# Using WeightIt to Estimate Balancing Weights

#### Noah Greifer

#### 2024-08-23

## 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.

## Balancing Weights for 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).

    library("cobalt")

    ##  cobalt (Version 4.5.5, Build Date: 2024-04-02)

    data("lalonde", package = "cobalt")
    head(lalonde)

treat | age | educ | race | married | nodegree | re74 | re75 | re78
--- | --- | --- | --- | --- | --- | --- | --- | ---
1 | 37 | 11 | black | 1 | 1 | 0 | 0 | 9930.0
1 | 22 | 9 | hispan | 0 | 1 | 0 | 0 | 3595.9
1 | 30 | 12 | black | 0 | 0 | 0 | 0 | 24909.5
1 | 27 | 11 | black | 0 | 1 | 0 | 0 | 7506.1
1 | 33 | 8 | black | 0 | 1 | 0 | 0 | 289.8
1 | 22 | 9 | black | 0 | 1 | 0 | 0 | 4056.5

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:

    bal.tab(treat ~ age + educ + race + married + nodegree + re74 + re75,
            data = lalonde, estimand = "ATT", thresholds = c(m = .05))

    ## Balance Measures
    ##                Type Diff.Un      M.Threshold.Un
    ## age         Contin.  -0.309 Not Balanced, >0.05
    ## educ        Contin.   0.055 Not Balanced, >0.05
    ## race_black   Binary   0.640 Not Balanced, >0.05
    ## race_hispan  Binary  -0.083 Not Balanced, >0.05
    ## race_white   Binary  -0.558 Not Balanced, >0.05
    ## married      Binary  -0.324 Not Balanced, >0.05
    ## nodegree     Binary   0.111 Not Balanced, >0.05
    ## re74        Contin.  -0.721 Not Balanced, >0.05
    ## re75        Contin.  -0.290 Not Balanced, >0.05
    ##
    ## Balance tally for mean differences
    ##                     count
    ## Balanced, <0.05         0
    ## Not Balanced, >0.05     9
    ##
    ## Variable with the greatest mean difference
    ##  Variable Diff.Un      M.Threshold.Un
    ##      re74  -0.721 Not Balanced, >0.05
    ##
    ## Sample sizes
    ##     Control Treated
    ## All     429     185

Based on this output, we can see that all variables are imbalanced in the sense that the standardized mean differences (for continuous variables) and differences in proportion (for binary variables) are greater than .05 for all 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 (`"glm"`) for generalized linear model propensity score weights).

    library("WeightIt")
    W.out <- weightit(treat ~ age + educ + race + married + nodegree + re74 + re75,
                      data = lalonde, estimand = "ATT", method = "glm")
    W.out #print the output

    ## A weightit object
    ##  - method: "glm" (propensity score weighting with GLM)
    ##  - 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         ||                    1.000
    ## control 0.0092 |---------------------------| 3.743
    ##
    ## - Units with the 5 most extreme weights by group:
    ##
    ##               5      4      3      2      1
    ##  treated      1      1      1      1      1
    ##             597    573    381    411    303
    ##  control 3.0301 3.0592 3.2397 3.5231 3.7432
    ##
    ## - Weight statistics:
    ##
    ##         Coef of Var   MAD Entropy # Zeros
    ## treated       0.000 0.000   0.000       0
    ## control       1.818 1.289   1.098       0
    ##
    ## - Effective Sample Sizes:
    ##
    ##            Control Treated
    ## Unweighted  429.       185
    ## Weighted     99.82     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, stats = c("m", "v"), thresholds = c(m = .05))

