Overadjustment for Post-Exposure Variables

Why the Timing of an Adjustment Matters

Regression adjustment is not simply a way to make a treatment comparison “fairer.” The meaning of an adjusted coefficient depends on which variables are controlled and when they occur. In particular, controlling for a variable that is a consequence of the exposure can remove part of the relationship between the exposure and outcome that a total-effect question is meant to include.

Suppose a treatment improves an intermediate health measure, which in turn improves a later outcome. A comparison that adjusts for the intermediate measure compares treated and untreated people at the same value of that measure. It therefore excludes the part of the treatment’s association that operates through changes in the intermediate measure. That may be appropriate if the question is about an effect not operating through that pathway, but it does not answer the same question as the total effect.

Definition: A post-exposure variable is a variable measured or occurring after the exposure. When the exposure can affect that variable, it is a consequence of the exposure. Overadjustment occurs when an adjustment includes such a consequence and thereby removes part of the exposure-outcome relationship that belongs in the target estimand, commonly the total effect. The term describes a mismatch between the adjustment and the question; adjustment for a post-exposure variable is not automatically wrong for every question.

Total Effects and Conditional Associations

Let $A$ denote an exposure or treatment, $M$ a post-exposure variable, and $Y$ an outcome. A simple directed acyclic graph for a mediated pathway is $A \rightarrow M \rightarrow Y$, possibly with an additional direct arrow $A \rightarrow Y$. Here, $M$ is a mediator: some of the exposure-outcome relationship may pass through it.

The total effect includes all causal pathways from $A$ to $Y$, including any pathway through $M$. By contrast, an association between $A$ and $Y$ conditional on $M$ compares exposure levels among observations with the same value of $M$, as represented by a regression coefficient for $A$ when $M$ is included as a predictor. These are different quantities. In general, the adjusted coefficient is not automatically a causal direct effect; a causal interpretation requires additional assumptions.

Linear-model result: Suppose the mediator and outcome follow the additive models $$ M = \alpha_0 + \alpha_A A + u $$ and $$ Y = \beta_0 + \tau A + \beta_M M + \varepsilon. $$ If the models are correctly specified and the errors satisfy the conditions needed for the causal interpretations under consideration, then the total effect per one-unit increase in $A$ is $\tau + \alpha_A\beta_M$. The coefficient $\tau$ is the outcome contrast for a one-unit increase in $A$ holding $M$ fixed. With a binary treatment, these quantities describe the treated-versus-untreated contrast under the same additive, no-interaction model.

The conditions matter. For a causal total-effect interpretation, the exposure comparison must be unconfounded (as in a properly randomized treatment, or under adequate adjustment for pre-exposure common causes), and the treatment and outcome must be defined consistently. For the mediator-based direct-effect interpretation, further requirements include no unmeasured confounding of the mediator-outcome relationship, no treatment-induced confounder of that relationship, and adequate overlap in mediator values across exposure groups. The additive models above also assume no exposure-by-mediator interaction. Without these conditions, regression coefficients remain model-based conditional associations, not guaranteed causal effects.

Why the Mediated Part Disappears

The linear result follows by substituting the mediator model into the outcome model. This makes explicit why the unadjusted and adjusted comparisons can differ.

\begin{aligned} Y &= \beta_0 + \tau A + \beta_M(\alpha_0 + \alpha_A A + u) + \varepsilon \\ &= (\beta_0 + \beta_M\alpha_0) + (\tau + \alpha_A\beta_M)A + \beta_M u + \varepsilon. \end{aligned}

The coefficient multiplying $A$ in the substituted, marginal outcome model is $\tau + \alpha_A\beta_M$: the direct component $\tau$ plus the pathway component $\alpha_A\beta_M$. In the model that includes $M$, the coefficient of $A$ is $\tau$, the contrast holding $M$ fixed. Thus, when $\alpha_A\beta_M$ is not zero, controlling for $M$ removes that pathway component from the coefficient for $A$. This substitution establishes the algebra; the causal interpretation still depends on the assumptions stated above.

