Updating Risk After a New Biomarker Measurement

Why a New Measurement Changes a Prediction

A patient’s predicted survival after a landmark time depends on what is known about that patient at the landmark. Baseline covariates and prior biomarker history provide information, but a newly observed biomarker value can provide additional evidence about the patient’s latent trajectory and associated event risk. The update has two linked parts: revise the distribution of the patient’s random effects using the new measurement, then average future survival over that revised distribution.

The measurement does not directly reveal the random effects. It is an imperfect observation generated by a longitudinal model. A value that is more likely under random effects associated with higher event risk will generally increase the predicted event risk, but the size of the change depends on measurement error, the information already available, and the survival model.

The Bayesian Update at the Landmark

Suppose the fitted joint model has parameters \(\theta\). Before observing the new value, the random-effects distribution for a patient who has survived to \(t_0\) is proportional to the prior random-effects density, the likelihood of the prior biomarker history, and the likelihood contribution from event-free follow-up to \(t_0\). For a patient with an observed event time, event-free follow-up contributes the conditional survival likelihood; the displayed form assumes the patient is event-free at the landmark.

Here \(S_i(t\mid b,x_i,\theta)=P(T_i>t\mid b,x_i,\theta)\). The prior-history likelihood and the survival term are already incorporated into \(q_i^-(b)\); they should not be inserted a second time when the new measurement is added. Let \(g(y_i^*\mid b,x_i,t_0,\theta)\) denote the longitudinal model’s density for the new value, conditional on the random effects and other relevant predictors. Bayes’ rule gives

The denominator is the normalizing constant: it ensures that \(q_i^+(b)\) integrates to one. In words, each possible random-effects value receives its earlier posterior weight multiplied by how well it predicts the new biomarker value. A Gaussian measurement model is common, but the likelihood factor must match the response model—for example, a probability rather than a density for a discrete observation.

From the Updated Posterior to Conditional Survival

Let \(S_i(t\mid b,x_i,\theta)\) be survival from the model’s time origin through time \(t\), conditional on \(b\) and \(x_i\). For a patient known to be event-free at \(t_0\), the conditional probability of surviving a further \(u\) time units at a fixed value of \(b\) is the survival ratio \(S_i(t_0+u\mid b,x_i,\theta)/S_i(t_0\mid b,x_i,\theta)\). Averaging this ratio over the updated posterior gives the patient-specific prediction:

This is a conditional survival probability, not survival from the time origin. Equivalently, if the model specifies a hazard \(h_i(t\mid b,x_i,\theta)\), the ratio inside the integral is \(\exp\{-\int_{t_0}^{t_0+u}h_i(s\mid b,x_i,\theta)\,ds\}\). The prediction averages the patient’s possible latent risks rather than pretending that the random effects are known exactly.

Worked Example: Updating Two Latent-Risk Weights

Worked Example: Calculating Updated Survival

Worked Example: How Measurement Error Limits the Update

Why the Averaging Step Matters

The posterior update and the survival calculation are related but distinct. The marker changes the weights assigned to possible random effects; the event model translates each possibility into a conditional survival probability. In general, averaging those survival probabilities is not the same as substituting a posterior mean random effect into the survival model. Nonlinear survival functions make these operations differ, and a point estimate can hide substantial posterior uncertainty.

The main alternatives have different meanings. A prediction that omits the new marker retains the pre-measurement weights and therefore does not use the new evidence. A plug-in prediction substitutes one estimated random-effects value and ignores uncertainty about \(b_i\). The posterior-integrated prediction uses the new value through its likelihood and averages over the resulting distribution. These distinctions matter most when the marker is noisy, the posterior is diffuse, or survival changes sharply across possible random effects.

Diagnostics, Limitations, and Common Mistakes

In computation, the posterior integral is often approximated by quadrature or Monte Carlo draws. With draws \(b^{(m)}\) from the pre-measurement posterior, calculate marker likelihood weights \(w_m=g(y_i^*\mid b^{(m)},x_i,t_0,\theta)\), normalize them to sum to one, and average the conditional survival ratios using those normalized weights. Inspect whether a small number of draws dominate the weights; that can signal a poorly supported update or unstable numerical approximation. Also check that the updated weights sum to one, that the survival prediction lies between zero and one, and that the likelihood is evaluated on the model’s correct response scale.

The prediction is conditional on the fitted joint model and the assumed measurement process. Model misspecification, an incorrectly handled event-free interval, or unmodeled informative measurement availability can distort the update. Treating fitted parameters as fixed also leaves parameter-estimation uncertainty outside these formulas. The update is not guaranteed to increase risk: the direction depends on which random-effects values the observation favors and how those values relate to conditional survival.

• Common mistake: reusing the survival-to-landmark term. The pre-measurement posterior already conditions on surviving to \(t_0\). Multiplying by \(S_i(t_0\mid b,x_i,\theta)\) again after updating the marker double-counts that evidence. Use the posterior survival ratio for future survival.
• Common mistake: plugging in the marker as if it were the true trajectory. The observed value contains measurement error. The likelihood update retains uncertainty about \(b_i\); replacing the latent trajectory with the raw observation does not.
• Common mistake: assuming a higher marker always means higher risk. That direction is not universal. It depends on the fitted longitudinal and event models and on the updated posterior weights.