Estimating Effects After Weighting

Noah Greifer

2023-05-09

Introduction

After assessing balance and deciding on a weighting specification, it comes time to estimate the effect of the treatment in the weighted sample. How the effect is estimated and interpreted depends on the desired estimand and the effect measure used. In addition to estimating effects, estimating the uncertainty of the effects is critical in communicating them and assessing whether the observed effect is compatible with there being no effect in the population. This guide explains how to estimate effects after weighting for point and longitudinal treatments and with various outcome types.

This guide is structured as follows: first, information on the concepts related to effect and standard error (SE) estimation is presented below. Then, instructions for how to estimate effects and SEs are described for the standard case (weighting for the ATE with a binary treatment and continuous outcome) and some other common circumstances. Finally, recommendations for reporting results and tips to avoid making common mistakes are presented.

(Note: much of this vignette is copied from vignette("estimating-effects", package = "MatchIt") and adapted to the context of weighting.)

Identifying the estimand

Before an effect is estimated, the estimand must be specified and clarified. Although some aspects of the estimand depend not only on how the effect is estimated after weighting but also on the weighting method itself, other aspects must be considered at the time of effect estimation and interpretation. Here, we consider three aspects of the estimand: the population the effect is meant to generalize to (the target population), the effect measure, and whether the effect is marginal or conditional.

The target population. Different weighting methods allow you to estimate effects that can generalize to different target populations. The most common estimand in weighting is the average treatment effect in the population (ATE), which is the average effect of treatment for the population from which the sample is a random sample. Other common estimands include the average treatment effect in the treated (ATT), the average treatment effect in the control (ATC), and the average treatment effect in the overlap (ATO). These are defined and explained in Greifer and Stuart (2021). The estimand for weighting is controlled by the estimand argument in the call to weightit(). Other allowable estimands for some weighting methods include the average treatment effect in the matched sample (ATM) and the average treatment effect in the optimal subset (ATOS). These are treated just like the ATO and will not be differentiated further.

Marginal and conditional effects. A marginal effect is a comparison between the expected potential outcome under treatment and the expected potential outcome under control. This is the same quantity estimated in randomized trials without blocking or covariate adjustment and is particularly useful for quantifying the overall effect of a policy or population-wide intervention. A conditional effect is the comparison between the expected potential outcomes in the treatment groups within strata. This is useful for identifying the effect of a treatment for an individual patient or a subset of the population.

Effect measures. The outcome types we consider here are continuous, with the effect measured by the mean difference; binary, with the effect measured by the risk difference (RD), risk ratio (RR), or odds ratio (OR); and time-to-event (i.e., survival), with the effect measured by the hazard ratio (HR). The RR, OR, and HR are noncollapsible effect measures, which means the marginal effect on that scale is not a (possibly) weighted average of the conditional effects within strata, even if the stratum-specific effects are of the same magnitude. For these effect measures, it is critical to distinguish between marginal and conditional effects because different statistical methods target different types of effects. The mean difference and RD are collapsible effect measures, so the same methods can be used to estimate marginal and conditional effects.

Our primary focus will be on marginal effects, which are appropriate for all effect measures, easily interpretable, and require few modeling assumptions.

G-computation

To estimate marginal effects, we use a method known as g-computation (Snowden, Rose, and Mortimer 2011) or regression estimation (Schafer and Kang 2008). This involves first specifying a model for the outcome as a function of the treatment and covariates. Then, for each unit, we compute their predicted values of the outcome setting their treatment status to treated, and then again for control, leaving us with two predicted outcome values for each unit, which are estimates of the potential outcomes under each treatment level. We compute the mean of each of the estimated potential outcomes across the entire sample, which leaves us with two average estimated potential outcomes. Finally, the contrast of these average estimated potential outcomes (e.g., their difference or ratio, depending on the effect measure desired) is the estimate of the treatment effect.

When doing g-computation after weighting, a few additional considerations are required. First, the outcome model should be fit incorporating the estimated weights (e.g., using weighted least squares or weighted maximum likelihood estimation). Second, when we take the average of the estimated potential outcomes under each treatment level, this must be a weighted average that incorporates the estimated weights. Third, if we want to target the ATT or ATC, we only estimate potential outcomes for the treated or control group, respectively (though we still generate predicted values under both treatment and control).

G-computation as a framework for estimating effects after weighting has a number of advantages over other approaches. It works the same regardless of the form of the outcome model or type of outcome (e.g., whether a linear model is used for a continuous outcome or a logistic model is used for a binary outcome); the only difference might be how the average expected potential outcomes are contrasted in the final step. In simple cases, the estimated effect is numerically identical to effects estimated using other methods; for example, if no covariates are included in the outcome model, the g-computation estimate is equal to the difference in means from a t-test or coefficient of the treatment in a linear model for the outcome. There are analytic and bootstrap approximations to the SEs of the g-computation estimate. The analytic approximation is computed using the delta method, a technique for computing the SE of a quantity derived from the regression model parameters, such as the g-computation estimate.

For the reasons above, we use weighted g-computation when possible for all effect estimates, even if there are simpler methods that would yield the same estimates. Using a single workflow (with some slight modifications depending on the context; see below) facilitates implementing best practices regardless of what choices a user makes.

