2 Notation, Assumptions, and Sequential G-estimation

We use $A$ to denote treatment, $M$ to denote the mediator, $Y$ to denote the observed outcome, and $Y(a,m)$ to denote the potential outcome under treatment $a$ and mediator $m$ . The CDE is defined as the average effect of changing treatment from $a$ to $a^{\prime }$ while fixing the mediator at a given level $m$ :

$$\begin{eqnarray}\text{CDE}(a,a^{\prime },m)=\mathbb{E}[Y(a,m)-Y(a^{\prime },m)].\end{eqnarray}$$

This quantity is identified under the assumption of sequential ignorability (Robins Reference Robins and Berkane1997, VanderWeele and Vansteelandt Reference VanderWeele and Vansteelandt2009), which can be formally expressed in two parts as follows:

1. (1) $Y(a,m)\bot \!\!\!\bot A|X,\forall a,m$ (i.e., no unmeasured treatment–outcome confounders).

2. (2) $Y(a,m)\bot \!\!\!\bot M|X,A,Z,\forall a,m$ (i.e., no unmeasured mediator–outcome confounders).

Here, $X$ denotes a vector of observed pretreatment confounders, while $Z$ denotes a vector of observed intermediate confounders that affect both the mediator and outcome and that may be affected by treatment. The sequential ignorability assumption is satisfied in Figure 1, which contains a directed acyclic graph summarizing a set of hypothesized causal relationships between the variables outlined previously. In this figure, $Z$ is affected by $A$ , and thus it is both an intermediate confounder and also a mediator. Because we focus on the CDE controlling for $M$ only, we henceforth refer to $Z$ exclusively as a confounder for clarity. Of course, it is possible to define a CDE controlling for $M$ and $Z$ jointly, which would illuminate the mediating role of both variables taken together. Estimands involving multiple mediators, however, are beyond the scope of this letter, although the methods we consider below can be generalized for more complex analyses of this type.

Figure 1.

Causal Relationships under Sequential Ignorability Shown in a Direct Acyclic Graph. Note: $A$ denotes the treatment, $M$ denotes the mediator, $Y$ denotes the outcome, $X$ denotes the pretreatment confounders, and $Z$ denotes the intermediate confounders.

The CDE is distinct from several other estimands considered in analyses of causal mediation. For example, it is distinct from the average direct effect (ADE) considered in Imai et al. (Reference Imai, Keele, Tingley and Yamamoto2011), which is defined as

$$\begin{eqnarray}\text{ADE}(a,a^{\prime })=\mathbb{E}[Y(a,M(a))-Y(a^{\prime },M(a))],\end{eqnarray}$$

where $M(a)$ denotes the potential outcome for the mediator under treatment $a$ . In contrast to the CDE, the ADE represents the average effect of changing treatment from $a$ to $a^{\prime }$ while fixing the mediator for each unit at its value under treatment $a$ . The ADE is equal to the difference between the total effect and the average causal mediation effect (ACME), which is defined as

$$\begin{eqnarray}\text{ACME}(a,a^{\prime })=\mathbb{E}[Y(a^{\prime },M(a))-Y(a^{\prime },M(a^{\prime }))].\end{eqnarray}$$

In general, the CDE differs from the ADE, and thus the difference between the total effect and CDE differs from the ACME, as long as the unit-level direct effect $Y(a,m)-Y(a^{\prime },m)$ depends on $m$ for some units. We focus on the CDE because it is identified under much weaker assumptions than the ADE and ACME. In particular, the CDE can still be identified in the presence of intermediate confounders affected by treatment, unlike the ADE and ACME (VanderWeele and Vansteelandt Reference VanderWeele and Vansteelandt2009).

