# A Guide to Using WeightIt for Estimating Balancing Weights

## Introduction

WeightIt contains several functions for estimating and assessing balancing weights for observational studies. These weights can be used to estimate the causal parameters of marginal structural models. I will not go into the basics of causal inference methods here. For good introductory articles, see Austin (2011), Austin and Stuart (2015), Robins, Hernán, and Brumback (2000), or Thoemmes and Ong (2016).

Typically, the analysis of an observation study might proceed as follows: identify the covariates for which balance is required; assess the quality of the data available, including missingness and measurement error; estimate weights that balance the covariates adequately; and estimate a treatment effect and corresponding standard error or confidence interval. This guide will go through all these steps for two observational studies: estimating the causal effect of a point treatment on an outcome, and estimating the causal parameters of a marginal structural model with multiple treatment periods. This is not meant to be a definitive guide, but rather an introduction to the relevant issues.

## Estimating the Effect of a Point Treatment

First we will use the Lalonde dataset to estimate the effect of a point treatment. We’ll use the version of the data set that resides within the cobalt package, which we will use later on as well. Here, we are interested in the average treatment effect on the treated (ATT).

data("lalonde", package = "cobalt")
head(lalonde)
treat age educ race married nodegree re74 re75 re78
1 37 11 black 1 1 0 0 9930.0460
1 22 9 hispan 0 1 0 0 3595.8940
1 30 12 black 0 0 0 0 24909.4500
1 27 11 black 0 1 0 0 7506.1460
1 33 8 black 0 1 0 0 289.7899
1 22 9 black 0 1 0 0 4056.4940

We have our outcome (re78), our treatment (treat), and the covariates for which balance is desired (age, educ, race, married, nodegree, re74, and re75). Using cobalt, we can examine the initial imbalance on the covariates:

library("cobalt")
##  cobalt (Version 3.9.0.9000, Build Date: )
##    Please read the documentation at ?bal.tab to understand the default outputs.
##    Submit bug reports and feature requests to https://github.com/ngreifer/cobalt/issues
##    Install the development version (not guaranteed to be stable) with:
##      devtools::install_github("ngreifer/cobalt")
##    Thank you for using cobalt!
bal.tab(treat ~ age + educ + race + married + nodegree + re74 + re75,
data = lalonde, estimand = "ATT", m.threshold = .1)
## Balance Measures
##                Type Diff.Un     M.Threshold.Un
## age         Contin. -0.3094 Not Balanced, >0.1
## educ        Contin.  0.0550     Balanced, <0.1
## race_black   Binary  0.6404 Not Balanced, >0.1
## race_hispan  Binary -0.0827     Balanced, <0.1
## race_white   Binary -0.5577 Not Balanced, >0.1
## married      Binary -0.3236 Not Balanced, >0.1
## nodegree     Binary  0.1114 Not Balanced, >0.1
## re74        Contin. -0.7211 Not Balanced, >0.1
## re75        Contin. -0.2903 Not Balanced, >0.1
##
## Balance tally for mean differences
##                    count
## Balanced, <0.1         2
## Not Balanced, >0.1     7
##
## Variable with the greatest mean difference
##  Variable Diff.Un     M.Threshold.Un
##      re74 -0.7211 Not Balanced, >0.1
##
## Sample sizes
##     Control Treated
## All     429     185

Based on this output, we can see that most variables are imbalanced in the sense that the standardized mean differences (for continuous variables) and differences in proportion (for binary variables) are greater than .1 for most variables. In particular, re74 and re75 are quite imbalanced, which is troubling given that they are likely strong predictors of the outcome. We will estimate weights using weightit() to try to attain balance on these covariates.

First, we’ll start simple, and use inverse probability weights from propensity scores generated through logistic regression. We need to supply weightit() with the formula for the model, the data set, the estimand (ATT), and the method of estimation ("ps") for propensity score weights).

library("WeightIt")
W.out <- weightit(treat ~ age + educ + race + married + nodegree + re74 + re75,
data = lalonde, estimand = "ATT", method = "ps")
W.out #print the ooutput
## A weightit object
##  - method: "ps" (propensity score weighting)
##  - number of obs.: 614
##  - sampling weights: none
##  - treatment: 2-category
##  - estimand: ATT (focal: 1)
##  - covariates: age, educ, race, married, nodegree, re74, re75