There are other methods to incorporate the outcome model into estimation of the treatment effect, the best studied of which is augmented inverse probability weighting (AIPW), which also involves a g-computation step. We only describe weighted g-computation here because of its conceptual simplicity, ease of implementation, and connection with best practices for estimating effects after matching.

Modeling the Outcome

The goal of the outcome model is to generate good predictions for use in the g-computation procedure described above. The type and form of the outcome model should depend on the outcome type. For continuous outcomes, one can use a linear model regressing the outcome on the treatment; for binary outcomes, one can use a generalized linear model with, e.g., a logistic link; for time-to-event outcomes, one can use a Cox proportional hazards model.

An additional decision to make is whether (and how) to include covariates in the outcome model. One may ask, why use weighting at all if you are going to model the outcome with covariates anyway? Weighting reduces the dependence of the effect estimate on correct specification of the outcome model; this is the central thesis of Ho et al. (2007) (though applied to matching in their case). Including covariates in the outcome model after weighting has several functions: it can increase precision in the effect estimate, reduce the bias due to residual imbalance, and make the effect estimate “doubly robust”, which means it is consistent if either the weighting reduces sufficient imbalance in the covariates or if the outcome model is correct. For these reasons, we recommend covariate adjustment after weighting when possible. There is some evidence that covariate adjustment is most helpful for covariates with standardized mean differences greater than .1 (Nguyen et al. 2017), so these covariates and covariates thought to be highly predictive of the outcome should be prioritized in treatment effect models if not all can be included due to sample size constraints.

Although there are many possible ways to include covariates (e.g., not just main effects but interactions, smoothing terms like splines, or other nonlinear transformations), it is important not to engage in specification search (i.e., trying many outcomes models in search of the “best” one). Doing so can invalidate results and yield a conclusion that fails to replicate. For this reason, we recommend only including the same terms included in the weighting model unless there is a strong a priori and justifiable reason to model the outcome differently.

It is important not to interpret the coefficients and tests of covariates in the outcome model. These are not causal effects and their estimates may be severely confounded. Only the treatment effect estimate can be interpreted as causal assuming the relevant assumptions about unconfoundedness are met. Inappropriately interpreting the coefficients of covariates in the outcome model is known as the Table 2 fallacy (Westreich and Greenland 2013). To avoid this, we only display the results of the g-computation procedure and do not examine or interpret the outcome models themselves.

Estimating Standard Errors and Confidence Intervals

Uncertainty estimation (i.e., of SEs, confidence intervals, and p-values) may consider the variety of sources of uncertainty present in the analysis, including (but not limited to!) estimation of the propensity score (if used) and estimation of the treatment effect (i.e., because of sampling error). For some methods, methods for analytically computing the correct asymptotic SE have been described. These often require custom coding or are limited to specific weighting methods and estimands1. Instead, we consider two approximations to the analytic SE: robust SEs and the bootstrap, described below.

Robust and Cluster-Robust Standard Errors

Robust standard errors. Also known as sandwich SEs (due to the form of the formula for computing them), heteroscedasticity-consistent SEs, or Huber-White SEs, robust SEs are an adjustment to the usual maximum likelihood or ordinary least squares SEs that are robust to violations of some of the assumptions required for usual SEs to be valid (MacKinnon and White 1985). Robust SEs have been shown to be conservative (i.e., yield overly large SEs and wide confidence intervals) for estimating the ATE after some forms of weighting (Robins, Hernán, and Brumback 2000), though they can be either conservative or not for other weighting methods and estimands, such as for the ATT (Reifeis and Hudgens 2020) or for entropy balancing (Chan, Yam, and Zhang 2016). Robust SEs treat the estimated weights as if they were fixed and known, ignoring uncertainty in their estimation. Although they are quick and simple to estimate using functionality in the sandwich and survey packages, they should be used with caution, and the bootstrap (described below) should be preferred in most cases.

Bootstrapping

Some problems with robust SEs include that they fail to take into account the estimation of the weights and are only an approximation when used to compute derived quantities from nonlinear models, which is often true when using g-computation to estimate effects. One solution to these problems is bootstrapping, which is a technique used to simulate the sampling distribution of an estimator by repeatedly drawing samples with replacement and estimating the effect in each bootstrap sample (Efron and Tibshirani 1986). From the bootstrap distribution, SEs and confidence intervals can be computed in several ways, including using the standard deviation of the bootstrap estimates as the SE estimate or using the 2.5 and 97.5 percentiles as 95% confidence interval bounds. Bootstrapping tends to be most useful when no analytic estimator of a SE is possible or has been derived yet. Bootstrapping has been found to be effective at estimating SEs and confidence intervals after weighting, often performing better even than the analytic asymptotic formula when it is available (Austin 2022).

Typically, bootstrapping involves performing the entire estimation process in each bootstrap sample, including estimation both of the weights and the effect. With bootstrapping, more bootstrap replications are always better but can take time and increase the chances that at least one error will occur within the bootstrap analysis (e.g., a bootstrap sample with zero treated units or zero units with an event). In general, numbers of replications upwards of 999 are recommended, with values one less than a multiple of 100 preferred to avoid interpolation when using the percentiles as confidence interval limits (MacKinnon 2006). There are several methods of computing bootstrap confidence intervals, but the bias-corrected accelerated (BCa) bootstrap confidence interval often performs well and is easy to implement, simply by setting type = "bca" in the call to boot::boot.ci() after running boot::boot()2.