    ## Balance Measures
    ##                 Type Diff.Adj         M.Threshold V.Ratio.Adj
    ## prop.score  Distance   -0.021     Balanced, <0.05       1.032
    ## age          Contin.    0.119 Not Balanced, >0.05       0.458
    ## educ         Contin.   -0.028     Balanced, <0.05       0.664
    ## race_black    Binary   -0.002     Balanced, <0.05           .
    ## race_hispan   Binary    0.000     Balanced, <0.05           .
    ## race_white    Binary    0.002     Balanced, <0.05           .
    ## married       Binary    0.019     Balanced, <0.05           .
    ## nodegree      Binary    0.018     Balanced, <0.05           .
    ## re74         Contin.   -0.002     Balanced, <0.05       1.321
    ## re75         Contin.    0.011     Balanced, <0.05       1.394
    ##
    ## Balance tally for mean differences
    ##                     count
    ## Balanced, <0.05         9
    ## Not Balanced, >0.05     1
    ##
    ## Variable with the greatest mean difference
    ##  Variable Diff.Adj         M.Threshold
    ##       age    0.119 Not Balanced, >0.05
    ##
    ## Effective sample sizes
    ##            Control Treated
    ## Unadjusted  429.       185
    ## Adjusted     99.82     185

For nearly all the covariates, these weights yielded very good balance. Only `age` remained imbalanced, with a standardized mean difference greater than .05 and a variance ratio greater than 2. Let’s see if we can do better. We’ll choose a different method: entropy balancing (Hainmueller 2012), 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    ||                         1.000
    ## control 0.0187 |---------------------------| 9.421
    ##
    ## - Units with the 5 most extreme weights by group:
    ##
    ##               5      4      3     2      1
    ##  treated      1      1      1     1      1
    ##             608    381    597   303    411
    ##  control 7.1272 7.5014 7.9998 9.036 9.4206
    ##
    ## - Weight statistics:
    ##
    ##         Coef of Var   MAD Entropy # Zeros
    ## treated       0.000 0.000   0.000       0
    ## control       1.834 1.287   1.101       0
    ##
    ## - Effective Sample Sizes:
    ##
    ##            Control Treated
    ## Unweighted  429.       185
    ## Weighted     98.46     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, stats = c("m", "v"), thresholds = c(m = .05))

    ## Balance Measures
    ##                Type Diff.Adj     M.Threshold V.Ratio.Adj
    ## age         Contin.        0 Balanced, <0.05       0.410
    ## educ        Contin.        0 Balanced, <0.05       0.664
    ## race_black   Binary        0 Balanced, <0.05           .
    ## race_hispan  Binary       -0 Balanced, <0.05           .
    ## race_white   Binary       -0 Balanced, <0.05           .
    ## married      Binary        0 Balanced, <0.05           .
    ## nodegree     Binary       -0 Balanced, <0.05           .
    ## re74        Contin.        0 Balanced, <0.05       1.326
    ## re75        Contin.       -0 Balanced, <0.05       1.335
    ##
    ## Balance tally for mean differences
    ##                     count
    ## Balanced, <0.05         9
    ## Not Balanced, >0.05     0
    ##
    ## Variable with the greatest mean difference
    ##  Variable Diff.Adj     M.Threshold
    ##      re75       -0 Balanced, <0.05
    ##
    ## Effective sample sizes
    ##            Control Treated
    ## Unadjusted  429.       185
    ## Adjusted     98.46     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. The functions `lm_weightit()` and `glm_weightit()` make it easy to fit (generalized) linear models that account for estimation of of the weights in their standard errors. We can then use functions in `marginaleffects` to perform g-computation to extract a treatment effect estimation from the outcome model.

    # Fit outcome model
    fit <- lm_weightit(re78 ~ treat * (age + educ + race + married +
                                         nodegree + re74 + re75),
                       data = lalonde, weightit = W.out)

    # G-computation for the treatment effect
    library("marginaleffects")
    avg_comparisons(fit, variables = "treat",
                    newdata = subset(lalonde, treat == 1))

    ##
    ##   Term          Contrast Estimate Std. Error    z Pr(>|z|)   S 2.5 % 97.5 %
    ##  treat mean(1) - mean(0)     1273        770 1.65   0.0983 3.3  -236   2783
    ##
    ## Columns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted
    ## Type:  response

Our confidence interval for `treat` contains 0, so there isn’t evidence that `treat` has an effect on `re78`. Several types of standard errors are available in `WeightIt`, including analytical standard errors that account for estimation of the weights using M-estimation, robust standard errors that treat the weights as fixed, and bootstrapping. All type are described in detail at `vignette("estimating-effects")`.

