Why Baseline Health May Not Be Enough
A baseline health measure describes a participant at one point in time. During follow-up, health can improve, worsen, or fluctuate. If an unobserved change in health affects both a biomarker and the chance of an event, then participants with different biomarker values may differ in event risk for reasons that arose after baseline. Adjusting for baseline health does not, by itself, remove that difference.
This matters when interpreting a time-varying biomarker as a predictor of an event, or when asking what would happen if the biomarker could be changed. A biomarker can track evolving health and therefore be associated with event risk even if changing the biomarker itself would not change that risk. The issue is not that baseline health is irrelevant; it is that a single baseline value may not capture the health history that develops later.
Let \(H_{i0}\) denote measured baseline health for participant \(i\), \(U_i(t)\) an unobserved health state that may change over time, \(M_i(t)\) the biomarker, and \(T_i\) the event time. A simple causal structure is that \(H_{i0}\) may affect later health, while \(U_i(t)\) affects both \(M_i(t)\) and the chance of an event after time \(t\). If \(U_i(t)\) is not fully determined by \(H_{i0}\), conditioning on \(H_{i0}\) leaves the later common cause in place.
The arrows summarize possible causal influence, not a claim that every study has this exact structure. The essential feature is the open path \(M_i(t) \leftarrow U_i(t) \rightarrow T_i\). Conditioning on baseline health does not close that path when the evolving \(U_i(t)\) still varies among participants with the same baseline health.
A Numerical Example Within Baseline Health Groups
Consider a hypothetical study that follows participants to a fixed follow-up horizon. For simplicity, classify the biomarker as high or low, and classify evolving health as deteriorated or stable. Suppose the baseline health groups have the same pattern shown below. Within each baseline group are 100 participants: 50 with deteriorated evolving health and 50 with stable evolving health. The event risk is 0.40 for deteriorated health and 0.10 for stable health. The biomarker is more often high when health has deteriorated.
Worked Example: A Residual Association After Baseline Stratification
In each baseline health group, suppose 40 of the 50 participants with deteriorated health have a high biomarker, while 10 of the 50 with stable health have a high biomarker. Thus, among the 100 participants in that baseline group, 50 have a high biomarker and 50 have a low biomarker.
For the high-biomarker group, the expected number of events is \(40(0.40)+10(0.10)=16+1=17\). Its event risk is therefore \(17/50=0.34\), or 34%.
For the low-biomarker group, the expected number of events is \(10(0.40)+40(0.10)=4+4=8\). Its event risk is \(8/50=0.16\), or 16%. The risk difference is \(0.34-0.16=0.18\), or 18 percentage points; the risk ratio is \(0.34/0.16=2.125\).
These calculations are identical within each baseline health group, so standardizing over the baseline groups also gives risks of 34% and 16%. Baseline adjustment has not removed the association. In this constructed example, the biomarker itself has no effect on event risk: within each evolving-health category, the event risk is the same for high and low biomarker values. The association arises because evolving health predicts both biomarker status and the event.
The example illustrates a general distinction. A baseline-adjusted association is a comparison among participants with similar measured baseline health; it is not automatically a comparison among participants with the same health history throughout follow-up. The latter would require adequate information about the evolving common causes, or additional assumptions that justify treating baseline health as sufficient.
Why the Residual Association Appears
The same point can be expressed using conditional covariance. In the example, let \(B=1\) indicate a high biomarker and \(Y=1\) indicate an event by the horizon. Within any fixed baseline health group, suppose \(P(U=1)=0.5\), where \(U=1\) denotes deteriorated health. The probabilities are \(P(B=1\mid U=1)=0.8\), \(P(B=1\mid U=0)=0.2\), \(P(Y=1\mid U=1)=0.4\), and \(P(Y=1\mid U=0)=0.1\). Assume \(B\) and \(Y\) are independent conditional on \(U\) and baseline health: there is no biomarker effect in this illustration.
Because of that conditional independence, the covariance within a baseline group comes from differences in the conditional means across the two health states. For a binary \(U\), the identity is \(\operatorname{Cov}(B,Y\mid H_0)=P(U=1\mid H_0)P(U=0\mid H_0)\Delta_B\Delta_Y\), where \(\Delta_B\) and \(\Delta_Y\) are the differences in the conditional means between \(U=1\) and \(U=0\). Substitution gives:
The positive covariance is present even though the conditional covariance of \(B\) and \(Y\) given both \(U\) and baseline health is zero. The shared evolving health state creates the association. This derivation also shows what would eliminate this particular source of bias: the common cause would need to be measured and appropriately controlled, or the assumptions would need to justify that no such residual common cause exists after conditioning on baseline health.
Worked Example: Checking the Marginal Risks From the Covariance Setup
Within a baseline group, \(P(B=1)=0.5(0.8)+0.5(0.2)=0.5\). The joint probability of a high biomarker and an event is \(P(B=1,Y=1)=0.5(0.8)(0.4)+0.5(0.2)(0.1)=0.16+0.01=0.17\). Therefore, \(P(Y=1\mid B=1)=0.17/0.5=0.34\).
