Why This Adjustment Is Not Like Adjusting for a Baseline Confounder
Suppose a treatment changes a clinical variable, such as an early side effect, which then affects both a later mediator and the outcome. The early side effect is a mediator–outcome confounder, but it is also caused by treatment. Adjusting for it may be necessary to compare mediator levels without confounding within its strata. Yet the same adjustment can remove part of the treatment effect that a natural direct or indirect effect is intended to include.
The difficulty is not simply that adjustment is “bad.” It is that an exposure-induced confounder has two causal roles: it helps determine the mediator and outcome, and it carries effects of the exposure. A method that treats it as if it were measured before treatment can control one role while mishandling the other.
For contrast, a baseline confounder is measured before treatment and can be used to control confounding without itself being a consequence of treatment. Under the relevant assumptions, the natural-effect mediation formula can standardize over baseline confounders while comparing mediator distributions across treatment levels. Substituting a post-treatment \(L\) into that role is not automatically valid: the distribution of \(L\) changes with treatment, and \(L\) itself lies on causal pathways from \(A\) to \(Y\).
Counterfactuals and the Adjustment Target
Write \(L(a)\) for the value of \(L\) under treatment level \(a\), and \(M(a,l)\) for the mediator value under treatment \(a\) and \(L\) set to \(l\). Then \(M(a)=M(a,L(a))\) is the mediator under treatment \(a\), allowing \(L\) to take its corresponding treatment-specific value. Write \(Y(a,l,m)\) for the outcome under treatment \(a\), \(L\) set to \(l\), and mediator set to \(m\). For compactness, define \(Y(a,m)=Y(a,L(a),m)\).
For binary treatment, one common definition of the natural direct effect (NDE) and natural indirect effect (NIE), on the mean-difference scale, is:
In \(Y(1,M(0))\), the outcome process uses \(L(1)\), while the mediator is the value \(M(0,L(0))\). Thus the natural effect combines counterfactuals from different treatment conditions. If one simply conditions on observed \(L\), the comparison instead holds its observed value fixed or averages over some selected distribution of \(L\). Neither operation automatically constructs the cross-treatment combination in the natural-effect definition.
The usual natural-effect formula with baseline covariates relies on the covariates being pre-treatment, along with appropriate consistency, positivity, and no-unmeasured-confounding conditions for treatment, mediator, and outcome. For an exposure-induced \(L\), conditioning on \(L\) does not satisfy the pre-treatment requirement. Even if treatment is randomized and all common causes are measured, ordinary regression adjustment for \(L\) does not, by itself, identify the natural effects.
Worked Example: A Binary Confounder Carries Treatment Effects
Worked Example: A Binary Confounder Carries Treatment Effects
Consider a hypothetical randomized study with binary \(A,L,M\). Assume independent uniform errors generate \(L\) and \(M\), and an independent mean-zero outcome error. The treatment-specific probabilities and outcome mean are:
These probabilities lie between zero and one for both treatment levels and both values of \(L\). Suppose the errors used to generate \(L\), \(M\), and \(Y\) are independent, so the example has no additional unmeasured common causes. From the specified model:
Because the outcome mean is additive, the total mean effect is:
For the NDE, hold the mediator at its control counterfactual value \(M(0)\), but allow \(L\) in the outcome process to change with treatment. The mediator contribution cancels in the contrast, giving \(2+3\{E[L(1)]-E[L(0)]\}=2+1.2=3.2\). For the NIE, hold treatment and its corresponding \(L(1)\) fixed and change the mediator from \(M(0)\) to \(M(1)\):
The effects sum to the total effect: \(3.2+1.8=5\). Now fit the conditional mean models implied by the example. Given \(A,L\), the change in mediator probability per unit change in \(A\) is \(0.2\); given \(A,L,M\), the outcome coefficient for \(M\) is \(5\), and the coefficient for \(A\) is \(2\). A conventional coefficient-product calculation therefore gives an indirect contrast \(0.2(5)=1\) and a conditional direct contrast of \(2\). These are not the natural effects \(1.8\) and \(3.2\). The conditional mediator contrast omits the \(A\to L\to M\) contribution, which is \(0.4(0.4)=0.16\) on the mediator scale, or \(5(0.16)=0.8\) on the outcome scale. The conditional direct contrast also omits the \(A\to L\to Y\) contribution, \(3(0.4)=1.2\).