## Balancing Weights for a Longitudinal Treatment

`WeightIt` can estimate weights for longitudinal treatment marginal structural models as well. This time, we’ll use the sample data set `msmdata` to estimate our weights. Data must be in “wide” format, with one row per unit.

    data("msmdata")
    head(msmdata)

X1_0 | X2_0 | A_1 | X1_1 | X2_1 | A_2 | X1_2 | X2_2 | A_3 | Y_B
--- | --- | --- | --- | --- | --- | --- | --- | --- | ---
2 | 0 | 1 | 5 | 1 | 0 | 4 | 1 | 0 | 0
4 | 0 | 1 | 9 | 0 | 1 | 10 | 0 | 1 | 1
4 | 1 | 0 | 5 | 0 | 1 | 4 | 0 | 0 | 1
4 | 1 | 0 | 4 | 0 | 0 | 6 | 1 | 0 | 1
6 | 1 | 1 | 5 | 0 | 1 | 6 | 0 | 0 | 1
5 | 1 | 0 | 4 | 0 | 1 | 4 | 0 | 1 | 0

We have a binary outcome variable (`Y_B`), pre-treatment time-varying variables (`X1_0` and `X2_0`, measured before the first treatment, `X1_1` and `X2_1` measured between the first and second treatments, and `X1_2` and `X2_2` measured between the second and third treatments), and three time-varying binary treatment variables (`A_1`, `A_2`, and `A_3`). 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(A_1 ~ X1_0 + X2_0,
                 A_2 ~ X1_1 + X2_1 +
                   A_1 + X1_0 + X2_0,
                 A_3 ~ X1_2 + X2_2 +
                   A_2 + X1_1 + X2_1 +
                   A_1 + X1_0 + X2_0),
            data = msmdata, stats = c("m", "ks"),
            which.time = .all)

    ## Balance by Time Point
    ##
    ##  - - - Time: 1 - - -
    ## Balance Measures
    ##         Type Diff.Un KS.Un
    ## X1_0 Contin.   0.690 0.276
    ## X2_0  Binary  -0.325 0.325
    ##
    ## Sample sizes
    ##     Control Treated
    ## All    3306    4194
    ##
    ##  - - - Time: 2 - - -
    ## Balance Measures
    ##         Type Diff.Un KS.Un
    ## X1_1 Contin.   0.874 0.340
    ## X2_1  Binary  -0.299 0.299
    ## A_1   Binary   0.127 0.127
    ## X1_0 Contin.   0.528 0.201
    ## X2_0  Binary  -0.060 0.060
    ##
    ## Sample sizes
    ##     Control Treated
    ## All    3701    3799
    ##
    ##  - - - Time: 3 - - -
    ## Balance Measures
    ##         Type Diff.Un KS.Un
    ## X1_2 Contin.   0.475 0.212
    ## X2_2  Binary  -0.594 0.594
    ## A_2   Binary   0.162 0.162
    ## X1_1 Contin.   0.573 0.237
    ## X2_1  Binary  -0.040 0.040
    ## A_1   Binary   0.100 0.100
    ## X1_0 Contin.   0.361 0.148
    ## X2_0  Binary  -0.040 0.040
    ##
    ## Sample sizes
    ##     Control Treated
    ## All    4886    2614
    ##  - - - - - - - - - - -

`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 = "glm"` and `stabilize = TRUE` for stabilized propensity score weights estimated using logistic regression.

    Wmsm.out <- weightitMSM(list(A_1 ~ X1_0 + X2_0,
                                 A_2 ~ X1_1 + X2_1 +
                                   A_1 + X1_0 + X2_0,
                                 A_3 ~ X1_2 + X2_2 +
                                   A_2 + X1_1 + X2_1 +
                                   A_1 + X1_0 + X2_0),
                            data = msmdata, method = "glm",
                            stabilize = TRUE)
    Wmsm.out

