(4), 1770-1780. doi: 10.1016/j.csda.2010.11.008

See Also

Match, MatchBalance

Examples

# a paper and pencil example with a few units

id  <- c(1,2,3,4,5, 6,7,8,9,10)
 x  <- c( 1,1,1.1,1.1,1.4, 2,1,1,1.3, 1.3 )
 t  <- c( 1,1,1,1,0, 0,0,0,0, 0 )
 g  <- c(1,1,2,2,1,1,2,2,2, 2 ) # two groups of four and six units
toy <- t(data.frame(id,g, t,x))

# reorder units by ascending group
 toyord <-toy[,order(g)] 
 x <-toyord["x",]
 t <-toyord["t",]
 g <- toyord["g",]

# pooled matching
pm <- Match(Y=NULL, Tr=t, X=x, caliper=2,ties=FALSE,replace=FALSE)
# quick look at matched dataset (matched pairs are vertically aligned)
pm$index.treated
pm$index.control


# within matching 
wm <- CMatch(type="within",Y=NULL, Tr=t, X=x, Group=g,caliper=2,ties=FALSE,replace=FALSE)
wm$index.treated
wm$index.control

# preferential-within matching
pwm <- CMatch(type="pwithin",Y=NULL, Tr=t, X=x, Group=g, caliper=2,ties=FALSE,replace=FALSE)
pwm$index.treated
pwm$index.control
 

Within and preferential-within cluster matching.

Description

This function implements multivariate and propensity score matching in clusters defined by the Group variable. It returns an object of class ”CMatch” which can be be summarized and used as input of the CMatchBalance function to examine how much the procedure resulted in improved covariate balance.

Usage

CMatch(type, Y = NULL, Tr, X, Group = NULL, estimand = "ATT", M = 1, 
exact = NULL, caliper = 0.25, weights = NULL, replace = TRUE, ties = TRUE, ...)

Arguments

Details

This function is meant to be a natural extension of the Match function to clustered data. It retains the main arguments of Match but it has additional output showing matching results cluster by cluster. It differs from wrapper Matchby in package Matching in the way standard errors are calculated and because the caliper is in standard deviation units of the covariates on the overall dataset (so the caliper is the same for all clusters). Moreover, observation weights are available.

Value

Note

The function returns an object of class CMatch. The CMatchBalance function can be used to examine the covariate balance before and after matching (see the examples below).

Author(s)

Massimo Cannas <massimo.cannas@unica.it>

References

Sekhon, Jasjeet S. 2011. Multivariate and Propensity Score Matching Software with Automated Balance Optimization. Journal of Statistical Software 42(7): 1-52. http://www.jstatsoft.org/v42/i07/

Arpino, B., and Cannas, M. (2016) Propensity score matching with clustered data. An application to the estimation of the impact of caesarean section on the Apgar score. Statistics in Medicine, 35: 2074–2091. doi: 10.1002/sim.6880.

See Also

See also Match, MatchBalance,cluster.vcov

Examples

data(schools)
	
# Kreft and De Leeuw, Introducing Multilevel Modeling, Sage (1988).   
# The data set is the subsample of NELS-88 data consisting of 10 handpicked schools
# from the 1003 schools in the full data set.
 
# Let us consider the following variables:

X<-schools$ses  # (socio economic status) 
Y<-schools$math #(mathematics score)
Tr<-ifelse(schools$homework > 1, 1 ,0)
Group<-schools$schid #(school ID)

# When Group is missing/NULL or there is only one group, CMatch returns 
# the output of the Match function (with a warning).

