Why a Numerical Counterexample Matters
A covariate measured between two treatment decisions can be both a consequence of the first treatment and a predictor of the second treatment and outcome. Adjusting for it may help describe the second treatment decision, but it also holds fixed part of the pathway through which the first treatment affects the outcome. The resulting conditional regression contrast need not equal the overall contrast between treatment histories.
We will construct a two-period example in which the outcome regression is exactly correct, treatment assignment at the second period depends on the intermediate covariate, and all relevant treatment probabilities are positive. Even so, the adjusted contrast for histories \((1,1)\) and \((0,0)\) is not their total effect. The discrepancy is not a failure of regression to recover its conditional coefficient: it is a mismatch between that coefficient contrast and the marginal regime contrast of interest.
A Model With Feedback and a Blocked Path
Consider the following structural model. \(A_1\) is randomized with \(P(A_1=1)=1/2\); \(U\sim N(0,1)\) and \(\varepsilon\sim N(0,1)\) are independent of \(A_1\) and of each other. The intermediate covariate and outcome are
The second treatment is assigned according to
Thus \(L_1\) is affected by \(A_1\), predicts \(A_2\), and has a direct effect on \(Y\). It is a time-varying confounder for the second treatment-outcome relationship that is itself affected by earlier treatment. The treatment probability is strictly between 0 and 1 at every finite \(L_1\). The outcome disturbance has conditional mean zero, so the conditional mean model given \(A_1,A_2,L_1\) is correctly specified.
Worked Example: The Total Contrast and the Adjusted Contrast
Under a fixed regime, \(A_2\) is set by intervention, so its natural assignment probability does not enter the outcome equation. Under history \((1,1)\), \(L_1=1+U\), and therefore
\(E[Y^{1,1}]=1+1+2E[1+U]+E[\varepsilon]=2+2=4\), since \(E[U]=E[\varepsilon]=0\).
Under history \((0,0)\), \(L_1=U\), and
\(E[Y^{0,0}]=0+0+2E[U]+E[\varepsilon]=0\).
The total regime contrast is consequently \(4-0=4\). Now fit the conditional outcome regression including \(A_1,A_2,L_1\). Its conditional mean is exactly \(A_1+A_2+2L_1\), so the coefficients are \(1,1,2\). Comparing histories while holding \(L_1\) fixed gives \(1+1=2\), not 4. The omitted amount is \(2\), the outcome effect \(2\) of the treatment-induced one-unit shift in \(L_1\).
Derivation: What the Regression Coefficients Estimate
The distinction can be stated generally for this model class. Suppose \(A_1\) is randomized, \(L_1=\delta A_1+U\), and
Assume the relevant expectations are finite and that the design matrix for the regression on \(1,A_1,A_2,L_1\) has full rank. These conditions make the population linear regression coefficients identifiable; the conditional-mean assumption makes them equal to the displayed model coefficients. For the regime interpretation, assume the structural equations for \(L_1\) and \(Y\) remain applicable under interventions setting \(A_1,A_2\).
At any common value \(l\), the conditional mean difference between \((A_1,A_2)=(1,1)\) and \((0,0)\) is \((\beta_1+\beta_2+\gamma l)-(\gamma l)=\beta_1+\beta_2\).
Under \(A_1=1\), \(E[L_1]=\delta+E[U]\); under \(A_1=0\), \(E[L_1]=E[U]\). Their difference is \(\delta\).
The common \(E[U]\) and mean-zero error cancel. The regime contrast is \(\beta_1+\beta_2+\gamma\{(\delta+E[U])-E[U]\}=\beta_1+\beta_2+\gamma\delta\).
This derivation proves the result: the conditional regression contrast removes the \(L_1\) contribution by comparing at the same \(l\), whereas the regime contrast allows the distribution of \(L_1\) to change in response to \(A_1\). The difference is \(\gamma\delta\). If either treatment has no effect on \(L_1\) (\(\delta=0\)), or \(L_1\) has no effect on the outcome (\(\gamma=0\)), this particular discrepancy vanishes. Otherwise, it need not. \(\square\)
Worked Example: Checking Treatment Feedback Numerically
In the first model, at \(L_1=0\), the probability of second-period treatment is \(\operatorname{expit}(0)=1/2\). At \(L_1=1\), it is \(\operatorname{expit}(1)=e/(1+e)\), approximately \(0.731\). The covariate therefore changes the chance of later treatment, while both treatment options remain possible at either covariate value.
The observed conditional mean is \(E[Y\mid A_1,A_2,L_1]=A_1+A_2+2L_1\). Substituting \((A_1,A_2,L_1)=(0,0,0)\) gives 0, and substituting \((1,1,0)\) gives 2. This is the adjusted contrast at \(L_1=0\). Under the regimes, however, the respective mean covariate values are \(E[L_1\mid do(A_1=0)]=0\) and \(E[L_1\mid do(A_1=1)]=1\). The latter shift contributes \(2(1-0)=2\) to the regime contrast, giving \(2+2=4\).