    ## A weightitMSM object
    ##  - method: "glm" (propensity score weighting with GLM)
    ##  - number of obs.: 7500
    ##  - sampling weights: none
    ##  - number of time points: 3 (A_1, A_2, A_3)
    ##  - treatment:
    ##     + time 1: 2-category
    ##     + time 2: 2-category
    ##     + time 3: 2-category
    ##  - covariates:
    ##     + baseline: X1_0, X2_0
    ##     + after time 1: X1_1, X2_1, A_1, X1_0, X2_0
    ##     + after time 2: X1_2, X2_2, A_2, X1_1, X2_1, A_1, X1_0, X2_0
    ##  - stabilized; stabilization factors:
    ##     + baseline: (none)
    ##     + after time 1: A_1
    ##     + after time 2: A_1, A_2, A_1:A_2

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)

    ##                         Time 1
    ##                   Summary of weights
    ##
    ## - Weight ranges:
    ##
    ##            Min                                 Max
    ## treated 0.1527 |---------------------------| 57.08
    ## control 0.1089 |--------|                    20.46
    ##
    ## - Units with the 5 most extreme weights by group:
    ##
    ##             4390    3440    3774   3593    5685
    ##  treated 22.1008 24.1278 25.6999 27.786 57.0794
    ##             6659    6284    1875   6163    2533
    ##  control 12.8943   13.09 14.5234 14.705  20.465
    ##
    ## - Weight statistics:
    ##
    ##         Coef of Var   MAD Entropy # Zeros
    ## treated       1.779 0.775   0.573       0
    ## control       1.331 0.752   0.486       0
    ##
    ## - Mean of Weights = 0.99
    ##
    ## - Effective Sample Sizes:
    ##
    ##            Control Treated
    ## Unweighted    3306    4194
    ## Weighted      1193    1007
    ##
    ##                         Time 2
    ##                   Summary of weights
    ##
    ## - Weight ranges:
    ##
    ##            Min                                 Max
    ## treated 0.1089 |---------------------------| 57.08
    ## control 0.1501 |--------|                    20.49
    ##
    ## - Units with the 5 most extreme weights by group:
    ##
    ##             4390    3440    3774    3593    5685
    ##  treated 22.1008 24.1278 25.6999  27.786 57.0794
    ##             1875    6163    6862    1286    6158
    ##  control 14.5234  14.705 14.8079 16.2311 20.4862
    ##
    ## - Weight statistics:
    ##
    ##         Coef of Var   MAD Entropy # Zeros
    ## treated       1.797 0.779   0.580       0
    ## control       1.359 0.750   0.488       0
    ##
    ## - Mean of Weights = 0.99
    ##
    ## - Effective Sample Sizes:
    ##
    ##            Control Treated
    ## Unweighted    3701  3799.
    ## Weighted      1300   898.2
    ##
    ##                         Time 3
    ##                   Summary of weights
    ##
    ## - Weight ranges:
    ##
    ##            Min                                 Max
    ## treated 0.1089 |---------------------------| 57.08
    ## control 0.2085 |-----------|                 25.70
    ##
    ## - Units with the 5 most extreme weights by group:
    ##
    ##             3576    4390    3440    3593    5685
    ##  treated 20.5828 22.1008 24.1278  27.786 57.0794
    ##             6163    6862     168    6158    3774
    ##  control  14.705 14.8079 16.9698 20.4862 25.6999
    ##
    ## - Weight statistics:
    ##
    ##         Coef of Var   MAD Entropy # Zeros
    ## treated       2.008 0.931   0.753       0
    ## control       1.269 0.672   0.407       0
    ##
    ## - Mean of Weights = 0.99
    ##
    ## - Effective Sample Sizes:
    ##
    ##            Control Treated
    ## Unweighted    4886  2614.
    ## Weighted      1871   519.8

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, stats = c("m", "ks"),
            which.time = .none)