Estimating Treatment Effects and Standard Errors After Weighting

Below, we describe effect estimation after weighting. The focus here is not on evaluating the methods but simply on demonstrating them. In all cases, the correct propensity score model is used. We will present both the faster and simpler method that uses robust SEs and the more complicated but accurate method using bootstrapping.

We’ll be using a simulated toy dataset d with several outcome and treatment types. Code to generate the dataset is at the end of this document. Below we display the first six rows of d:

head(d)
##   A Am      Ac      X1      X2      X3       X4 X5      X6      X7      X8       X9     Y_C Y_B    Y_S
## 1 0 C1 -2.2185  0.1725 -1.4283 -0.4103 -2.36059  1 -1.1199  0.6398 -0.4840 -0.59385 -3.5907   0  857.7
## 2 0 C2 -2.2837 -1.0959  0.8463  0.2456 -0.12333  1 -2.2687 -1.4491 -0.5514 -0.31439 -1.5481   0  311.6
## 3 0 C1 -1.1362  0.1768  0.7905 -0.8436  0.82366  1 -0.2221  0.2971 -0.6966 -0.69516  6.0714   0  241.2
## 4 0 C1 -0.8865 -0.4595  0.1726  1.9542 -0.62661  1 -0.4019 -0.8294 -0.5384  0.20729  2.4906   1  142.4
## 5 1  T  0.8613  0.3563 -1.8121  0.8135 -0.67189  1 -0.8297  1.7297 -0.6439 -0.02648 -0.6687   0  206.8
## 6 0 C2 -2.1697 -2.4313 -1.7984 -1.2940  0.04609  1 -1.2419 -1.1252 -1.8659 -0.56513 -9.8504   0 1962.9

A is a binary treatment variable, X1 through X9 are covariates, Y_C is a continuous outcome, Y_B is a binary outcome, and Y_S is a survival outcome. Am and Ac are multi-category and continuous treatment variables, respectively.

We will need to the following packages to perform the analyses:

Of course, we also need WeightIt to perform the weighting.

library("WeightIt")
library("marginaleffects")

Effect estimates will be computed using marginaleffects::avg_comparions(), even when its use may be superfluous (e.g., for comparing the weighted difference in means). As previously mentioned, this is because it is useful to have a single workflow that works no matter the situation, perhaps with very slight modifications to accommodate different contexts. Using avg_comparisons() has several advantages, even when the alternatives are simple: it only provides the effect estimate, and not other coefficients; it automatically incorporates robust SEs if requested; and it always produces average marginal effects for the correct population if requested.

Other packages may be of use but are not used here. There are alternatives to the marginaleffects package for computing average marginal effects, including margins and stdReg. The survey package can be used to estimate robust SEs incorporating weights and provides functions for survey-weighted generalized linear models and Cox-proportional hazards models. Much of the code here can be adapted to be used with survey, and we will demonstrate that as well.

The Standard Case: Binary Treatment with Robust SEs

For most weighting methods, estimating the effect after weighting is straightforward and involves fitting a model for the outcome that incorporates the estimated weights, then estimating the treatment effect using g-computation (i.e., using marginaleffects::avg_comparisons()) with a robust SE to account for pair membership. This procedure is the same for continuous and binary outcomes with and without covariates.

There are a few adjustments that need to be made for certain scenarios, which we describe in the section “Adjustments to the Standard Case”. These adjustments include for the following cases: when weighting for the ATT or ATC, for estimating effects with binary outcomes, and for estimating effects with survival outcomes. Estimation for all estimands other than the ATT and ATC proceeds as it does for the ATE. You must read the Standard Case to understand the basic procedure before reading about these special scenarios. We also demonstrate how to estimate effects for multi-category and continuous treatments.

Here, we demonstrate the faster analytic approach to estimating confidence intervals; for the bootstrap approach, see the section “Using Bootstrapping to Estimate Confidence Intervals” below.

First, we will perform propensity score weighting for the ATE. Remember, all weighting methods use this exact procedure or a slight variation, so this section is critical even if you are using a different weighting method.

#PS weighting for the ATE with a logistic regression PS
W <- weightit(A ~ X1 + X2 + X3 + X4 + X5 + 
                X6 + X7 + X8 + X9, data = d,
              method = "glm", estimand = "ATE")
W
## A weightit object
##  - method: "glm" (propensity score weighting with GLM)
##  - number of obs.: 2000
##  - sampling weights: none
##  - treatment: 2-category
##  - estimand: ATE
##  - covariates: X1, X2, X3, X4, X5, X6, X7, X8, X9

Typically one would assess balance and ensure that this weighting specification works, but we will skip that step here to focus on effect estimation. See vignette("WeightIt") and vignette("cobalt", package = "cobalt") for more information on this necessary step.

First, we fit a model for the outcome given the treatment and (optionally) the covariates. It’s usually a good idea to include treatment-covariate interactions, which we do below, but this is not always necessary, especially when excellent balance has been achieved.

