Why the Adjustment Set Changes the Effect Being Estimated
In a longitudinal study, an early treatment can change a later covariate, which in turn predicts the outcome. That covariate may also help determine a later treatment. It is therefore tempting to include it in every regression as a useful predictor or confounder. But if the question concerns the overall effect of the initial treatment, conditioning on a covariate caused by that treatment can remove part of the effect being sought.
The algebra makes the issue visible: substituting the covariate equation into the outcome equation reveals the contribution carried along the path from initial treatment through the covariate to the outcome. In contrast, the treatment coefficient in a regression that holds the covariate fixed describes a contrast that excludes that pathway under suitable model assumptions.
A Two-Equation Model and Its Path Decomposition
Consider a binary initial treatment \(A\), an intermediate covariate \(L\), and a continuous outcome \(Y\). Suppose their relationships follow the additive linear equations
Here \(U\) and \(V\) are mean-zero errors. For the derivations below, assume treatment is randomized, \(P(A=1)\) is strictly between zero and one, and \(U\) and \(V\) are independent of \(A\) and of each other. Assume also that \(\operatorname{Var}(U)>0\), so treatment and \(L\) are not perfectly collinear. These are deliberately strong assumptions: they isolate the pathway algebra without confounding or model misspecification.
To derive the marginal contrast, substitute the equation for \(L\) into the equation for \(Y\):
Since the errors have mean zero and are independent of \(A\), taking conditional expectations gives
The product \(\alpha_A\beta_L\) is the path contribution through \(L\): a one-unit change in \(A\) changes the mean of \(L\) by \(\alpha_A\), and a one-unit change in \(L\) changes the mean of \(Y\) by \(\beta_L\). The direct component is \(\beta_A\). Their sum is the total mean contrast in this additive model.
For the adjusted regression, the outcome equation itself gives the conditional mean \(E[Y\mid A,L]=\beta_0+\beta_A A+\beta_L L\). It is therefore the population linear regression function. Because \(A\) and \(L\) are not perfectly collinear, its coefficients are uniquely identified, so the coefficient of \(A\) is \(\beta_A\). This proves the result. \(\square\)
Worked Example: A Mediated Component in a Randomized Study
Suppose the intermediate covariate follows \(L=2+0.8A+U\), and the outcome follows \(Y=10+1.2A+2.5L+V\), with the assumptions above. Calculate the total treatment contrast and the adjusted treatment coefficient.
The treatment changes the mean of \(L\) by \(\alpha_A=0.8\). The change in \(Y\) associated with a one-unit change in \(L\) is \(\beta_L=2.5\). Hence the pathway through \(L\) contributes
The component not passing through \(L\) is \(\beta_A=1.2\). Substitution gives the total contrast:
The same calculation from group means checks the result. Under \(A=0\), \(E[L\mid A=0]=2\), so \(E[Y\mid A=0]=10+2.5(2)=15\). Under \(A=1\), \(E[L\mid A=1]=2.8\), so \(E[Y\mid A=1]=10+1.2+2.5(2.8)=18.2\). The difference is \(18.2-15=3.2\). A regression of \(Y\) on both \(A\) and \(L\), however, has treatment coefficient \(1.2\): it compares treatment levels at the same \(L\), blocking the \(2.0\)-unit component through \(L\).
When the Path Contribution Is Not a Simple Product
The product formula depends on the additive linear equations and the absence of an \(A\)-by-\(L\) interaction. More generally, the total contrast can still be written as a change in the outcome’s conditional mean at fixed \(L\), plus a change caused by the treatment-related shift in the distribution of \(L\). The second part need not be a constant product.
For a discrete \(L\), write \(m(a,l)=E[Y\mid A=a,L=l]\) and \(p_a(l)=P(L=l\mid A=a)\). Under randomization and consistency, the group mean is \(\sum_l m(a,l)p_a(l)\). Adding and subtracting \(\sum_l m(1,l)p_0(l)\) gives the exact identity
The first term compares treatment levels while using the covariate distribution under \(A=0\); the second reflects the change in that distribution, evaluated using the outcome mean under \(A=1\). This is an algebraic decomposition, not a claim that either term is a uniquely defined causal effect under every model. The reference distribution matters, especially when treatment effects vary with \(L\).
Worked Example: A Binary Intermediate Covariate
Let \(A\) be randomized, with \(P(L=1\mid A=0)=0.2\) and \(P(L=1\mid A=1)=0.6\). Suppose \(E[Y\mid A,L]=1+3A+5L\). Find the group means and separate the contrast into its two terms using the distribution under \(A=0\) as the reference.
When \(A=0\), the mean outcome is \(1+5L\), so
When \(A=1\), the mean outcome is \(4+5L\), so
The total contrast is \(7-2=5\). In the decomposition, the first term is the fixed-\(L\) treatment difference \(3\), averaged over the \(A=0\) covariate distribution, so it equals \(3\). The second term is
Thus the total contrast is \(3+2=5\). The adjusted treatment coefficient is \(3\), while the two-unit difference comes from the treatment-associated change in the distribution of \(L\).