Printing the output of weightit() displays a summary of how the weights were estimated. Let’s examine the quality of the weights using summary(). Weights with low variability are desirable because they improve the precision of the estimator. This variability is presented in several ways: by the ratio of the largest weight to the smallest in each group, the coefficient of variation (standard deviation divided by the mean) of the weights in each group, and the effective sample size computed from the weights. We want a small ratio, a smaller coefficient of variation, and a large effective sample size (ESS). What constitutes these values is mostly relative, though, and must be balanced with other constraints, including covariate balance. These metrics are best used when comparing weighting methods, but the ESS can give a sense of how much information remains in the weighted sample on a familiar scale.

summary(W.out)
## Summary of weights:
##
## - Weight ranges:
##            Min                                  Max
## treated 1.0000         ||                    1.0000
## control 0.0092 |---------------------------| 3.7432
##
## - Units with 5 greatest weights by group:
##
##               1      3      4      6      7
##  treated      1      1      1      1      1
##             597    573    411    381    303
##  control 3.0301 3.0592 3.2397 3.5231 3.7432
##
##            Ratio Coef of Var
## treated   1.0000      0.0000
## control 408.4971      1.8181
## overall 408.4971      1.1737
##
## - Effective Sample Sizes:
##            Control Treated
## Unweighted 429.000     185
## Weighted    99.815     185

These weights have quite high variability, and yield an ESS of close to 100 in the control group. Let’s see if these weights managed to yield balance on our covariates.

bal.tab(W.out, m.threshold = .1, disp.v.ratio = TRUE)
## Call
##  weightit(formula = treat ~ age + educ + race + married + nodegree +
##     re74 + re75, data = lalonde, method = "ps", estimand = "ATT")
##
## Balance Measures
## prop.score  Distance  -0.0205                         1.0324
## age          Contin.   0.1188 Not Balanced, >0.1      0.4578
## educ         Contin.  -0.0284     Balanced, <0.1      0.6636
## race_black    Binary  -0.0022     Balanced, <0.1
## race_hispan   Binary   0.0002     Balanced, <0.1
## race_white    Binary   0.0021     Balanced, <0.1
## married       Binary   0.0186     Balanced, <0.1
## nodegree      Binary   0.0184     Balanced, <0.1
## re74         Contin.  -0.0021     Balanced, <0.1      1.3206
## re75         Contin.   0.0110     Balanced, <0.1      1.3938
##
## Balance tally for mean differences
##                    count
## Balanced, <0.1         8
## Not Balanced, >0.1     1
##
## Variable with the greatest mean difference
##       age   0.1188 Not Balanced, >0.1
##
## Effective sample sizes
##            Control Treated
## Adjusted    99.815     185

For nearly all the covariates, these weights yielded very good balance. Only age remained imbalanced, with a standardized mean difference greater than .1 and a variance ratio greater than 2. Let’s see if we can do better. We’ll choose a different method: entropy balancing, which guarantees perfect balance on specified moments of the covariates while minimizing the entropy (a measure of dispersion) of the weights.

W.out <- weightit(treat ~ age + educ + race + married + nodegree + re74 + re75,
data = lalonde, estimand = "ATT", method = "ebal")
summary(W.out)
## Summary of weights:
##
## - Weight ranges:
##            Min                                  Max
## treated 1.0000        ||                     1.0000
## control 0.0081 |---------------------------| 4.0624
##
## - Units with 5 greatest weights by group:
##
##               1      2      3      4      5
##  treated      1      1      1      1      1
##             608    597    411    381    303
##  control 3.0735 3.2348 3.4498 3.8965 4.0624
##
##            Ratio Coef of Var
## treated   1.0000      0.0000
## control 502.3782      1.8344
## overall 502.3782      1.1794
##
## - Effective Sample Sizes:
##            Control Treated
## Unweighted 429.000     185
## Weighted    98.459     185

The variability of the weights has not changed much, but let’s see if there are any gains in terms of balance:

bal.tab(W.out, m.threshold = .1, disp.v.ratio = TRUE)
## Call
##  weightit(formula = treat ~ age + educ + race + married + nodegree +
##     re74 + re75, data = lalonde, method = "ebal", estimand = "ATT")
##
## Balance Measures
## age         Contin.       -0 Balanced, <0.1      0.4096
## educ        Contin.       -0 Balanced, <0.1      0.6635
## race_black   Binary        0 Balanced, <0.1
## race_hispan  Binary        0 Balanced, <0.1
## race_white   Binary       -0 Balanced, <0.1
## married      Binary       -0 Balanced, <0.1
## nodegree     Binary        0 Balanced, <0.1
## re74        Contin.       -0 Balanced, <0.1      1.3265
## re75        Contin.       -0 Balanced, <0.1      1.3351
##
## Balance tally for mean differences
##                    count
## Balanced, <0.1         9
## Not Balanced, >0.1     0
##
## Variable with the greatest mean difference
##  race_white       -0 Balanced, <0.1
##
## Effective sample sizes
##            Control Treated
## Adjusted    98.459     185