#Bring weights into the dataset
d$weights <- W$weights

#Linear model with covariates
fit <- lm(Y_C ~ A * (X1 + X2 + X3 + X4 + X5 + 
                        X6 + X7 + X8 + X9),
           data = d, weights = weights)

Next, we use marginaleffects::avg_comparisons() to estimate the ATE.

avg_comparisons(fit,
                variables = "A",
                vcov = "HC3",
                wts = "weights")
## 
##  Term Contrast Estimate Std. Error   z Pr(>|z|) 2.5 % 97.5 %
##     A    1 - 0     1.81      0.477 3.8   <0.001 0.875   2.74
## 
## Columns: term, contrast, estimate, std.error, statistic, p.value, conf.low, conf.high

Let’s break down the call to avg_comparisons(): to the first argument, we supply the model fit, fit; to the variables argument, the name of the treatment ("A"); to the vcov argument, the name of the robust SE we want to request (in this case, HC3, which perform well relative to other robust SEs); and to the wts argument, the names of the variable in d containing the matching weights ("weights") to ensure they are included in the analysis. Some of these arguments differ depending on the specifics of the outcome type; see the sections below for information.

If, in addition to the effect estimate, we want the average estimated potential outcomes, we can use marginaleffects::avg_predictions(), which we demonstrate below. Note the interpretation of the resulting estimates as the expected potential outcomes is only valid if all covariates present in the outcome model (if any) are interacted with the treatment.

avg_predictions(fit,
                variables = "A",
                vcov = "HC3",
                wts = "weights")
## 
##  A Estimate Std. Error     z Pr(>|z|) 2.5 % 97.5 %
##  0     1.42      0.136 10.44   <0.001  1.15   1.69
##  1     3.17      0.380  8.35   <0.001  2.42   3.91
## 
## Columns: A, estimate, std.error, statistic, p.value, conf.low, conf.high

We can see that the difference in potential outcome means is equal to the average treatment effect computed previously3. All of the arguments to avg_predictions() are the same as those to avg_comparisons().

Adjustments to the Standard Case

This section explains how the procedure might differ if any of the following special circumstances occur.

Weighting for the ATT or ATC

When weighting for the ATT, everything is identical to the Standard Case except that in the calls to avg_comparisons() and avg_predictions(), the newdata argument must additionally be supplied to avg_comparisons() and avg_predictions() as

newdata = subset(d, A == 1)

This requests that g-computation be done only for the treated units. For the ATC, replace 1 with 0.

Binary outcomes

Estimating effects on binary outcomes is essentially the same as for continuous outcomes. The main difference is that there are several measures of the effect one can consider, which include the odds ratio (OR), risk ratio/relative risk (RR), and risk difference (RD), and the syntax to avg_comparisons() depends on which one is desired. The outcome model should be one appropriate for binary outcomes (e.g., logistic regression) but is unrelated to the desired effect measure because we can compute any of the above effect measures using avg_comparisons() after the logistic regression.

To fit a logistic regression model, change lm() to glm() and set family = quasibinomial4. To compute the marginal RD, we can use exactly the same syntax as in the Standard Case; nothing needs to change5.

To compute the marginal RR, we need to add comparison = "lnratioavg" to avg_comparisons(); this computes the marginal log RR. To get the marginal RR, we need to add transform = "exp" to avg_comparisons(), which exponentiates the marginal log RR and its confidence interval. The code below computes the effects and displays the statistics of interest:

#Logistic regression model with covariates
fit <- glm(Y_B ~ A * (X1 + X2 + X3 + X4 + X5 + 
                        X6 + X7 + X8 + X9),
           data = d, weights = weights,
           family = quasibinomial)

#Compute effects; RR and confidence interval
avg_comparisons(fit,
                variables = "A",
                vcov = "HC3",
                wts = "weights",
                comparison = "lnratioavg",
                transform = "exp")
## 
##  Term              Contrast Estimate Pr(>|z|) 2.5 % 97.5 %
##     A ln(mean(1) / mean(0))     1.71   <0.001  1.42   2.05
## 
## Columns: term, contrast, estimate, p.value, conf.low, conf.high, predicted, predicted_hi, predicted_lo

The output displays the marginal RR, its Z-value, the p-value for the Z-test of the log RR against 0, and its confidence interval. (Note that even though the Contrast label still suggests the log RR, the RR is actually displayed.) To view the log RR and its standard error, omit the transform argument.

For the marginal OR, the only thing that needs to change is that comparison should be set to "lnoravg".

Survival outcomes

There are several measures of effect size for survival outcomes, some of which are described by Mao et al. (2018). When using the Cox proportional hazards model, the quantity of interest is the hazard ratio (HR) between the treated and control groups. As with the OR, the HR is non-collapsible, which means the estimated HR will only be a valid estimate of the marginal HR when no other covariates are included in the model. Other effect measures, such as the difference in mean survival times or probability of survival after a given time, can be treated just like continuous and binary outcomes as previously described.

For the HR, we cannot compute average marginal effects and must use the coefficient on treatment in a Cox model fit without covariates. This means that we cannot use the procedures from the Standard Case. Here we describe estimating the marginal HR using coxph() from the survival package. (See help("coxph", package = "survival") for more information on this model.) Robust SEs for HRs were studied by Austin (2016) and were found to be conservative; they are requested automatically when weights are supplied to coxph(). Other formulas have been developed for estimating standard errors more accurately (Mao et al. 2018; Hajage et al. 2018), though Austin (2016) also found the bootstrap to be adequate.

library("survival")

#Cox Regression for marginal HR
coxph(Surv(Y_S) ~ A, data = d, weights = weights)
## Call:
## coxph(formula = Surv(Y_S) ~ A, data = d, weights = weights)
## 
##   coef exp(coef) se(coef) robust se z     p
## A 0.39      1.47     0.03      0.10 4 6e-05
## 
## Likelihood ratio test=151  on 1 df, p=<2e-16
## n= 2000, number of events= 2000

The coef column contains the log HR, and exp(coef) contains the HR. Remember to always use the robust se for the SE of the log HR. The displayed z-test p-value results from using the robust SE.