This calculation illustrates the central failure: adjustment isolates treatment contrasts holding \(L\) fixed, while the natural effects include changes in \(L\) under treatment. The coefficients are well defined in this hypothetical model, but labeling them as natural effects would be incorrect.
Worked Example: Adjusting Can Help With Mediator–Outcome Confounding
Worked Example: Adjusting Can Help With Mediator–Outcome Confounding
Use the same hypothetical model and focus on participants with \(A=1\). Among them, \(P(L=1)=0.6\), and the mediator probabilities are \(P(M=1\mid L=0,A=1)=0.3\) and \(P(M=1\mid L=1,A=1)=0.7\). The joint probabilities for \(L\) and \(M=1\) are \(0.4(0.3)=0.12\) when \(L=0\), and \(0.6(0.7)=0.42\) when \(L=1\). Thus \(P(M=1\mid A=1)=0.54\), and:
For \(M=0\), the corresponding joint probabilities are \(0.4(0.7)=0.28\) and \(0.6(0.3)=0.18\), so:
The outcome mean under \(A=1\) is \(12+3L+5M\). Therefore the unadjusted difference in observed outcome means between mediator groups is:
This exceeds the causal effect of changing \(M\) from 0 to 1 while holding \(A\) and \(L\) fixed, which is 5 in the specified outcome model. The difference arises because the \(M=1\) group contains a larger proportion with \(L=1\), and \(L\) raises the outcome. Adjusting for \(L\) correctly removes this mediator–outcome confounding in this model: within either \(L\) stratum, the outcome contrast for \(M\) is \(5\). But that success does not make the adjusted contrast a natural indirect effect. It addresses the mediator–outcome comparison, not the treatment-induced changes in \(L\) that the natural effects also involve.
Why a Naive Mediation Formula Misses the Target
With baseline confounders \(C\), a natural-effect mediation formula can standardize outcome and mediator distributions over the same pre-treatment covariate distribution. The logic depends on \(C\) not changing when \(A\) changes. If \(L\) is inserted as though it were such a covariate, a formula might compare mediator distributions across treatment levels within \(L=l\), then average those comparisons over a chosen distribution of \(L\).
That calculation answers a conditional or standardized question, depending on the chosen averaging distribution. It does not generally reproduce \(M(0,L(0))\) together with the outcome-side value \(L(1)\) in \(Y(1,M(0))\). Averaging over \(L\) under control, under treatment, or a pooled distribution can yield different quantities. Selecting one of these distributions does not resolve the counterfactual mismatch merely by making the computation explicit.
A short algebraic decomposition makes clear why the natural effects add to the total effect under their definitions. Add and subtract \(E\{Y(1,M(0))\}\):
This is an algebraic identity for the stated counterfactual definitions; it is not a proof that either effect is identified from observed data. Identification requires assumptions and a valid observed-data formula. In particular, the identity does not license treating an exposure-induced \(L\) as baseline.
Common Mistake and Practical Interpretation
A common mistake is: “\(L\) confounds the mediator–outcome relation, so I should adjust for \(L\); after adjustment, the mediation formula identifies natural effects.” The first clause can be right while the conclusion is not. Adjustment may be needed to remove confounding in a conditional mediator–outcome comparison, but because \(L\) is affected by treatment, conditioning on it can block paths such as \(A\to L\to Y\) and \(A\to L\to M\to Y\). A conditional coefficient or product of coefficients then need not equal either natural effect.
The distinction also matters when interpreting a treatment coefficient from a regression that includes \(L\). That coefficient compares treatment levels at fixed \(L\) (and at any other included covariates); it is not automatically the total effect, which includes treatment-induced changes in \(L\). Depending on the causal structure, conditioning on a post-treatment variable can also introduce selection bias, for example when it is a common effect of treatment and another cause of the outcome. The direction and size of such bias are not determined by the word “adjustment.”
Check Your Understanding
Use the definitions and calculations above to check your reasoning.
- What makes \(L\) an exposure-induced mediator–outcome confounder rather than a baseline confounder?
- In the binary example, which treatment pathways account for the difference between the conditional coefficient-product result and the natural effects?
- Why does adjustment for \(L\) help with the mediator–outcome comparison in the second example?
- In \(Y(1,M(0))\), which treatment-specific value of \(L\) enters the outcome, and which treatment-specific value enters the mediator counterfactual?
- Why does the algebraic decomposition of the total effect not, by itself, establish that the natural effects are identified?