Baseline Mediator-Outcome Confounding

Why Baseline Confounding Matters in Mediation

A mediator–outcome association does not, by itself, show that changing the mediator would change the outcome. A pretreatment variable may influence both. For example, baseline disease severity may affect whether a patient receives a treatment-related behavioral change and may also predict the later clinical outcome. If severity is ignored, the mediator distribution and the outcome distribution can be combined in a way that does not identify a natural effect.

The remedy for a measured baseline common cause is to condition on it when relating the mediator to the outcome and when describing the mediator distribution under each treatment level. Then average the resulting conditional quantities over the baseline covariate distribution in the target population. This is standardization: adjustment is done within covariate strata first, not by treating the pooled mediator and outcome distributions as if the covariate were absent.

The word “baseline” matters. The formula below concerns a covariate \(C\) that precedes treatment. It does not cover a mediator–outcome confounder that is itself changed by treatment. The standard mediation notation is \(M_a\), the mediator value that would occur under treatment \(a\), and \(Y_{a,m}\), the outcome that would occur under treatment \(a\) if the mediator were set to \(m\). The nested outcome \(Y_{a,M_{a'}}\) combines treatment \(a\) with the mediator value that would occur under treatment \(a'\).

The Covariate-Adjusted Mediation Formula

For treatment levels \(a\) and \(a'\), define the nested mean \(E[Y_{a,M_{a'}}]\). The mediation formula with baseline covariates expresses this mean using observed conditional outcome means, observed conditional mediator distributions, and the covariate distribution in the target population.

For continuous \(C\) or \(M\), the corresponding sums are integrals. The weights \(P(C=c)\) refer to the baseline covariate distribution of the population for which the effect is being reported. This distribution may be the study population or another clearly specified target population.

The assumptions have distinct roles. Treatment exchangeability requires that, conditional on \(C\), treatment assignment does not leave uncontrolled common causes of treatment and the relevant potential outcomes or mediator. Mediator–outcome exchangeability requires that, conditional on treatment and \(C\), there are no unmeasured common causes of mediator and outcome. The cross-world condition connects the potential outcome under one treatment and mediator value to the mediator under another treatment; it is an additional assumption used to identify natural effects. Consistency links observed outcomes and mediators to their corresponding potential outcomes. Positivity requires that the treatment and mediator values needed for the formula occur with positive probability within the covariate strata being averaged.

If treatment is randomized, treatment exchangeability is supported by randomization, though positivity and the other assumptions are still needed. A randomized treatment does not automatically remove mediator–outcome confounding: baseline \(C\) can still cause both \(M\) and \(Y\). Also, conditioning on \(C\) is useful only to the extent that the relevant confounders are measured and the conditional comparisons have adequate support.

Why the Formula Standardizes Within Strata

The formula can be understood by separating two operations. First, within each \(C=c\) stratum, use the mediator distribution under \(a'\) to average the outcome mean under treatment \(a\). Second, average those stratum-specific results using the target population’s \(P(C=c)\). The mediator distribution and outcome mean are both conditional on the same covariate value; this is what prevents the baseline common cause from creating a spurious mediator–outcome comparison.

A short derivation makes clear where each part enters. For discrete \(M\), first expand the nested mean over the possible mediator values. The cross-world independence lets us replace the conditional potential outcome mean among those with \(M_{a'}=m\) by its mean conditional on \(C=c\). Exchangeability and consistency then link that mean and the mediator probability to observed conditional quantities.

The second line uses \(Y_{a,m} \perp M_{a'}\mid C\). The final line uses the exchangeability and consistency assumptions to identify the potential-outcome quantities from observed data. This derivation also shows why a pooled calculation generally does not work: it drops the shared stratum \(C=c\) before combining the mediator and outcome pieces.

Natural direct and indirect effects are contrasts between these nested means. For example, on an additive scale, one common indirect contrast holds treatment at \(a\) and changes the mediator distribution from that under \(a'\) to that under \(a\):

The corresponding direct contrast holds the mediator distribution at that under \(a'\) and changes treatment from \(a'\) to \(a\). The covariate-adjusted formula applies to each nested mean in the contrast. These contrasts are on the outcome’s scale and retain the chosen target distribution of \(C\).

Common Mistakes and Practical Checks

A common mistake is to calculate a pooled outcome mean for each mediator value and combine it with a pooled mediator distribution, reasoning that the treatment groups or the whole sample are large enough to average out baseline differences. That calculation can compare people with different severity distributions across mediator levels. Because severity predicts both mediator and outcome, the pooled mediator–outcome relationship can partly reflect severity rather than the mediator pathway. The correction is to calculate the outcome and mediator quantities conditional on \(C\), combine them within each stratum, and only then average using the intended \(P(C)\).

Another mistake is to report a conditional regression coefficient for the mediator as though it were a natural indirect effect. A coefficient from a model conditional on \(C\) is not automatically the standardized nested mean contrast. The formula requires both the conditional outcome model and the conditional mediator distribution, followed by averaging over the target covariate distribution. Treatment–mediator interaction can also make the relevant outcome means depend jointly on \(A\) and \(M\); the conditional formula accommodates that dependence by retaining both values in the outcome mean.

Before interpreting an adjusted mediation result, check that \(C\) truly precedes treatment, that the important mediator–outcome common causes have been measured, and that the necessary treatment and mediator combinations occur within the covariate strata. Sparse or empty strata undermine positivity and make the formula unsupported by the observed data. Finally, standardization controls measured baseline confounding only under the stated assumptions; it does not turn an unmeasured mediator–outcome common cause into a measured one.