# Multivariate Matching on covariates in X 
# default parameters: one-to-one matching on X with replacement with a caliper of 0.25

### Matching within schools
 mw <- CMatch(type="within",Y=Y, Tr=Tr, X=X, Group=Group, caliper=0.1)
 
 # compare balance before and after matching
 bmw  <- CMatchBalance(Tr~X, data=schools, match.out = mw)
 
 # calculate proportion of matched observations
  (mw$orig.treated.nobs-mw$ndrops)/mw$orig.treated.nobs 
  
 # check number of drops by school
 mw$orig.dropped.nobs.by.group
 
 # examine output
 mw           # complete output                 
 summary(mw)  # basic output statistics
 
### Match preferentially within school 
# i.e. first match within schools
# then tries to match remaining units between schools
 mpw <- CMatch(type="pwithin",Y=schools$math, Tr=Tr, X=schools$ses, 
 Group=schools$schid, caliper=0.1)

# examine covariate balance
  bmpw<- CMatchBalance(Tr~ses,data=schools,match.out = mpw)

# proportion of matched observations
  (mpw$orig.treated.nobs-mpw$ndrops) / mpw$orig.treated.nobs 
# check drops by school
  mpw$orig.dropped.nobs.by.group.after.pref.within
# proportion of matched observations after match-within only
 (mpw$orig.treated.nobs-sum(mpw$orig.dropped.nobs.by.group.after.within)) / mpw$orig.treated.nobs

# see complete output
   mpw
# or use summary method for main results
   summary(mpw) 

#### Propensity score matching

# estimate the ps model

mod <- glm(Tr~ses+parented+public+sex+race+urban,
family=binomial(link="logit"),data=schools)
eps <- fitted(mod)

# eg 1: within school propensity score matching
psmw <- CMatch(type="within",Y=schools$math, Tr=Tr, X=eps, 
Group=schools$schid, caliper=0.1)

# eg 2: preferential within school propensity score matching
psmw <- CMatch(type="pwithin",Y=schools$math, Tr=Tr, X=eps, Group=schools$schid, caliper=0.1)

# eg 3: propensity score matching using ps estimated from a logit model with dummies for hospitals

mod <- glm(Tr ~ ses + parented + public + sex + race + urban 
+schid - 1,family=binomial(link="logit"),data=schools)
eps <- fitted(mod)

dpsm <- CMatch(type="within",Y=schools$math, Tr=Tr, X=eps, Group=NULL, caliper=0.1)
# this is equivalent to run Match with X=eps

# eg4: propensity score matching using ps estimated from multilevel logit model 
# (random intercept at the hospital level); see Arpino and Mealli

require(lme4)
mod <- glmer(Tr ~ ses + parented + public + sex + race + urban + (1 | schid),
family=binomial(link="logit"), data=schools)
eps <- fitted(mod)

mpsm <- CMatch(type="within",Y=schools$math, Tr=Tr, X=eps, Group=NULL, caliper=0.1)
# note: equivalent to run Match with X=eps

 

Analyze covariate balance before and after matching.

Description

Generic function for analyzing covariate balance. If match.out is NULL only balance statistics for the unmatched data are returned otherwise both before and after matching balance are given. The function is a wrapper calling MatchBalance, possibly after coercing the class of match.out. See MatchBalance for more detailed description.

Usage

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

Arguments

Details

The function is a wrapper of the MatchBalance function. If match.out is of class Match (or NULL) then it calls MatchBalance. If match.out is of classCMatch then it coerces the class to Match before calling MatchBalance. This function is meant to exploit MatchBalance for CMatch objects for which MatchBalance would not work.

Value

Balance statistics for the covariates specified in the right side of formula argument. Statistics are compared between the two groups specified by the binary variable in the left side of formula.

Author(s)

Massimo Cannas <massimo.cannas@unica.it> and a special thanks to Thomas W. Yee for his help.

References

Sekhon, Jasjeet S. 2011. Multivariate and Propensity Score Matching Software with Automated Balance Optimization. Journal of Statistical Software 42(7): 1-52. http://www.jstatsoft.org/v42/i07/

See Also

MatchBalance

Examples

data(schools)
 
# Kreft and De Leeuw, Introducing Multilevel Modeling, Sage (1988).   
# The data set is the subsample of NELS-88 data consisting of 10 handpicked schools
# from the 1003 schools in the full data set.
 
