Following a Biomarker-Guided Dose Sequence
A biomarker may be measured between two treatment decisions. The first dose can change the biomarker; the biomarker can then influence the next dose; and both the biomarker and the doses can predict the eventual outcome. This ordering matters: the biomarker is not simply a baseline covariate to adjust for in every analysis. Its role depends on which treatment decision is under consideration.
Consider a hypothetical clinical protocol in which clinicians choose an initial dose, measure a biomarker several weeks later, and use that measurement to guide a later dose. The outcome is assessed after the later dose. Let \(C\) denote baseline disease severity, \(A_0\) the initial dose, \(L_1\) the intermediate biomarker, \(A_1\) the later dose, and \(Y\) the final outcome.
The diagram shows the central time order, not every possible cause. Baseline severity may affect both treatment choices and the outcome; the biomarker may also predict the outcome directly. The later dose may affect the outcome as well. A careful analysis must therefore ask what was measured before each decision and what could have influenced that decision.
First Establish the Measurement Order
The labels “baseline,” “intermediate,” and “outcome” are not enough by themselves. The study protocol must establish when each quantity is measured relative to each treatment decision. For example, a biomarker drawn after the later dose cannot be treated as a pre-decision cause of that dose. A useful record for the intended sequence is:
- Measure baseline severity \(C\) before treatment begins.
- Choose and record the initial dose \(A_0\).
- Measure the biomarker \(L_1\) after \(A_0\) but before the later dose decision.
- Choose and record the later dose \(A_1\), using information available at that decision.
- Measure the outcome \(Y\) after the later dose.
Worked Example: Classifying Each Measurement in the Protocol
Suppose a hypothetical clinic records baseline disease severity, an initial dose on day 1, a biomarker on day 14, a dose change on day 15, and a symptom score on day 60. Classify the variables in relation to the two treatment decisions.
Before the initial decision: Baseline disease severity is measured before \(A_0\). It can help explain why the initial dose differs between patients and may also predict the day-60 symptom score.
Between the decisions: The day-14 biomarker is measured after \(A_0\) and before \(A_1\). If the initial dose changes the biomarker, it is treatment-affected. If clinicians use the biomarker to choose the day-15 dose and it predicts the symptom score, it also confounds the comparison of later doses.
After both decisions: The day-60 symptom score is the outcome \(Y\). It cannot be a cause of the earlier doses in this timeline.
The ordering supports the proposed feedback structure only if it matches the actual protocol. If the day-14 specimen was collected after the day-15 dose change, the biomarker cannot be a pre-decision guide to \(A_1\) as represented here.
What a Hypothetical Data Set Shows
The following invented data illustrate the pattern. Disease severity \(C\) is coded as lower or higher; the biomarker is measured in hypothetical units; dose levels are coded 0 or 1; and the outcome is a symptom score in points, with lower scores indicating fewer symptoms. These numbers are for instruction only, not a real study.
| Patient | Baseline severity \(C\) | Initial dose \(A_0\) | Biomarker \(L_1\) | Later dose \(A_1\) | Symptom score \(Y\) |
|---|---|---|---|---|---|
| 1 | Lower | 0 | 8 | 1 | 42 |
| 2 | Lower | 0 | 6 | 0 | 35 |
| 3 | Lower | 1 | 4 | 1 | 28 |
| 4 | Lower | 1 | 3 | 0 | 24 |
| 5 | Higher | 0 | 13 | 1 | 70 |
| 6 | Higher | 0 | 11 | 1 | 63 |
| 7 | Higher | 1 | 9 | 1 | 52 |
| 8 | Higher | 1 | 7 | 0 | 46 |
Worked Example: Calculating the Observed Patterns
Use the table to compare the biomarker and outcome across initial-dose groups, then compare outcome across later-dose groups. Each comparison is descriptive: the table alone does not establish a causal dose effect.
Biomarker by initial dose. For \(A_0=0\), the biomarker values are 8, 6, 13, and 11. Their mean is
For \(A_0=1\), the values are 4, 3, 9, and 7, so the mean is
In this hypothetical table, the group with initial dose 1 has a biomarker mean 3.75 units lower than the group with initial dose 0: \(5.75-9.5=-3.75\). This is consistent with the proposed idea that the initial dose can change the biomarker, but it is not by itself proof of that causal effect.
Outcome by later dose. The five patients with \(A_1=1\) have symptom scores totaling \(42+28+70+63+52=255\), for a mean of \(255/5=51\) points. The three with \(A_1=0\) have scores totaling \(35+24+46=105\), for a mean of \(105/3=35\) points. The observed mean is therefore 16 points higher in the \(A_1=1\) group.
That difference does not show that the later dose worsens symptoms. The higher later dose is more common among patients with higher biomarker values and higher baseline severity, both of which are associated with worse outcomes in these data. Clinicians may have increased the dose because a patient appeared more unwell. This is treatment by indication: the reason for receiving more treatment can make the treated group look worse even if the treatment itself is beneficial.
