Why Baseline Adjustment Misses Evolving Confounding

Why Baseline Information Can Become Out of Date

At the start of follow-up, baseline covariates may describe patients well enough to compare treatment decisions. But a later decision is made after patients have changed: symptoms may have worsened, a laboratory measure may have shifted, or a new complication may have appeared. If that updated status helps determine treatment and also predicts the outcome, patients receiving different later treatments are not comparable merely because their baseline covariates match.

The issue is not that baseline covariates are useless. They can be important predictors of treatment and outcome. The problem is that baseline adjustment cannot account for differences in a relevant covariate that arises or is measured later. The following argument isolates this issue at the later treatment decision.

Holding \(A_0\) fixed in \(B\) makes the point precise: any failure below is not simply due to comparing different earlier treatment histories. Let \(Y^a\) denote the outcome that would be observed if the later treatment were set to \(a\), for \(a\in\{0,1\}\). A common identification condition at this decision is conditional exchangeability, \(Y^a\perp A_1\mid (B,L_1)\), together with consistency and positivity. Consistency means that \(Y=Y^a\) for a person with \(A_1=a\); positivity means that each treatment value has positive probability within covariate histories being compared. These conditions do not imply exchangeability given \(B\) alone.

How Updated Status Produces a Baseline-Adjusted Association

The next result spells out the source of the association. It uses discrete \(L_1\) for a transparent sum; with continuous covariates, the same argument uses conditional expectations and integrals, provided the relevant expectations exist.

Consequently, if the later treatment has no effect within these histories, so that \(m_1(b,l)=m_0(b,l)=m(b,l)\), then the baseline-adjusted contrast is

Proof. For either treatment value \(a\), consistency gives \(Y=Y^a\) among those with \(A_1=a\). Conditional exchangeability gives \(E[Y^a\mid A_1=a,B=b,L_1=l]=E[Y^a\mid B=b,L_1=l]=m_a(b,l)\). Averaging this conditional mean over the distribution of \(L_1\) among people with \(A_1=a,B=b\), by the law of total expectation, yields the first equation. Subtracting the equation for \(a=0\) from the one for \(a=1\), and setting both conditional means equal to \(m(b,l)\) under the no-effect condition, yields the second equation. \(\square\)

The equation shows why a baseline-adjusted contrast can be nonzero even when treatment has no effect: the two treatment groups can contain different mixes of updated patient status, and that status can predict \(Y\). It also gives two sufficient ways this particular source of bias disappears: either the distribution of \(L_1\) is the same across later treatment groups within \(B\), or the outcome mean \(m(b,l)\) does not vary with \(l\). Neither condition follows merely from having measured and adjusted for \(L_0\).

Worked Examples

A Practical Condition and Its Limits

The preceding result also provides a useful sufficient condition for baseline-only adjustment to work for a later treatment comparison. If \(Y^a\perp A_1\mid B\), with consistency and positivity given \(B\), then \(E[Y\mid A_1=a,B=b]=E[Y^a\mid B=b]\). The observed conditional means then identify the corresponding potential-outcome means within \(B\). The difficulty is that exchangeability given \(B\) is an assumption, not a consequence of exchangeability given the richer history \((B,L_1)\).

Proof. In the no-effect contrast formula, independence gives equal conditional distributions of \(L_1\) across \(A_1\) at each \(B=b\), making every difference of probabilities zero. Alternatively, if \(m(b,l)=c\) for every \(l\), the contrast equals \(c\) times the difference between two probability sums, each of which is one; that difference is zero. \(\square\)

These are sufficient conditions, not a checklist that can be verified from an outcome table alone. In applied work, ask whether clinical knowledge suggests that the updated measure both influences treatment choice and predicts the outcome. Examine treatment frequencies across levels of the updated measure within baseline strata, and check whether meaningful status groups have little or no chance of receiving one treatment. Such patterns indicate possible confounding or a positivity problem; a fitted model cannot create comparisons absent from the data.

The main alternatives are not interchangeable. Baseline stratification or regression adjustment can control baseline differences, but cannot by itself balance updated status. A time-updated comparison conditions on relevant measured history, subject to suitable assumptions and careful choice of estimand. Inverse-probability weighting and marginal structural models are standard approaches in longitudinal causal inference when treatment-confounder feedback complicates ordinary adjustment; their construction and assumptions require separate development. No approach repairs unmeasured confounding, poor measurement of status, or a lack of treatment overlap.

Common Mistakes and Diagnostics

• Assuming baseline adjustment covers all later confounding. It does not balance covariates that vary among people with the same baseline values. Check whether updated status predicts both the later decision and the outcome.
• Calling an observed contrast a treatment effect. The examples show that a large contrast can occur when treatment has no effect. State the identifying assumptions before interpreting it causally.
• Ignoring treatment overlap. If some status-history groups almost always receive one treatment, conditional comparisons rely on little information. Inspect treatment probabilities or cell counts by relevant history.
• Treating an adjusted model as proof of no confounding. Adjustment only addresses variables measured and represented adequately. Unmeasured severity, measurement error, and model misspecification can leave bias.

A further limitation matters in longitudinal settings: \(L_1\) may itself have been affected by \(A_0\). The examples above isolate confounding of the later decision; they do not settle how to estimate the effect of an entire treatment history when earlier treatment changes later status. The key conclusion here is narrower: baseline covariates alone do not generally make later treatment groups comparable.