The overall event risk is \(P(Y=1)=0.5(0.4)+0.5(0.1)=0.25\). Thus, \(P(B=1)P(Y=1)=0.5(0.25)=0.125\), and the covariance is \(0.17-0.125=0.045\), matching the conditional-covariance calculation. For the low-biomarker group, the joint event probability is \(0.5(0.2)(0.4)+0.5(0.8)(0.1)=0.08\), giving \(P(Y=1\mid B=0)=0.08/0.5=0.16\).
Both calculations describe the same hypothetical data. The risk contrast is an observational association, not evidence that changing the biomarker would change risk.
What Baseline Adjustment Assumes
A baseline-adjusted analysis can support a causal interpretation only under demanding conditions. In plain terms, after accounting for the variables in the analysis, biomarker differences must not remain associated with unmeasured causes of later event risk. Baseline health can be enough if it captures all relevant health differences that jointly affect the biomarker and the event throughout follow-up. It is not enough merely because it was measured before the biomarker or event.
For example, a baseline measure may be a useful predictor of later health while still leaving substantial variation in subsequent deterioration. A model containing \(H_{i0}\) can adjust for measured differences at baseline, but it cannot directly adjust for \(U_i(t)\) if that state is unobserved. Adding a flexible function of baseline health does not solve this problem unless baseline health actually contains enough information to account for the evolving common cause.
Worked Example: The Same Baseline Score, Different Follow-Up Health
Suppose two hypothetical groups each have a mean baseline health score of 80 points. By month 6, one group remains near 80, while the other has deteriorated to 60. Assume the biomarker is high in the deteriorated group and low in the stable group. For illustration, suppose the event hazard is 0.02 per month for stable health and twice as large, 0.04 per month, for deteriorated health, with no direct biomarker effect.
A model that adjusts for the common baseline score of 80 does not distinguish the groups by their month-6 health. The observed hazard ratio comparing the high- with low-biomarker group is \(0.04/0.02=2\), despite the stipulated absence of a biomarker effect. The units here are events per person-month for the hazards; the ratio is unitless.
This example is illustrative, not a method for estimating a hazard ratio from two group averages. It shows why identical baseline scores cannot guarantee comparable health during follow-up. It also emphasizes that a time-specific biomarker and event risk can share an evolving cause that baseline adjustment leaves uncontrolled.
Modeling Choices, Diagnostics, and Limits
A baseline-adjusted analysis is often useful for describing an association conditional on measured starting health. Its limitation is that it cannot represent health changes that were not measured. Measuring health repeatedly and adjusting for relevant time-updated health can address observed evolving differences, provided those measures are sufficiently accurate and appropriately ordered in time. It does not automatically address unobserved health or measurement error.
Joint longitudinal and time-to-event models can represent dependence between a biomarker trajectory and event risk, including dependence through shared latent features under specified model assumptions. Such a model is not a guarantee of causal identification: its conclusions depend on the model structure and assumptions about the unobserved components. A standard baseline-adjusted event model, a model with time-updated measured health, and a joint model therefore answer different questions and rely on different information.
| Approach | What it can account for | What it does not guarantee |
|---|---|---|
| Baseline adjustment | Measured differences in health at baseline | Control of unobserved health changes after baseline |
| Time-updated adjustment | Measured health differences during follow-up, if modeled appropriately | Control of unmeasured or poorly measured time-varying common causes |
| Joint modeling | Dependence between modeled biomarker trajectories and event processes | Causal interpretation without suitable identification assumptions |
Diagnostics can reveal problems but generally cannot prove that all evolving confounding has been removed. Compare the amount of follow-up health information with what the analysis adjusts for. Check whether conclusions change across reasonable baseline-health specifications, and examine whether biomarker patterns track clinical deterioration or changing care. Large changes across models signal sensitivity to assumptions; little change does not prove that unobserved confounding is absent. Also verify that the biomarker is measured before the event time being predicted and that its values are available in the population relevant to the question.
Common Mistake
- Wrong reasoning: “Health was measured before follow-up and included in the model, so the biomarker–event association is no longer confounded by health.”
- Correction: Baseline adjustment addresses measured baseline health. If unobserved health changes during follow-up and affects both the biomarker and event risk, the association can remain confounded within baseline health levels.
- Another overstatement: “The residual association proves the biomarker causes the event.” The numerical examples show the opposite is possible: a common evolving cause can produce an association when the biomarker has no effect in the constructed model.
Check Your Understanding
Answer the questions, then compare your reasoning with the solutions below.
- If an unobserved health change raises both the biomarker and event risk, why may adjustment for baseline health leave an association?
- In the numerical example, what is the event risk for the high-biomarker group within a baseline health group?
- Does a positive biomarker–event association establish that intervening to change the biomarker would change event risk? Explain.
- Name one kind of follow-up information that could help address the problem, and state a limitation that remains.
Solutions. (1) Participants with the same baseline health can still differ in their later unobserved health; that health affects both the biomarker and event risk, leaving a common-cause path open. (2) The expected event count is \(40(0.40)+10(0.10)=17\) among 50 high-biomarker participants, so the risk is \(17/50=0.34\), or 34%. (3) No. The association may be due to evolving health rather than a biomarker effect; a causal claim requires additional assumptions. (4) Repeated measurements of health can help account for observed changes, but they may be incomplete or inaccurate and cannot by themselves control unmeasured changes.