Indeed, we have achieved perfect balance on the means of the covariates. However, the variance ratio of age is still quite high. We could continue to try to adjust for this imbalance, but if there is reason to believe it is unlikely to affect the outcome, it may be best to leave it as is. (You can try adding I(age^2) to the formula and see what changes this causes.)

Now that we have our weights stored in W.out, let’s extract them and estimate our treatment effect. We’ll use the survey package to incorporate the weights into our analysis.

library("survey")
d.w <- svydesign(ids = ~1, weights = W.out$weights, data = lalonde) fit <- svyglm(re78 ~ treat, design = d.w) coef(fit) ## (Intercept) treat ## 5075.888 1273.255 Now let’s do some inference. Although some authors recommend using “robust” sandwich standard errors to adjust for the weights (Robins et al., 2000; Hainmueller, 2012), others believe these can misleading and recommend bootstrapping instead (e.g., Chan, Yam, & Zhang, 2016). We’ll examine both approaches. svyglm() in the survey package produces robust standard errors. We’ll use the jtools package to provide clean summaries of our results. Similar output can be generated using summary() without installing jtools. #Robust standard errors and confidence intervals library("jtools") summ(fit, confint = TRUE, model.fit = FALSE, model.info = FALSE)  Est. 2.5% 97.5% t val. p (Intercept) 5075.89 3918.36 6233.41 8.61 0.00 treat 1273.26 -347.03 2893.54 1.54 0.12 Standard errors: Robust Our confidence interval for treat contains 0, so there isn’t evidence that treat has an effect on re78. Next let’s use bootstrapping to estimate confidence intervals. We don’t need to use svyglm() and can simply use glm() to compute the effect estimates in each bootstrapped sample because we are not computing standard errors, and the estimated treatment effect estimates will be the same. #Bootstrapping library("boot") est.fun <- function(data, index) { W.out <- weightit(treat ~ age + educ + race + married + nodegree + re74 + re75, data = data[index,], estimand = "ATT", method = "ebal") fit <- glm(re78 ~ treat, data = data[index,], weights = W.out$weights)
return(coef(fit)["treat"])
}
boot.out <- boot(est.fun, data = lalonde, R = 999)
boot.ci(boot.out, type = "bca") #type shouldn't matter so much
## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
## Based on 999 bootstrap replicates
##
## CALL :
## boot.ci(boot.out = boot.out, type = "bca")
##
## Intervals :
## Level       BCa
## 95%   (-343, 2864 )
## Calculations and Intervals on Original Scale

In this case, our confidence intervals were similar. Bootstrapping can take some time, especially with weight estimation methods that take longer, such as generalized boosted modeling (method = "gbm"), covariate balancing propensity score estimation (method = "cbps"), or entropy balancing with stabilize = TRUE.

If we wanted to produce a “doubly-robust” treatment effect estimate, we could add baseline covariates to the glm() (or svyglm()) model (in both the original effect estimation and the confidence interval estimation).

## Estimating the Effect of a Longitudinal Treatment

WeightIt can estimate weights for longitudinal treatment marginal structural models as well. This time, we’ll use the sample data set from twang to estimate our weights. Data must be in “wide” format; to go from long to wide, see the example at ?weightitMSM.

data("iptwExWide", package = "twang")
head(iptwExWide)
outcome gender age use0 use1 use2 tx1 tx2 tx3
-0.2782802 0 43 1.1349651 0.4674825 0.3174825 1 1 1
0.5319329 0 50 1.1119318 0.4559659 0.4059659 1 0 1
-0.8173614 1 36 -0.8707776 -0.5353888 -0.5853888 1 0 0
-0.1530853 1 63 0.2107316 0.0053658 -0.1446342 1 1 1
-0.7344267 0 24 0.0693956 -0.0653022 -0.1153022 1 0 1
-0.8519376 1 20 -1.6626489 -0.9313244 -1.0813244 1 1 1

We have our outcome variable (outcome), our time-stable baseline variables (gender and age), our pre-treatment time-varying variables (use0, measured before the first treatment, use1, and use2), and our three time-varying treatment variables (tx1, tx2, and tx3). We are interested in the joint, unique, causal effects of each treatment period on the outcome. At each treatment time point, we need to achieve balance on all variables measured prior to that treatment, including previous treatments.