Although the CDE is identified under sequential ignorability, additional modeling assumptions are needed to estimate the CDE in finite samples. Sequential g-estimation, for example, relies on a linear model for the conditional mean of the outcome given $A$ , $M$ , $X$ , and $Z$ . Moreover, because sequential g-estimation is difficult to implement in the presence of intermediate interactions, its application in practice also typically relies on an additional simplifying assumption that the effect of the mediator on the outcome is not moderated by intermediate confounders, which can be formally expressed as follows:

$$\begin{eqnarray}\displaystyle & & \displaystyle \mathbb{E}[Y(a,m)-Y(a,m^{\prime })|X=x,A=a,Z=z]\nonumber\\ \displaystyle & & \displaystyle \qquad =\mathbb{E}[Y(a,m)-Y(a,m^{\prime })|X=x,A=a]\quad \text{for any }a,m,m^{\prime },x\text{ and }z.\nonumber\end{eqnarray}$$

In words, this assumption states that among the units exposed to treatment $a$ , the effect of the mediator on the outcome would not differ across the subgroups defined by the postreatment confounders within levels of the pretreatment confounders.

Under this assumption, Acharya, Blackwell, and Sen (Reference Acharya, Blackwell and Sen2016) illustrate sequential g-estimation of the CDE using the following model for the outcome:

(1) $$\begin{eqnarray}\mathbb{E}[Y|X=x,A=a,Z=z,M=m]=\unicode[STIX]{x1D6FD}_{0}+\unicode[STIX]{x1D6FD}_{1}^{T}x+\unicode[STIX]{x1D6FD}_{2}a+\unicode[STIX]{x1D6FD}_{3}^{T}z+m(\unicode[STIX]{x1D6FE}_{0}+\unicode[STIX]{x1D6FE}_{1}^{T}x+\unicode[STIX]{x1D6FE}_{2}a).\end{eqnarray}$$

With this model, sequential g-estimation proceeds in three steps:

1. (1) Compute least squares estimates for equation (1) and save $\hat{\unicode[STIX]{x1D6FE}}_{2}$ .

2. (2) Construct a “de-mediated” outcome defined as $Y_{d}=Y-M(\hat{\unicode[STIX]{x1D6FE}}_{0}+\hat{\unicode[STIX]{x1D6FE}}_{1}^{T}X+\hat{\unicode[STIX]{x1D6FE}}_{2}A)$ .

3. (3) Compute least squares estimates for a linear regression of $Y_{d}$ on $X$ and $A$ , which can be expressed as ${\hat{Y}}_{d}=\hat{\unicode[STIX]{x1D705}}_{0}+\hat{\unicode[STIX]{x1D705}}_{1}^{T}X+\hat{\unicode[STIX]{x1D705}}_{2}A$ .

The sequential g-estimate of the CDE is then given by

(2) $$\begin{eqnarray}\widehat{\text{CDE}}_{\text{SG}}(a,a^{\prime },m)=(\hat{\unicode[STIX]{x1D705}}_{2}+\hat{\unicode[STIX]{x1D6FE}}_{2}m)(a-a^{\prime }).\end{eqnarray}$$

This estimator is consistent under the assumptions of sequential ignorability and a correctly specified linear model for the outcome, which here requires that there must not be any effect moderation by the intermediate confounders. Standard errors can be obtained via the nonparametric bootstrap or a consistent variance estimator derived in Acharya, Blackwell, and Sen (Reference Acharya, Blackwell and Sen2016).

Figure 2.

The Logic of Sequential G-estimation. Note: $A$ denotes the treatment, $M$ denotes the mediator, $Y$ denotes the outcome, $X$ denotes the pretreatment confounders, and $Z$ denotes the intermediate confounders.

In Figure 2, we illustrate the logic of sequential g-estimation. First, under the identification and modeling assumptions outlined previously, the regression in step 1 identifies the causal effect of $M$ on $Y$ . Then, the “de-mediation” calculation in step 2 neutralizes the causal path from $M$ to $Y$ , while all other causal paths remain intact. Finally, the regression of the de-mediated outcome, $Y_{d}$ , on $X$ and $A$ in step 3 identifies the controlled direct effect of $A$ when $M=0$ , and because $\hat{\unicode[STIX]{x1D6FE}}_{2}$ is a consistent estimate of the treatment–mediator interaction effect, the CDE when $M=m$ can be estimated with equation (2).

