## 31.1 Selection on Observables

### 31.1.1 MatchIt

Procedure typically involves (proposed by Noah Freifer using MatchIt)

1. planning
2. matching
3. checking (balance)
4. estimating the treatment effect
library(MatchIt)
data("lalonde")

examine treat on re78

1. Planning
• select type of effect to be estimated (e.g., mediation effect, conditional effect, marginal effect)

• select the target population

• select variables to match/balance

1. Check Initial Imbalance
# No matching; constructing a pre-match matchit object
m.out0 <- matchit(
formula(treat ~ age + educ + race
+ married + nodegree + re74 + re75, env = lalonde),
data = data.frame(lalonde),
method = NULL,
# assess balance before matching
distance = "glm" # logistic regression
)

# Checking balance prior to matching
summary(m.out0)
1. Matching
# 1:1 NN PS matching w/o replacement
m.out1 <- matchit(treat ~ age + educ,
data = lalonde,
method = "nearest",
distance = "glm")
m.out1
#> A matchit object
#>  - method: 1:1 nearest neighbor matching without replacement
#>  - distance: Propensity score
#>              - estimated with logistic regression
#>  - number of obs.: 614 (original), 370 (matched)
#>  - target estimand: ATT
#>  - covariates: age, educ
1. Check balance

Sometimes you have to make trade-off between balance and sample size.

# Checking balance after NN matching
summary(m.out1, un = FALSE)
#>
#> Call:
#> matchit(formula = treat ~ age + educ, data = lalonde, method = "nearest",
#>     distance = "glm")
#>
#> Summary of Balance for Matched Data:
#>          Means Treated Means Control Std. Mean Diff. Var. Ratio eCDF Mean
#> distance        0.3080        0.3077          0.0094     0.9963    0.0033
#> age            25.8162       25.8649         -0.0068     1.0300    0.0050
#> educ           10.3459       10.2865          0.0296     0.5886    0.0253
#>          eCDF Max Std. Pair Dist.
#> distance   0.0432          0.0146
#> age        0.0162          0.0597
#> educ       0.1189          0.8146
#>
#> Sample Sizes:
#>           Control Treated
#> All           429     185
#> Matched       185     185
#> Unmatched     244       0

# examine visually
plot(m.out1, type = "jitter", interactive = FALSE)


plot(
m.out1,
type = "qq",
interactive = FALSE,
which.xs = c("age")
)

Try Full Match (i.e., every treated matches with one control, and every control with one treated).

# Full matching on a probit PS
m.out2 <- matchit(treat ~ age + educ,
data = lalonde,
method = "full",
distance = "glm",
m.out2
#> A matchit object
#>  - method: Optimal full matching
#>  - distance: Propensity score
#>              - estimated with probit regression
#>  - number of obs.: 614 (original), 614 (matched)
#>  - target estimand: ATT
#>  - covariates: age, educ

Checking balance again

# Checking balance after full matching
summary(m.out2, un = FALSE)
#>
#> Call:
#> matchit(formula = treat ~ age + educ, data = lalonde, method = "full",
#>     distance = "glm", link = "probit")
#>
#> Summary of Balance for Matched Data:
#>          Means Treated Means Control Std. Mean Diff. Var. Ratio eCDF Mean
#> distance        0.3082        0.3081          0.0023     0.9815    0.0028
#> age            25.8162       25.8035          0.0018     0.9825    0.0062
#> educ           10.3459       10.2315          0.0569     0.4390    0.0481
#>          eCDF Max Std. Pair Dist.
#> distance   0.0270          0.0382
#> age        0.0249          0.1110
#> educ       0.1300          0.9805
#>
#> Sample Sizes:
#>               Control Treated
#> All            429.       185
#> Matched (ESS)  145.23     185
#> Matched        429.       185
#> Unmatched        0.         0

plot(summary(m.out2))

Exact Matching

# Full matching on a probit PS
m.out3 <-
matchit(
treat ~ age + educ,
data = lalonde,
method = "exact"
)
m.out3
#> A matchit object
#>  - method: Exact matching
#>  - number of obs.: 614 (original), 332 (matched)
#>  - target estimand: ATT
#>  - covariates: age, educ