Using cobalt, we can examine the initial imbalance at each time point and overall:

library("cobalt") #if not already attached
bal.tab(list(tx1 ~ age + gender + use0,
tx2 ~ tx1 + use1 + age + gender + use0,
tx3 ~ tx2 + use2 + tx1 + use1 + age + gender + use0),
data = iptwExWide, m.threshold = .1, disp.ks = TRUE)
## Balance by Time Point
##
##  - - - Time: 1 - - -
## Balance Measures
##           Type Diff.Un     M.Threshold.Un  KS.Un
## age    Contin.  0.3799 Not Balanced, >0.1 0.2099
## gender  Binary  0.2945 Not Balanced, >0.1 0.2945
## use0   Contin.  0.2668 Not Balanced, >0.1 0.1681
##
## Balance tally for mean differences
##                    count
## Balanced, <0.1         0
## Not Balanced, >0.1     3
##
## Variable with the greatest mean difference
##  Variable Diff.Un     M.Threshold.Un
##       age  0.3799 Not Balanced, >0.1
##
## Sample sizes
##     Control Treated
## All     294     706
##
##  - - - Time: 2 - - -
## Balance Measures
##           Type Diff.Un     M.Threshold.Un  KS.Un
## age    Contin.  0.2240 Not Balanced, >0.1 0.1331
## gender  Binary  0.1927 Not Balanced, >0.1 0.1927
## use0   Contin.  0.1169 Not Balanced, >0.1 0.0913
## tx1     Binary  0.1695 Not Balanced, >0.1 0.1695
## use1   Contin.  0.0848     Balanced, <0.1 0.0763
##
## Balance tally for mean differences
##                    count
## Balanced, <0.1         1
## Not Balanced, >0.1     4
##
## Variable with the greatest mean difference
##  Variable Diff.Un     M.Threshold.Un
##       age   0.224 Not Balanced, >0.1
##
## Sample sizes
##     Control Treated
## All     492     508
##
##  - - - Time: 3 - - -
## Balance Measures
##           Type Diff.Un     M.Threshold.Un  KS.Un
## age    Contin.  0.3431 Not Balanced, >0.1 0.1863
## gender  Binary  0.1532 Not Balanced, >0.1 0.1532
## use0   Contin.  0.1859 Not Balanced, >0.1 0.1350
## tx1     Binary  0.1071 Not Balanced, >0.1 0.1071
## use1   Contin.  0.1662 Not Balanced, >0.1 0.1397
## tx2     Binary  0.2423 Not Balanced, >0.1 0.2423
## use2   Contin.  0.1087 Not Balanced, >0.1 0.1161
##
## Balance tally for mean differences
##                    count
## Balanced, <0.1         0
## Not Balanced, >0.1     7
##
## Variable with the greatest mean difference
##  Variable Diff.Un     M.Threshold.Un
##       age  0.3431 Not Balanced, >0.1
##
## Sample sizes
##     Control Treated
## All     415     585
##  - - - - - - - - - - -
##
## Balance summary across all time points
##          Times    Type Max.Diff.Un     M.Threshold.Un Max.KS.Un
## age    1, 2, 3 Contin.      0.3799 Not Balanced, >0.1    0.2099
## gender 1, 2, 3  Binary      0.2945 Not Balanced, >0.1    0.2945
## use0   1, 2, 3 Contin.      0.2668 Not Balanced, >0.1    0.1681
## tx1       2, 3  Binary      0.1695 Not Balanced, >0.1    0.1695
## use1      2, 3 Contin.      0.1662 Not Balanced, >0.1    0.1397
## tx2          3  Binary      0.2423 Not Balanced, >0.1    0.2423
## use2         3 Contin.      0.1087 Not Balanced, >0.1    0.1161
##
## Sample sizes
##  - Time 1
##     Control Treated
## All     294     706
##  - Time 2
##     Control Treated
## All     492     508
##  - Time 3
##     Control Treated
## All     415     585

bal.tab() indicates significant imbalance on most covariates at most time points, so we need to do some work to eliminate that imbalance in our weighted data set. We’ll use the weightitMSM() function to specify our weight models. The syntax is similar both to that of weightit() for point treatments and to that of bal.tab() for longitudinal treatments. We’ll use method = "ps" for propensity score weights estimated using logistic regression.