3 Regression-with-Residuals Estimation

RWR estimation was originally developed to assess how time-varying covariates moderate the effect of time-varying treatments (Almirall, Ten Have, and Murphy Reference Almirall, Ten Have and Murphy2010, Wodtke and Almirall Reference Wodtke and Almirall2017). In this section, we show how RWR can be adapted to estimate CDEs while properly adjusting for intermediate confounders. Specifically, RWR estimation of the CDE based on a model without intermediate interactions, such as equation (1), proceeds in two steps:

1. (1) For each of the intermediate confounders, compute least squares estimates for a linear regression of $Z$ on $X$ and $A$ , and save the residuals, which we denote by $Z_{\bot }$ .

2. (2) Compute least squares estimates for a model similar to equation (1) but with $Z$ replaced by $Z_{\bot }$ , which can be expressed as ${\hat{Y}}=\tilde{\unicode[STIX]{x1D6FD}}_{0}+\tilde{\unicode[STIX]{x1D6FD}}_{1}^{T}X+\tilde{\unicode[STIX]{x1D6FD}}_{2}A+\tilde{\unicode[STIX]{x1D6FD}}_{3}^{T}Z_{\bot }+M(\tilde{\unicode[STIX]{x1D6FE}}_{0}+\tilde{\unicode[STIX]{x1D6FE}}_{1}^{T}X+\tilde{\unicode[STIX]{x1D6FE}}_{2}A)$ .

The RWR estimate of the CDE is then given by

(3) $$\begin{eqnarray}\widehat{\text{CDE}}_{\text{RWR}}(a,a^{\prime },m)=(\tilde{\unicode[STIX]{x1D6FD}}_{2}+\tilde{\unicode[STIX]{x1D6FE}}_{2}m)(a-a^{\prime }).\end{eqnarray}$$

As shown in Supplementary Material A, RWR and sequential g-estimation are algebraically equivalent (i.e., $\hat{\unicode[STIX]{x1D705}}_{2}=\tilde{\unicode[STIX]{x1D6FD}}_{2}$ ; $\hat{\unicode[STIX]{x1D6FE}}_{2}=\tilde{\unicode[STIX]{x1D6FE}}_{2}$ ) when there are no intermediate interactions. They rely on the same identification and modeling assumptions, and they share the same statistical properties.

In Figure 3, we illustrate the logic of RWR. First, residualizing the intermediate confounders in step 1 neutralizes the causal paths emanating from $X$ and $A$ to $Z$ . Then, the residualized confounders can be included in an outcome regression to adjust for mediator–outcome confounding while avoiding the bias that normally results from conditioning on posttreatment variables. RWR estimation avoids posttreatment bias because $Z_{\bot }$ is no longer a consequence of $A$ , and it avoids omitted variable bias because all confounders have been appropriately controlled in a model for the outcome.

Figure 3.

The Logic of Regression-with-residuals. Note: $A$ denotes the treatment, $M$ denotes the mediator, $Y$ denotes the outcome, $X$ denotes the pretreatment confounders, and $Z$ denotes the intermediate confounders.

4 Intermediate Interactions

In the models considered previously, the effect of the mediator on the outcome is assumed to be invariant across all intermediate confounders. This is a strong and arguably implausible assumption in many social science applications. When it is not satisfied, estimates of the CDE may be biased and inconsistent. Thus, methods that accommodate, rather than naively assume away, intermediate interactions will make analyses of causal mediation more robust. The main advantage of RWR over sequential g-estimation is the ease with which RWR can accommodate intermediate interactions (Wodtke, Alaca, and Zhou Reference Wodtke, Alaca and Zhou2018). Consider the following model, which extends equation (1) by including an interaction term between $M$ and $Z$ :

(4) $$\begin{eqnarray}\mathbb{E}[Y|X=x,A=a,Z=z,M=m]=\unicode[STIX]{x1D6FD}_{0}+\unicode[STIX]{x1D6FD}_{1}^{T}x+\unicode[STIX]{x1D6FD}_{2}a+\unicode[STIX]{x1D6FD}_{3}^{T}z+m(\unicode[STIX]{x1D6FE}_{0}+\unicode[STIX]{x1D6FE}_{1}^{T}x+\unicode[STIX]{x1D6FE}_{2}a+\unicode[STIX]{x1D6FE}_{3}^{T}z).\end{eqnarray}$$