Subclassfication

m.out4 <- matchit(
treat ~ age + educ,
data = lalonde,
method = "subclass"
)
m.out4
#> A matchit object
#>  - method: Subclassification (6 subclasses)
#>  - distance: Propensity score
#>              - estimated with logistic regression
#>  - number of obs.: 614 (original), 614 (matched)
#>  - target estimand: ATT
#>  - covariates: age, educ

# Or you can use in conjunction with "nearest"
m.out4 <- matchit(
treat ~ age + educ,
data = lalonde,
method = "nearest",
option = "subclass"
)
m.out4
#> A matchit object
#>  - method: 1:1 nearest neighbor matching without replacement
#>  - distance: Propensity score
#>              - estimated with logistic regression
#>  - number of obs.: 614 (original), 370 (matched)
#>  - target estimand: ATT
#>  - covariates: age, educ

Optimal Matching

m.out5 <- matchit(
treat ~ age + educ,
data = lalonde,
method = "optimal",
ratio = 2
)
m.out5
#> A matchit object
#>  - method: 2:1 optimal pair matching
#>  - distance: Propensity score
#>              - estimated with logistic regression
#>  - number of obs.: 614 (original), 555 (matched)
#>  - target estimand: ATT
#>  - covariates: age, educ

Genetic Matching

m.out6 <- matchit(
treat ~ age + educ,
data = lalonde,
method = "genetic"
)
m.out6
#> A matchit object
#>  - method: 1:1 genetic matching without replacement
#>  - distance: Propensity score
#>              - estimated with logistic regression
#>  - number of obs.: 614 (original), 370 (matched)
#>  - target estimand: ATT
#>  - covariates: age, educ
1. Estimating the Treatment Effect
# get matched data
m.data1 <- match.data(m.out1)

#>      treat age educ   race married nodegree re74 re75       re78  distance
#> NSW1     1  37   11  black       1        1    0    0  9930.0460 0.2536942
#> NSW2     1  22    9 hispan       0        1    0    0  3595.8940 0.3245468
#> NSW3     1  30   12  black       0        0    0    0 24909.4500 0.2881139
#> NSW4     1  27   11  black       0        1    0    0  7506.1460 0.3016672
#> NSW5     1  33    8  black       0        1    0    0   289.7899 0.2683025
#> NSW6     1  22    9  black       0        1    0    0  4056.4940 0.3245468
#>      weights subclass
#> NSW1       1        1
#> NSW2       1       98
#> NSW3       1      109
#> NSW4       1      120
#> NSW5       1      131
#> NSW6       1      142
library("lmtest") #coeftest
library("sandwich") #vcovCL

# imbalance matched dataset
fit1 <- lm(re78 ~ treat + age + educ ,
data = m.data1,
weights = weights)

coeftest(fit1, vcov. = vcovCL, cluster = ~subclass)
#>
#> t test of coefficients:
#>
#>              Estimate Std. Error t value Pr(>|t|)
#> (Intercept)  -174.902   2445.013 -0.0715 0.943012
#> treat       -1139.085    780.399 -1.4596 0.145253
#> age           153.133     55.317  2.7683 0.005922 **
#> educ          358.577    163.860  2.1883 0.029278 *
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

treat coefficient = estimated ATT

# balance matched dataset
m.data2 <- match.data(m.out2)

fit2 <- lm(re78 ~ treat + age + educ ,
data = m.data2, weights = weights)

coeftest(fit2, vcov. = vcovCL, cluster = ~subclass)
#>
#> t test of coefficients:
#>
#>             Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 2151.952   3141.152  0.6851  0.49355
#> treat       -725.184    703.297 -1.0311  0.30289
#> age          120.260     53.933  2.2298  0.02612 *
#> educ         175.693    241.694  0.7269  0.46755
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

When reporting, remember to mention

1. the matching specification (method, and additional options)
2. the distance measure (e.g., propensity score)
3. other methods, and rationale for the final chosen method.
4. balance statistics of the matched dataset.
5. number of matched, unmatched, discarded
6. estimation method for treatment effect.

### 31.1.2 designmatch

This package includes

• distmatch optimal distance matching

• bmatch optimal bipartile matching

• cardmatch optimal cardinality matching

• profmatch optimal profile matching

• nmatch optimal nonbipartile matching

library(designmatch)

### 31.1.3 MatchingFrontier

As mentioned in MatchIt, you have to make trade-off (also known as bias-variance trade-off) between balance and sample size. An automated procedure to optimize this trade-off is implemented in MatchingFrontier , which solves this joint optimization problem.