# Let us consider the following variables:

X<-schools$ses  # (socio economic status) 
Y<-schools$math #(mathematics score)
Tr<-ifelse(schools$homework > 1, 1 ,0)
Group<-schools$schid #(school ID)

# Multivariate Matching on covariates X 

### Matching within schools
 mw <- CMatch(type="within",Y=Y, Tr=Tr, X=X, Group=Group, caliper=0.1)
 
 # Balance statistics for X variables(s) before and after matching within schools. 
 CMatchBalance(Tr~X,data=schools,match.out = mw)
 
 
### Match preferentially within school 
# i.e. first match within schools
# then tries to match remaining units between schools

 mpw <- CMatch(type="pwithin",Y=schools$math, Tr=Tr, X=schools$ses, 
 Group=schools$schid, caliper=0.1)

# examine covariate balance of variable(s) X before and after preferential matching within schools
  CMatchBalance(Tr~X, data=schools, match.out = mpw)

Preferential Within-cluster Matching

Description

This function implements preferential within-cluster matching. In other words, units that do not match within clusters (as defined by the Group variable) can match between cluster in the second step.

Usage

MatchPW(Y = NULL, Tr, X, Group = NULL, estimand = "ATT", M = 1,
 exact = NULL, caliper = 0.25, replace = TRUE, ties = TRUE, weights = NULL, ...)

Arguments

Details

The function performs preferential within-cluster matching in the clusters defined by the variable Group. In the first phase matching within clusters is performed (see MatchW) and in the second the unmatched treated (or controls if estimand="ATC") are matched with all controls (treated) units. This can be helpful to avoid dropping many units in small clusters.

Value

Note

The function returns an object of class CMatch. The CMatchBalance function can be used to examine the covariate balance before and after matching. See the examples below.

Author(s)

Massimo Cannas <massimo.cannas@unica.it>

References

Sekhon, Jasjeet S. 2011. Multivariate and Propensity Score Matching Software with Automated Balance Optimization. Journal of Statistical Software, 42(7): 1-52. http://www.jstatsoft.org/v42/i07/

Arpino, B., and Cannas, M. (2016) Propensity score matching with clustered data. An application to the estimation of the impact of caesarean section on the Apgar score. Statistics in Medicine, 35: 2074-2091 doi: 10.1002/sim.6880.

See Also

See also Match, MatchBalance,cluster.vcov

Examples

data(schools)
	
# Kreft and De Leeuw, Introducing Multilevel Modeling, Sage (1988).   
# The data set is the subsample of NELS-88 data consisting of 10 handpicked schools 
# from the 1003 schools in the full data set.

X<-schools$ses  # (socio economic status) 
Y<-schools$math #(mathematics score)
Tr<-ifelse(schools$homework > 1, 1 ,0)
Group<-schools$schid #(school ID)
# Note that when Group is missing, NULL or there is only one Group, 
# MatchPW returns the same output of the Match function (with a warning).


# Matching math scores between group of students. X are confounders.


### Match preferentially within-school 
# first match students within schools
# then tries to match remaining students between schools
 mpw <- MatchPW(Y=schools$math, Tr=Tr, X=schools$ses, Group=schools$schid, caliper=0.1)


# examine covariate balance
  bmpw<- CMatchBalance(Tr~ses,data=schools,match.out=mpw)

# proportion of matched observations
  (mpw$orig.treated.nobs-mpw$ndrops) / mpw$orig.treated.nobs 
# check drops by school
  mpw$orig.ndrops.by.group  
  
# estimate the math score difference (default is ATT)
  mpw$estimand

# complete results
   mpw
# or use summary method for main results
   summary(mpw) 


#### Propensity score matching

# estimate the propensity score (eps)

mod <- glm(Tr~ses+parented+public+sex+race+urban,
family=binomial(link="logit"),data=schools)
eps <- fitted(mod)

# eg 1: preferential within-school propensity score matching
MatchPW(Y=schools$math, Tr=Tr, X=eps, Group=schools$schid, caliper=0.1)

# eg 2: standard propensity score matching using eps
# from a logit model with dummies for schools

mod <- glm(Tr ~ ses + parented + public + sex + race + urban 
+schid - 1,family=binomial(link="logit"),data=schools)
eps <- fitted(mod)