With this model, sequential g-estimation can still be used to estimate the CDE at $m=0$ . The only modification to the sequential g-estimator in this situation is that the de-mediated outcome, $Y_{d}$ , is obtained by subtracting $M(\hat{\unicode[STIX]{x1D6FE}}_{0}+\hat{\unicode[STIX]{x1D6FE}}_{1}^{T}X+\hat{\unicode[STIX]{x1D6FE}}_{2}A+\hat{\unicode[STIX]{x1D6FE}}_{3}^{T}Z)$ instead of $M(\hat{\unicode[STIX]{x1D6FE}}_{0}+\hat{\unicode[STIX]{x1D6FE}}_{1}^{T}X+\hat{\unicode[STIX]{x1D6FE}}_{2}A)$ from the observed outcome. Then, $\widehat{\text{CDE}}_{\text{ SG}}(a,a^{\prime },0)=\hat{\unicode[STIX]{x1D705}}_{2}(a-a^{\prime })$ , where $\hat{\unicode[STIX]{x1D705}}_{2}$ is the coefficient on treatment from the regression of $Y_{d}$ on $X$ and $A$ .

Although sequential g-estimation can still be used to estimate the CDE at $m=0$ in the presence of intermediate interactions, we can no longer estimate the CDE at any general value of $m$ using equation (2). This is because $\hat{\unicode[STIX]{x1D6FE}}_{2}m(a-a^{\prime })$ is no longer a consistent estimate of the treatment–mediator interaction effect, as the inclusion of the term $\unicode[STIX]{x1D6FE}_{3}^{T}z$ in equation (4) leads to posttreatment bias in $\unicode[STIX]{x1D6FE}_{2}a$ . In other words, the de-mediation step only removes posttreatment bias in $\unicode[STIX]{x1D6FD}_{2}a$ but not in $\unicode[STIX]{x1D6FE}_{2}a$ .Footnote ¹

RWR estimation, by contrast, easily accommodates intermediate interactions, and its implementation in their presence remains almost exactly the same as before:

1. (1) For each of the intermediate confounders, compute least squares estimates for a linear regression of $Z$ on $X$ and $A$ , and save the residuals, denoted by $Z_{\bot }$ .

2. (2) Compute least squares estimates for a model similar to equation (4) but with $Z$ replaced by $Z_{\bot }$ , which can be expressed as

   $$\begin{eqnarray}{\hat{Y}}=\tilde{\unicode[STIX]{x1D6FD}}_{0}+\tilde{\unicode[STIX]{x1D6FD}}_{1}^{T}X+\tilde{\unicode[STIX]{x1D6FD}}_{2}A+\tilde{\unicode[STIX]{x1D6FD}}_{3}^{T}Z_{\bot }+M(\tilde{\unicode[STIX]{x1D6FE}}_{0}+\tilde{\unicode[STIX]{x1D6FE}}_{1}^{T}X+\tilde{\unicode[STIX]{x1D6FE}}_{2}A+\tilde{\unicode[STIX]{x1D6FE}}_{3}^{T}Z_{\bot }).\end{eqnarray}$$

The RWR estimate of the CDE is then given by

(5) $$\begin{eqnarray}\widehat{\text{CDE}}_{\text{RWR}}(a,a^{\prime },m)=(\tilde{\unicode[STIX]{x1D6FD}}_{2}+\tilde{\unicode[STIX]{x1D6FE}}_{2}m)(a-a^{\prime }),\end{eqnarray}$$

where $\tilde{\unicode[STIX]{x1D6FE}}_{2}$ remains a consistent estimate of the treatment–mediator interaction effect.Footnote ²

As shown in Supplementary Material B, equation (5) is a consistent estimator of the CDE under the assumptions of sequential ignorability and no model misspecification. RWR estimation remains consistent even in the presence of intermediate interactions because, by appropriately residualizing the intermediate confounders, it removes any posttreatment bias from the main effect of treatment and from the treatment–mediator interaction effect. RWR can also accommodate “baseline interactions” between treatment $A$ and the pretreatment confounders $X$ . In this situation, we need only recenter the pretreatment confounders at their sample means and then include the appropriate interaction terms in the outcome regression. Standard errors can be computed using the nonparametric bootstrap.