Following MatchingFrontier guide

# library(devtools)
# install_github('ChristopherLucas/MatchingFrontier')
library(MatchingFrontier)
data("lalonde")
# choose var to match on
match.on <-
colnames(lalonde)[!(colnames(lalonde) %in% c('re78', 'treat'))]
match.on

# Mahanlanobis frontier (default)
mahal.frontier <-
makeFrontier(
dataset = lalonde,
treatment = "treat",
match.on = match.on
)
mahal.frontier

# L1 frontier
L1.frontier <-
makeFrontier(
dataset = lalonde,
treatment = 'treat',
match.on = match.on,
QOI = 'SATT',
metric = 'L1',
ratio = 'fixed'
)
L1.frontier

# estimate effects along the frontier

# Set base form
my.form <-
as.formula(re78 ~ treat + age + black + education
+ hispanic + married + nodegree + re74 + re75)

# Estimate effects for the mahalanobis frontier
mahal.estimates <-
estimateEffects(
mahal.frontier,
're78 ~ treat',
mod.dependence.formula = my.form,
continuous.vars = c('age', 'education', 're74', 're75'),
prop.estimated = .1,
means.as.cutpoints = TRUE
)

# Estimate effects for the L1 frontier
L1.estimates <-
estimateEffects(
L1.frontier,
're78 ~ treat',
mod.dependence.formula = my.form,
continuous.vars = c('age', 'education', 're74', 're75'),
prop.estimated = .1,
means.as.cutpoints = TRUE
)

# Plot covariates means
# plotPrunedMeans()

# Plot estimates (deprecated)
# plotEstimates(
#     L1.estimates,
#     ylim = c(-10000, 3000),
#     cex.lab = 1.4,
#     cex.axis = 1.4,
#     panel.first = grid(NULL, NULL, lwd = 2,)
# )

# Plot estimates
plotMeans(L1.frontier)

# parallel plot
parallelPlot(
L1.frontier,
N = 400,
variables = c('age', 're74', 're75', 'black'),
treated.col = 'blue',
control.col = 'gray'
)

# export matched dataset
# take 400 units
matched.data <- generateDataset(L1.frontier, N = 400) 

### 31.1.4 Propensity Scores

Even though I mention the propensity scores matching method here, it is no longer recommended to use such method in research and publication because it increases

• imbalance

• inefficiency

• model dependence: small changes in the model specification lead to big changes in model results

• bias

note

• The initial estimation of the propensity score influences the large sample distribution of the estimators.

• Adjustments are made to the large sample variances of these estimators for both ATE and ATT.

• The adjustment for the ATE estimator is either negative or zero, indicating greater efficiency when matching on an estimated propensity score versus the true score in large samples.

• For the ATET estimator, the sign of the adjustment depends on the data generating process. Neglecting the estimation error in the propensity score can lead to inaccurate confidence intervals for the ATT estimator, making them either too large or too small.

PSM tries to accomplish complete randomization while other methods try to achieve fully blocked. Hence, you probably better off use any other methods.

Propensity is “the probability of receiving the treatment given the observed covariates.”

Equivalently, it can to understood as the probability of being treated.

$e_i (X_i) = P(T_i = 1 | X_i)$

Estimation using

• logistic regression

• Non parametric methods:

• boosted CART

• generalized boosted models (gbm)

Steps by Gary King’s slides

• reduce k elements of X to scalar

• $$\pi_i \equiv P(T_i = 1|X) = \frac{1}{1+e^{X_i \beta}}$$

• Distance ($$X_c, X_t$$) = $$|\pi_c - \pi_t|$$

• match each treated unit to the nearest control unit

• control units: not reused; pruned if unused

• prune matches if distances > caliper

In the best case scenario, you randomly prune, which increases imbalance

Other methods dominate because they try to match exactly hence

• $$X_c = X_t \to \pi_c = \pi_t$$ (exact match leads to equal propensity scores) but

• $$\pi_c = \pi_t \nrightarrow X_c = X_t$$ (equal propensity scores do not necessarily lead to exact match)

Notes:

• Do not include/control for irrelevant covariates because it leads your PSM to be more random, hence more imbalance

• Do not include for instrumental variable in the predictor set of a propensity score matching estimator. More generally, using variables that do not control for potential confounders, even if they are predictive of the treatment, can result in biased estimates

What you left with after pruning is more important than what you start with then throw out.

