The Competing Roles of a Time-Varying Confounder
With treatment at two visits, a variable measured between visits can play two roles at once. Earlier treatment may change that variable; the variable may then help determine later treatment and predict the outcome. It is therefore both a mediator of some earlier treatment effects and a confounder of the later treatment-outcome relationship. This feedback creates a dilemma for ordinary regression: adjusting for the variable can block part of the earlier treatment effect, while omitting it can leave later treatment confounded.
The issue is not that regression arithmetic fails. Rather, a coefficient from a single outcome regression is conditional on the covariates included in that model. Its meaning depends on the adjustment set and the model assumptions; it does not automatically become the marginal contrast between two treatment policies.
A path sketch captures the competing roles: \(A_0 \rightarrow L_1 \rightarrow Y\) is one pathway for an earlier treatment effect, while \(L_1 \rightarrow A_1\) and \(L_1 \rightarrow Y\) make \(L_1\) a common cause of later treatment and outcome. There may also be paths from \(A_0\) through \(L_1\) and \(A_1\) to \(Y\). Adjusting for \(L_1\) is useful for controlling the later-treatment association, but it conditions away pathways carrying earlier-treatment effects.
A Linear Model That Makes the Dilemma Visible
The following model is deliberately small enough to calculate by hand. It is an illustration of the regression problem, not a claim that real longitudinal data are linear. Suppose \(A_0\) is randomized with \(P(A_0=1)=P(A_0=0)=1/2\), and let \(U,V,\varepsilon\) be mutually independent, mean-zero random variables, independent of \(A_0\), each with variance 1. Let the observed variables satisfy
Here \(A_0\) shifts \(L_1\); \(L_1\) shifts the average level of \(A_1\); and both \(L_1\) and \(A_1\) predict \(Y\). The full-rank linear regression of \(Y\) on \(A_0,L_1,A_1\) has the stated coefficients: the outcome equation is exactly linear, and \(\varepsilon\) is independent of the regressors with mean zero. The two calculations below show why neither coefficient selection nor the word “adjusted” resolves the target question by itself.
Worked Example: The Later-Treatment Coefficient When \(L_1\) Is Omitted
Fit the ordinary regression of \(Y\) on \(A_0\) and \(A_1\), leaving \(L_1\) out. What coefficient does this model assign to \(A_1\)?
Because \(A_0\) is included, first subtract the part explained by \(A_0\). From the structural equations,
After removing the linear projection on \(A_0\), the residuals for \(A_1\) and \(Y\) are \(U+V\) and \(7U+4V+\varepsilon\), respectively. Their covariance is
by independence and the stated variances. The residual variance of \(A_1\) is \(\operatorname{Var}(U+V)=2\). Therefore the regression coefficient for \(A_1\) is \(11/2=5.5\), not the coefficient 4 in the outcome equation. The difference arises because the omitted \(L_1\) is associated with both \(A_1\) and \(Y\), even after controlling for \(A_0\). In this model, omitting \(L_1\) leaves confounding of the later-treatment association.
Worked Example: Adjusting for \(L_1\) Changes the Earlier-Treatment Coefficient
Now fit the regression of \(Y\) on all three variables \(A_0,L_1,A_1\). The coefficients are \(2,3,4\), directly from the correctly specified outcome equation. In particular, the adjusted coefficient for \(A_0\) is 2.
To see what that coefficient leaves out, compare two ways of changing \(A_0\). First, hold \(L_1\) and \(A_1\) fixed in the outcome equation. Increasing \(A_0\) by one unit changes the fitted outcome by \(2\). This is the conditional contrast represented by its coefficient.
Alternatively, consider the change induced by the model's equations when \(A_0\) is raised by one unit and the natural assignment rule \(A_1=L_1+V\) is left in place. For fixed \(U,V,\varepsilon\), \(L_1\) rises by one and \(A_1\) also rises by one. The outcome change is therefore
The conditional coefficient 2 is not this policy-mediated change of 9: conditioning on \(L_1\) and \(A_1\) holds fixed variables that would otherwise respond to \(A_0\). The example does not say that 2 is an incorrect regression coefficient. It shows that the coefficient answers a different question from the total change under the specified assignment process.
Two Derivations, and What They Establish
The numerical results follow from general algebra. The first derivation states precisely how an omitted covariate changes a regression slope in a linear model. The second traces the change under the structural equations. Both rely on the assumptions in the example; neither establishes that an ordinary regression identifies a causal effect without those assumptions.