5 The Effect of Media Framing on Support for Immigration

To illustrate RWR, we reanalyze data from Brader, Valentino, and Suhay (Reference Brader, Valentino and Suhay2008) to estimate the CDE of negative media framing on public support for immigration, controlling for respondent anxiety potentially triggered by negative media cues.Footnote ³ With a nationally representative sample of 354 white non-Hispanic adults, Brader, Valentino, and Suhay (Reference Brader, Valentino and Suhay2008) conducted a survey experiment in which respondents were asked to read a mock news report on immigration. In this report, both the ethnicity of the featured immigrant and the tone of the story were randomly manipulated using a $2\times 2$ design. Specifically, respondents were presented with a story that featured either a white European immigrant or a Latino immigrant and that focused on either the benefits or the costs of immigration. After reading the story, respondents were asked to report their beliefs about the harms of immigration, their feelings about increased immigration, and their support for immigration. With these data, Brader, Valentino, and Suhay (Reference Brader, Valentino and Suhay2008) found that stories featuring both a Latino immigrant and a negative frame emphasizing the costs of immigration had a large negative effect on support for immigration. They also reported that a substantial proportion of this effect is mediated by respondents’ anxiety about increased immigration and that beliefs about the harms of immigration, as opposed to negative emotions, do not play an important mediating role. However, Brader, Valentino, and Suhay (Reference Brader, Valentino and Suhay2008) assessed the mediating role of beliefs and emotions separately under the assumption that respondent anxiety is not affected by perceptions of the harms associated with immigration, which seems unlikely and appears to be inconsistent with their own data (Imai and Yamamoto Reference Imai and Yamamoto2013). Thus, we treat beliefs about the harm of immigration as an intermediate confounder and reassess the mediating role of respondent anxiety using RWR and, for comparative purposes, sequential g-estimation.

Specifically, we estimate the CDE of negative media framing on support for immigration, controlling for respondent anxiety, using several variants of the following model:

(6) $$\begin{eqnarray}\mathbb{E}[Y|X=x,A=a,Z=z,M=m]=\unicode[STIX]{x1D6FD}_{0}+\unicode[STIX]{x1D6FD}_{1}^{T}x+\unicode[STIX]{x1D6FD}_{2}a+\unicode[STIX]{x1D6FD}_{3}z+m(\unicode[STIX]{x1D6FE}_{0}+\unicode[STIX]{x1D6FE}_{1}^{T}x+\unicode[STIX]{x1D6FE}_{2}a),\end{eqnarray}$$

where the outcome, $Y$ , is a measure of support for immigration on a five-point scale; the treatment, $A$ , denotes receipt of a news story featuring both a Latino immigrant and a negative frame emphasizing the costs of immigration; the mediator, $M$ , is the level of anxiety expressed by the respondent on a ten-point scale; the intermediate confounder, $Z$ , is a measure of the perceived harm of immigration on a seven-point scale; and finally, the vector of pretreatment covariates, $X$ , includes measures of gender, age, education, and income.Footnote ⁴ We control for a set of pretreatment covariates because, although treatment is randomly assigned, the mediator–outcome relationship may still be confounded by baseline factors in these data. To simplify interpretation, all variables except the treatment and outcome are centered at their sample means.

Table 1.

Estimated CDE of Media Framing on Support for Immigration using Sequential G-estimation, RWR, and RWR with Intermediate Interactions.

Note: Numbers in parentheses are bootstrapped standard errors (500 replications). For ease of interpretation, all predictors except the treatment are centered at their means. Coefficients of pretreatment covariates are omitted. Supplementary Material C presents the R code used to generate the results.

