From a Patient’s History to a Prediction
A dynamic prediction is made for a patient who has remained event-free up to a specified time. It uses the marker measurements available by that time, along with the patient’s covariates and a fitted joint model, to estimate the chance of an event in a defined future interval. A complete prediction therefore requires more than inserting the latest marker value into a hazard equation: it must specify the information available at the landmark, condition on survival to that landmark, and account for uncertainty about the patient’s latent marker trajectory.
Let \(s\) be the landmark time and \(h>0\) the prediction horizon. Write \(T_i\) for patient \(i\)’s event time and \(\mathcal H_i(s)\) for the patient’s observed longitudinal history through \(s\). The target is the probability of an event by \(s+h\), given that the patient has survived event-free through \(s\) and has history \(\mathcal H_i(s)\). This is a patient-specific conditional probability, not the patient’s marginal probability of surviving from the beginning of follow-up.
Model and Calculation
For a concrete case study, suppose the fitted joint model has longitudinal measurements \(Y_{ij}\) and latent marker trajectory \(m_i(t)\):
where \(b_i\) is a patient-specific latent effect and measurement errors \(\epsilon_{ij}\) are conditionally independent with a specified distribution, such as \(N(0,\sigma^2)\). Suppose the event hazard, conditional on \(b_i\), is
Here \(h_0(t)\) is the baseline hazard, \(w_i(t)\) denotes event-model covariates, and \(\alpha\) links the latent marker to the hazard. The fitted parameters are denoted collectively by \(\theta\). For a fixed \(b_i\) and specified covariate path, define \(S_i(t\mid b_i;\theta)\) as the event-free survival probability through time \(t\). The model must give a valid hazard, the marker and event models must be correctly specified enough for the intended prediction, and the measurement and event processes must be conditionally independent given the modeled latent effects and covariates. The measurement process must also be ignorable under the model, or its observation mechanism must be modeled.
The patient’s latent effect is not observed. Its distribution after seeing the marker history and knowing the patient survived to \(s\) is proportional to three contributions: its model distribution, the likelihood of the observed marker history, and the probability of surviving to \(s\). In particular,
The survival contribution matters: latent effects associated with greater event hazard become less plausible among patients who are known to have survived to the landmark. Given this updated distribution, the dynamic risk is
For each possible latent effect, the expression in braces is the conditional risk over \((s,s+h]\). The integral averages those conditional risks over latent effects compatible with the patient’s history and survival. In practice, numerical integration or posterior simulation is often used.
Choose the event, landmark \(s\), horizon \(h\), and covariate information available or assumed after \(s\).
Include the patient’s eligible marker observations through \(s\), with their measurement times and recorded values. Do not use future measurements.
Combine the latent-effect distribution, the marker-history likelihood, and survival through \(s\), then normalize.
For each latent effect, calculate the conditional risk from \(s\) to \(s+h\), then average using the normalized weights.
Account for uncertainty in fitted parameters as appropriate, and state the event, landmark, horizon, and assumptions with the estimate.
Worked Case Study
Worked Example: Constructing a Patient’s Landmark History
Suppose a hypothetical patient is followed for a clinical event. The fitted model uses years since enrollment as its time scale. The patient’s marker values are \(8.1\) at year \(0\), \(7.4\) at year \(1\), and \(6.8\) at year \(3\). The patient is known to be event-free at year \(4\), and a prediction is requested for the next \(2\) years.
The landmark is \(s=4\) years and the horizon is \(h=2\) years, so the target is the probability of the event during \((4,6]\), conditional on being event-free at year \(4\) and on the marker history available through that time. The history is \(\mathcal H_i(4)=\{(0,8.1),(1,7.4),(3,6.8)\}\). All three measurements precede the landmark; no measurement after year \(4\) is available or included. The fact of surviving to year \(4\) is also part of the conditioning information.
Suppose a future covariate is not known at the landmark. The prediction must then use a stated assumption about its future path, such as holding it at its year-4 value, or average over a model for its future path. It would not be valid to use the patient’s actual future value as though it were known at year \(4\). For this example, assume the relevant covariates are fixed over the prediction horizon.
Worked Example: Updating Latent-Effect Weights and Calculating Risk
For this hypothetical patient and fitted model, suppose numerical integration uses three representative latent effects. After multiplying each prior integration weight by the marker-history likelihood and survival probability through year \(4\), the unnormalized weights are \(0.2023\), \(0.3933\), and \(0.1905\). These values are the model-based contributions for this patient; their sum is
Dividing each contribution by the sum gives the normalized survival-updated weights:
The displayed weights sum to \(0.9999\) because of rounding; the unrounded normalized weights sum to one. The third weight is \(0.1905/0.7861\approx0.2423356\), which rounds to \(0.2423\).