Worked Example: A Treatment That Improves an Intermediate Measure

Suppose a hypothetical randomized treatment is studied for its effect on a later health score. Let $A=1$ denote treatment and $A=0$ denote control. Let $M$ be an intermediate measure, with higher values favorable, and $Y$ the later health score, also with higher values favorable. Suppose the models are $$ M = 2A + u,\qquad Y = 3A + 2M + \varepsilon. $$ Assume $E(u\mid A)=0$, $E(\varepsilon\mid A,M)=0$, the treatment was randomized, and the additive models are appropriate. Assume also that the mediator-outcome causal conditions described above hold.

Step 1: Find the treatment’s effect on the intermediate measure. The coefficient of $A$ in the mediator model is $\alpha_A=2$. On average, treatment increases $M$ by 2 units.

Step 2: Find the outcome change through $M$. The outcome model gives $\beta_M=2$ outcome-score points per one-unit increase in $M$. The pathway through $M$ therefore contributes $$ \alpha_A\beta_M = 2(2)=4 $$ outcome-score points per treated-versus-control contrast.

Step 3: Compare the two treatment contrasts. The coefficient of $A$ holding $M$ fixed is $\tau=3$. The total effect is $$ \tau+\alpha_A\beta_M = 3+4=7 $$ outcome-score points. The adjusted treatment coefficient describes a 3-point contrast at the same value of $M$; the total treatment contrast is 7 points because it also includes the 4-point pathway through $M$. If the scientific question concerns the overall effect of offering treatment, adjusting for $M$ discards part of the target effect.

Worked Health-Outcome Applications

Worked Example: Pain Improvement and Later Function

In a hypothetical trial, $A$ indicates receipt of a pain treatment. Let $M$ be improvement in pain by an early follow-up, measured in points, and let $Y$ be a later physical-function score, where higher is better. Suppose treatment increases early pain improvement by 2 points, and each additional point of early improvement contributes 1.5 points to later function. Suppose treatment also has a direct 1-point effect on later function in the additive model.

These values give $\alpha_A=2$, $\beta_M=1.5$, and $\tau=1$. The pathway through early pain improvement contributes $$ \alpha_A\beta_M = 2(1.5)=3 $$ later function-score points. Thus, under the stated linear model, the total effect is $$ \tau+\alpha_A\beta_M = 1+3=4 $$ points. The coefficient of treatment in a regression that also controls for early pain improvement is 1 point, not 4. That coefficient asks about treatment contrasts among people with the same early improvement; it does not capture the treatment’s benefit from producing more early improvement.

The timing and causal structure make this a plausible overadjustment for the total-effect question: treatment precedes and can change early pain improvement, which precedes later function. If the question were instead about a treatment effect not operating through early pain improvement, a conditional or direct-effect analysis could be relevant, but it would need the additional mediator assumptions. Randomization of treatment alone does not establish those assumptions.

Worked Example: Early Blood Pressure and a Later Outcome

Consider a hypothetical randomized medication study. Let $M$ be the change in systolic blood pressure shortly after treatment begins, in mmHg, and let $Y$ be systolic blood pressure at a later visit, also in mmHg. Negative changes mean reductions. Suppose the medication lowers early systolic pressure by 5 mmHg, each 1 mmHg early reduction predicts a 0.8 mmHg later reduction, and the medication has a further direct effect of a 2 mmHg reduction in the additive model.

Using signed changes, $\alpha_A=-5$, $\beta_M=0.8$, and $\tau=-2$. The pathway contribution is $$ \alpha_A\beta_M = (-5)(0.8)=-4\text{ mmHg}. $$ Therefore, the total effect is $$ \tau+\alpha_A\beta_M = -2+(-4)=-6\text{ mmHg}. $$ An analysis that adjusts for early blood pressure estimates the conditional treatment contrast of $-2$ mmHg under the model, whereas the total reduction is $-6$ mmHg. The adjusted coefficient is not evidence that the medication’s overall effect is only a 2 mmHg reduction: it excludes the portion associated with the early pressure change.