Worked Example: Effect Modification Changes the Conditional Contrast
Suppose \(P(L=1\mid A=0)=0.25\), \(P(L=1\mid A=1)=0.75\), and the conditional mean is \(E[Y\mid A,L]=2A+3L+4AL\). Calculate the group means and the treatment contrasts at each value of \(L\).
For \(A=0\), the conditional mean is \(3L\), giving \(E[Y\mid A=0]=0.75(0)+0.25(3)=0.75\). For \(A=1\), it is \(2+7L\), giving \(E[Y\mid A=1]=0.25(2)+0.75(9)=0.5+6.75=7.25\). The total group-mean contrast is \(7.25-0.75=6.5\).
At \(L=0\), the treatment contrast is \(2-0=2\). At \(L=1\), it is \(9-3=6\). There is no single constant adjusted treatment effect across the two covariate strata. A regression that includes \(A\), \(L\), and \(A L\) represents these contrasts; one that includes only \(A\) and \(L\) imposes a common treatment contrast and is misspecified for these conditional means.
What Adjustment Does—and Does Not—Establish
The calculations identify a specific issue: an intermediate covariate can be both a predictor of later treatment and a carrier of an earlier treatment effect. Including it in an outcome regression may help describe conditional outcome means, but the resulting treatment coefficient does not automatically represent the total effect of the initial treatment.
This is distinct from adjusting for a baseline confounder. A baseline variable precedes initial treatment and may be included to control confounding without removing a pathway caused by that treatment. By contrast, \(L\) in the model above is downstream of \(A\). The introductory material on time-varying confounder feedback motivates why \(L\) may also be important for later treatment assignment; that role does not change the algebraic fact that conditioning on \(L\) removes its contribution to the earlier treatment contrast in this model.
| Analysis choice | What it compares in the simple model | Main caution |
|---|---|---|
| Regress \(Y\) on \(A\) without \(L\) | The marginal treatment-group means | In observational data, confounding may remain; this is not a general adjustment strategy. |
| Regress \(Y\) on \(A\) and intermediate \(L\) | Treatment levels at fixed \(L\) | Can exclude the pathway through \(L\); the coefficient also depends on correct model form. |
| Adjust for baseline confounders | Treatment comparisons conditional on pre-treatment variables | Requires adequate confounder measurement and appropriate modeling. |
| Marginal structural model | A marginal treatment-effect target using a longitudinal weighting framework | Requires its own identification assumptions and diagnostics; it is not equivalent to adding every covariate to an outcome regression. |
Diagnostics and Common Pitfalls
Before interpreting an adjusted coefficient, draw the temporal order and ask whether each adjusted variable could have been changed by the treatment whose effect is being estimated. Then check whether the outcome model includes plausible nonlinearities or treatment-by-covariate interactions. Compare conditional treatment contrasts across covariate levels when such interactions are plausible, and inspect the distribution of the intermediate covariate across treatment groups. Sparse or non-overlapping covariate values make fixed-\(L\) comparisons unstable or unsupported by the data.
- Wrong reasoning: “\(L\) predicts \(Y\), so including \(L\) must remove bias and improve the estimate of the total effect.”
- Correction: Prediction alone does not determine whether adjustment is appropriate. If treatment changes \(L\), conditioning on it can remove the \(A \rightarrow L \rightarrow Y\) component. The adjustment set must match the target contrast and causal structure.
- Wrong reasoning: “The adjusted treatment coefficient is always the direct effect.”
- Correction: That interpretation relies on the model and assumptions. With effect modification, the conditional contrast varies with \(L\); with confounding, measurement error, or misspecification, the coefficient may not equal a causal direct effect at all.
The path algebra is a diagnostic for interpretation, not a universal prescription to omit \(L\). In observational longitudinal data, omitting a covariate that confounds later treatment and outcome can also be problematic. The central limitation is that a single ordinary regression coefficient may not preserve the pathway contributions needed for a marginal total-effect question. The causal target, temporal ordering, and assumptions must be stated before selecting an adjustment strategy.
Check Your Understanding
Use the equations and distinctions above to answer each question. Brief solutions follow.
- If \(\alpha_A=1.5\), \(\beta_L=2\), and \(\beta_A=0.5\), what are the mediated component and total contrast?
- In the binary-covariate example, what would happen to the total contrast if treatment changed the distribution of \(L\) but \(L\) had no association with the outcome conditional on \(A\)?
- Why can including an \(A\)-by-\(L\) interaction change the interpretation of the coefficient of \(A\)?
- Explain why adjusting for a pre-treatment confounder is conceptually different from adjusting for an intermediate covariate.
Solutions. 1. The mediated component is \(1.5(2)=3\), and the total contrast is \(0.5+3=3.5\). 2. If \(L\) has no outcome association conditional on \(A\), its distributional shift contributes zero to the outcome contrast; changing \(L\) alone does not change the mean outcome. 3. With an interaction, the conditional treatment contrast depends on \(L\); the coefficient of \(A\) is the contrast at \(L=0\), not a common contrast for all covariate values. 4. A pre-treatment confounder is not a pathway caused by the initial treatment, whereas an intermediate covariate may carry part of that treatment’s effect to the outcome. Conditioning on the latter can therefore remove part of the total effect.