Conditional Interventional Effects Across Baseline Strata

Why Condition Interventional Effects on Baseline Strata?

An average mediation effect can conceal important differences between groups. For example, an exposure may change the mediator distribution more in one baseline group than another, or the outcome may respond differently to the mediator across groups. Stratum-specific interventional effects make these patterns visible while defining the mediator intervention within each baseline stratum.

Let \(A\) denote an exposure with levels \(a\) and \(a^\ast\), \(M\) a mediator, \(Y\) an outcome, and \(C\) a baseline variable defining the stratum of interest. Let \(L\) denote other measured baseline covariates. The target subgroup is \(C=c\). The analysis standardizes over the distribution of \(L\) in that subgroup rather than treating the observed covariate mix as incidental.

Definition: The conditional interventional mean \(\psi_c(a,a')\) is the expected outcome in the baseline stratum \(C=c\) if exposure were set to \(a\) and, within each covariate profile \(L=l\), the mediator were drawn from its distribution under exposure \(a'\). Thus, the mediator draw uses the conditional distribution of \(M\) given \(A=a'\), \(C=c\), and \(L=l\), and the result is averaged over the distribution of \(L\) among people with \(C=c\).

In potential-outcome language, this intervention assigns \(A=a\), then draws a mediator value from the specified conditional distribution independently of the individual’s own mediator potential value, given \(L\) and \(C\). Write \(Y(a,G_{a'|c,l})\) for the resulting outcome, where \(G_{a'|c,l}\) denotes that conditional random draw. The conditional interventional mean is \(E\{Y(a,G_{a'|c,L})\mid C=c\}\). It is a stochastic-intervention estimand, not a contrast involving the same person’s two incompatible mediator potential values.

\psi_c(a,a') = \int \int E(Y\mid A=a,M=m,C=c,L=l)\, dF(m\mid A=a',C=c,L=l)\, dF(l\mid C=c).

For a binary mediator, the inner integral is a weighted sum over its two values. For a continuous mediator, it is an integral over the conditional mediator density. The outer integral averages over baseline covariates \(L\) in the target stratum. If \(C\) contains all baseline covariates needed for this standardization, \(L\) can be omitted.

Stratum-Specific Direct and Indirect Effects

The interventional direct effect compares exposure levels while holding the mediator distribution at the reference exposure level. The interventional indirect effect compares mediator distributions while holding exposure at \(a\). These definitions use the additive scale of the outcome mean.

Definition: With \(a\) as the exposure level of interest and \(a^\ast\) as the reference level, the stratum-specific interventional direct effect is \(\psi_c(a,a^\ast)-\psi_c(a^\ast,a^\ast)\). The stratum-specific interventional indirect effect is \(\psi_c(a,a)-\psi_c(a,a^\ast)\). Their sum is the stratum-specific total effect \(\psi_c(a,a)-\psi_c(a^\ast,a^\ast)\).

\begin{aligned} \text{IDE}_c &=\psi_c(a,a^\ast)-\psi_c(a^\ast,a^\ast),\\ \text{IIE}_c &=\psi_c(a,a)-\psi_c(a,a^\ast),\\ \text{TE}_c &=\psi_c(a,a)-\psi_c(a^\ast,a^\ast) =\text{IDE}_c+\text{IIE}_c. \end{aligned}

The identity follows by adding the first two contrasts: the intermediate term \(\psi_c(a,a^\ast)\) appears once with a positive sign and once with a negative sign, so it cancels. This is an algebraic decomposition on the mean-difference scale; it does not assert that the direct and indirect effects are mechanisms that can be isolated without assumptions.

Identification and Estimation

The formula for \(\psi_c(a,a')\) expresses the target causal mean in terms of observed-data distributions. Its use requires consistency (observed outcomes and mediators agree with the corresponding interventions), no relevant interference between individuals, and adequate positivity for the exposure and mediator distributions in the covariate profiles being standardized. It also requires sufficient control of confounding.

Exposure exchangeability: conditional on \(C,L\), exposure assignment is independent of the relevant potential mediator and outcome values. Randomization can support this condition; in observational data it requires adequate adjustment for baseline confounders.
Mediator-outcome exchangeability: conditional on \(A,C,L\), there are no unmeasured common causes of \(M\) and \(Y\). The stated formula assumes the covariates used for this control are baseline covariates; a post-exposure mediator-outcome confounder requires a different longitudinal formulation.
Positivity: within covariate profiles represented in \(C=c\), the required exposure levels have support, and the mediator values assigned by the reference conditional distribution have support for estimating the outcome regression at exposure \(a\).

