Why This Example Matters
Disease severity often affects the next treatment decision: patients who are doing worse may be more likely to receive intensified treatment. But severity may itself have been changed by the earlier treatment. In that case, severity is both a predictor of later treatment and a consequence of prior treatment. An ordinary regression that adjusts for severity can therefore answer a different question from the effect of following a sustained treatment strategy.
The example below makes that distinction numerical. It uses continuous treatment intensity and a continuous severity score so the calculations can be shown directly. The same issue arises with other outcome types, although the regression calculations will differ.
A Structural Model for the Example
Suppose the data-generating process is
Assume \(A_0,U,V,\varepsilon\) are mutually independent. The variables \(U,V,\varepsilon\) represent unmeasured causes of severity, later treatment, and the outcome, respectively. In this model, greater severity tends to prompt more later treatment, as represented by the positive coefficient \(0.5\) in the treatment-assignment equation. Severity also predicts the final outcome.
The model is deliberately simple: it assumes linear relationships, constant effects, and no treatment-by-severity interactions. Its purpose is to expose the difference between a conditional regression comparison and a comparison of treatment histories, not to claim that these assumptions usually hold in clinical data.
Worked Example: Calculating the Sustained-Treatment Contrast
Compare the mean outcome if treatment is set to 1 at both visits with the mean outcome if it is set to 0 at both visits. Under an intervention setting \(A_0=a_0\) and \(A_1=a_1\), the severity equation becomes \(L_1(a_0)=2a_0+U\). Substituting this into the outcome equation gives
Since \(E[U]=E[\varepsilon]=0\), it follows that \(E[Y(1,1)]=5+1.5=6.5\), while \(E[Y(0,0)]=0\). The sustained-treatment contrast is therefore \(6.5-0=6.5\) outcome units.
The increase includes the effect of initial treatment operating through the later severity score. In particular, setting \(A_0\) from 0 to 1 raises \(L_1\) by 2 units, which contributes \(2\times2=4\) outcome units through the severity term. The direct contribution of \(A_0\) is 1, and the contribution of changing \(A_1\) from 0 to 1 is 1.5; \(1+4+1.5=6.5\).
What Ordinary Adjustment Estimates
Consider the population linear regression of \(Y\) on \(A_0,A_1,L_1\), including an intercept. The outcome equation already has exactly this form. Its error \(\varepsilon\) is independent of all three regressors and has mean zero. Consequently, the conditional mean is
The regression coefficients are thus 1 for \(A_0\), 1.5 for \(A_1\), and 2 for \(L_1\). There is no exact linear dependence among the regressors: \(A_0\), \(U=L_1-2A_0\), and \(V=A_1-0.5L_1\) are independent variables with positive variances, and they determine the three regressors. The population regression coefficients are therefore uniquely defined.
Worked Example: Deriving the Adjusted Regression Contrast
Use the fitted conditional mean \(A_0+1.5A_1+2L_1\). At a fixed severity value \(L_1=l\), the predicted outcome for \((A_0,A_1)=(1,1)\) is \(1+1.5+2l=2.5+2l\). For \((A_0,A_1)=(0,0)\) at that same severity value, it is \(2l\). Subtracting gives \(2.5\).
This calculation holds severity at the same value in both predictions. Under the sustained interventions, however, the expected severity is \(E[L_1(1)]=2\) for the first history and \(E[L_1(0)]=0\) for the second. Allowing severity to change adds \(2(2-0)=4\) outcome units, so the regime contrast is \(2.5+4=6.5\), in agreement with the direct derivation.
The distinction can be stated as a general derivation for this structural model. Under an intervention, the difference in expected outcomes between histories \((a_0,a_1)\) and \((a'_0,a'_1)\) is
The difference between these expressions is \(4(a_0-a'_0)\): the part of the initial-treatment effect transmitted through severity. For the sustained contrast from \((0,0)\) to \((1,1)\), this difference is 4.
Confounding by Indication in a Later-Treatment Regression
A different ordinary regression can fail in another way. If one regresses \(Y\) only on \(A_1\), the coefficient mixes the effect of later treatment with the association between later treatment and severity. In the specified model, severity is also associated with initial treatment, so that association contributes as well.
