Why the Biomarker Representation Matters
At a landmark, a prediction model can summarize a patient's biomarker history in different ways. The most recent measurement gives the patient's current observed level; the difference between that measurement and baseline gives the observed change. These summaries answer different questions. A high latest value may reflect a high starting level, a substantial rise, or both. A change score separates movement from baseline, but it does not by itself reveal the current level.
The distinction matters when predicting an event over a specified future period among patients who have reached the landmark without the event. The predictors must be calculated using information available at that time. A measurement collected after the landmark cannot be used to predict risk from the landmark, even if it is the next measurement in the patient's record.
For a prediction at \(s\), the eligible risk set consists of patients who are event-free and under observation at \(s\). A patient who experienced the event before \(s\) is not eligible to receive a prediction for future risk from that landmark. If the prediction horizon is \(\tau\), the target is the probability of an event during \((s,s+\tau]\), conditional on being event-free at \(s\). In a fixed-horizon logistic model, write this risk as \(r_i(s,\tau)\), with
A latest-value model might use \(\operatorname{logit}\{r_i(s,\tau)\}=\alpha_s+\beta_L L_i(s)+\boldsymbol{\gamma}^{T}\mathbf{Z}_i(s)\), where \(\mathbf{Z}_i(s)\) denotes any other predictors specified for the model. A change-only model replaces \(L_i(s)\) with \(\Delta_i(s)\). These are alternative model specifications: their coefficients are not automatically interchangeable. The examples below use a single fixed landmark and omit other covariates to make the biomarker calculations clear.
Constructing the Predictors at a Landmark
Worked Example: Finding Latest Values at Month 12
Suppose three hypothetical patients have baseline biomarker values and visit records shown below. The landmark is month 12. Assume the recorded visits occurred before any event at that visit time.
| Patient | Baseline | Measurements after baseline | Status at month 12 |
|---|---|---|---|
| A | 6 mg/L | Month 3: 7; month 8: 9; month 12: 8 mg/L | Event-free |
| B | 10 mg/L | Month 4: 9; month 10: 7 mg/L | Event at month 11 |
| C | 4 mg/L | Month 7: 5 mg/L | Event-free |
Patient A is eligible at month 12. The latest measurement is the month-12 value, so \(L_A(12)=8\) mg/L. The change is \(8-6=2\) mg/L, giving \(\Delta_A(12)=2\) mg/L.
Patient B experienced the event before month 12, so B is not in the risk set at this landmark. B does not receive a month-12 prediction, even though the record contains measurements from before the event.
Patient C is eligible. The latest measurement available by month 12 is from month 7; no later value is invented or carried forward as if measured. Thus \(L_C(12)=5\) mg/L and \(\Delta_C(12)=5-4=1\) mg/L. The predictors use the recorded value and its actual time, not a value observed after the landmark.
When a fixed-horizon logistic model is fitted, the binary outcome must also be known for each included patient: an event during the horizon is a positive outcome, and remaining event-free through its end is a negative outcome. A patient censored before the horizon without an event does not have a known binary outcome for that horizon. Such a patient should not simply be coded as a non-event; ordinary binary logistic regression does not resolve this incomplete outcome information.
What the Coefficients Mean
In a latest-value model with linear predictor \(\alpha_s+\beta_L L_i(s)\), \(\beta_L\) is the change in log odds of the horizon event for a one-unit higher latest biomarker value, at the same landmark and with other model terms held fixed. The corresponding odds ratio is \(\exp(\beta_L)\) per one-unit increase. In a change-only model, \(\beta_\Delta\) has the corresponding interpretation for a one-unit greater change from baseline. Neither coefficient is a change in risk itself; the change in probability depends on the starting risk.
A change-only model does not adjust for baseline level merely because its predictor is calculated by subtracting baseline. Two patients with the same change can have different latest values, and two patients with the same latest value can have different changes. If the distinction between starting level and subsequent change matters, both pieces of information can be represented in the model.
The result follows directly from the definition of change: replace each \(\Delta_i(s)\) by \(L_i(s)-B_i\), distribute \(\beta_\Delta\), and collect the coefficients of \(B_i\) and \(L_i(s)\). Since the transformation between \((B_i,\Delta_i(s))\) and \((B_i,L_i(s))\) is one-to-one, the two combined-predictor forms yield the same fitted linear predictor when they use the same observations and corresponding coefficients.
Worked Example: Comparing a Latest-Value and a Change-Only Prediction
Consider a hypothetical patient with baseline biomarker value 5 mg/L and latest value 8 mg/L at the landmark. The observed change is \(8-5=3\) mg/L. Suppose two separately fitted models for the same future horizon are
These coefficients are illustrative fitted values, not universal effects. For the latest-value model, substitute \(L=8\) mg/L:
The predicted event risk is approximately 26.1% over the specified horizon. In this model, a one mg/L higher latest value is associated with an odds multiplier of \(\exp(0.12)\approx1.13\), conditional on the model terms.