Under these conditions, the outcome regression \(E(Y\mid A=a,M=m,C=c,L=l)\) represents the mean outcome under the corresponding exposure and mediator values, and the observed conditional mediator distribution represents the specified stochastic intervention. Integrating their product as in the formula identifies the interventional mean. In practice, estimate the outcome regression and conditional mediator distribution, calculate the inner expectation for each target covariate profile, and average those values over the \(L\) distribution in the stratum. This is covariate-specific g-computation.

Fit the conditional models.
Estimate the outcome mean given exposure, mediator, and baseline covariates, and estimate the mediator distribution given exposure and baseline covariates.

Calculate each intervention mean.
For each target individual or covariate profile in \(C=c\), average the predicted outcome over the mediator distribution for \(a'\), while setting exposure to \(a\).

Standardize and contrast.
Average over the target stratum’s baseline covariate distribution to obtain \(\psi_c(a,a')\), then form the direct, indirect, or total contrast.

Worked Calculations

Worked Example: Binary Mediator Effects in Two Strata

Suppose a hypothetical study has binary exposure \(A\), binary mediator \(M\), and binary outcome \(Y\). There are no additional covariates \(L\). The following entries give mediator probabilities and outcome risks within two baseline strata. Assume the identification conditions stated above hold.

Stratum	\(P(M=1\mid A=0,C)\)	\(P(M=1\mid A=1,C)\)	\(E(Y\mid A=0,M=0,C)\)	\(E(Y\mid A=0,M=1,C)\)	\(E(Y\mid A=1,M=0,C)\)	\(E(Y\mid A=1,M=1,C)\)
\(C=0\)	0.20	0.60	0.10	0.30	0.20	0.50
\(C=1\)	0.50	0.80	0.20	0.50	0.30	0.70

For \(C=0\), the mean under \(A=1\) with the mediator distribution from \(A=0\) is

\psi_0(1,0) =(1-0.20)(0.20)+(0.20)(0.50) =0.16+0.10 =0.26.

Under \(A=1\) with the mediator distribution from \(A=1\), the mean is \(\psi_0(1,1)=(1-0.60)(0.20)+(0.60)(0.50)=0.08+0.30=0.38\). Under \(A=0\) with the mediator distribution from \(A=0\), it is \(\psi_0(0,0)=(1-0.20)(0.10)+(0.20)(0.30)=0.08+0.06=0.14\). Therefore, the direct effect is \(0.26-0.14=0.12\), and the indirect effect is \(0.38-0.26=0.12\). The total effect is \(0.38-0.14=0.24\), equal to their sum.

For \(C=1\), \(\psi_1(1,0)=(1-0.50)(0.30)+(0.50)(0.70)=0.15+0.35=0.50\). Also, \(\psi_1(1,1)=(1-0.80)(0.30)+(0.80)(0.70)=0.06+0.56=0.62\), and \(\psi_1(0,0)=(1-0.50)(0.20)+(0.50)(0.50)=0.10+0.25=0.35\). Thus, the direct effect is \(0.50-0.35=0.15\), the indirect effect is \(0.62-0.50=0.12\), and the total effect is \(0.62-0.35=0.27\). These effects are risk differences: for example, the \(C=0\) total effect is a 0.24 increase in outcome risk, or 24 percentage points.

The indirect effects happen to match in these hypothetical data, while the direct effects and total effects differ between strata. This is evidence of differing estimated effects on the risk-difference scale, not by itself evidence about why the differences occur.

Worked Example: Why the Mediator Distribution Must Track Baseline Covariates

Now consider one target stratum \(C=c\), with a binary baseline covariate \(L\) equally represented in that stratum. Suppose the mediator probabilities under \(A=0\) are 0.10 when \(L=0\) and 0.50 when \(L=1\). Under \(A=1\), they are 0.50 and 0.90, respectively. The hypothetical outcome risks are:

\(L\)	\(E(Y\mid A=1,M=0,L)\)	\(E(Y\mid A=1,M=1,L)\)	\(E(Y\mid A=0,M=0,L)\)	\(E(Y\mid A=0,M=1,L)\)
0	0.10	0.50	0.05	0.25
1	0.40	0.60	0.30	0.50

Use the \(A=0\) mediator distribution separately within each \(L\) level. For \(\psi_c(1,0)\), the \(L=0\) mean is \((1-0.10)(0.10)+(0.10)(0.50)=0.14\); the \(L=1\) mean is \((1-0.50)(0.40)+(0.50)(0.60)=0.50\). Standardizing equally over \(L\) gives \(\psi_c(1,0)=(0.14+0.50)/2=0.32\).