Using the survey package

The survey package has often been recommended for use in estimating treatment effects after propensity score weighting. When combined with marginaleffects functions, it yields identical estimates and similar standard errors to the methods described above that use lm() or glm() and manually request a robust SE. The main benefits of using survey are that you don’t need to request robust SEs in the call to avg_comparisons() and you can incorporate other survey design information, e.g., if your study comes from a complex survey.

Below is how you would adjust the standard case to use survey instead:

library("survey")

#Declare a survey design using the estimated weights
des <- svydesign(~1, weights = ~weights, data = d)

#Fit the outcome model
fit <- svyglm(Y_C ~ A * (X1 + X2 + X3 + X4 + X5 + 
                           X6 + X7 + X8 + X9),
              design = des)

#G-computation for the difference in means
avg_comparisons(fit,
                variables = "A",
                wts = "(weights)")
## 
##  Term Contrast Estimate Std. Error    z Pr(>|z|) 2.5 % 97.5 %
##     A    1 - 0     1.81      0.356 5.08   <0.001  1.11   2.51
## 
## Columns: term, contrast, estimate, std.error, statistic, p.value, conf.low, conf.high

Note that we need to supply "(weights)" to the wts argument for them to be correctly incorporated. You might notice that the standard errors computed using survey are different from those using lm() or glm() and HC3 robust SEs; to get similar SEs using lm() or glm(), set vcov = "HC0" in the call to avg_comparisons(). This approach is not recommended because HC3 standard errors tend to perform better, which is one reason to avoid using survey. In many cases, though, the results will be similar.

Multi-Category Treatments

Multi-category treatments work essentially the same way as binary treatments. The main practical differences are in choosing the estimand and estimating the weights. The ATE and ATO are straightforward. The ATT requires choosing one group to be the treated or “focal” group. Effects are the estimated for members of that group. The contrast in the focal group between the expected potential outcomes under a non-focal treatment and the expected potential outcomes for the focal (actual) treatment can be interpreted similarly to how ATTs are interpreted for binary treatments. The contrasts in the focal group between expected potential outcomes under a pair of non-focal treatments can be interpreted as the contrast between the ATTs of the non-focal treatments. Weights may be estimated differently for multi-category treatments from binary treatments; see the individual methods pages for how they differ.

Below, we’ll estimate the ATTs of the multi-category treatment Am for a focal level.

table(d$Am)
## 
##  C1  C2   T 
## 751 801 448

We have a focal treatment group, "T", and two control groups "C1" and "C2". We expect the ATTs for the two control groups to be the same since we assigned them randomly from within the original control group.

First, we estimate our weights using entropy balancing (Hainmueller 2012), identifying the focal group using focal:

W <- weightit(Am ~ X1 + X2 + X3 + X4 + X5 + 
                X6 + X7 + X8 + X9, data = d,
              method = "ebal", estimand = "ATT",
              focal = "T")
W
## A weightit object
##  - method: "ebal" (entropy balancing)
##  - number of obs.: 2000
##  - sampling weights: none
##  - treatment: 3-category (C1, C2, T)
##  - estimand: ATT (focal: T)
##  - covariates: X1, X2, X3, X4, X5, X6, X7, X8, X9

Typically one would assess the performance of the weights (balance and effective sample size) but we will skip that for now. Next, we fit the outcome model and perform weighted g-computation. We use avg_predictions() first to compute the expected potential outcome under each treatment for the focal group, and the use hypotheses() to test all pairwise comparisons.

#Bring weights into the dataset
d$weights <- W$weights

#Fit the outcome model
fit <- lm(Y_C ~ Am * (X1 + X2 + X3 + X4 + X5 + 
                        X6 + X7 + X8 + X9),
          data = d, weights = weights)

#G-computation
p <- avg_predictions(fit,
                     variables = "Am",
                     vcov = "HC3",
                     newdata = subset(boot_data, A == 1),
                     wts = "weights")
p
## 
##  Am Estimate Std. Error    z Pr(>|z|) 2.5 % 97.5 %
##  C1     1.73      0.275 6.27   <0.001 1.187   2.27
##  C2     1.40      0.233 6.00   <0.001 0.942   1.86
##  T      2.98      0.320 9.30   <0.001 2.350   3.60
## 
## Columns: Am, estimate, std.error, statistic, p.value, conf.low, conf.high
hypotheses(p, "revpairwise")
## 
##     Term Estimate Std. Error      z Pr(>|z|)  2.5 % 97.5 %
##  C2 - C1   -0.327      0.353 -0.925  0.35479 -1.020  0.366
##  T - C1     1.251      0.410  3.055  0.00225  0.448  2.054
##  T - C2     1.578      0.396  3.984  < 0.001  0.802  2.355
## 
## Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high

We find significant ATTs between the focal treatment and control levels (T - C1 and T - C2), but the difference between control levels (C2 - C1), which can be interpreted as the difference between these ATTs, is nonsignificant, as expected. Remember, for the ATT, bootstrapping usually provides more valid inference than the robust SEs used here.