To make the risk calculation explicit, suppose the fitted hazard for this illustration is exponential conditional on the latent marker level: \(h_i(t\mid b_i)=0.04\exp\{0.5m_i\}\) events per year. The three representative effects imply constant future marker levels of approximately \(-1.346\), \(0.938\), and \(2.989\). The event risk for a constant rate \(r\) over two years is \(1-\exp(-2r)\). Thus the three conditional risks are approximately \(0.0400\), \(0.1200\), and \(0.2999\), respectively. Averaging with the updated weights gives
The model therefore estimates a \(14.30\%\) chance of the event during years \(4\) to \(6\), conditional on this patient’s specified history and being event-free at year \(4\), under the stated covariate and model assumptions. It is not a \(14.30\%\) probability that the model is correct, nor does it say that the event will occur for this patient.
Uncertainty, Interpretation, and Checks
Worked Example: Reporting Uncertainty in the Individual Risk
The value \(0.1430\) above treats the fitted model parameters as fixed. To reflect parameter uncertainty, suppose a hypothetical parametric bootstrap refits the joint model in each of \(1{,}000\) resampled data sets and recalculates the patient’s risk using the same landmark, history rule, and horizon. Assume the resulting percentile interval is \(0.081\) to \(0.223\).
A contextual report is: “For this hypothetical patient, the estimated risk of the event during the two years after year \(4\), conditional on survival to year \(4\) and the observed marker history, is \(14.3\%\); a bootstrap 95% uncertainty interval is \(8.1\%\) to \(22.3\%\).” The interval describes uncertainty in estimating this conditional risk under the fitted modeling and resampling procedure. It is not a statement that there is a 95% probability the patient’s fixed risk lies in this particular interval. This parameter-uncertainty interval also differs from a statement about whether the patient will experience the event: an individual outcome remains uncertain even if the risk were known exactly.
A prediction is only as credible as its inputs and assumptions. Before reporting it, check that the marker values and times are correctly ordered, that no post-landmark information entered the history, and that the patient was event-free at the landmark. Check that the fitted model supports the patient’s covariate and marker ranges; extrapolation beyond the data can make a numerically precise prediction unreliable. For numerical integration, verify that posterior weights are nonnegative and normalize to one before averaging risks. Inspect whether predictions are stable to increased integration accuracy or simulation size. A very small effective number of posterior draws, unstable estimates across runs, or large sensitivity to a few integration points signals that the numerical calculation needs attention.
A common mistake is to update latent-effect weights using the marker likelihood alone. That ignores that the patient is known to have survived to \(s\). The correct update includes \(S_i(s\mid b_i;\theta)\), as in the formula above; otherwise latent effects that imply high event risk may receive too much weight among survivors. Another mistake is to report \(1-S_i(s+h)\) as the patient’s risk. That quantity is survival-model risk from the time origin for a fixed latent effect, not the required conditional risk after surviving to \(s\). The conditional quantity uses the survival ratio \(S_i(s+h\mid b_i)/S_i(s\mid b_i)\), followed by averaging over the updated latent-effect distribution.
A plug-in prediction based only on the most likely latent effect is simpler, but it discards uncertainty about the latent trajectory and generally differs from integrating the conditional risk. A calculation using only the latest observed marker value is also a shortcut: it can fail to account for measurement error and the pattern of earlier measurements. A landmark regression can directly model risk from a fixed set of landmark predictors, but it does not automatically use the joint model’s latent trajectory and survival update. These approaches answer related but not identical questions; whichever is used, the target and conditioning information must be made explicit.
Even a carefully calculated prediction has limitations. It depends on the fitted longitudinal and event models, the assumed relationship between latent marker and hazard, the handling of measurement error and missingness, and the future covariate path. If those assumptions are poor, integration does not repair misspecification. Individual predictions can also be imprecise when a patient has sparse or noisy measurements. A prediction describes risk under the model and available information; it does not establish that changing the marker would change the event risk.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A patient has measurements at years \(1\), \(2\), and \(5\), and is event-free at year \(4\). Which measurements belong in a prediction made at year \(4\), and why?
- Why does the posterior weight for a latent effect include survival through the landmark?
- Suppose two latent-effect values have updated weights \(0.6\) and \(0.4\), and their conditional risks over the chosen horizon are \(0.10\) and \(0.25\). What is the integrated risk?
- What does a 95% bootstrap uncertainty interval for a patient’s predicted risk quantify, and what does it not mean?
Solutions. (1) Use only the measurements at years \(1\) and \(2\). The year-5 measurement is future information relative to the year-4 landmark and cannot be used. (2) The patient is known to have survived event-free to the landmark; latent effects associated with lower survival to that time should therefore be less compatible with the observed data. The likelihood contribution \(S_i(s\mid b_i;\theta)\) encodes that information. (3) The integrated risk is \(0.6(0.10)+0.4(0.25)=0.06+0.10=0.16\), or \(16\%\). (4) It quantifies uncertainty in the estimated conditional risk under the specified bootstrap procedure and model. It does not assign a 95% probability that the patient’s fixed risk lies in the observed interval, and it does not predict with certainty whether the patient will experience the event.