When the Biomarker and Event Move Together
Consider a hypothetical study following adults with chronic heart failure. At regular visits, the study measures a blood biomarker that tends to rise as cardiac function worsens. The event of interest is hospital admission for heart failure. A rising biomarker may help describe a patient's changing prognosis, but it does not automatically follow that the biomarker itself is driving the admission risk.
A useful way to understand the problem is to distinguish the observed biomarker from the underlying disease process. Worsening disease can alter the biomarker and increase event risk at the same time. The biomarker is then a changing indicator of a changing health state, rather than an externally assigned exposure. This is the clinical setting in which a time-varying biomarker can be an endogenous predictor.
The distinction matters even when the marker is measured before the event interval being predicted. Measuring it first establishes the order of observation; it does not make the marker independent of underlying health. The earlier material on endogenous time-varying covariates gives the general concern. Here, the clinical example makes the feedback between disease progression, biomarker values, and event risk explicit.
A Simple Model for the Clinical Story
Let \(D_i(t)\) denote the underlying disease severity for person \(i\) at time \(t\). It is a conceptual health state and may not be measured directly. Let \(Y_i(t)\) be the biomarker, with larger values indicating worse disease in this illustration, and let \(T_i\) be the time to hospital admission. A simple measurement model and event model are
Here, \(\varepsilon_i(t)\) is measurement variation, \(\lambda_0(t)\) is the baseline event hazard, and \(\gamma>0\) means that greater disease severity corresponds to a higher event hazard. For this illustration, assume the measurement errors have mean zero and are independent of disease severity and the event process conditional on disease severity. Most importantly, the event model depends on \(D_i(t)\), not directly on \(Y_i(t)\). Thus, even if the biomarker is strongly associated with admission, this model does not say that changing the measured biomarker would change admission risk.
This is a deliberately simplified model, not a universal account of a clinical biomarker. Real biomarkers may measure disease imperfectly, reflect several processes, or also participate in disease mechanisms. The point is to show that a biomarker–event relationship can arise without a direct biomarker effect.
Worked Example: A Worsening Biomarker and a Higher Monthly Risk
Suppose a hypothetical patient's disease severity increases from \(D=0.5\) to \(D=1.5\). Use a constant monthly baseline hazard \(\lambda_0=0.02\) and \(\gamma=0.7\). Assume the biomarker equals disease severity in this simplified example, so its value rises from 0.5 to 1.5. Calculate the event risk over a one-month interval at each severity.
For a constant hazard \(\lambda\) over an interval of length \(\Delta\), the probability of an event during that interval, conditional on being event-free at its start, is \(q=1-\exp(-\lambda\Delta)\). Here, \(\Delta=1\) month and \(\lambda=0.02\exp(0.7D)\). At \(D=0.5\),
so the one-month event risk is
That is approximately 2.80% over one month. At \(D=1.5\),
and therefore \(q=1-\exp(-0.05717)\approx0.0556\), or 5.56% over one month. In this model, the biomarker rises as disease worsens and the event risk rises as well. The calculation shows how the two can move together; it does not establish that lowering the biomarker would lower the risk.
Why an Association Appears
The shared disease process can produce a biomarker–event association even when the biomarker has no direct effect in the event model. A short derivation makes the direction of the association clear. Consider one interval and write \(q(D)\) for the event probability given severity \(D\). Because the hazard increases with \(D\), \(q(D)\) also increases with \(D\). Under the measurement assumptions above, \(Y=D+\varepsilon\), with mean-zero error independent of \(D\) and the event conditional on \(D\). Consequently, the covariance between the biomarker and the interval event indicator \(E\) is
The last equality follows because the independent mean-zero measurement error contributes no covariance with the event indicator. To see why \(\operatorname{Cov}(D,q(D))\geq0\), let \(D'\) be an independent copy of \(D\). Then
The inequality holds because \(q\) is increasing: \(D-D'\) and \(q(D)-q(D')\) always have the same sign or one is zero. Thus, under these assumptions, larger biomarker values tend to accompany higher event risk even though \(Y\) does not appear in the event model. Measurement error can weaken the association, but it does not remove the shared cause.
This derivation is a justification for the illustration, not a claim that every observed biomarker association must be positive or that every biomarker is only a proxy. Other factors, measurement patterns, and treatment decisions can alter the observed association.
Worked Example: Comparing Two Groups With Different Disease Mixtures
Suppose a hypothetical cohort is divided into a high-biomarker group and a low-biomarker group at the start of a one-month interval. In the high-biomarker group, 60% have high disease severity and 40% have low severity. In the low-biomarker group, 20% have high severity and 80% have low severity. Assume the one-month event probability is 0.07688 for high severity and 0.01980 for low severity, and assume these probabilities are the same in both biomarker groups at a given severity. Calculate each group's one-month event risk.
The high-biomarker group's risk is the weighted average of its severity-specific risks:
Its one-month risk is therefore about 5.405%. For the low-biomarker group,
or about 3.122%. The risk ratio is \(0.054048/0.031216\approx1.731\). Thus, the high-biomarker group has about 1.73 times the one-month event risk in this hypothetical comparison. Yet within each severity level the event probability was specified to be identical across biomarker groups. The group difference arises because high biomarker values are more common among people with high underlying severity. This is a prognostic association, not evidence that changing the biomarker would change risk.
Following One Patient Over Time
A time-varying predictor updates as a person's measured health changes. To interpret an updated value as a predictor for a future interval, the value must be recorded before that interval begins. If an event occurs, the event process also ends the opportunity to collect later biomarker measurements. The observed sequence therefore combines a changing health state with a measurement schedule that cannot continue after the event.
