Likelihood-Based Integration for Individual Prediction

Why integrate over random effects?

A patient’s observed biomarker history gives information about that patient’s latent trajectory, but it does not reveal the trajectory exactly. In a joint model, the trajectory depends on latent random effects, and the event hazard may depend on that same trajectory. An individual prediction therefore needs to account for which random effects are plausible given the patient’s data, rather than treating one estimated value as certain.

Likelihood-based integration does this by weighting each possible random-effect value according to its model probability and its compatibility with the patient’s longitudinal observations and survival to the prediction landmark. The resulting prediction averages conditional survival over that weighted distribution. This is a frequentist plug-in prediction when the population-model parameters are set to their fitted estimates; the integration over random effects does not, by itself, integrate uncertainty in those parameter estimates.

From the patient-specific likelihood to a prediction

For clarity, suppose the patient has measured covariates \(x_i\), that their longitudinal measurements up to \(t\) have a density conditional on \(b_i\), and that their event-free status through \(t\) is known. A common joint-model assumption is that, conditional on \(b_i\) and the covariates, the longitudinal process and event time are independent. Under this conditional independence, the joint contribution of the observed longitudinal history and survival to the landmark, conditional on \(b_i\), is the product \(f\{y_i(t)\mid b_i;\theta\}S_i(t\mid b_i;\theta)\). Multiplying by \(f(b_i;\theta)\) gives the patient-specific contribution before integrating out \(b_i\).

The corresponding conditional density for the latent effects is

The denominator normalizes the weights so they integrate to one. At a later prediction time \(u>t\), the individual’s predicted survival is the average of their random-effects-conditional survival from \(t\) to \(u\), using this landmark distribution:

Equivalently, substituting the definition of \(q_i\) cancels the landmark survival term inside the integral:

The event risk over \((t,u]\), conditional on being event-free at \(t\) and on the observed history, is one minus this survival probability. In an actual frequentist analysis, \(\theta\) is commonly replaced by its fitted estimate \(\hat\theta\). The expressions then describe a plug-in prediction conditional on the fitted population parameters.

Worked calculations

Justification and assumptions

The integration formula follows directly from conditioning on the latent random effects. Given \(b_i\), the probability of surviving beyond \(u\), conditional on survival to \(t\), is \(S_i(u\mid b_i)/S_i(t\mid b_i)\). Averaging that conditional probability over the random effects compatible with the patient’s observed history gives the first integral formula. Substituting the normalized density \(q_i\) yields the ratio of integrals: the factor \(S_i(t\mid b_i)\) in the landmark weights cancels the denominator in the conditional survival ratio, leaving \(S_i(u\mid b_i)\) in the numerator and the landmark likelihood normalizer in the denominator. This establishes the desired conditional probability without treating a single estimated random effect as known.

The calculation relies on a correctly specified joint model for the longitudinal measurements, random effects, and event process. In particular, the conditional-independence assumption must be reasonable: after conditioning on the random effects and modeled covariates, the longitudinal and event processes have no remaining dependence that the joint model needs to represent. The random-effects distribution, longitudinal measurement model, and event hazard must also be adequate for the data. If the hazard depends on the latent trajectory, its conditional survival \(S_i(s\mid b_i;\theta)\) must use that dependence consistently.

The observed history must be represented by the appropriate longitudinal likelihood, including the actual measurement times when the model requires them. The formula also assumes that the observation or visit process is ignorable for the likelihood being used, or that it has been modeled. A missingness or visit process that depends on unobserved marker values or event risk can otherwise change the information the observed history carries about \(b_i\). Predictions beyond \(t\) additionally require future covariates to be fixed, known, or handled under an explicit model; the formulas do not make unknown future covariates disappear.

Diagnostics, alternatives, and limitations

Likelihood-based integration is a construction, not a guarantee of a reliable prediction. Examine whether the fitted model reproduces longitudinal trajectories and event patterns, especially in the parts of the covariate and marker ranges relevant to the patient. Check that the patient’s observed measurements and event-free time have non-negligible likelihood under the fitted model. Very concentrated weights on a narrow region of random-effects space may be appropriate for a highly informative history, but can also signal a poor fit, sparse data, or numerical difficulty. Predictions that change sharply when a measurement is modestly perturbed deserve scrutiny.

A common mistake is to use only \(f\{y_i(t)\mid b_i\}f(b_i)\) to weight random effects and then condition on survival afterward. That omits the fact that this particular patient is known to have survived to \(t\). The correction is to include \(S_i(t\mid b_i)\) in the landmark weights, or equivalently to use the ratio-of-integrals expression with \(S_i(t\mid b_i)\) in the denominator.

Another tempting alternative is to plug in a single patient-specific estimate of \(b_i\) and calculate conditional survival there. That is simpler, but it discards uncertainty about the patient’s latent trajectory and can produce a different risk, particularly when the observed history is short or noisy. The earlier material on separating parameter and random-effects uncertainty distinguishes uncertainty in \(b_i\) from uncertainty in fitted population parameters: the integral here addresses the former conditional on \(\theta\), not the latter.

The likelihood-weighted prediction is also distinct from a population-average prediction. It is tailored to one patient’s history and survival to the landmark; a population-average quantity averages over a population distribution rather than that patient’s updated random-effects distribution. Neither target is universally preferable: the target must match the clinical question. Finally, integration can be computationally demanding when \(b_i\) is high-dimensional or the integrand is concentrated. The exact target remains the same, but numerical methods are needed to evaluate the integrals; selecting and assessing those methods is a separate computational issue.