Why Baseline Adjustment Can Be Incomplete
In a longitudinal study, treatment is often decided more than once. A patient's condition may change after an initial treatment, and that change may inform the next treatment decision. If the evolving condition also predicts the outcome, patients receiving different later treatments can differ in their prognosis even when they have the same measured baseline characteristics.
Adjusting for baseline characteristics can make treatment groups more comparable with respect to those characteristics. It cannot, by itself, balance a later covariate that varies within baseline groups and helps determine subsequent treatment. The key issue is not simply that a variable was measured after baseline: it is that, at a particular treatment decision, the variable predicts both the treatment choice and the outcome.
For notation, let \(X\) denote baseline covariates, \(A_0\) an initial treatment, \(L_1\) a covariate measured later, \(A_1\) the subsequent treatment, and \(Y\) the outcome. The relevant time ordering is \(X, A_0, L_1, A_1, Y\). When \(L_1\) influences the choice of \(A_1\) and is prognostic for \(Y\), adjusting only for \(X\) leaves the relationship between \(A_1\) and \(Y\) potentially confounded.
This is a statement about what baseline adjustment can achieve, not a claim that every later covariate must be adjusted for in every analysis. A covariate that predicts the outcome but does not help predict treatment, given the other information, is not a confounder for that treatment decision. Likewise, a variable associated with treatment but unrelated to the outcome under the treatment being studied does not create this form of confounding.
What Baseline Adjustment Assumes
For a single treatment \(A\), baseline adjustment is justified for a causal contrast when, among other requirements, treatment groups are exchangeable conditional on the measured baseline covariates. In potential-outcomes notation, this condition is \(Y(a) \perp A \mid X\): within each level of \(X\), treatment assignment is independent of the outcome that would occur under treatment level \(a\). Consistency links the observed outcome to the potential outcome under the treatment actually received, and positivity requires a nonzero probability of each treatment being received at the relevant baseline covariate levels.
In a longitudinal setting, a baseline-only analysis needs a corresponding condition for the later decision: after conditioning on baseline information, there must be no remaining common causes of \(A_1\) and the outcome under the treatment sequence of interest. If the evolving \(L_1\) is such a common cause and is not determined by \(X\), that condition is not met merely because \(X\) has been adjusted for.
Worked Example: Finding the Later Decision Point
Consider a hypothetical study of a chronic condition. Researchers record baseline disease severity \(X\), assign an initial therapy \(A_0\), measure symptom severity \(L_1\) at week four, choose a second therapy \(A_1\), and record a later outcome \(Y\). Clinicians are more likely to intensify therapy when symptoms at week four are severe, and severe symptoms also predict a poor later outcome.
Suppose that, among 100 patients with the same baseline severity and initial therapy, 60 have severe symptoms at week four and 40 have mild symptoms. In this group, 48 of the 60 patients with severe symptoms receive the intensified second therapy, compared with 8 of the 40 patients with mild symptoms. The treatment probabilities are
Thus, even within the same baseline-severity and initial-treatment group, the later treatment is strongly associated with week-four symptoms. Because those symptoms also predict the outcome, the treated and untreated patients can have different prognoses before the second therapy's effect is considered. Adjusting for baseline severity alone does not remove that difference: \(L_1\) still varies within the baseline group and helps guide \(A_1\).
This example identifies a potential source of confounding; the treatment probabilities alone do not quantify its size or establish the causal effect. To make a causal interpretation, one would also need appropriate measurement, treatment positivity, consistency, and control of other common causes at the relevant decision.
How Baseline-Stratified Comparisons Can Remain Confounded
The basic reason can be seen by expanding the outcome risk within a baseline stratum over levels of the evolving covariate. The observed outcome risk among patients receiving \(A_1=a\), at baseline level \(X=x\), is a mixture of risks across \(L_1\). The mixture weights are the distribution of \(L_1\) among patients who received that treatment.
If the distribution of \(L_1\) differs by \(A_1\), and outcome risk varies with \(L_1\), the observed treatment-group risks can differ because the groups contain different mixes of prognostic states. Stratifying or standardizing on \(X\) does not change those within-stratum mixes unless \(X\) fully determines \(L_1\) or otherwise makes it irrelevant to the treatment–outcome comparison. This identity explains why a baseline-adjusted contrast can retain confounding; it does not, on its own, determine the direction or magnitude of the bias.
Worked Example: Baseline Adjustment Does Not Remove a Reversal
Suppose a hypothetical cohort has one baseline stratum, so all patients have the same \(X\) and the same earlier treatment history. At the later decision, patients with severe evolving disease are more likely to receive treatment \(A_1=1\). Assume the following hypothetical outcome risks under each treatment level:
| Week-four disease state \(L_1\) | Risk of outcome if \(A_1=1\) | Risk of outcome if \(A_1=0\) |
|---|---|---|
| Severe | 0.40 | 0.50 |
| Mild | 0.10 | 0.20 |
Within either disease state, treatment is associated in this hypothetical setup with a 0.10 lower outcome risk. But treatment groups have different disease-state distributions: 40 of 50 treated patients are severe, while 10 of 50 untreated patients are severe. The corresponding severe proportions are \(40/50=0.80\) and \(10/50=0.20\).