The biomarker is prognostic in the table: patients with higher values tend to have higher symptom scores. For instance, the four values at least 8 are 8, 13, 11, and 9, with symptom scores totaling \(42+70+63+52=227\), or \(227/4=56.75\) points on average. The four values below 8 have scores totaling \(35+28+24+46=133\), or \(133/4=33.25\) points on average. This association is compatible with the biomarker predicting the outcome; it does not determine whether changing the biomarker would change the outcome.
Tracing the Total Effect in a Simple Model
A small structural model can make the pathways explicit. The following equations are deliberately hypothetical and deterministic: they omit baseline severity and random variation, and their coefficients are assumed for illustration rather than estimated from the table. Suppose \(A_0\) is either 0 or 1, the biomarker is \(L_1=10-2A_0\), clinicians set \(A_1=1\) when \(L_1>9\) and \(A_1=0\) otherwise, and the symptom score is \(Y=30+2L_1-4A_0-3A_1\). The outcome equation says that, holding its other inputs fixed, a one-unit higher biomarker adds 2 symptom points, the initial dose subtracts 4 points, and the later dose subtracts 3 points.
Worked Example: Following the Biomarker-Guided Dose Rule
Compare the outcome under initial dose \(A_0=0\) with the outcome under \(A_0=1\), allowing the specified biomarker rule to determine the later dose. Apply the model in order.
If \(A_0=0\): \(L_1=10-2(0)=10\). Since \(10>9\), the rule sets \(A_1=1\). The outcome is
If \(A_0=1\): \(L_1=10-2(1)=8\). Since \(8\) is not greater than 9, the rule sets \(A_1=0\). The outcome is
Under this model and this dose rule, changing the initial dose from 0 to 1 changes the symptom score by \(42-47=-5\) points. The decomposition is \(-4\) points through the direct initial-dose term, \(-4\) through the biomarker change from 10 to 8, and \(+3\) because the rule assigns no later dose rather than dose 1. These contributions sum to \(-4-4+3=-5\) points.
This calculation is complete for the stated model, not an estimate from clinical data. Its purpose is to show why the total effect of an initial dose includes downstream changes in the biomarker and in the later dose. In a real analysis, those pathways would not be learned merely by substituting observed group means into these equations.
Interpreting the Roles Without Confusing Them
For the initial dose, \(L_1\) lies downstream: \(A_0\) can change \(L_1\), and \(L_1\) can contribute to the outcome. Adjusting for \(L_1\) in a conventional outcome regression can therefore remove part of the initial dose’s effect that operates through the biomarker. For the later dose, the timing is different: \(L_1\) is measured before \(A_1\), can guide \(A_1\), and predicts \(Y\). It is then a relevant confounder of the later-dose comparison.
This is the treatment-confounder feedback problem in a concrete form: a covariate affected by earlier treatment becomes relevant to a later treatment decision. A single coefficient for \(A_0\) or \(A_1\) in a regression that includes all measured variables does not automatically equal the total causal effect of either dose. The coefficient’s meaning depends on the variables included and the assumptions that would justify a causal interpretation.
Verify that \(L_1\) is measured after \(A_0\), before \(A_1\), and before \(Y\).
Ask whether \(A_0\) can change \(L_1\), and whether \(L_1\) affects the later dose decision and predicts \(Y\).
Calculate observed means accurately, then ask whether differences could reflect severity or other reasons for treatment.
A total effect of the initial dose includes downstream changes; a comparison of later doses must account for information available before that decision.
Common Mistake
Wrong reasoning: “The later-dose group has a mean symptom score 16 points higher, so the later dose increases symptoms by 16 points.” This treats a crude association as a treatment effect and ignores why clinicians chose each dose.
Correction: The observed contrast is \(51-35=16\) points, but patients receiving the higher later dose also tend to have worse biomarker values and greater baseline severity. The comparison describes the groups as observed; it does not isolate what would have happened to comparable patients under different later doses. Likewise, automatically adjusting for the biomarker when estimating the initial dose’s total effect can block part of the pathway through which \(A_0\) operates.
Check Your Understanding
Use the sequence and hypothetical data above to answer the following questions.
- Why is \(L_1\) a post-treatment covariate for \(A_0\), but potentially a confounder for \(A_1\)?
- Calculate the observed mean symptom score for \(A_0=0\) and \(A_0=1\). What does their difference describe, and what does it not establish?
- Why can the later dose appear associated with worse symptoms even if a higher dose could be beneficial?
- In the illustrative model, which pathways contribute to the five-point decrease in symptom score under \(A_0=1\) and the biomarker-guided dose rule?
- What timing error would undermine the claim that the biomarker guided the later dose?