Diagnostics:

• balance of the covariates

• no need to concern about collinearity

• can’t use c-stat or stepwise because those model fit stat do not apply

### 31.1.5 Mahalanobis Distance

Approximates fully blocked experiment

Distance $$(X_c,X_t)$$ = $$\sqrt{(X_c - X_t)'S^{-1}(X_c - X_t)}$$

where $$S^{-1}$$ standardize the distance

In application we use Euclidean distance.

Prune unused control units, and prune matches if distance > caliper

### 31.1.6 Coarsened Exact Matching

Steps from Gray King’s slides International Methods Colloquium talk 2015

• Temporarily coarsen $$X$$

• Apply exact matching to the coarsened $$X, C(X)$$

• sort observation into strata, each with unique values of $$C(X)$$

• prune stratum with 0 treated or 0 control units

• Pass on original (uncoarsened) units except those pruned

Properties:

• Monotonic imbalance bounding (MIB) matching method

• maximum imbalance between the treated and control chosen ex ante
• meets congruence principle

• robust to measurement error

• can be implemented with multiple imputation

• works well for multi-category treatments

Assumptions:

• Ignorability (i.e., no omitted variable bias)

More detail in

Example by package’s authors

library(cem)
data(LeLonde)

Le <- data.frame(na.omit(LeLonde)) # remove missing data
# treated and control groups
tr <- which(Le$treated==1) ct <- which(Le$treated==0)
ntr <- length(tr)
nct <- length(ct)

# unadjusted, biased difference in means
mean(Le$re78[tr]) - mean(Le$re78[ct])
#> [1] 759.0479

# pre-treatment covariates
vars <-
c(
"age",
"education",
"black",
"married",
"nodegree",
"re74",
"re75",
"hispanic",
"u74",
"u75",
"q1"
)

# overall imbalance statistics
imbalance(group=Le$treated, data=Le[vars]) # L1 = 0.902 #> #> Multivariate Imbalance Measure: L1=0.902 #> Percentage of local common support: LCS=5.8% #> #> Univariate Imbalance Measures: #> #> statistic type L1 min 25% 50% 75% #> age -0.252373042 (diff) 5.102041e-03 0 0 0.0000 -1.0000 #> education 0.153634710 (diff) 8.463851e-02 1 0 1.0000 1.0000 #> black -0.010322734 (diff) 1.032273e-02 0 0 0.0000 0.0000 #> married -0.009551495 (diff) 9.551495e-03 0 0 0.0000 0.0000 #> nodegree -0.081217371 (diff) 8.121737e-02 0 -1 0.0000 0.0000 #> re74 -18.160446880 (diff) 5.551115e-17 0 0 284.0715 806.3452 #> re75 101.501761679 (diff) 5.551115e-17 0 0 485.6310 1238.4114 #> hispanic -0.010144756 (diff) 1.014476e-02 0 0 0.0000 0.0000 #> u74 -0.045582186 (diff) 4.558219e-02 0 0 0.0000 0.0000 #> u75 -0.065555292 (diff) 6.555529e-02 0 0 0.0000 0.0000 #> q1 7.494021189 (Chi2) 1.067078e-01 NA NA NA NA #> max #> age -6.0000 #> education 1.0000 #> black 0.0000 #> married 0.0000 #> nodegree 0.0000 #> re74 -2139.0195 #> re75 490.3945 #> hispanic 0.0000 #> u74 0.0000 #> u75 0.0000 #> q1 NA # drop other variables that are not pre - treatmentt matching variables todrop <- c("treated", "re78") imbalance(group=Le$treated, data=Le, drop=todrop)
#>
#> Multivariate Imbalance Measure: L1=0.902
#> Percentage of local common support: LCS=5.8%
#>
#> Univariate Imbalance Measures:
#>
#>               statistic   type           L1 min 25%      50%       75%
#> age        -0.252373042 (diff) 5.102041e-03   0   0   0.0000   -1.0000
#> education   0.153634710 (diff) 8.463851e-02   1   0   1.0000    1.0000
#> black      -0.010322734 (diff) 1.032273e-02   0   0   0.0000    0.0000
#> married    -0.009551495 (diff) 9.551495e-03   0   0   0.0000    0.0000
#> nodegree   -0.081217371 (diff) 8.121737e-02   0  -1   0.0000    0.0000
#> re74      -18.160446880 (diff) 5.551115e-17   0   0 284.0715  806.3452
#> re75      101.501761679 (diff) 5.551115e-17   0   0 485.6310 1238.4114
#> hispanic   -0.010144756 (diff) 1.014476e-02   0   0   0.0000    0.0000
#> u74        -0.045582186 (diff) 4.558219e-02   0   0   0.0000    0.0000
#> u75        -0.065555292 (diff) 6.555529e-02   0   0   0.0000    0.0000
#> q1          7.494021189 (Chi2) 1.067078e-01  NA  NA       NA        NA
#>                  max
#> age          -6.0000
#> education     1.0000
#> black         0.0000
#> married       0.0000
#> nodegree      0.0000
#> re74      -2139.0195
#> re75        490.3945
#> hispanic      0.0000
#> u74           0.0000
#> u75           0.0000
#> q1                NA