As a benchmark, the first two columns of Table 1 present an estimate of the total treatment effect from a regression of $Y$ on $X$ and $A$ as well as a “naive” estimate of the CDE from a regression model similar to equation (6) but without adjustments for the intermediate confounder $Z$ . Consistent with results reported by Brader, Valentino, and Suhay (Reference Brader, Valentino and Suhay2008), the estimated total effect indicates that negative media framing reduces support for immigration, and the naive estimate of the CDE suggests that about half of the total treatment effect is due to heightened anxiety.

The third and fourth columns of Table 1 present sequential g-estimates and RWR estimates, respectively, for the CDE based on model (6). As expected, the estimates given by these two methods are exactly the same ( $-0.33$ ) because there are no intermediate interactions in this model. Contrary to the naive estimate of the CDE discussed previously, these results suggest that less than one-quarter of the total treatment effect may be due to heightened respondent anxiety. This finding is consistent with estimates of the ADE reported by Imai and Yamamoto (Reference Imai and Yamamoto2013).

With sequential g-estimation, only the CDE at $m=0$ (i.e., when the level of respondent anxiety is set at its sample mean) is reported in the final step. To construct the CDE at other levels of the mediator, the analyst must extract the coefficient on the treatment–mediator interaction from the regression in step 1 of the procedure. With RWR, by contrast, all the coefficients required for constructing the CDE at any level of the mediator are reported in the single regression for the outcome. This allows an analyst to construct any CDE of interest directly from the results in Table 1. For example, when respondent anxiety is one standard deviation (2.77) above the sample mean, the CDE is estimated to be $(-0.33+2.77\ast 0.064)=-0.15$ .

Thus far, the effect of anxiety on support for immigration has been assumed to be invariant across levels of the intermediate confounder, but if this effect is in fact moderated by beliefs about the harms of immigration, then estimates reported previously may be biased. We now relax this assumption by additionally including an interaction term between the level of anxiety and the perceived harm of immigration when implementing RWR. Results from this analysis are shown in the last column of Table 1. The estimated CDE from this model at $m=0$ is $-0.31$ , which is similar to that obtained from the model without intermediate interactions. In this example, it appears that our findings are fairly robust to the exclusion of intermediate interactions. Nevertheless, it is the flexibility of RWR that allows us to easily assess the sensitivity of results to different specifications.

6 Concluding Remarks

In this letter, we introduced RWR for estimating controlled direct effects. In the absence of intermediate interactions, RWR is algebraically equivalent to the sequential g-estimator. But unlike the sequential g-estimator, RWR can easily accommodate several different types of effect moderation, including intermediate interactions, which are likely common in the social sciences. In general, models with less stringent parametric constraints can be estimated more easily with RWR than with sequential g-estimation.

Nevertheless, RWR is still premised on a number of strong modeling assumptions. In particular, RWR requires a correctly specified linear model for the outcome. In applications with many confounders or complex patterns of effect heterogeneity, the modeling assumptions required of RWR may be difficult to satisfy, and when they are violated, RWR is biased. Moreover, in applications where a linear model may be inappropriate (e.g., in analyses with binary outcomes), RWR does not generalize in a straightforward manner for use with nonlinear models. Thus, semiparametric methods, such as IPW estimation of MSMs or certain types of sensitivity analysis (e.g., Imai and Yamamoto Reference Imai and Yamamoto2013), may be preferable in applications with a large number of confounders, complex effect heterogeneity, or categorical outcomes.

These limitations notwithstanding, simulation studies indicate that g- and RWR estimation can still outperform IPW estimation even when the outcome model is misspecified, especially in applications with continuous treatments and/or mediators (Vansteelandt Reference Vansteelandt2009, Wodtke Reference Wodtke2018). RWR estimation can also be combined with a sensitivity analysis to assess the robustness of estimates to different violations of its motivating assumptions. Given its simplicity, flexibility, and relative efficiency, we expect that RWR will be widely used in causal mediation analyses.