Worked Current-Value Risk Update

Why a New Reading Changes a Risk Prediction

A prediction made at a landmark time \(s\) uses the patient’s biomarker history and the fact that the patient remained event-free through \(s\). When another biomarker value arrives, it provides new information about the patient’s latent marker trajectory. A current-value joint model connects that trajectory to the event hazard: at each time, the hazard depends on the marker’s latent value at that time, not simply on the most recently observed measurement.

The update therefore has two distinct parts. First, revise the distribution of the latent marker trajectory using the new, potentially noisy observation. Second, compute future risk conditional on each possible trajectory and average those risks using the revised distribution. Plugging the observed reading directly into the hazard skips the first part’s uncertainty and can misstate risk.

The reading \(Y_i(s)\) is not generally equal to \(m_i(s;b_i)\). For example, a measurement model may be \(Y_i(s)=m_i(s;b_i)+\epsilon_i(s)\), with mean-zero measurement error having variance \(\sigma^2\). The posterior distribution of \(b_i\), and hence of the latent marker, accounts for that error. It should also account for survival to the landmark: a patient who is still event-free at \(s\) contributes information about \(b_i\) through the event model.

The Update in Two Steps

Suppose the prediction made just before a new measurement is based on the distribution of latent effects conditional on the old history and survival to \(s\). Once the new measurement \(y_i(s)\) arrives, Bayes’ rule reweights that distribution by the measurement likelihood. In density notation,

The proportionality constant is chosen so the updated density integrates to one. If the new reading is already included in \(\mathcal H_i(s)\), it must not be multiplied into the likelihood a second time. The event-free condition must also remain in the conditioning set; dropping it would change the prediction target.

For a fixed latent trajectory and covariate path, define the cumulative hazard over the prediction interval by \(A_i(s,s+\Delta;b_i)=\int_s^{s+\Delta}h_i(u\mid b_i)\,du\). Conditional survival over the interval is \(\exp\{-A_i(s,s+\Delta;b_i)\}\), so conditional event risk is one minus that quantity. Averaging over the updated latent-effects distribution gives the patient-specific prediction:

If the marker can change in the future in ways not fixed by \(b_i\), the risk inside the integral must also be averaged over possible future marker paths, conditional on the observed history and latent effects. The prediction is then an expectation of the path-specific risk, not the risk calculated from an average marker path. This distinction is important because the exponential hazard is nonlinear.

Worked Updates With Latent Marker Values

When the Marker Changes During the Prediction Horizon

A current-value hazard can respond to the marker throughout the future interval. If the marker trajectory is uncertain and varies over time, calculating risk from its value at the landmark alone can be inadequate. The following example integrates the hazard along each of two possible paths, then averages their risks.

Computation, Diagnostics, and Alternatives

In realistic models, the updated posterior and the integral over latent effects are rarely available in closed form. Monte Carlo integration can approximate the prediction: draw latent effects from the posterior conditional on the old history and landmark survival, reweight or sample those draws using the new measurement likelihood, calculate each draw’s interval risk, and average. If future marker paths are uncertain, simulate or integrate those paths as well. Numerical integration is another option when the latent dimension is small. The same posterior draws must be used consistently in the measurement update and risk calculation.

Several checks help identify a faulty update. Verify that the updated posterior weights sum to one and that the new observation has been included exactly once. Confirm that the risk is between zero and one, and that increasing \(\alpha m_i(t)\) increases the hazard when \(\alpha\) is positive. Check that numerical integration is stable as the number of draws or integration points increases. Also inspect whether a small number of draws dominate the posterior weights, which can indicate poor Monte Carlo overlap or a highly informative reading. Sensitivity to the measurement-error variance and association parameter is especially relevant when the new reading is extreme.

A landmark prediction that carries the last observed biomarker value forward is simpler, but it treats that value as the future latent marker and ignores measurement error and trajectory uncertainty. A plug-in approach that uses the posterior mean marker is also simpler, but in general it does not equal the posterior average risk because exponentiation and averaging do not commute. A full current-value update retains the model’s latent-marker uncertainty, at the cost of relying on the measurement, longitudinal, and event-model assumptions. It cannot correct misspecification merely by performing the integration accurately.

The calculation assumes a correctly specified association structure, baseline hazard, longitudinal measurement model, and relevant covariate histories. It also assumes that the new measurement is available at the stated landmark and that the patient is event-free then. If the measurement process is informative beyond the modeled history, or the marker dynamics are poorly described, the updated risk can be biased. As in the earlier material on measurement error versus latent biomarker values, an observed value should not be treated as the latent state without justification.