Wmsm.out <- weightitMSM(list(tx1 ~ age + gender + use0,
tx2 ~ tx1 + use1 + age + gender + use0,
tx3 ~ tx2 + use2 + tx1 + use1 + age + gender + use0),
data = iptwExWide, method = "ps")
Wmsm.out
## A weightitMSM object
##  - method: "ps" (propensity score weighting)
##  - number of obs.: 1000
##  - sampling weights: none
##  - number of time points: 3 (tx1, tx2, tx3)
##  - treatment:
##     + time 1: 2-category
##     + time 2: 2-category
##     + time 3: 2-category
##  - covariates:
##     + baseline: age, gender, use0
##     + after time 1: tx1, use1, age, gender, use0
##     + after time 2: tx2, use2, tx1, use1, age, gender, use0

No matter which method is selected, weightitMSM() estimates separate weights for each time period and then takes the product of the weights for each individual to arrive at the final estimated weights. Printing the output of weightitMSM() provides some details about the function call and the output. We can take a look at the quality of the weights with summary(), just as we could for point treatments.

summary(Wmsm.out)
## Summary of weights:
##
##  - - - - - - - - - - Time 1 - - - - - - - - - -
## - Weight ranges:
##            Min                                   Max
## treated 1.6667 |---|                         15.5292
## control 2.6123 |---------------------------| 82.2349
##
## - Units with 5 greatest weights by group:
##
##              642     350     324     299     228
##  treated 13.4464 13.6233 13.7059  14.436 15.5292
##              288     176     147      72      52
##  control 44.8027 46.1355 58.4164 79.8919 82.2349
##
##           Ratio Coef of Var
## treated  9.3171      0.4808
## control 31.4795      0.7679
## overall 49.3385      0.8569
##
## - Effective Sample Sizes:
##            Control Treated
## Unweighted  294.00 706.000
## Weighted    185.18 573.598
##
##  - - - - - - - - - - Time 2 - - - - - - - - - -
## - Weight ranges:
##            Min                                   Max
## treated 1.6667 |---------------------------| 82.2349
## control 2.6123 |--------------------------|  79.8919
##
## - Units with 5 greatest weights by group:
##
##              497     338     290     286      22
##  treated  33.371 34.7405 42.9442 58.4164 82.2349
##              265     139     103      47      39
##  control 39.7862 42.2256 44.8027 46.1355 79.8919
##
##           Ratio Coef of Var
## treated 49.3385      0.9605
## control 30.5826      0.7375
## overall 49.3385      0.8569
##
## - Effective Sample Sizes:
##            Control Treated
## Unweighted 492.000 508.000
## Weighted   318.903 264.487
##
##  - - - - - - - - - - Time 3 - - - - - - - - - -
## - Weight ranges:
##            Min                                   Max
## treated 1.6667 |-------------|               44.8027
## control 2.6123 |---------------------------| 82.2349
##
## - Units with 5 greatest weights by group:
##
##              548     450     382     124      67
##  treated  26.003  26.471 28.0582 39.7862 44.8027
##              403     264     226     224     110
##  control 42.9442 46.1355 58.4164 79.8919 82.2349
##
##           Ratio Coef of Var
## treated 26.8803      0.7731
## control 31.4795      0.8734
## overall 49.3385      0.8569
##
## - Effective Sample Sizes:
##            Control Treated
## Unweighted 415.000 585.000
## Weighted   235.672 366.399

Displayed are summaries of how the weights perform at each time point with respect to variability. Next, we’ll examine how well they perform with respect to covariate balance.

bal.tab(Wmsm.out, m.threshold = .1, disp.ks = TRUE, which.time = .none)
## Balance summary across all time points
## prop.score 1, 2, 3 Distance       0.0251                    0.0935
## age        1, 2, 3  Contin.       0.0703 Balanced, <0.1     0.0796
## gender     1, 2, 3   Binary       0.0263 Balanced, <0.1     0.0263
## use0       1, 2, 3  Contin.       0.0558 Balanced, <0.1     0.0938
## tx1           2, 3   Binary       0.0171 Balanced, <0.1     0.0171
## use1          2, 3  Contin.       0.0316 Balanced, <0.1     0.0692
## tx2              3   Binary       0.0085 Balanced, <0.1     0.0085
## use2             3  Contin.       0.0315 Balanced, <0.1     0.0659
##
## Effective sample sizes
##  - Time 1
##            Control Treated
##  - Time 2
##            Control Treated
##  - Time 3
##            Control Treated
## Adjusted   235.672 366.399