The calculation is not claiming that the second-treatment assignment probability is irrelevant to all analyses. It establishes that, in this structural model, the outcome regression’s conditional contrast is not the total contrast between the two fixed histories.
A Second Parameter Choice: The Error Can Go the Other Way
The problem is not restricted to underestimating a positive effect. Changing the sign of the covariate’s outcome effect can make adjustment overstate the total regime contrast.
Worked Example: An Adjusted Contrast Larger Than the Total Effect
Keep \(A_1\), \(U\), and the treatment assignment mechanism as above, but set \(L_1=2A_1+U\) and \(Y=A_1+A_2-L_1+\varepsilon\). Here \(\delta=2\), \(\beta_1=\beta_2=1\), and \(\gamma=-1\).
Under \((1,1)\), \(E[L_1]=2\), so \(E[Y^{1,1}]=1+1-2=0\). Under \((0,0)\), \(E[L_1]=0\), so \(E[Y^{0,0}]=0\). The total regime contrast is \(0-0=0\). The conditional regression contrast holding \(L_1\) fixed remains \(1+1=2\). The discrepancy is \(\gamma\delta=(-1)(2)=-2\), and \(2+(-2)=0\), as required. Thus adjustment has not merely attenuated the total effect; it has produced a nonzero contrast where the total contrast is zero.
Assumptions, Diagnostics, and Alternatives
The counterexample isolates a target mismatch, not every possible source of bias. The conditional regression calculation relies on a correct conditional mean, mean-zero outcome errors given the included variables, and sufficient variation for the regression coefficients to be identified. Its causal interpretation also relies on well-defined treatment interventions and on the structural relations used for the regime calculation applying under those interventions. In observational applications, additional confounding, measurement error, or model misspecification can add further discrepancies.
Useful warning signs include a covariate measured after an earlier treatment, strong prediction of later treatment by that covariate, and a plausible effect of the covariate on the outcome. A large change in the earlier-treatment coefficient when that covariate is included is also worth investigating, although it does not by itself diagnose bias: coefficients can change for several reasons. A good diagnostic is to write down the target first and ask whether the adjustment holds fixed a variable whose distribution would change under the regimes being compared.
| Approach | What it compares | Key caution |
|---|---|---|
| Conditional outcome regression including \(L_1\) | Histories at a common value of \(L_1\), for the specified model | May remove the pathway from \(A_1\) through \(L_1\) |
| Standardization or g-computation | Predicted outcomes averaged under specified treatment histories | Requires correctly specified models and suitable confounding control |
| Marginal structural model with inverse-probability weighting | A marginal treatment-history or regime contrast under its assumptions | Requires exchangeability and positivity; extreme weights can make estimates unstable |
The introductory material on marginal structural models provides the broad motivation for targeting marginal effects. The point here is narrower: including a time-varying covariate as an ordinary outcome-regression adjustment does not automatically turn a conditional coefficient contrast into a total regime contrast. Neither the numerical example nor the simple linear derivation says that adjustment is always wrong. If the target is a controlled direct contrast at a common covariate value, conditioning may be appropriate. The mistake is to interpret that different target as the total effect of changing the treatment history.
Common Mistake
- Wrong reasoning: “The regression includes the covariate that predicts the second treatment, so the adjusted treatment coefficients must give the overall effect of the treatment history.”
- Correction: Predictive adjustment and target definition are different issues. If the covariate was changed by the first treatment, holding it fixed removes its contribution to the total regime contrast. In the first example, the adjusted contrast is 2, while the total contrast is 4.
- Another pitfall: Calling the regression coefficient “biased” without specifying its target. Under the stated model, it correctly estimates a conditional contrast. It is biased as an estimate of the total regime contrast only when it is interpreted as estimating that different quantity.
Check Your Understanding
Answer each question, then compare with the short solutions below.
- In the general model, let \(\beta_1=2\), \(\beta_2=1\), \(\gamma=3\), and \(\delta=2\). What are the conditional contrast and total regime contrast?
- If \(\delta=0\) but \(\gamma\ne 0\), what is the difference between the two contrasts in this model?
- For the first numerical example, why does the second-treatment assignment mechanism not change the calculation of \(E[Y^{1,1}]\)?
- Give one reason why an observed change in a treatment coefficient after adding \(L_1\) is not, by itself, proof of the counterexample’s mechanism.
Solutions. (1) The conditional contrast is \(\beta_1+\beta_2=2+1=3\). The total contrast is \(3+\gamma\delta=3+(3)(2)=9\). (2) The difference is \(\gamma\delta=0\), so these contrasts agree in this model. (3) Under the fixed regime, the intervention sets \(A_2=1\); its observed assignment probability is replaced by that intervention, and \(E[U]=E[\varepsilon]=0\). (4) The change could also reflect confounding, misspecification, measurement error, or other model changes; the coefficient shift alone does not establish that \(L_1\) mediates an effect of \(A_1\).