Derivation. By the Frisch–Waugh–Lovell theorem, the coefficient on \(A_1\) in the reduced regression is the slope from regressing the residualized outcome \(Y^\perp\) on \(A_1^\perp\), where \(Y^\perp\) is the residual after its linear projection on an intercept and \(A_0\). Residualizing the stated outcome equation gives
Since \(E[\varepsilon\mid A_0,L_1,A_1]=0\), its covariance with \(A_1^\perp\) is zero: \(A_1^\perp\) is a linear combination of \(A_0,A_1\), so \(E[\varepsilon A_1^\perp]=0\); its projection component on the intercept and \(A_0\) is also orthogonal to \(A_1^\perp\). Taking covariance with \(A_1^\perp\) and dividing by its positive variance gives the displayed formula. In the worked model, \(\beta_L=3\), \(\operatorname{Cov}(A_1^\perp,L_1^\perp)=1\), and \(\operatorname{Var}(A_1^\perp)=2\), so the coefficient is \(4+3(1/2)=5.5\). \(\square\)
Derivation. Under the retained assignment rule, a unit increase in \(A_0\) changes \(L_1=A_0+U\) by one and changes \(A_1=L_1+V\) by one. Substitution in the outcome equation gives change \(2+3+4=9\). When \(L_1\) and \(A_1\) are held fixed, only the term \(2A_0\) changes, giving 2. The distinction is a direct consequence of which variables are allowed to respond; it does not require a regression approximation. \(\square\)
Worked Example: Comparing Three Adjustment Choices
Suppose the scientific question concerns the effect of an earlier treatment policy, including any consequences it has through the later covariate and later treatment. In the example's model, compare these ordinary regressions:
| Regression covariates | What the example gives | What remains unresolved |
|---|---|---|
| \(A_0,A_1\), omitting \(L_1\) | Coefficient on \(A_1\): 5.5 | \(L_1\) confounds the later-treatment association. |
| \(A_0,L_1,A_1\) | Coefficient on \(A_0\): 2 | Conditioning on descendants of \(A_0\) does not retain its full change through \(L_1\) and \(A_1\). |
| Neither coefficient nor adjustment set alone | Model-based change under the retained later-treatment rule: 9 | Requires stating the policy and the structural assumptions used to calculate it. |
The third row is not a recommendation to add regression coefficients mechanically. It illustrates that a marginal policy contrast may involve pathways through more than one visit and must be defined before deciding which model quantity answers it. Under the exact equations here, one can calculate that contrast because the structural rules and coefficients were specified. In applied settings, those ingredients are not generally known just because a regression was fitted.
What to Check, and What Ordinary Regression Does Not Guarantee
A routine outcome regression can still be appropriate for a clearly defined conditional association or for a causal target under a suitable, justified model and adjustment strategy. The difficulty under feedback is treating a single covariate list as if it simultaneously controlled later-treatment confounding and preserved every earlier-treatment pathway. The earlier material on time-varying confounding and regression adjustment motivates why those aims can conflict; the example here makes the conflict explicit in one two-visit calculation.
Several diagnostics can reveal that this problem deserves attention, although none proves that a chosen model is causally valid:
- Check the measurement timeline: is a covariate recorded after earlier treatment but before a later treatment decision?
- Ask whether that covariate predicts both the later treatment and the outcome, and whether earlier treatment could have changed it.
- Compare estimates from models that include and omit the evolving covariate. A change is a warning about dependence on the adjustment choice, not a test that identifies the correct estimate.
- Inspect whether the fitted model represents treatment history, relevant interactions, and nonlinear relationships plausibly. A poorly specified outcome model can create additional errors beyond the adjustment dilemma.
- Check whether important common causes of later treatment and outcome were measured. Regression cannot adjust for a cause absent from the data.
There are important limitations. A correctly fitted conditional regression does not, by itself, establish the assumptions needed for a causal interpretation. The simple numerical model uses a linear outcome equation, independent errors, and an explicit treatment-assignment rule. With binary outcomes, nonlinear links, interactions, or poorly measured covariates, coefficient comparisons can be more complicated. In particular, a coefficient need not be a marginal contrast even without feedback. The general lesson is narrower and more dependable: under feedback, “adjust for everything” and “do not adjust for a treatment-affected variable” are both inadequate rules of thumb.
When the target is a contrast between longitudinal treatment strategies, the analyst must use a method whose estimand and assumptions address the treatment process over time. The g-formula and inverse-probability-weighted marginal structural models are standard alternatives developed for this setting; their definitions and identification conditions require separate treatment. Naming them does not remove the need to specify the target, assess assumptions, and diagnose model fit.
Common Pitfall
Check Your Understanding
Use the two-visit model and the distinctions developed above. Brief solutions are included for checking your reasoning.
- Which variable is affected by \(A_0\) and also predicts \(A_1\) and \(Y\)?
Solution: \(L_1\); it is the time-varying confounder affected by earlier treatment. - In the worked model, what is the coefficient of \(A_1\) when \(L_1\) is omitted but \(A_0\) is included? Show the covariance ratio.
Solution: \(4+3(1/2)=5.5\), since the residual covariance of \(A_1\) and \(L_1\) given \(A_0\) is 1 and the residual variance of \(A_1\) is 2. - Why is the \(A_0\) coefficient 2 in the regression adjusted for \(L_1,A_1\), while the model-based change under the retained assignment rule is 9?
Solution: The regression coefficient holds \(L_1,A_1\) fixed; under the assignment rule, both rise by one when \(A_0\) rises by one, adding \(3+4\) to the direct change of 2. - Does a large difference between adjusted and unadjusted estimates establish which one is causal? Why or why not?
Solution: No. It shows sensitivity to adjustment, but does not establish the target estimand, model correctness, or causal assumptions. - Name one standard alternative family for estimating longitudinal treatment-strategy contrasts under feedback.
Solution: The g-formula or inverse-probability-weighted marginal structural models; each requires its own assumptions and implementation choices.