Continuous Treatments

For continuous treatments, one estimand of interest is the average dose-response function (ADRF), which links the value of the treatment to the expected potential outcome under that treatment value across the full sample. We do not provide a detailed account of the ADRF but will demonstrate how to estimate weights that balance covariates with respect to a continuous treatment and how to estimate and plot the ADRF in the weighted sample. We’ll use the continuous treatment Ac, which ranges from -10.37 to 6.26, and estimate its effect on the continuous outcome Y_C.

First, we’ll estimate distance covariance optimal weights (Huling, Greifer, and Chen 2023) using weightit().

W <- weightit(Ac ~ X1 + X2 + X3 + X4 + X5 + 
                X6 + X7 + X8 + X9, data = d,
              method = "energy")
W
## A weightit object
##  - method: "energy" (energy balancing)
##  - number of obs.: 2000
##  - sampling weights: none
##  - treatment: continuous
##  - covariates: X1, X2, X3, X4, X5, X6, X7, X8, X9

Typically one would assess the performance of the weights (balance and effective sample size) but we will skip that for now. Next, we fit the outcome model and perform weighted g-computation. For the outcome model, we will use a natural cubic spline on Ac with 4 df. You can use any transformation of the treatment, e.g., a polynomial transformation using poly(), as long as it is flexible enough to capture any possible ADRF; purely nonparametric methods like kernel regression can be used as well, but inference for them is more challenging. Covariates can be included in the model, but with many covariates, many basis functions for the treatment, and a full set of treatment-covariate interactions, the resulting estimates may be imprecise.

#Bring weights into the dataset
d$weights <- W$weights

#Fit the outcome model
fit <- lm(Y_C ~ splines::ns(Ac, df = 4) *
            (X1 + X2 + X3 + X4 + X5 + 
               X6 + X7 + X8 + X9),
          data = d, weights = weights)

Next we use avg_predictions() first to compute the expected potential outcome under a representative set of treatment values. We’ll examine 51 treatment values from the 10th to 90th percentiles of Ac because estimates outside those ranges tend to be imprecise.

#Represenative values of Ac:
values <- with(d, seq(quantile(Ac, .1),
                      quantile(Ac, .9),
                      length.out = 51))

#G-computation
p <- avg_predictions(fit,
                     variables = list(Ac = values),
                     vcov = "HC3",
                     wts = "weights")

Although one can examine the expected potential outcomes, it is often more useful to see them plotted. We can generate a plot of the ADRF and its pointwise confidence band using ggplot2:

library("ggplot2")
ggplot(p, aes(x = Ac)) +
  geom_line(aes(y = estimate)) +
  geom_ribbon(aes(ymin = conf.low, ymax = conf.high),
              alpha = .3) +
  labs(x = "Ac", y = "E[Y|A]") +
  theme_bw()

We can see that the ADRF has an elbow-shape, with a zone of no evidence of treatment effect followed a zone of increasing values of the outcome as the treatment increases.

Another way to characterize the effect of continuous treatments is to examine the average marginal effect function (AMEF), which is a function that relates the value of treatment to the derivative of the ADRF. When this derivative is different from zero, there is evidence of a treatment effect at the corresponding value of treatment. Below, we use avg_slopes() to compute the pointwise derivatives of the ADRF across levels of Ac and then plot it.

# Estimate the pointwise derivatives at representative
# values of Ac
s <- avg_slopes(fit,
                variables = "Ac",
                newdata = datagridcf(Ac = values),
                by = "Ac",
                vcov = "HC3",
                wts = "weights")

# Plot the AMEF
ggplot(s, aes(x = Ac)) +
  geom_line(aes(y = estimate)) +
  geom_ribbon(aes(ymin = conf.low, ymax = conf.high),
              alpha = .3) +
  geom_hline(yintercept = 0, linetype = "dashed") +
  labs(x = "Ac", y = "dE[Y|A]/dA") +
  theme_bw()

We can see that between values of -1 and .5, there is evidence of a positive effect of Ac on Y_C (i.e., because the confidence intervals for the slope of the ADRF at those values of Ac exclude 0). This is in line with our observation above that the treatment only appears to have an effect at higher values of Ac.

Longitudinal Treatments

Coming soon! For now see the example at vignette("WeightIt").

Using Bootstrapping to Estimate Confidence Intervals

The bootstrap is an alternative to the delta method for estimating confidence intervals for estimated effects. See the section Bootstrapping above for details. Here, we’ll demonstrate the standard bootstrap, which involves resampling units and estimating the weights and treatment effect within each bootstrap sample. For both, we will use functionality in the boot package. It is critical to set a seed using set.seed() prior to performing the bootstrap in order for results to be replicable.

For the standard bootstrap, we need a function that takes in the original dataset and a vector of sampled unit indices and returns the estimated quantity of interest. This function should estimate the weights in the bootstrap sample, fit the outcome model, and estimate the treatment effect using g-computation. We’ll consider the marginal RR ATT of A on the binary outcome Y_B.