MatchPW(Y=schools$math, Tr=Tr, X=eps, caliper=0.1)

# eg3: standard propensity score matching using ps estimated from 
# multilevel logit model (random intercept at the school level)

require(lme4)
mod<-glmer(Tr ~ ses + parented + public + sex + race + urban + (1|schid),
family=binomial(link="logit"), data=schools)
eps <- fitted(mod)

MatchPW(Y=schools$math, Tr=Tr, X=eps, Group=NULL, caliper=0.1)

 

Within-cluster Matching

Description

This function implements multivariate and propensity score matching within clusters defined by the Group variable.

Usage

MatchW(Y = NULL, Tr, X, Group = NULL, estimand = "ATT", M = 1, 
exact = NULL, caliper = 0.25, weights = NULL, replace = TRUE, ties = TRUE, ...)

Arguments

Details

This function is meant to be a natural extension of the Match function to clustered data. It retains the main arguments of Match but it has additional output showing matching results cluster by cluster. It differs from wrapper Matchby in package Matching in the way standard errors are calculated and because the caliper is in standard deviation units of the covariates on the overall dataset (so the caliper is the same for all clusters). Moreover, observation weights are available.

Value

Note

The function returns an object of class CMatch. The CMatchBalance function can be used to examine the covariate balance before and after matching (see the examples below).

Author(s)

Massimo Cannas <massimo.cannas@unica.it>

References

Sekhon, Jasjeet S. 2011. Multivariate and Propensity Score Matching Software with Automated Balance Optimization. Journal of Statistical Software 42(7): 1-52. http://www.jstatsoft.org/v42/i07/

Arpino, B., and Cannas, M. (2016) Propensity score matching with clustered data. An application to the estimation of the impact of caesarean section on the Apgar score. Statistics in Medicine, 35: 2074–2091. doi: 10.1002/sim.6880.

See Also

See also See also Match, MatchBalance,cluster.vcov

Examples

data(schools)
	
# Kreft and De Leeuw, Introducing Multilevel Modeling, Sage (1988).   
# The data set is the subsample of NELS-88 data consisting of 10 handpicked schools 
# from the 1003 schools in the full data set.
 
# Let us consider the following variables:

X<-schools$ses 
Y<-schools$math
Tr<-ifelse(schools$homework>1,1,0)
Group<-schools$schid

# Note that when Group is missing / NULL or there is only one group the function MatchW returns
# the output of the Match function with a warning.

# Matching math scores between gropus of students. X are covariate(s) we wish to match on. 

### Matching within schools
 mw <- MatchW(Y=Y, Tr=Tr, X=X, Group=Group, caliper=0.1)
 
 # compare balance before and after matching
   CMatchBalance(Tr~X,data=schools,match.out=mw)
 
 # find proportion of matched observations
  (mw$orig.treated.nobs-mw$ndrops)/mw$orig.treated.nobs 
  
 # check number of drops by school
 mw$orig.ndrops.by.group 
 
 # estimate the math score difference (default is ATT)
  mw$estimand
 
 # examine output
 mw                   # complete results                 
 summary(mw)          # main results
 
 
#### Propensity score matching

# estimate the propensity score (ps) model

mod <- glm(Tr~ses+parented+public+sex+race+urban,
family=binomial(link="logit"),data=schools)
eps <- fitted(mod)

# eg 1: within-school propensity score matching
psmw <- MatchW(Y=schools$math, Tr=Tr, X=eps, Group=schools$schid, caliper=0.1)

# We can use other strategies for controlling unobserved cluster covariates
# by using different specifications of ps:

# eg 2: standard propensity score matching using ps estimated
# from a logit model with dummies for schools

mod <- glm(Tr ~ ses + parented + public + sex + race + urban 
+schid - 1,family=binomial(link="logit"),data=schools)
eps <- fitted(mod)



dpsm <- MatchW(Y=schools$math, Tr=Tr, X=eps, caliper=0.1)
# this is equivalent to run Match with X=eps