This interpretation relies on the hypothetical model and mediator assumptions, not just on the fact that treatment was randomized. For example, if other factors arising after treatment affect both early pressure and later pressure, the simple causal interpretation may fail. In practice, the time order and subject-matter reasoning must support the assumed pathway.

Worked Example: Choosing Between a Baseline and a Post-Treatment Measure

A hypothetical study compares a rehabilitation program with usual care for its effect on walking distance after three months. Researchers recorded baseline mobility before assignment and attendance at therapy sessions after assignment. Suppose their stated target is the total effect of offering the program, and the program can increase attendance, which can in turn improve walking distance.

Baseline mobility occurs before the program and may predict both participation and later walking distance. It may therefore be a relevant pre-exposure adjustment variable, subject to the study’s causal assumptions. Attendance occurs after assignment and can be changed by the program. If the program increases attendance by 3 sessions on average, and each additional session contributes 4 metres to walking distance in the assumed additive model, the attendance pathway contributes $$ 3\text{ sessions}\times 4\text{ metres per session}=12\text{ metres}. $$ A treatment coefficient adjusted for attendance would hold attendance fixed and would omit this 12-metre pathway component under the model. For the stated total-effect question, the post-assignment attendance variable should not be included merely because it predicts the outcome. If the question concerns an effect not operating through attendance, then adjustment addresses a different target and requires additional assumptions.

A Practical Check Before Adjusting

Before adding a covariate to a regression, define the exposure-outcome quantity the analysis is intended to describe. The following sequence helps prevent a post-exposure variable from silently changing that target.

Set the target.
Decide whether the question concerns the total exposure effect or an association conditional on a specified variable.

Check the timing.
Establish whether the candidate variable was measured before or after the exposure. A post-exposure measurement may still reflect a process that began earlier, so timing should be interpreted substantively.

Ask whether exposure can change it.
If the exposure can affect the candidate variable, draw the possible path from exposure through that variable to outcome. Controlling for it can remove that pathway from the exposure coefficient.

Match the model to the target.
For a total-effect question, do not adjust for a mediator simply because it predicts the outcome. For a direct-effect question, state that target and assess the stronger assumptions needed to interpret it causally.

This check does not imply that every post-exposure variable must always be excluded. A descriptive conditional association may deliberately condition on it, and a direct-effect question may require considering a mediator. The key is to state what the coefficient means after adjustment and avoid labeling it as the total effect when it excludes a pathway included in that total.

Common Mistake

Wrong reasoning: “Early response predicts the final health outcome, so adding early response to the regression must improve the treatment-effect estimate.”

Correction: Prediction alone does not determine whether adjustment is appropriate. If treatment can change early response and early response affects the later outcome, controlling for it removes part of the treatment pathway. For a total-effect question, that is a change in the quantity being estimated, not automatically an improvement. State whether the goal is the total effect or a conditional/direct effect, then choose and interpret the adjustment accordingly.

Key takeaway: A post-exposure consequence can lie on a pathway from treatment to outcome. Adjusting for it changes the treatment coefficient from a contrast that may include that pathway to a contrast holding the consequence fixed. Whether that is overadjustment depends on the target question and the assumptions supporting the interpretation.

Check Your Understanding

Answer each question by identifying the target quantity and the role of the post-exposure variable.

A treatment improves an early biomarker, and the biomarker predicts a later outcome. Why can adjusting for the biomarker make the treatment coefficient smaller in magnitude than the total effect?
In the additive models $M=\alpha_0+\alpha_A A+u$ and $Y=\beta_0+\tau A+\beta_M M+\varepsilon$, write the total effect and the coefficient conditional on $M$.
A study adjusts for a measurement taken after treatment because it strongly predicts the outcome. What key question should be answered before deciding that this adjustment is appropriate?
Why does random assignment of treatment, by itself, not guarantee that an adjusted treatment coefficient has a causal direct-effect interpretation?
Give one example of a question for which adjustment for a mediator could be relevant, and explain why it differs from a total-effect question.