automated coarsening

mat <-
cem(
treatment = "treated",
data = Le,
drop = "re78",
keep.all = TRUE
)
#>
#> Using 'treated'='1' as baseline group
mat
#>            G0  G1
#> All       392 258
#> Matched    95  84
#> Unmatched 297 174

# mat$w coarsening by explicit user choice # categorial variables levels(Le$q1) # grouping option
#> [1] "agree"             "disagree"          "neutral"
#> [4] "no opinion"        "strongly agree"    "strongly disagree"
q1.grp <-
list(
c("strongly agree", "agree"),
c("neutral", "no opinion"),
c("strongly disagree", "disagree")
) # if you want ordered categories

# continuous variables
table(Le\$education)
#>
#>   3   4   5   6   7   8   9  10  11  12  13  14  15
#>   1   5   4   6  12  55 106 146 173 113  19   9   1
educut <- c(0, 6.5, 8.5, 12.5, 17)  # use cutpoints

mat1 <-
cem(
treatment = "treated",
data = Le,
drop = "re78",
cutpoints = list(education = educut),
grouping = list(q1 = q1.grp)
)
#>
#> Using 'treated'='1' as baseline group
mat1
#>            G0  G1
#> All       392 258
#> Matched   158 115
#> Unmatched 234 143
• Can also use progressive coarsening method to control the number of matches.

• cem can also handle some missingness.

### 31.1.7 Genetic Matching

• GM uses iterative checking process of propensity scores, which combines propensity scores and Mahalanobis distance.

• GenMatch
• GM is arguably “superior” method than nearest neighbor or full matching in imbalanced data

• Use a genetic search algorithm to find weights for each covariate such that we have optimal balance.

• Implementation

• could use with replacement

• balance can be based on

• paired $$t$$-tests (dichotomous variables)

• Kolmogorov-Smirnov (multinomial and continuous)

Packages

Matching

library(Matching)
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 http://sekhon.berkeley.edu/papers/MatchingJSS.pdf.
#
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
)

### 31.1.8 Entropy Balancing

• Entropy balancing is a method for achieving covariate balance in observational studies with binary treatments.

• It uses a maximum entropy reweighting scheme to ensure that treatment and control groups are balanced based on sample moments.

• This method adjusts for inequalities in the covariate distributions, reducing dependence on the model used for estimating treatment effects.

• Entropy balancing improves balance across all included covariate moments and removes the need for repetitive balance checking and iterative model searching.

### 31.1.9 Matching for high-dimensional data

One could reduce the number of dimensions using methods such as:

• Lasso

• Penalized logistic regression

• PCA (Principal Component Analysis)

• Locality Preserving Projections (LPP)

• Random projection

• Autoencoders

Additionally, one could jointly does dimension reduction while balancing the distributions of the control and treated groups .