# eg3: standard propensity score matching using ps estimated from 
# multilevel logit model (random intercept at the school level)

require(lme4)
mod<-glmer(Tr ~ ses + parented + public + sex + race + urban + (1|schid),
family=binomial(link="logit"), data=schools)
eps <- fitted(mod)

mpsm<-MatchW(Y=schools$math, Tr=Tr, X=eps, Group=NULL, caliper=0.1)
# this is equivalent to run Match with X=eps
 

Schools data set (NELS-88)

Description

Data set used by Kreft and De Leeuw in their book Introducing Multilevel Modeling, Sage (1988) to analyse the relationship between math score and time spent by students to do math homework. The data set is a subsample of NELS-88 data consisting of 10 handpicked schools from the 1003 schools in the full data set. Students are nested within schools and information is available both at the school and student level.

Usage

data("schools")

Format

A data frame with 260 observations on the following 19 variables.

schid

School ID: a numeric vector identyfing each school.

stuid

The student ID.

ses

Socioeconomic status.

meanses

Mean ses for the school.

homework

The number of hours spent weekly doing homeworks.

white

A dummy for white race (=1) versus non-white (=0).

parented

Parents highest education level.

public

Public school: 1=public, 0=non public.

ratio

Student-teacher ratio.

percmin

Percent minority in school.

math

Math score

sex

Sex: 1=male, 2=female.

race

Race of student, 1=asian, 2=Hispanic, 3=Black, 4=White, 5=Native American.

sctype

Type of school: 1=public, 2=catholic, 3= Private other religion, 4=Private non-r.

cstr

Classroom environment structure: ordinal from 1=not accurate to 5=very much accurate.

scsize

School size: ordinal from 1=[1,199) to 7=[1200+).

urban

Urbanicity: 1=Urban, 2=Suburban, 3=Rural.

region

Geographic region of the school: NE=1,NC=2,South=3,West=4.

schnum

Standardized school ID.

Details

The data set is used in the example section to illustrate the use of functions MatchW and MatchPW.

Source

Ita G G Kreft, Jan De Leeuw 1988. Introducing Multilevel Modeling, Sage National Education Longitudinal Study of 1988 (NELS:88): https://nces.ed.gov/surveys/nels88/

See Also

See also MatchW, MatchPW

Examples

data(schools)
 
# Kreft and De Leeuw, Introducing Multilevel Modeling, Sage (1988).   
# The data set is the subsample of NELS-88 data consisting of 10 handpicked schools
# from the 1003 schools in the full data set.
 
# To study the effect of the homeworks on the outcome math score, conditional on
# confounder(s) X and unobserved school features, we can define the following variables:

X<-schools$ses 
# or define a vector for more than one confounder
X<-as.matrix(schools[,c("ses","white","public")])
Y<-schools$math
Tr<-ifelse(schools$homework>1,1,0)
Group<-schools$schid

Summarizing output from MatchW and MatchPW functions

Description

Summary method for MatchW and MatchPW

Usage

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

Arguments

Details

If Group contains only one value the output is the same of the summary method of package Matching. Otherwise the output shows also the distribution of treated, control and possibly drop units, by group.

Value

A list giving a summary of the output from a "CMatch" object. The list includes the size of the original and the matched dataset, the number of treated and control observations in each group and the estimate (if Y is not NULL).

Note

Naive standard errors are not available when there is more than one group so the full parameter is ineffective in that case.

Author(s)

Massimo Cannas <massimo.cannas@unica.it>

References

Sekhon, Jasjeet S. 2011. Multivariate and Propensity Score Matching Software with Automated Balance Optimization. Journal of Statistical Software 42(7): 1-52. http://www.jstatsoft.org/v42/i07/

Arpino, B., and Cannas, M. (2016) Propensity score matching with clustered data. An application to the estimation of the impact of caesarean section on the Apgar score. Statistics in Medicine, 35: 2074–2091. doi: 10.1002/sim.6880.

See Also

See also CMatch, CMatchBalance