The expected outcome risk among treated patients is
The expected risk among untreated patients is
The observed treated-minus-untreated risk difference is \(0.34-0.26=0.08\), or 8 percentage points higher risk in the treated group. Yet within each \(L_1\) state, the treatment contrast is \(-0.10\), or 10 percentage points lower risk. Since everyone has the same baseline \(X\), baseline adjustment cannot alter the 8-percentage-point contrast. The example shows how confounding by an evolving prognostic covariate can even reverse the direction of an observed association.
The risks are hypothetical and illustrate the mechanism, not a real study result. Interpreting the within-state contrasts causally would require assumptions that make treatment groups comparable within those states, as well as a clearly defined treatment contrast. The example's main point is narrower: equal baseline covariates do not imply equal distributions of later disease severity.
Not Every Baseline-Adjusted Contrast Is Equally Misleading
Residual confounding need not reverse an association. It may attenuate a treatment contrast, exaggerate it, or leave its direction unchanged. The direction depends on how the evolving covariate affects treatment selection and outcome risk, together with the treatment's effect. A baseline-adjusted estimate is not automatically closer to the causal effect just because some baseline predictors have been controlled.
Worked Example: Baseline Standardization Can Leave Attenuation
Consider a second hypothetical setting with two equally sized baseline groups, low-risk and high-risk. There are 100 treated and 100 untreated patients in each baseline group. In both groups, 60% of treated patients but 20% of untreated patients have severe evolving disease. Suppose outcome risks are as follows:
- In the low-risk baseline group, risks for treated patients are 0.30 when \(L_1\) is severe and 0.10 when it is mild; corresponding untreated risks are 0.40 and 0.20.
- In the high-risk baseline group, add 0.10 to each of those four risks.
For the low-risk baseline group, the treated risk is \(0.60(0.30)+0.40(0.10)=0.22\), and the untreated risk is \(0.20(0.40)+0.80(0.20)=0.24\). Its observed risk difference is \(0.22-0.24=-0.02\).
For the high-risk group, the treated risk is \(0.60(0.40)+0.40(0.20)=0.32\), and the untreated risk is \(0.20(0.50)+0.80(0.30)=0.34\). Its risk difference is also \(0.32-0.34=-0.02\). Standardizing equally across the two baseline groups gives
so the baseline-standardized risk difference is \(0.27-0.29=-0.02\), or 2 percentage points lower risk. By contrast, within each evolving-disease state, the specified treatment risks are 0.10 lower than the corresponding untreated risks. The baseline-standardized contrast is therefore attenuated in this illustration, not reversed. The reason remains the same: treated patients have more severe evolving disease within each baseline group.
The equal baseline-group weights make the standardization transparent. In other settings, the target population determines the weights, but changing baseline weights still does not itself equalize the treatment groups' distributions of \(L_1\) within baseline strata.
When Baseline Adjustment Can Be Enough
Baseline adjustment can be adequate if the measured baseline covariates are sufficient to control common causes of the treatment and outcome for the causal contrast under study. In a longitudinal process, this could hold if later treatment decisions do not depend on any additional outcome-predictive information after conditioning on baseline covariates and the relevant treatment history. It could also hold if an evolving variable is fully determined by the conditioned information, or if it does not predict the outcome under the treatment being compared.
These are substantive assumptions, not conclusions that follow from including many baseline variables in a model. A long list of baseline predictors cannot guarantee control of information that only becomes available during follow-up. Study protocols, clinical decision rules, and the timing of measurement help determine what clinicians knew when they chose later treatment.
A practical diagnostic is to ask the same two questions at each treatment decision: What information available before this decision influenced treatment choice? and Which of that information also predicts the outcome? If an evolving covariate answers both questions and is not captured by baseline variables, a baseline-only adjustment set is inadequate for that decision.
Common Mistake
Wrong reasoning: “The treatment groups were adjusted to have the same baseline severity, so their later treatment comparison is unconfounded.” This conclusion does not follow. Patients with the same baseline severity can develop different symptoms, biomarkers, or disease states, and those evolving states can influence later treatment and predict outcomes.
Correction: Assess confounding at the time each treatment is chosen. Baseline adjustment controls differences represented by baseline covariates; it does not automatically control later differences in evolving covariates. At the same time, do not label every post-baseline prognostic variable a confounder: it must also be relevant to treatment assignment at the decision being studied.
Check Your Understanding
Use the timing of covariate measurement and treatment decisions to answer each question.
- A biomarker measured after initial treatment predicts the next treatment and the outcome. Why might adjusting only for baseline severity leave confounding of the next treatment decision?
- Give one reason a later covariate that predicts the outcome might not be a confounder of a particular treatment decision.
- In the reversal example, calculate the severe-disease proportion in the treated group and in the untreated group. What feature of those proportions matters for baseline adjustment?
- Does an observed baseline-adjusted risk difference necessarily have the same direction as the causal effect? Explain using the hypothetical risks in the tutorial.
- Name two conditions, involving the evolving covariate or treatment decision, under which baseline adjustment could still be sufficient.