The first step is to write the estimation function, we call boot_fun. This function returns the marginal RR. In it, we estimate the weights and the effect and return the estimate of interest.

boot_fun <- function(data, i) {
  boot_data <- data[i,]
  
  #PS weighting for the ATE
  W <- weightit(A ~ X1 + X2 + X3 + X4 + X5 + 
                 X6 + X7 + X8 + X9,
               data = boot_data,
               method = "glm", estimand = "ATT")
  
  #Bring weights into the dataset
  boot_data$weights <- W$weights
  
  #Fit outcome model
  fit <- glm(Y_B ~ A * (X1 + X2 + X3 + X4 + X5 + 
                 X6 + X7 + X8 + X9),
             data = boot_data, weights = weights,
             family = quasibinomial)
  
  #G-computation
  comp <- avg_comparisons(fit,
                          variables = "A",
                          vcov = FALSE,
                          newdata = subset(boot_data, A == 1),
                          wts = "weights",
                          comparison = "lnratioavg",
                          transform = "exp")
  
  comp$estimate
}

Next, we call boot::boot() with this function and the original dataset supplied to perform the bootstrapping. We’ll request 199 bootstrap replications here, but in practice you should use many more, upwards of 999. More is always better. Using more also allows you to use the bias-corrected and accelerated (BCa) bootstrap confidence intervals (which you can request by setting type = "bca" in the call to boot.ci()), which are known to be the most accurate. See ?boot.ci for details. Here, we’ll just use a percentile confidence interval.

library("boot")
set.seed(54321)
boot_out <- boot(d, boot_fun, R = 199)

boot_out
## 
## ORDINARY NONPARAMETRIC BOOTSTRAP
## 
## 
## Call:
## boot(data = d, statistic = boot_fun, R = 199)
## 
## 
## Bootstrap Statistics :
##     original    bias    std. error
## t1*    1.477 -0.008648      0.1108
boot.ci(boot_out, type = "perc")
## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
## Based on 199 bootstrap replicates
## 
## CALL : 
## boot.ci(boot.out = boot_out, type = "perc")
## 
## Intervals : 
## Level     Percentile     
## 95%   ( 1.274,  1.693 )  
## Calculations and Intervals on Original Scale
## Some percentile intervals may be unstable

We find a RR of 1.477 with a confidence interval of (1.274, 1.693). If we had wanted a risk difference, we could have adjusted the arguments to avg_comparisons() (i.e., by removing the comparison and transform arguments).

To estimate the bootstrap confidence for the HR, we can do the same as above except our boot_fun will look like the following:

boot_fun <- function(data, i) {
  boot_data <- data[i,]
  
  #PS weighting for the ATE
  W <- weightit(A ~ X1 + X2 + X3 + X4 + X5 + 
                 X6 + X7 + X8 + X9,
               data = boot_data,
               method = "glm", estimand = "ATT")
  
  #Bring weights into the dataset
  boot_data$weights <- W$weights
  
  #Fit outcome model
  fit <- coxph(Surv(Y_S) ~ A, data = boot_data,
               weights = weights)
  
  #Return the coefficient on treatment
  coef(fit)["A"]
}

(Remember that you will need to load in the survival package before doing so.)

Reporting Results

It is important to be as thorough and complete as possible when describing the methods of estimating the treatment effect and the results of the analysis. This improves transparency and replicability of the analysis. Results should at least include the following:

All this is in addition to information about the weighting method, estimand, propensity score estimation procedure (if used), balance assessment, etc.

References

Austin, Peter C. 2016. “Variance Estimation When Using Inverse Probability of Treatment Weighting (IPTW) with Survival Analysis.” Statistics in Medicine 35 (30): 5642–55. https://doi.org/10.1002/sim.7084.
———. 2022. “Bootstrap Vs Asymptotic Variance Estimation When Using Propensity Score Weighting with Continuous and Binary Outcomes.” Statistics in Medicine 41 (22): 4426–43. https://doi.org/10.1002/sim.9519.
Chan, Kwun Chuen Gary, Sheung Chi Phillip Yam, and Zheng Zhang. 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. https://doi.org/10.1111/rssb.12129.
Efron, B., and R. Tibshirani. 1986. “Bootstrap Methods for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy.” Statistical Science 1 (1): 54–75. https://www.jstor.org/stable/2245500.
Greifer, Noah, and Elizabeth A. Stuart. 2021. “Choosing the Estimand When Matching or Weighting in Observational Studies.” arXiv:2106.10577 [Stat], June. https://arxiv.org/abs/2106.10577.
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.
Hajage, David, Guillaume Chauvet, Lisa Belin, Alexandre Lafourcade, Florence Tubach, and Yann De Rycke. 2018. “Closed-Form Variance Estimator for Weighted Propensity Score Estimators with Survival Outcome.” Biometrical Journal 60 (6): 1151–63. https://doi.org/10.1002/bimj.201700330.
Ho, Daniel E., Kosuke Imai, Gary King, and Elizabeth A. Stuart. 2007. “Matching as Nonparametric Preprocessing for Reducing Model Dependence in Parametric Causal Inference.” Political Analysis 15 (3): 199–236. https://doi.org/10.1093/pan/mpl013.
Huling, Jared D., Noah Greifer, and Guanhua Chen. 2023. “Independence Weights for Causal Inference with Continuous Treatments.” arXiv. https://doi.org/10.48550/arXiv.2107.07086.
MacKinnon, James G. 2006. “Bootstrap Methods in Econometrics.” Economic Record 82 (s1): S2–18. https://doi.org/10.1111/j.1475-4932.2006.00328.x.
MacKinnon, James G, and Halbert White. 1985. “Some Heteroskedasticity-Consistent Covariance Matrix Estimators with Improved Finite Sample Properties.” Journal of Econometrics 29 (3): 305–25. https://doi.org/10.1016/0304-4076(85)90158-7.
Mao, Huzhang, Liang Li, Wei Yang, and Yu Shen. 2018. “On the Propensity Score Weighting Analysis with Survival Outcome: Estimands, Estimation, and Inference.” Statistics in Medicine 37 (26): 3745–63. https://doi.org/10.1002/sim.7839.
Nguyen, Tri-Long, Gary S. Collins, Jessica Spence, Jean-Pierre Daurès, P. J. Devereaux, Paul Landais, and Yannick Le Manach. 2017. “Double-Adjustment in Propensity Score Matching Analysis: Choosing a Threshold for Considering Residual Imbalance.” BMC Medical Research Methodology 17: 78. https://doi.org/10.1186/s12874-017-0338-0.
Reifeis, Sarah A., and Michael G. Hudgens. 2020. “On Variance of the Treatment Effect in the Treated Using Inverse Probability Weighting.” arXiv:2011.11874 [Stat], November. https://arxiv.org/abs/2011.11874.
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.
Schafer, Joseph L., and Joseph Kang. 2008. “Average Causal Effects from Nonrandomized Studies: A Practical Guide and Simulated Example.” Psychological Methods 13 (4): 279–313. https://doi.org/10.1037/a0014268.
Snowden, Jonathan M., Sherri Rose, and Kathleen M. Mortimer. 2011. “Implementation of G-Computation on a Simulated Data Set: Demonstration of a Causal Inference Technique.” American Journal of Epidemiology 173 (7): 731–38. https://doi.org/10.1093/aje/kwq472.
Westreich, Daniel, and Sander Greenland. 2013. “The Table 2 Fallacy: Presenting and Interpreting Confounder and Modifier Coefficients.” American Journal of Epidemiology 177 (4): 292–98. https://doi.org/10.1093/aje/kws412.