    ## Balance summary across all time points
    ##        Times    Type Max.Diff.Adj Max.KS.Adj
    ## X1_0 1, 2, 3 Contin.        0.033      0.018
    ## X2_0 1, 2, 3  Binary        0.018      0.018
    ## X1_1    2, 3 Contin.        0.087      0.039
    ## X2_1    2, 3  Binary        0.031      0.031
    ## A_1     2, 3  Binary        0.130      0.130
    ## X1_2       3 Contin.        0.104      0.054
    ## X2_2       3  Binary        0.007      0.007
    ## A_2        3  Binary        0.154      0.154
    ##
    ## Effective sample sizes
    ##  - Time 1
    ##            Control Treated
    ## Unadjusted    3306    4194
    ## Adjusted      1193    1007
    ##  - Time 2
    ##            Control Treated
    ## Unadjusted    3701  3799.
    ## Adjusted      1300   898.2
    ##  - Time 3
    ##            Control Treated
    ## Unadjusted    4886  2614.
    ## Adjusted      1871   519.8

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 KS statistics. Now we can estimate our treatment effects.

First, we fit a marginal structural model for the outcome using `glm()` with the weights included:

    # Fit outcome model
    fit <- glm_weightit(Y_B ~ A_1 * A_2 * A_3 * (X1_0 + X2_0),
                        data = msmdata,
                        weightit = Wmsm.out,
                        family = binomial)

Then, we compute the average expected potential outcomes under each treatment regime using `marginaleffects::avg_predictions()`:

    library("marginaleffects")
    (p <- avg_predictions(fit,
                          variables = c("A_1", "A_2", "A_3"),
                          type = "response"))

    ##
    ##  A_1 A_2 A_3 Estimate Std. Error    z Pr(>|z|)     S 2.5 % 97.5 %
    ##    0   0   0    0.687     0.0166 41.4   <0.001   Inf 0.654  0.719
    ##    0   0   1    0.521     0.0379 13.7   <0.001 140.3 0.447  0.595
    ##    0   1   0    0.491     0.0213 23.1   <0.001 389.1 0.449  0.532
    ##    0   1   1    0.438     0.0295 14.8   <0.001 163.2 0.380  0.496
    ##    1   0   0    0.602     0.0211 28.5   <0.001 590.8 0.561  0.644
    ##    1   0   1    0.544     0.0314 17.3   <0.001 221.0 0.482  0.605
    ##    1   1   0    0.378     0.0163 23.2   <0.001 393.1 0.346  0.410
    ##    1   1   1    0.422     0.0261 16.1   <0.001 192.3 0.371  0.473
    ##
    ## Columns: A_1, A_2, A_3, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
    ## Type:  response

We can compare the expected potential outcomes under each regime using `marginaleffects::hypotheses()`. To get all pairwise comparisons, supply the `avg_predictions()` output to `hypotheses(., "pairwise")`. To compare individual regimes, we can use `hypotheses()`, identifying the rows of the `avg_predictions()` output. For example, to compare the regimes with no treatment for all three time points vs. the regime with treatment for all three time points, we would run

    hypotheses(p, "b8 - b1 = 0")

    ##
    ##     Term Estimate Std. Error     z Pr(>|z|)    S  2.5 % 97.5 %
    ##  b8-b1=0   -0.265     0.0308 -8.61   <0.001 56.9 -0.325 -0.204
    ##
    ## Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
    ## Type:  response

These results indicate that receiving treatment at all time points reduces the risk of the outcome relative to not receiving treatment at all.

## References

Austin, Peter C. 2011. “An Introduction to Propensity Score Methods for Reducing the Effects of Confounding in Observational Studies.” Multivariate Behavioral Research 46 (3): 399–424. https://doi.org/10.1080/00273171.2011.568786.
Austin, Peter C., and Elizabeth A. Stuart. 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): 3661–79. https://doi.org/10.1002/sim.6607.
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. https://doi.org/10.1093/pan/mpr025.
Robins, James M., Miguel Ángel Hernán, and Babette Brumback. 2000. “Marginal Structural Models and Causal Inference in Epidemiology.” Epidemiology 11 (5): 550–60. https://doi.org/10.1097/00001648-200009000-00011.
Thoemmes, Felix J., and Anthony D. Ong. 2016. “A Primer on Inverse Probability of Treatment Weighting and Marginal Structural Models.” Emerging Adulthood 4 (1): 40–59. https://doi.org/10.1177/2167696815621645.