Worked Example: Updating Risk Across Two Months
Compare two hypothetical patients who are both event-free at the start of month 1. Use the model in the earlier example, with \(\lambda_0=0.02\) per month, \(\gamma=0.7\), and a one-month interval. Patient A has disease severity \(D=0.5\) in month 1 and \(D=1.5\) in month 2; Patient B remains at \(D=0.5\) in both months. Suppose a biomarker measured at each month’s start tracks severity exactly, and use each measurement to describe risk during the following month.
The monthly event risks are \(q(0.5)\approx0.0280\) and \(q(1.5)\approx0.0556\). If the monthly hazards are conditionally constant and the two intervals are evaluated under these specified severity paths, Patient A's probability of remaining event-free through both months is
Thus, Patient A's two-month event risk is approximately \(1-0.9180=0.0820\), or 8.20%. Patient B's probability of remaining event-free through both months is
so the two-month event risk is approximately \(1-0.9448=0.0552\), or 5.52%. The different risks reflect the different disease-severity paths, which are also reflected in the biomarker measurements. These calculations assume the stated severity path and interval hazards; they do not identify a causal effect of the biomarker.
In an actual study, measured biomarker values will not perfectly reveal the latent path. Measurement error, visit timing, and differences in patients' health can all matter. The example also assumes the marker is measured at the start of each interval; using a value measured after the interval or after an event to predict that interval would not match the stated temporal setup.
Assumptions, Alternatives, and Limitations
The calculations above rely on specific assumptions. The hazard model treats disease severity as the relevant driver of event risk and assumes a log-linear relationship with severity. The measurement equation treats the biomarker as a noisy indicator of severity. The group example assumes the stated severity mixtures and risks apply within each biomarker group. These are transparent teaching assumptions, not facts established by observing a biomarker and an event.
Several analysis choices answer different questions:
| Approach | What it uses | Key limitation in this setting |
|---|---|---|
| Baseline-only biomarker analysis | A single value measured at study entry | Does not represent later biomarker changes; underlying severity can still relate to both baseline value and event risk. |
| Time-dependent event analysis | Biomarker values updated over follow-up | Can describe how updated values relate to subsequent event risk, but the association may still reflect unmeasured evolving disease. |
| Joint longitudinal and event modeling | The repeated biomarker measurements and event process in a linked model | Can represent shared latent trajectories and event risk under chosen assumptions, but does not make those assumptions automatically correct or prove causality. |
A time-dependent analysis is useful when the scientific aim is to describe or predict risk using values available at successive times. Its interpretation depends on the model and on how the values were generated. A joint model can provide a framework for connecting the longitudinal trajectory with event risk, but conclusions still depend on the specified trajectory, event, and dependence structures. Neither approach turns an observational biomarker association into a causal effect without additional assumptions and design information.
The simple illustration also leaves out treatment changes, multiple causes of biomarker variation, irregular visit schedules, and uncertainty about disease severity. In practice, treatment may respond to prior biomarker values, and treatment may then affect later biomarker values and event risk. Those feedbacks make the relationship more complex than the two-equation example. The present example motivates why a changing marker needs careful modeling; it does not specify a complete clinical analysis strategy.
Common Pitfall: Treating a Prognostic Marker as a Cause
A common mistake is to reason, “The risk was higher when the biomarker was higher, so lowering the biomarker would lower the risk.” The worked examples do not support that conclusion. In the first model, disease severity causes both the biomarker to rise and the event hazard to increase, while the biomarker itself has no direct role in the event model. An intervention that changes the measured value without changing underlying disease could therefore leave event risk unchanged in that model.
A safer interpretation is: “In these hypothetical data, a higher biomarker identifies intervals or groups with higher event risk, consistent with the biomarker tracking worsening disease.” This is a prognostic statement. A causal statement would require a clearly defined intervention and a justified strategy for separating the effect of that intervention from the evolving health factors that affect both marker values and events.
Check Your Understanding
Answer each question using the definitions and assumptions in this tutorial. Full solutions follow the questions.
- In the simple clinical model, what causes the biomarker and event risk to be associated if the biomarker does not appear in the hazard model?
- Suppose \(q(D)\) increases with disease severity and \(Y=D+\varepsilon\), where the mean-zero error is independent of disease severity and the event conditional on severity. What is the sign of \(\operatorname{Cov}(Y,E)\), and why?
- A biomarker has a value measured at the start of each month. When can that value be used as a predictor in the interval-risk calculation described here?
- Why does the high-biomarker group in the group comparison have a higher event risk even though risks were set equal across biomarker groups within each severity level?
- Give one limitation of a time-dependent event analysis and one way joint longitudinal and event modeling addresses the motivation for that analysis.
Solutions. (1) Underlying disease severity affects both the observed biomarker and the event hazard, creating an association through a shared cause. (2) The covariance is nonnegative because \(\operatorname{Cov}(Y,E)=\operatorname{Cov}(D,q(D))\), and an increasing \(q\) makes \((D-D')\{q(D)-q(D')\}\geq0\) for independent copies \(D,D'\). (3) It can describe risk during the following interval if measured before that interval begins and if the model uses the stated timing. (4) The high-biomarker group contains a larger fraction of people with high severity, whose specified event risk is greater; its weighted average risk is therefore higher. (5) A time-dependent analysis can retain association with unmeasured evolving disease; joint modeling links the repeated measurements and event process to represent their dependence under explicit assumptions, but it does not by itself establish causality.