Why Compare Predictions Across Landmarks?
A prediction made today is conditional on what is known today. If a patient remains event-free and has another biomarker measurement next month, the information set changes. The patient’s predicted risk should therefore be recalculated using both the additional follow-up and the new measurement, rather than carried forward unchanged.
Comparing predictions across landmark times helps clarify what a joint model is doing for an individual. A change may reflect the fact that the patient survived event-free to a later time, a biomarker history that now suggests a different latent trajectory, or simply that the prediction window has shifted. These are distinct sources of change. A useful comparison states the landmark, the event horizon, and the information used for each prediction.
The conditioning event matters: this is not the patient’s original risk from study entry. Nor is it an unconditional population risk. It is a patient-specific prediction among those who have reached the landmark without the event, using that patient’s information available by then.
How the Information Set Changes
In a joint model, the longitudinal and event processes are linked through latent patient-specific features, often represented by random effects \(b_i\). A concise expression for the prediction is $$ R_i(s,u)=1-\int \frac{S(s+u\mid b_i)}{S(s\mid b_i)}\,p(b_i\mid T_i>s,\mathcal{H}_i(s))\,db_i, $$ where \(S(t\mid b_i)\) is the event survival function conditional on the latent features and relevant covariates. The ratio is the probability of remaining event-free through \(s+u\), conditional on having reached \(s\), for a given \(b_i\). Averaging that ratio over the latent-feature distribution informed by survival and biomarker history gives the conditional survival probability; subtracting from one gives risk.
As the landmark moves from \(s_1\) to \(s_2>s_1\), two kinds of evidence can update the distribution \(p(b_i\mid T_i>s,\mathcal{H}_i(s))\). First, the patient has survived event-free through the additional interval. Second, measurements obtained by \(s_2\) add information about the latent trajectory. The event model’s survival probabilities over the prediction window may also differ at the new landmark.
Worked Updates Across Landmarks
Worked Example: Survival and a New Marker Update Latent Risk
Consider a hypothetical two-class illustration. “High” and “low” denote latent event-risk classes. Before the patient’s interval of event-free follow-up, the class probabilities are 0.40 and 0.60. The probabilities of remaining event-free through that interval are 0.81875 for the high class and 0.95117 for the low class. Assume, for this illustration, that these probabilities and the later marker likelihoods are correctly specified and that the marker is conditionally independent of event status given class.
The unnormalized class probabilities among survivors are \(0.40(0.81875)=0.32750\) and \(0.60(0.95117)\approx0.57070\). Therefore, before observing the new marker, the high-class probability among survivors is
Now suppose the marker result is more likely in the high-risk class: its likelihood is 0.800 in the high class and 0.200 in the low class. Updating the survivor probabilities gives
For a one-year prediction from the new landmark, suppose the class-specific event hazards are constant over that year at 0.150 per year and 0.0400 per year. The corresponding one-year conditional survival probabilities are \(e^{-0.150}\) and \(e^{-0.0400}\). Thus the updated risk is
The estimated one-year event risk at this landmark is about 10.89%. The calculation separates two updates: event-free follow-up alone reduced the high-class probability to about 0.3646, while the new marker increased it to about 0.6965. Neither update is a causal effect; each is a change in conditional prediction under the assumed model.
Worked Example: Event-Free Follow-Up Alone Can Lower Risk
Suppose a hypothetical patient’s latent high-class probability at a landmark, after incorporating all information available then, is 0.4000. During subsequent follow-up, the probability of remaining event-free is 0.5185 in the high class and 0.9000 in the low class. Assume these class-specific survival probabilities correctly describe the interval and that no new marker is observed.
The updated high-class probability among survivors is
At the later landmark, suppose the one-year event hazards are 0.150 per year in the high class and 0.0400 per year in the low class. The one-year risk based on survival information alone is
The later landmark’s estimated one-year risk is approximately 6.70%. In this example, survival to the landmark shifted the latent class distribution toward the lower-risk class. This is a conditional update among survivors, not evidence that surviving follow-up caused a reduction in the patient’s underlying risk.
Worked Example: A Later Prediction Can Be Higher
Suppose a hypothetical patient has a high-class probability of 0.6000 at landmark \(s_1\). The one-year event probabilities from that landmark are 0.200 in the high class and 0.050 in the low class. The predicted one-year risk is
The patient remains event-free until \(s_2\). For the intervening period, assume the class-specific event-free probabilities are 0.800 in the high class and 0.950 in the low class. The high-class probability after this survival update is \(\{0.6000(0.800)\}/\{0.6000(0.800)+0.4000(0.950)\}=0.5581\). A new marker then has likelihood 0.900 in the high class and 0.300 in the low class. Updating again gives
Suppose the one-year event probabilities from \(s_2\) are now 0.180 in the high class and 0.0400 in the low class. The new one-year risk is
The predicted risk has changed from 14.00% to 15.08%, despite the patient’s event-free survival to \(s_2\). The marker shifted the class probability upward, enough to outweigh the downward survivor update in this illustration. The two predictions concern one-year windows beginning at different landmarks, so they are not predictions for the same calendar-time interval.
