Why Treatment History Matters
A clinician may increase, reduce, start, or stop treatment after reviewing a patient’s biomarker results. That decision can make treatment exposure informative: treated patients may differ from untreated patients because they have different biomarker histories, and those histories may also predict an event. Consequently, a treatment–event association can reflect both treatment and the clinical reasons for changing it.
The key issue is the order of events. A biomarker measured before a treatment decision can influence that decision. If the biomarker also signals worsening health, it is related to the patient’s subsequent event risk. Treatment status is then not simply an exposure that patients happened to receive independently of their prognosis. It is partly a response to that prognosis.
Write Down the Timeline First
Let \(B_k\) denote a biomarker value measured at visit time \(t_k\), \(A_k\) the treatment decision made after that measurement, and \(T\) the event time. The intended sequence is \(B_k\), then the decision about \(A_k\), then event follow-up under that treatment. A simple history may look like
The arrows indicate possible dependencies, not guaranteed effects. For example, \(B_1\) may reflect both the patient’s health and the effect of \(A_0\); it may then inform the decision to change treatment to \(A_1\). If worsening biomarker values predict the event, \(B_1\) helps explain why patients with \(A_1=1\) may have different event risks from patients with \(A_1=0\), even before considering the effect of \(A_1\).
A time-dependent event model might describe the event hazard using current treatment and a measured biomarker:
Here \(X_i\) denotes measured baseline covariates and \(\lambda_0(t)\) is the baseline hazard. The treatment value used at time \(t\) must represent treatment known just before that time; using future treatment information to classify earlier follow-up reverses the order of cause and outcome. In this model, \(\beta_A\) describes a conditional association in the specified hazard model. It is not automatically a causal treatment effect. That interpretation would require additional assumptions about treatment assignment, measurement, model form, and censoring.
Worked Example: Following a Treatment Change Through Time
Consider a hypothetical patient whose biomarker is measured at months 0, 3, and 6. The values are 4, 7, and 5 units. At month 3, the clinician reviews the value of 7 and increases treatment. The patient has an event at month 9.
Before month 3: The value of 7 is a prior biomarker measurement for the treatment decision at month 3. If it reflects worsening health, it may predict both the clinician’s decision to increase treatment and the patient’s subsequent event risk.
After month 3: The patient contributes follow-up under the increased treatment from month 3 until the next treatment change, event, or censoring. The value of 5 at month 6 occurs after the treatment change. It can help inform the month-6 decision, but it cannot be used as information available at month 3.
Interpretation: If patients whose biomarker rises are more likely to receive intensified treatment, a comparison of event rates by current treatment can make intensified treatment look less effective—or even harmful—because the treated group may have worse biomarker histories. This one patient’s timeline illustrates the ordering problem; it does not estimate a treatment effect.
How Different Biomarker Histories Change the Comparison
The basic arithmetic is that an observed treatment-group risk averages over the biomarker histories found in that group. Those histories need not have the same distribution in treated and untreated patients. Suppose \(Y\) indicates an event by a fixed follow-up time and \(B\) is a measured biomarker category. Then
This identity is a decomposition of an observed risk, not a causal formula. The weights \(P(B=b\mid A=a)\) can differ across treatment groups. Thus, even if the event risk is lower with treatment within each biomarker category, the overall treated risk can be higher when treated patients are more concentrated in categories with high underlying event risk.
Worked Example: A Conditional Advantage Hidden by Treatment Selection
Suppose a hypothetical cohort has two biomarker-history categories, high and low. The following event counts are stipulated for illustration:
| Biomarker history | Treated: events / patients | Untreated: events / patients |
|---|---|---|
| High | 48 / 400 | 16 / 100 |
| Low | 2 / 100 | 16 / 400 |
Within the high-history category, the treated risk is \(48/400=0.12\), or 12%, and the untreated risk is \(16/100=0.16\), or 16%. The treated-minus-untreated risk difference is \(0.12-0.16=-0.04\), a 4-percentage-point lower risk among treated patients in this category.
Within the low-history category, the treated risk is \(2/100=0.02\), or 2%, and the untreated risk is \(16/400=0.04\), or 4%. The difference is \(0.02-0.04=-0.02\), a 2-percentage-point lower risk among treated patients in this category.
But the treated group has 500 patients in total, with \(48+2=50\) events, for an overall risk of \(50/500=0.10\), or 10%. The untreated group also has 500 patients, with \(16+16=32\) events, for an overall risk of \(32/500=0.064\), or 6.4%. The crude difference is \(0.10-0.064=0.036\), or 3.6 percentage points higher in the treated group.
This reversal occurs because 400 of the 500 treated patients have high biomarker history, compared with only 100 of 500 untreated patients. The calculation shows why the crude association can differ from the within-category comparisons. It does not, on its own, prove that treatment causes lower risk: such a conclusion needs assumptions about all relevant differences between treatment groups, not only the two categories shown.