By setting which.time = .none in bal.tab(), we can focus on the overall balance assessment, which displays the greatest imbalance for each covariate across time points. We can see that our estimated weights balance all covariates all time points with respect to means and variances. Now we can estimate our treatment effects. We’ll sequentially simplify our model by checking whether interaction terms are needed (implying that specific patterns of treatment yield different outcomes), then by checking whether different coefficients are needed for the treatments (implying that outcomes depend on which treatments are received).

library("survey")
d.w.msm <- svydesign(ids = ~1, weights = Wmsm.out\$weights,
data = iptwExWide)
full.fit <- svyglm(outcome ~ tx1*tx2*tx3, design = d.w.msm)
main.effects.fit <- svyglm(outcome ~ tx1 + tx2 + tx3, design = d.w.msm)
anova(full.fit, main.effects.fit)
## Working (Rao-Scott+F) LRT for tx1:tx2 tx1:tx3 tx2:tx3 tx1:tx2:tx3
##  in svyglm(formula = outcome ~ tx1 * tx2 * tx3, design = d.w.msm)
## Working 2logLR =  7.798048 p= 0.10163
## (scale factors:  1.2 1.1 0.91 0.72 );  denominator df= 992

Based on the non-significant p-value, we don’t have to assume specific treatment patterns yield different outcomes, but rather only that which treatments received or the number of treatments received are sufficient to explain variation in the outcome. Next we’ll narrow down these options by comparing the main effects fit to one that constrains the coefficients to be equal (implying that the cumulative number of treatments received is what matters), as Robins et al. (2000) describe.

cum.fit <- svyglm(outcome ~ I(tx1+tx2+tx3), design = d.w.msm)
anova(main.effects.fit, cum.fit)
## Working (Rao-Scott+F) LRT for tx1 tx2 tx3 - I(tx1 + tx2 + tx3)
##  in svyglm(formula = outcome ~ tx1 + tx2 + tx3, design = d.w.msm)
## Working 2logLR =  3.756596 p= 0.15446
## (scale factors:  1.1 0.94 );  denominator df= 996

Based on the non-significant p-value, we can assume the effects of each treatment are close enough to be treated as the same, indicating that the number of treatments received is the relevant predictor of the outcome. Now we can examine what that treatment effect is with summ() in jtools (or summary()).

summ(cum.fit, confint = TRUE,
model.fit = FALSE, model.info = FALSE) 
Est. 2.5% 97.5% t val. p
(Intercept) 0.11 0.00 0.22 1.96 0.05
I(tx1 + tx2 + tx3) -0.15 -0.20 -0.09 -5.18 0.00
Standard errors: Robust

For each additional treatment received, the outcome is expected to decrease by 0.15 points. The confidence interval excludes 0, so there is evidence of a treatment effect in the population.

There is more we can do, as well. We could have fit different types of models to estimate the weights, and we could have stabilized the weights with stabilize = TRUE or by including stabilization factors in our weights using num.formula (see Cole & Hernán, 2008, for more details on doing so). There are other ways of computing confidence intervals for our effect estimates (although model comparison is the most straightforward with the method we used).

## References

Austin, P. C. (2011). An Introduction to Propensity Score Methods for Reducing the Effects of Confounding in Observational Studies. Multivariate Behavioral Research, 46(3), 399–424.

Austin, P. C., & Stuart, E. A. (2015). Moving towards best practice when using inverse probability of treatment weighting (IPTW) using the propensity score to estimate causal treatment effects in observational studies. Statistics in Medicine, 34(28), 3661–3679.

Chan, K. C. G., Yam, S. C. P., & Zhang, Z. (2016). Globally efficient non-parametric inference of average treatment effects by empirical balancing calibration weighting. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 78(3), 673–700.

Cole, S. R., & Hernán, M. A. (2008). Constructing Inverse Probability Weights for Marginal Structural Models. American Journal of Epidemiology, 168(6), 656–664.

Hainmueller, J. (2012). Entropy Balancing for Causal Effects: A Multivariate Reweighting Method to Produce Balanced Samples in Observational Studies. Political Analysis, 20(1), 25–46.

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