Worked Example: Calculating the Later-Treatment-Only Slope
First calculate the needed moments. Since \(L_1=2A_0+U\), \(\operatorname{Var}(L_1)=4+1=5\) and \(\operatorname{Cov}(L_1,A_0)=2\). Since \(A_1=0.5L_1+V\),
The outcome equation then gives
For a population regression of \(Y\) on a single variable \(A_1\) with an intercept, the slope is \(\operatorname{Cov}(A_1,Y)/\operatorname{Var}(A_1)\). Thus the slope is \(9.375/2.25=4.166\overline{6}\), not the structural effect 1.5 of intervening on \(A_1\) while holding the other causes fixed. The excess reflects the positive association of \(A_1\) with severity and prior treatment. This is a numerical example of confounding by indication.
Assumptions, Diagnostics, and Limitations
The calculations above are exact for the stated structural model. Their causal interpretation relies on clearly defined interventions on both treatment variables, consistency between observed outcomes and the corresponding potential outcomes, and no interference between individuals. The model also assumes the displayed structural equations are correct and that the exogenous variables are mutually independent. Independence of \(\varepsilon\) from the regressors is what makes the adjusted outcome regression recover its stated conditional mean.
In applied work, a regression coefficient does not verify those assumptions. Useful checks include examining whether severity and treatment have plausible temporal ordering, whether the outcome model has important nonlinearity or interactions, and whether treatment groups overlap across the observed severity range. A fitted regression can look stable while relying on extrapolation if, for example, very severe patients almost always receive high treatment. Residual and functional-form diagnostics can flag some model failures, but cannot establish absence of unmeasured confounding.
The key comparison is between three different choices:
| Analysis | What it compares in this example | Main concern |
|---|---|---|
| Regression of outcome on later treatment alone | Patients with different observed \(A_1\) | Mixes treatment effect with differences in severity and prior treatment |
| Regression adjusted for \(A_0,A_1,L_1\) | Treatment differences at a fixed observed severity | Does not by itself include the severity change caused by initial treatment |
| Explicit comparison of sustained histories | Outcomes under interventions setting both treatment visits | Requires a valid causal model or identification strategy, not just a coefficient label |
A common mistake is to say, “Severity predicts the outcome and later treatment, so adjusting for it must remove bias and give the total treatment effect.” The first part of that reasoning identifies a reason to account for severity when studying later treatment. The conclusion does not follow: severity is also downstream of \(A_0\), so fixing it removes the pathway through which initial treatment changes severity. In this example, the adjusted contrast is 2.5 whereas the sustained contrast is 6.5.
This does not mean severity should never be included in an outcome regression. It means the estimand must be explicit. A conditional contrast at fixed severity, a direct effect, and a sustained-treatment contrast are different targets. Ordinary regression can be useful for its specified conditional comparison, but its coefficients do not automatically become regime effects merely because treatment history appears among the regressors.
Check Your Understanding
Answer each question using the model in this tutorial. Brief solutions follow the questions.
- What is \(E[L_1(1)]-E[L_1(0)]\)?
- What is the sustained-treatment contrast from \((0,0)\) to \((1,1)\), and how much of it is contributed through severity?
- At fixed \(L_1\), what contrast does the adjusted regression predict for changing \((A_0,A_1)\) from \((0,0)\) to \((1,1)\)?
- Why is the later-treatment-only slope not the structural effect of \(A_1\)?
Solutions. (1) Since \(L_1(a_0)=2a_0+U\) and \(E[U]=0\), the difference is \(2-0=2\). (2) The contrast is \(6.5\); the severity pathway contributes \(2\times2=4\). (3) The regression gives \(1+1.5=2.5\), because severity is held fixed in both predictions. (4) The observed \(A_1\) is associated with \(L_1\) and \(A_0\), both of which predict \(Y\). The single-predictor slope therefore combines those associations with the effect of intervening on \(A_1\), which is 1.5 in the model.