Code to Generate Data used in Examples

#Generating data similar to Austin (2009) for demonstrating treatment effect estimation
gen_X <- function(n) {
  X <- matrix(rnorm(9 * n), nrow = n, ncol = 9)
  X[,5] <- as.numeric(X[,5] < .5)
  X
}

gen_Ac <- function(X) {
  LP_A <- -1.2 + log(2)*X[,1] - log(1.5)*X[,2] + log(2)*X[,4] - log(2.4)*X[,5] + log(2)*X[,7] - log(1.5)*X[,8]
  LP_A + rlogis(nrow(X))
}

#~20% treated
gen_A <- function(Ac) {
  1 * (Ac > 0)
}

gen_Am <- function(A) {
  factor(ifelse(A == 1, "T", sample(c("C1", "C2"), length(A), TRUE)))
}

# Continuous outcome
gen_Y_C <- function(A, X) {
  2*A + 2*X[,1] + 2*X[,2] + 2*X[,3] + 1*X[,4] + 2*X[,5] + 1*X[,6] + rnorm(length(A), 0, 5)
}
#Conditional:
#  MD: 2
#Marginal:
#  MD: 2

# Binary outcome
gen_Y_B <- function(A, X) {
  LP_B <- -2 + log(2.4)*A + log(2)*X[,1] + log(2)*X[,2] + log(2)*X[,3] + log(1.5)*X[,4] + log(2.4)*X[,5] + log(1.5)*X[,6]
  P_B <- plogis(LP_B)
  rbinom(length(A), 1, P_B)
}
#Conditional:
#  OR:   2.4
#  logOR: .875
#Marginal:
#  RD:    .144
#  RR:   1.54
#  logRR: .433
#  OR:   1.92
#  logOR  .655

# Survival outcome
gen_Y_S <- function(A, X) {
  LP_S <- -2 + log(2.4)*A + log(2)*X[,1] + log(2)*X[,2] + log(2)*X[,3] + log(1.5)*X[,4] + log(2.4)*X[,5] + log(1.5)*X[,6]
  sqrt(-log(runif(length(A)))*2e4*exp(-LP_S))
}
#Conditional:
#  HR:   2.4
#  logHR: .875
#Marginal:
#  HR:   1.57
#  logHR: .452

set.seed(19599)

n <- 2000
X <- gen_X(n)
Ac <- gen_Ac(X)
A <- gen_A(Ac)
Am <- gen_Am(A)

Y_C <- gen_Y_C(A, X)
Y_B <- gen_Y_B(A, X)
Y_S <- gen_Y_S(A, X)

d <- data.frame(A, Am, Ac, X, Y_C, Y_B, Y_S)

  1. These can be computed with M-estimation using the geex package or automatically using the PSweight package.↩︎

  2. Sometimes, an error will occur with this method, which usually means more bootstrap replications are required. The number of replicates must be greater than the original sample size.↩︎

  3. To verify that they are equal, supply the output of avg_predictions() to hypotheses(), e.g., avg_predictions(…) |> hypotheses("revpairwise"); this explicitly compares the average potential outcomes and should yield identical estimates to the avg_comparisons() call.↩︎

  4. We use quasibinomial() instead of binomial() simply to avoid a spurious warning; the results will be identical regardless.↩︎

  5. Note that for low or high average expected risks computed with predictions(), the confidence intervals may go below 0 or above 1; this is because an approximation is used. To avoid this problem, bootstrapping or simulation-based inference can be used instead.↩︎