Estimating Effects After 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) (Vansteelandt and Keiding 2011). Second, when we take the average of the estimated potential outcomes under each treatment level, if an estimand other than the ATT, ATC, or ATE is used or sampling weights are included, this must be a weighted average that incorporates the weights¹. 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) (Wang, Nianogo, and Arah 2017).

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. Note that the purpose of including the outcome model is not to arrive at a doubly robust estimator (i.e., one that is consistent if either the outcome or propensity score model is correct); rather, it is simply to increase the precision of the weighted estimate essentially for free. To take advantage of this feature, it is important to use a canonical link (i.e., the default link for a given family), as recommended by Gabriel et al. (2024).

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 and are implement in WeightIt when available. These methods rely on M-estimation (Stefanski and Boos 2002; Ross et al. 2024), a method of combining estimation of multiple models to adjust the standard errors for their joint estimation. For other methods, one must rely on an approximation that treats the weights as fixed and known, which can either yield conservative or anti-conservative standard errors depending on the method and estimand (Austin 2022; Reifeis and Hudgens 2020). Alternatively, one can use the bootstrap, which tends to have good performance regardless of the situation but can be computationally intensive for large datasets or slow weight estimation methods (Austin 2022). We describe the correct asymptotic standard errors, the robust standard errors treating the weights as fixed, and the bootstrap below.

The Standard Case: Binary Treatment with Asymptotically Correct SEs

For many weighting methods, estimating the effect after weighting is straightforward and involves fitting a model for the outcome that incorporates the estimated weights using lm_weightit() or glm_weightit(), then estimating the treatment effect using g-computation (i.e., using marginaleffects::avg_comparisons()). This procedure is the same for continuous and binary outcomes with and without covariates. This method uses the asymptotically correct M-estimation-based SEs when available and robust SEs otherwise or bootstrap standard errors if requested.

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, continuous, and sequential 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.

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

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

avg_comparisons(fit, variables = "A")

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")

We can see that the difference in potential outcome means is equal to the average treatment effect computed previously². 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.

newdata = subset(d, A == 1)

wts = W$weights

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

#Compute effects; RR and confidence interval
avg_comparisons(fit,
                variables = "A",
                comparison = "lnratioavg",
                transform = "exp")

library("survival")

#Cox Regression for marginal HR
coxph(Surv(Y_S) ~ A, data = d, weights = W$weights)

## Call:
## coxph(formula = Surv(Y_S) ~ A, data = d, weights = W$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

#Estimate the balancing weights, with sampling weights called "sw"
W <- weightit(A ~ X1 + X2 + X3 + X4 + X5 + 
                X6 + X7 + X8 + X9, data = d,
              method = "glm", estimand = "ATE",
              s.weights = "sw")

#Fit the outcome model, with clustering variable "clu"
fit <- glm_weightit(Y_C ~ A * (X1 + X2 + X3 + X4 + X5 + 
                           X6 + X7 + X8 + X9),
                    data = d, weightit = W,
                    cluster = ~clu)

#Compute the ATE, include sampling weights in the estimation
avg_comparisons(fit,
                variables = "A",
                wts = W$s.weights)

library(survey)

#Multiply sampling weights and estimated weights
d$weights <- W$weights * d$sw

#Declare a survey design using the combined weights with
#appropriate clustering
des <- svydesign(~clu, 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, including sampling weights
avg_comparisons(fit,
                variables = "A",
                wts = d$sw)

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.

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

#G-computation
p <- avg_predictions(fit,
                     variables = "Am",
                     newdata = subset(d, Am == "T"))
p

hypotheses(p, "revpairwise")

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.

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 entropy balancing weights (Vegetabile et al. 2021) using weightit().

W <- weightit(Ac ~ X1 + X2 + X3 + X4 + X5 + 
                X6 + X7 + X8 + X9, data = d,
              moments = 2, int = TRUE,
              method = "ebal")
W

## A weightit object
##  - method: "ebal" (entropy 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.

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

Next we use avg_predictions() first to compute the expected potential outcome under a representative set of treatment values. We’ll examine 31 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 = 31))

#G-computation
p <- avg_predictions(fit,
                     variables = list(Ac = values))

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^[You can also use plot_predictions(), though after requesting the predictions in the prior step it is quicker to use ggplot().]:

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 by 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 it⁴.

# Estimate the pointwise derivatives at representative
# values of Ac
s <- avg_slopes(fit,
                variables = "Ac",
                newdata = datagrid(Ac = values,
                                   grid_type = "counterfactual"),
                by = "Ac")

# 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 -1.8 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

For longitudinal treatments, estimation proceeds as usual, except that we estimate weights using weightitMSM() and fit a weighted outcome model that includes treatment at all time periods (and, optionally, and covariates measured prior to the first treatment). It is important not to include any covariates possibly caused by any of the treatments to avoid any bias; this is the whole point of using weighting to estimate the marginal structural model in the first place.

glm_weightit() makes it easy to incorporate the weights and account for their estimation in computing the parameter variance matrix, if desired. As with other methods, when the components required for M-estimation are included in the weightitMSM() output, they will be used to estimate standard errors that adjust for estimation of the weights. Otherwise, the weights can be treated as fixed or the whole estimation can be done through bootstrapping.

Below, we demonstrate using the usual inverse probability weights for a marginal structural model. Because these weights are computed from propensity scores estimated with logistic regression, we can use M-estimation to adjust for their estimation in computing the parameter covariance matrix. This also includes estimation of the standardization factor, if any. We’ll use the toy dataset msmdata that comes with WeightIt.

data("msmdata")

Wmsm <- 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)

Next we’ll fit the outcome model using glm_weightit(), which includes the baseline covariates (i.e., only those measured prior the first treatment).

fit <- glm_weightit(Y_B ~ A_1 * A_2 * A_3 * (X1_0 + X2_0),
                    data = msmdata, weightit = Wmsm,
                    family = binomial)

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

(p <- avg_predictions(fit,
                      variables = c("A_1", "A_2", "A_3")))

We can compare individual predictions using marginaleffects::hypotheses(). For example, to compare all treatment histories to just the first treatment history (i.e., in which all units are untreated for all time periods), we can run the following:

hypotheses(p, "reference")