How Adjusting For A Confounder Can Block An Earlier Effect

Why a Later Covariate Can Change an Earlier Treatment Contrast

In a two-period study, an earlier treatment can change a covariate measured before the next treatment decision. That covariate may then predict the final outcome and influence the later treatment. It is therefore a confounder of the later treatment–outcome relationship, but it is also potentially part of the pathway from the earlier treatment to the outcome.

This creates a specific problem for ordinary adjustment: holding the later covariate fixed removes the part of the earlier treatment’s effect that operates through changes in that covariate. The algebra below isolates this pathway. It does not say that adjustment is always wrong: the adjusted comparison answers a different question, and its causal interpretation depends on assumptions about treatment assignment and the outcome model.

To focus on that pathway, consider comparisons made while \(A_1\) is held at a fixed value \(a_1\). The model is

Here \(U\) is the part of \(L_1\) not explained by \(A_0\), and \(\varepsilon\) is the part of \(Y\) not explained by the displayed terms. Assume \(E[U\mid A_0]=0\), \(E[\varepsilon\mid A_0,L_1]=0\), finite second moments, and that \(A_0\) varies in the population. These assumptions make the conditional means in the model correct. For a causal interpretation, also assume consistency and that the intervention on \(A_0\) being compared is well-defined. Additional assumptions are needed to identify the causal contrast from observational data.

Deriving the Effect That Adjustment Removes

Substituting the equation for \(L_1\) into the outcome equation gives

Taking the conditional expectation given \(A_0\), the assumptions on \(U\) and \(\varepsilon\) imply

Thus, for this model and a fixed \(a_1\), the expected outcome contrast per one-unit increase in \(A_0\), allowing \(L_1\) to respond as the model specifies, is \(\beta_A+\beta_L\alpha_A\). The term \(\beta_L\alpha_A\) is the contribution transmitted through \(L_1\). In contrast, holding \(L_1\) fixed leaves only the coefficient \(\beta_A\).

Proof. The outcome equation itself is the conditional mean given \(A_0,L_1\), because \(E[\varepsilon\mid A_0,L_1]=0\). It is linear in \(A_0,L_1\), so its population regression coefficients are \(\beta_A\) and \(\beta_L\), provided these regressors are not perfectly collinear. The earlier substitution and conditional expectation show that the slope when \(L_1\) is allowed to vary with \(A_0\) is \(\beta_A+\beta_L\alpha_A\). Subtracting the adjusted coefficient gives

This difference is the modeled contribution through \(L_1\). If \(\alpha_A=0\), \(A_0\) does not change \(L_1\); if \(\beta_L=0\), \(L_1\) does not affect the outcome in this model. Either way, the pathway contribution is zero. \(\square\)

Worked Calculations

Assumptions, Diagnostics, and Alternatives

The algebra identifies what adjustment does in the stated model; it does not, by itself, guarantee that a regression coefficient in observational data is a causal effect. The linear conditional mean must be adequate for the intended contrast, the treatment and covariate must have sufficient variation to estimate the regression, and the error must have conditional mean zero given the included regressors. For causal interpretation, treatment assignment must also be adequately controlled for by the design or assumptions. In particular, unmeasured causes of \(L_1\) and \(Y\), misspecification, or poor overlap can invalidate a causal reading.

The fixed-\(A_1\) setup is deliberate: it isolates the \(A_0 \to L_1 \to Y\) pathway. In practice, \(A_1\) is often assigned in response to \(L_1\). Conditioning on \(A_1\) as well as \(L_1\) can change the comparison further, and may introduce additional bias if the treatment assignment structure is not handled appropriately. The calculation here should not be mistaken for a general formula for the effect of a treatment regime.

Useful diagnostics include checking whether \(L_1\) differs across levels of \(A_0\), examining the relationship between \(L_1\) and \(Y\), and assessing overlap in \(L_1\) across treatment groups. In the linear model, these correspond to asking whether \(\alpha_A\) and \(\beta_L\) may be nonzero. Plotting or fitting flexible conditional mean models can also reveal nonlinearity or treatment-by-covariate variation that a single linear coefficient conceals. Diagnostics cannot establish the absence of unmeasured confounding, but they can expose obvious failures of the working model or lack of empirical support.

A regression that adjusts only for baseline covariates does not hold this post-treatment covariate fixed, so it does not remove this particular pathway by that adjustment. It may nevertheless leave confounding of the later treatment unresolved. Marginal structural models and the g-formula are alternative approaches designed for longitudinal treatment and covariate histories; their role is not to reinterpret the ordinary adjusted coefficient as a total effect, but to target specified treatment comparisons under their own identification and modeling assumptions.