For \(\psi_c(1,1)\), the corresponding means are \((1-0.50)(0.10)+(0.50)(0.50)=0.30\) and \((1-0.90)(0.40)+(0.90)(0.60)=0.58\), so \(\psi_c(1,1)=(0.30+0.58)/2=0.44\). For \(\psi_c(0,0)\), the means are \((1-0.10)(0.05)+(0.10)(0.25)=0.07\) and \((1-0.50)(0.30)+(0.50)(0.50)=0.40\), giving \(\psi_c(0,0)=(0.07+0.40)/2=0.235\). The direct effect is \(0.32-0.235=0.085\), the indirect effect is \(0.44-0.32=0.12\), and the total effect is \(0.44-0.235=0.205\).

If instead one used a single pooled mediator probability under \(A=0\), it would be \((0.10+0.50)/2=0.30\). Applying that probability to both \(L\) groups would give the \(A=1\) outcome means \(0.70(0.10)+0.30(0.50)=0.22\) and \(0.70(0.40)+0.30(0.60)=0.46\), averaging to 0.34 rather than 0.32. The pooled draw defines a different intervention: it does not preserve the conditional mediator distribution associated with each baseline covariate profile.

Effect Heterogeneity and Target-Population Composition

A stratum-specific effect is a conditional causal contrast for a named baseline subgroup. Effect heterogeneity means that these contrasts differ across strata, on the chosen outcome scale. A marginal effect is different: it averages stratum-specific effects using the stratum proportions in a specified target population. If those proportions change, the marginal effect can change even when every stratum-specific effect stays fixed.

For a target population with stratum weights \(q(c)\), where \(q(c)\geq 0\) and \(\sum_c q(c)=1\), the standardized direct effect is \(\sum_c q(c)\text{IDE}_c\); the indirect and total effects are standardized in the same way. The weights describe the target population, not the strength of the causal effects within strata.

Worked Example: Changing Target Weights Without Changing Stratum Effects

Use the effects calculated in the first example: direct effects of 0.12 and 0.15 and indirect effects of 0.12 and 0.12 in \(C=0\) and \(C=1\), respectively. Suppose a first target population has 80% in \(C=0\) and 20% in \(C=1\). Its standardized direct effect is \(0.80(0.12)+0.20(0.15)=0.126\). Its indirect effect is \(0.80(0.12)+0.20(0.12)=0.12\), and its total effect is \(0.80(0.24)+0.20(0.27)=0.246\).

For a second target population with 20% in \(C=0\) and 80% in \(C=1\), the standardized direct effect is \(0.20(0.12)+0.80(0.15)=0.144\), the indirect effect remains \(0.12\), and the total effect is \(0.20(0.24)+0.80(0.27)=0.264\). No stratum-specific effect changed. The marginal direct and total effects differ because the target populations give different weights to strata with different effects.

Common Pitfall: Treating a Marginal Difference as Effect Modification

A common mistake is to see different marginal mediation estimates in two populations and conclude that the causal effect differs by baseline stratum. That conclusion does not follow: the populations may simply have different proportions in the strata. The correction is to first estimate and compare the stratum-specific contrasts using the same definitions and outcome scale. Then, if a marginal result is wanted, state the target weights used to standardize those contrasts. Conversely, comparing subgroup effects requires checking that each stratum has adequate support and that the same assumptions and estimand definitions apply.

Key takeaway: Conditional interventional effects combine two choices: the mediator distribution used within each baseline covariate profile and the baseline distribution over which results are averaged. Keep both choices explicit. Differences between stratum-specific effects concern effect heterogeneity; differences caused by averaging with different stratum weights concern target-population composition.

Check Your Understanding

Use the definitions and calculations above to answer the following.

In words, what does \(\psi_c(a,a')\) hold fixed, and what does it change?
Why is the mediator distribution conditioned on \(L\) in the g-formula?
For a binary mediator, explain how the inner integral in \(\psi_c(a,a')\) becomes a weighted sum.
If two target populations have different stratum proportions but identical stratum-specific effects, can their marginal effects differ? Explain.
What identification concern arises if a mediator-outcome confounder is affected by exposure?