Why Risk Need Not Move in One Direction
There is no general rule that repeated landmark predictions must increase or must decrease. Event-free follow-up can lower estimated risk when higher-risk latent profiles are less likely to survive to the next landmark. A new biomarker value can move the prediction in either direction, depending on how it updates the latent trajectory and how that trajectory enters the event model. The event model may also imply different near-term hazards at different times.
A prediction can therefore rise even though the patient remains event-free, as the third example shows. Conversely, a new measurement can lower the estimated risk even when the measured biomarker is higher, if the model’s trajectory, slope, or cumulative-exposure association implies a different risk pattern. The interpretation must follow the specified joint model, not an informal expectation that “more follow-up” always means lower risk.
To make comparisons interpretable, first say whether the horizon length is fixed or the calendar-time endpoint is fixed. A one-year risk at \(s_1\) and a one-year risk at \(s_2\) cover different intervals. If the question is about a shared endpoint \(\tau\), compare probabilities conditional on the relevant landmark, such as \(P(T_i\leq\tau\mid T_i>s,\mathcal{H}_i(s))\), when \(\tau>s\). Even then, the conditioning information differs across landmarks.
Assumptions, Diagnostics, and Alternatives
The calculations above rely on a correctly specified joint model and a coherent update of the longitudinal and event information. In particular, the event-free status through each landmark must be correctly recorded; biomarker measurements used at a landmark must have been available by that time; and the association between the latent trajectory and event risk must be represented adequately. If measurement timing or dropout is informative beyond what the model accounts for, predictions can be biased. Unmodeled competing events also change the meaning of event risk and require an appropriate competing-risk formulation.
When inspecting sequential predictions, check that each prediction uses only information available at its landmark, that the stated horizon is clear, and that the survival and marker updates are being applied in the intended order. Large abrupt changes may be legitimate, but they are also a reason to check data coding, measurement units, time scales, and whether a new observation lies far outside the history used to fit the model. Calibration can be examined separately at prespecified landmark times by comparing predicted and observed event frequencies among patients eligible at those landmarks. Such checks assess predictive performance; they do not establish that a biomarker change causes a change in risk.
A simple alternative is to report a single prediction from the most recent landmark rather than display a trajectory of predictions. That is easier to communicate but hides how information changed the forecast. A baseline-only prediction avoids sequential biomarker updates but discards later patient history. A time-updated survival model can use observed covariates directly, while a joint model is designed to link a longitudinal process with event risk through latent features. The appropriate comparison depends on the scientific question and assumptions; none removes the need to define the landmark and prediction window.
| Comparison choice | What is held fixed | Interpretive caution |
|---|---|---|
| Same horizon length at each landmark | Duration, such as one year | Calendar-time intervals differ |
| Same calendar-time endpoint | Target date \(\tau\) | Remaining prediction duration shrinks as the landmark advances |
| One prediction at the latest landmark | Only the latest information set is reported | Does not show how earlier forecasts changed |
Common Mistakes
- Expecting risk to decline automatically. Survival to a later landmark can shift latent risk downward, but a new biomarker or changing hazard can offset or reverse that update.
- Comparing unlike windows as if they were identical. A one-year risk from \(s_1\) and a one-year risk from \(s_2\) refer to different time intervals. State the horizons before interpreting a difference.
- Using future information in an earlier prediction. A measurement collected after \(s\) cannot be included in \(\mathcal{H}_i(s)\). Doing so creates information leakage and an invalid landmark prediction.
- Treating a prediction change as a causal effect. The change reflects conditioning and updating under a model; it does not show that a marker change caused the patient’s event risk to change.
Check Your Understanding
Answer each question, then compare with the concise solutions below.
- In one sentence, define the risk predicted at landmark \(s\) over horizon \(u\).
- A patient’s estimated high-risk-class probability falls after event-free follow-up but rises after a new marker. Is that internally inconsistent? Explain.
- At a landmark, the high-class probability is 0.3000. One-year event probabilities are 0.120 in the high class and 0.0300 in the low class. Calculate the one-year risk.
- Why is a one-year risk at a later landmark not a prediction for the same interval as a one-year risk at an earlier landmark?
- Name one reason sequential predictions might change abruptly that warrants a data or model check.
Solutions. (1) It is \(P(T_i\leq s+u\mid T_i>s,\mathcal{H}_i(s))\), the event probability in the next \(u\) time units conditional on being event-free and on history through \(s\). (2) No. Survival and the new marker provide different information and may shift the latent-risk distribution in opposite directions. (3) \(0.3000(0.120)+0.7000(0.0300)=0.0360+0.0210=0.0570\), or 5.70%. (4) The windows begin at different landmarks, so their calendar-time intervals differ. (5) Examples include a time-scale or unit error, a miscoded biomarker, or a new value far outside the range represented in model fitting.