Worked Example With a Biomarker and Event Endpoint

Why carry uncertainty from a biomarker to an event endpoint?

In a multi-arm, multi-stage trial, a continuous biomarker may be measured before enough cardiovascular events have accrued to compare arms directly on the final endpoint. The biomarker can therefore inform an interim decision about which arms continue. But a biomarker contrast is not itself an event treatment effect. Turning one into the other requires a calibrated relationship, and that relationship is uncertain. A responsible ranking uses both sources of uncertainty rather than treating the biomarker as a perfect substitute for the event endpoint.

The setup below focuses on a randomized comparison of each active arm with a common control. Define biomarker improvement so that larger values always represent a more favorable change, regardless of whether the raw marker is expected to rise or fall. For example, if lower biomarker values are favorable, reverse the sign of the treatment-control contrast. For the event endpoint, use a proportional-hazards summary when that is the prespecified estimand: a hazard ratio below 1 favors treatment.

This is a relationship between treatment effects: it asks whether treatments that improve the biomarker more tend also to reduce event hazard more. It is not established merely by showing that, among individuals, people with better biomarker values tend to have fewer events. Individual-level association and treatment-effect prediction are different claims. Calibration should therefore use relevant randomized treatment comparisons, preferably from a setting sufficiently similar in population, treatment mechanism, biomarker measurement, and event definition.

Model and assumptions for a predictive ranking

At the interim look, let \(\hat B_j\) be the estimated direction-adjusted biomarker contrast and \(SE_j\) its standard error. A working sampling model is \(\hat B_j\mid B_j\sim N(B_j,SE_j^2)\). The calibration model is \(E_j=\gamma B_j+\epsilon_j\), with mean-zero residuals of variance \(\tau^2\). The calibration slope \(\gamma\) is itself uncertain; represent that uncertainty by a distribution with mean \(g\) and variance \(v_g\). In a simple calculation, assume the biomarker estimate and calibration slope are independent, and that the residual is independent of both.

These assumptions are useful approximations, not automatic properties of a trial. The biomarker must be measured consistently, with missingness and assay timing handled appropriately. The calibration data must support the direction and scale of the relationship, and residual variation across treatments must not be ignored. If calibration comes from the same trial, or shares control data with the interim biomarker contrasts, dependence must be represented in a joint model rather than set to zero. For a hazard-ratio interpretation, proportional hazards must be reasonable for the estimand; if it is not, a prespecified alternative event summary is needed.

When the stated independence assumptions hold, the predictive mean and variance of event benefit have a simple form. The variance includes biomarker sampling uncertainty, calibration-slope uncertainty, their product, and unexplained treatment-effect variation.

The third term is sometimes omitted when a first-order approximation is used. It follows from uncertainty in the product of two uncertain quantities. The calculation is still a moment summary: the product of two normal variables is not generally normal. A normal predictive distribution based on these moments is most defensible when uncertainty is moderate; with a highly uncertain slope or a small calibration set, simulate from the full model instead.

Deriving the uncertainty calculation

Write \(B_j=\hat B_j+u\) and \(\gamma=g+w\), where \(u\) and \(w\) have means zero and variances \(SE_j^2\) and \(v_g\), respectively. Under independence, the product is \(g\hat B_j+gu+\hat B_jw+uw\). The three random terms have zero means and zero pairwise covariances, so their variances add. The residual \(\epsilon_j\) contributes the additional \(\tau^2\).

This derivation also makes the assumptions visible. If \(u\) and \(w\) are correlated, the covariance terms need to be added. If residuals across arms are correlated, that covariance matters when ranking arms by their difference. These are modeling choices to justify, not details that can be fixed by rounding.

Ranking two arms with shared calibration uncertainty

When two arms use the same calibration slope, uncertainty in that slope is shared. It should not be counted as if each arm had an independent slope. Let \(D=B_A-B_B\) be the biomarker contrast between arms, with estimate \(\hat D=\hat B_A-\hat B_B\) and standard error \(SE_D\). Under the same product model, the predicted difference in event benefit has mean \(g\hat D\) and variance \(g^2SE_D^2+\hat D^2v_g+SE_D^2v_g+\operatorname{Var}(\epsilon_A-\epsilon_B)\). If the arms’ residuals are independent, the final term is \(2\tau^2\). If not, use the residual covariance implied by the calibration model.

The standard error of a biomarker difference must reflect the design. With a common control, \(\hat B_A\) and \(\hat B_B\) are generally correlated. Specifically, \(SE_D^2=SE_A^2+SE_B^2-2\operatorname{Cov}(\hat B_A,\hat B_B)\). Ignoring the shared control can therefore misstate ranking uncertainty. A ranking probability such as \(P(E_A>E_B\mid\text{interim data})\) can be approximated from the predictive distribution of the difference; it is not the probability that an arm is truly efficacious or that the null hypothesis is false.

Updating a calibration relationship

If suitable randomized calibration data are available, slope uncertainty can be updated rather than fixed at a single estimate. A deliberately simple example uses a zero-intercept model \(y_k=\gamma x_k+\epsilon_k\), with independent \(\epsilon_k\sim N(0,\tau^2)\), known \(\tau\), and known calibration treatment-effect contrasts \(x_k\). A normal prior \(\gamma\sim N(m_0,s_0^2)\) then gives a normal posterior. The zero intercept is a scientific modeling choice: it says zero biomarker treatment effect corresponds, on average, to zero event benefit. It should not be imposed if unsupported.

Diagnostics, alternatives, and limitations

Before applying a biomarker-based ranking rule, inspect whether the calibration is informative for the decision at hand. Useful checks include a plot of randomized treatment-effect estimates on the biomarker against event-benefit estimates, uncertainty intervals for each point, and sensitivity of the predicted ranking to the slope, residual variation, and calibration population. A few imprecise points can produce an apparently strong slope. If estimates share controls, their covariance belongs in the calibration analysis. Also check whether the biomarker’s measurement scale, direction, timing, and missingness are compatible across trials.

A linear calibration may be inadequate if the relationship bends, differs by treatment mechanism, or has substantial unexplained heterogeneity. A slope estimated near zero, a wide slope distribution, or a prediction interval spanning both benefit and harm indicates weak ranking information. A nonpositive slope may be scientifically plausible in some contexts; it should prompt scrutiny of the assumed direction, evidence, and model rather than being silently truncated. When uncertainty is material, simulation from a joint model can propagate shared control, calibration, and residual uncertainty more reliably than a normal approximation.

The main alternatives have different strengths. Direct ranking on the interim biomarker is simpler, but does not express uncertainty about whether biomarker effects predict event effects. A fixed-slope plug-in prediction is easy to communicate, but can be overconfident because it treats the calibration as known. Waiting to rank by observed event outcomes avoids the biomarker-to-event translation, but may delay selection until more events accrue. A joint Bayesian or likelihood-based model can represent dependence and nonlinear or treatment-specific relationships, at the cost of more modeling assumptions and greater data requirements. Whatever approach is chosen, the final cardiovascular event analysis remains based on the prespecified event endpoint; an intermediate-based selection is not a substitute for that analysis.

A selection rule should be prespecified in terms of the quantity it uses: for example, predicted event benefit, a predictive interval, or a probability that one arm exceeds another by a meaningful margin. The threshold should reflect the consequences of dropping a potentially useful arm and the trial’s operating characteristics. A ranking probability is a decision summary conditional on the model and data, not a conventional test p-value. If treatment effects on the biomarker and final endpoint can disagree, no calibration formula removes that possibility; residual variation and sensitivity analysis are essential.