### 31.1.10 Matching for time series-cross-section data

Examples: and

Identification strategy:

• Within-unit over-time variation

• within-time across-units variation

See DID with in and out treatment condition for details of this method

### 31.1.11 Matching for multiple treatments

In cases where you have multiple treatment groups, and you want to do matching, it’s important to have the same baseline (control) group. For more details, see

• : also for continuous treatment

If you insist on using the MatchIt package, then see this answer

### 31.1.12 Matching for multi-level treatments

See

Package in R shuyang1987/multilevelMatching on Github

### 31.1.13 Matching for repeated treatments

https://cran.r-project.org/web/packages/twang/vignettes/iptw.pdf

package in R twang

### References

Abadie, Alberto, and Guido W Imbens. 2016. “Matching on the Estimated Propensity Score.” Econometrica 84 (2): 781–807.
Acemoglu, Daron, Suresh Naidu, Pascual Restrepo, and James A Robinson. 2019. “Democracy Does Cause Growth.” Journal of Political Economy 127 (1): 47–100.
Austin, Peter C. 2011. “Optimal Caliper Widths for Propensity-Score Matching When Estimating Differences in Means and Differences in Proportions in Observational Studies.” Pharmaceutical Statistics 10 (2): 150–61.
Bapna, Ravi, Jui Ramaprasad, and Akhmed Umyarov. 2018. “Monetizing Freemium Communities.” Mis Quarterly 42 (3): 719–A4.
Bhattacharya, Jay, and William B Vogt. 2007. “Do Instrumental Variables Belong in Propensity Scores?” National Bureau of Economic Research Cambridge, Mass., USA.
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): 932–45.
Eckles, Dean, and Eytan Bakshy. 2021. “Bias and High-Dimensional Adjustment in Observational Studies of Peer Effects.” Journal of the American Statistical Association 116 (534): 507–17.
Gordon, Brett R, Florian Zettelmeyer, Neha Bhargava, and Dan Chapsky. 2019. “A Comparison of Approaches to Advertising Measurement: Evidence from Big Field Experiments at Facebook.” Marketing Science 38 (2): 193–225.
Hainmueller, Jens. 2012. “Entropy Balancing for Causal Effects: A Multivariate Reweighting Method to Produce Balanced Samples in Observational Studies.” Political Analysis 20 (1): 25–46.
Iacus, Stefano M, Gary King, and Giuseppe Porro. 2012. “Causal Inference Without Balance Checking: Coarsened Exact Matching.” Political Analysis 20 (1): 1–24.
King, Gary, Christopher Lucas, and Richard A Nielsen. 2017. “The Balance-Sample Size Frontier in Matching Methods for Causal Inference.” American Journal of Political Science 61 (2): 473–89.
King, Gary, and Richard Nielsen. 2019. “Why Propensity Scores Should Not Be Used for Matching.” Political Analysis 27 (4): 435–54.
Li, Sheng, Nikos Vlassis, Jaya Kawale, and Yun Fu. 2016. “Matching via Dimensionality Reduction for Estimation of Treatment Effects in Digital Marketing Campaigns.” In IJCAI, 16:3768–74.
Lopez, Michael J, and Roee Gutman. 2017. “Estimation of Causal Effects with Multiple Treatments: A Review and New Ideas.” Statistical Science, 432–54.
McCaffrey, Daniel F, Beth Ann Griffin, Daniel Almirall, Mary Ellen Slaughter, Rajeev Ramchand, and Lane F Burgette. 2013. “A Tutorial on Propensity Score Estimation for Multiple Treatments Using Generalized Boosted Models.” Statistics in Medicine 32 (19): 3388–3414.
Ramachandra, Vikas. 2018. “Deep Learning for Causal Inference.” arXiv Preprint arXiv:1803.00149.
———. 1985. “The Bias Due to Incomplete Matching.” Biometrics, 103–16.
Scheve, Kenneth, and David Stasavage. 2012. “Democracy, War, and Wealth: Lessons from Two Centuries of Inheritance Taxation.” American Political Science Review 106 (1): 81–102.
VanderWeele, Tyler J. 2019. “Principles of Confounder Selection.” European Journal of Epidemiology 34: 211–19.
Yang, Shu, Guido W Imbens, Zhanglin Cui, Douglas E Faries, and Zbigniew Kadziola. 2016. “Propensity Score Matching and Subclassification in Observational Studies with Multi-Level Treatments.” Biometrics 72 (4): 1055–65.
Yao, Liuyi, Sheng Li, Yaliang Li, Mengdi Huai, Jing Gao, and Aidong Zhang. 2018. “Representation Learning for Treatment Effect Estimation from Observational Data.” Advances in Neural Information Processing Systems 31.
Zhao, Qin-Yu, Jing-Chao Luo, Ying Su, Yi-Jie Zhang, Guo-Wei Tu, and Zhe Luo. 2021. “Propensity Score Matching with r: Conventional Methods and New Features.” Annals of Translational Medicine 9 (9).