Worked Example: Checking a Biomarker-Dependent Risk Calculation
Suppose, purely hypothetically, that the one-period event probabilities over the stated biomarker ranges follow
For treated patients, the smallest probability occurs at \(b=4\): \(0.01+0.01(4)=0.05\). The largest occurs at \(b=10\): \(0.01+0.01(10)=0.11\). Thus the treated probabilities range from 0.05 to 0.11. For untreated patients, they range from \(0.02+0.01(0)=0.02\) to \(0.02+0.01(8)=0.10\). Both formulas give valid probabilities over their stipulated ranges.
At a biomarker value in the overlap, such as \(b=6\), the treated probability is \(0.01+0.01(6)=0.07\), and the untreated probability is \(0.02+0.01(6)=0.08\). The conditional difference is \(-0.01\), or 1 percentage point lower for treated patients at that value. Yet treated patients could have higher overall risk if their biomarker values are generally higher. Comparisons outside the overlap of biomarker ranges would require extrapolation rather than a direct within-range comparison.
Assumptions, Diagnostics, and Alternatives
A conventional time-dependent event analysis can be useful for describing how treatment status and event risk are associated. Its interpretation depends on careful timing and adequate measurement. For a causal comparison, the more demanding issue is whether patients receiving different treatment histories are comparable after accounting for measured history. If unrecorded health severity influences both treatment changes and event risk, adjustment for the recorded biomarker alone may not suffice.
- Check the decision timeline. Confirm that each biomarker value used to explain a treatment change was measured before that change, and that treatment status is updated at the correct time.
- Check the treatment pattern. Tabulate treatment starts, stops, and intensifications against prior biomarker levels or trends. Strong concentration of changes after deterioration is evidence that treatment exposure is history-dependent.
- Check overlap. If some biomarker histories occur only under one treatment strategy, the data provide little or no direct comparison for those histories. A model may still produce a number, but that part of the estimate relies on extrapolation.
- Check what is measured. A single latest value may miss a trend, duration of worsening, or clinical information that shaped the decision. Clinical records can contain treatment indications not captured by the biomarker.
- Check model adequacy. A simple linear biomarker term or constant treatment association may not represent nonlinear relationships or changes over time. These checks can reveal poor description; they cannot guarantee that all confounding has been removed.
Baseline adjustment alone cannot account for treatment decisions made in response to later biomarker history. A time-dependent event model can place observed treatment and biomarker values on the correct timeline, but ordinary adjustment for a time-varying biomarker can still be problematic when that biomarker is affected by prior treatment and then influences later treatment. Methods designed for time-varying treatment-confounder feedback, such as the g-formula or marginal structural models, target that causal structure under their own assumptions. A joint model can represent the biomarker trajectory and event process together, but it does not automatically model or remove confounding in the treatment-decision process. The treatment assignment mechanism and the scientific estimand still need to be specified.
These approaches answer different questions. A descriptive time-dependent model summarizes association conditional on its included covariates. A causal analysis asks what event risk would be under specified treatment strategies and requires assumptions such as consistency, adequate treatment-history measurement, positivity, and no unmeasured confounding given the measured history. Randomization can address treatment assignment at baseline, but treatment changes after randomization may again depend on evolving health and need their own careful analysis.
Common Pitfall
Wrong reasoning: “Treatment was included as a time-varying covariate, so the treatment coefficient is the causal effect.” Updating treatment over time fixes neither the reasons treatment changed nor the fact that earlier treatment may have influenced later biomarker values. The coefficient remains a model-based association unless the causal assumptions and analysis support a causal interpretation.
Correction: First establish the order \(B_k\) then \(A_k\), identify which parts of biomarker history predict treatment and events, and state the comparison being targeted. Then decide whether the chosen method and its assumptions address that history. Report the observed association as an association when those causal requirements are not justified.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A biomarker measured at month 4 leads to a treatment increase at month 5. Which measurement is prior history for the treatment decision, and why should the month-6 measurement not be used to define treatment at month 5?
- In the worked example with the two biomarker categories, what were the treated and untreated overall risks, and why did their ordering differ from the within-category ordering?
- At \(b=5\), calculate the treated and untreated probabilities from the formulas in the risk-calculation example. Are both within the stipulated ranges?
- Name one reason that a time-dependent treatment model can still fail to identify a causal effect.
Solutions. (1) The month-4 value is prior history because it was observed before the month-5 decision. The month-6 value occurs after that decision and is unavailable at month 5; using it to define month-5 treatment would use future information. (2) The treated risk was \(50/500=10\%\), and the untreated risk was \(32/500=6.4\%\). Treated patients were much more concentrated in the high-history category, so their overall risk was higher despite lower treated risks within each category. (3) At \(b=5\), treated risk is \(0.01+0.01(5)=0.06\), or 6%; untreated risk is \(0.02+0.01(5)=0.07\), or 7%. Since 5 lies in both stated ranges, both calculations apply. (4) For example, unmeasured health severity may influence both treatment changes and event risk; updating treatment in the model does not remove that unmeasured confounding.