For the change-only model, substitute \(\Delta=3\) mg/L:
This model predicts approximately 23.1% risk. Its coefficient corresponds to an odds multiplier of \(\exp(0.20)\approx1.22\) per one mg/L greater change. The two probabilities need not agree: they come from different fitted models, with different coefficients and intercepts. Their difference does not establish which predictor is better; that requires an appropriate comparison of predictive performance in relevant data.
Using Both Baseline and Change
Worked Example: Reading a Combined-Predictor Model
Suppose a hypothetical fitted model uses both baseline and change, with values measured in mg/L:
For a patient with baseline \(B=5\) mg/L and latest value \(L=8\) mg/L, the change is \(\Delta=3\) mg/L. The predicted log odds and risk are
The model predicts approximately 18.8% risk over its specified horizon. Holding baseline fixed, a one mg/L greater change is associated with an odds multiplier of \(\exp(0.18)\approx1.20\). Equivalently, rewrite the model in terms of baseline and latest value:
Using \(B=5\) and \(L=8\) gives \(-2.40-0.10(5)+0.18(8)=-1.46\), the same linear predictor and risk. In this representation, the latest-value coefficient describes a one mg/L higher latest value at fixed baseline. This conditional interpretation differs from the coefficient in a latest-only model.
Comparing What the Predictors Capture
The following hypothetical patients illustrate why neither summary contains all the information in the other:
| Patient | Baseline \(B\) (mg/L) | Latest \(L\) (mg/L) | Change \(\Delta=L-B\) (mg/L) |
|---|---|---|---|
| A | 4 | 8 | 4 |
| B | 7 | 8 | 1 |
| C | 4 | 6 | 2 |
| D | 8 | 10 | 2 |
A latest-only model assigns A and B the same biomarker predictor because both have a latest value of 8 mg/L, despite their different changes. A change-only model assigns C and D the same change of 2 mg/L, despite different latest values. Whether the model should emphasize current level, movement from baseline, or both is a substantive modeling choice. The choice should be made for the prediction question and evaluated with data, not selected solely because one representation produces a preferred result in the fitting sample.
Assumptions, Checks, and Limitations
Interpretation of these predictions depends on the landmark setup and the model. The risk set must include only patients event-free at \(s\), and the predictors must be calculated from measurements available by \(s\). The event status for the full prediction horizon must be ascertainable for the patients used in ordinary fixed-horizon logistic regression. Within the model, the log odds are assumed to follow the specified relationship with the predictors; for example, a linear term assumes a constant change in log odds per mg/L. Standard model-based inference also relies on an appropriate independent-observation structure, or a method that accounts for dependence when observations are not independent.
Check that baseline and latest values use consistent measurement units and definitions, that the landmark precedes the prediction horizon, and that no future measurement has entered a predictor. Examine whether predicted risks are sensible across the observed predictor range and whether the fitted model's calibration is adequate for the intended use. Sparse data, extreme predictor values, or very few events can make estimates unstable. A strong association in the fitted sample is not enough to establish useful prediction.
Latest value and change are observed summaries, not error-free measurements of a patient's underlying biomarker trajectory. A change score also combines information from two measurements, so its variation reflects variation in both baseline and latest measurements. These predictors can be useful summaries, but they do not by themselves describe the timing or pattern of all intervening measurements. Finally, their coefficients describe associations within the specified prediction model; they do not establish that changing a patient's biomarker would cause a change in event risk.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- At a landmark, a patient's baseline value is 3 mg/L, and the latest available value is 5 mg/L. What are the latest-value and change-from-baseline predictors?
- A patient experiences the event before the landmark. Should that patient be included in the landmark risk set for predicting events after that time? Why?
- A change-only logistic model has coefficient \(0.10\) per mg/L. What is the odds multiplier for a one mg/L greater change, and what does it not mean?
- A model contains baseline and change with coefficients \(0.06\) and \(0.14\), respectively. What are the equivalent coefficients on baseline and latest value?
- Why should a patient censored halfway through a fixed prediction horizon not automatically be coded as having no event during the horizon?
Solutions. (1) The latest value is \(L=5\) mg/L, and the change is \(\Delta=5-3=2\) mg/L. (2) No. The patient is not event-free at the landmark, so is not eligible for prediction of future risk conditional on reaching that landmark event-free. (3) The odds multiplier is \(\exp(0.10)\approx1.11\) per mg/L greater change. It is an odds ratio, not an increase of 10 percentage points in risk or a causal effect. (4) Using \(\beta_B B+\beta_\Delta(L-B)=(\beta_B-\beta_\Delta)B+\beta_\Delta L\), the equivalent baseline coefficient is \(0.06-0.14=-0.08\), and the latest-value coefficient is \(0.14\). (5) The patient could experience an event after censoring but before the horizon ends; without further information, the fixed-horizon event status is unknown rather than known to be negative.