Choosing Prediction Times and Horizons

Why Prediction Times and Horizons Must Be Chosen Together

A dynamic prediction answers a question about a particular subject at a particular time, using information available up to that time. To evaluate such predictions, the target must specify both when the prediction is made and how far ahead it looks. A prediction made at month 6 for the next 12 months is not aimed at the same outcome as a prediction made at month 18 for the next 12 months, even if both are reported as a probability of an event.

The choice also determines who is eligible for evaluation. A subject who has already had the event by the prediction time cannot be asked for their future event risk from that time. Thus each prediction time has its own event-free risk set. The prediction times and horizons should be chosen to reflect the intended use of the model, not selected after inspecting which choices make its performance look best.

The condition \(T>s\) is substantive, not a notational detail. It says the prediction is conditional on having remained event-free to the landmark. For a joint model, \(H(s)\) can include longitudinal measurements observed up to \(s\), and the prediction uses that history to update risk. It must not include measurements collected after \(s\). Whether a subject is eligible is determined separately at each landmark.

Choosing a Useful Evaluation Grid

A study may evaluate predictions at one landmark and one horizon, or at several prespecified combinations \((s,w)\). Each combination defines a distinct target. For example, evaluating 6-month risk at landmarks 6, 12, and 18 months asks about three conditional risks, each among subjects still event-free at its respective landmark. It does not ask for one common risk in a fixed cohort.

A defensible choice starts with the clinical or operational decision. Ask when a prediction could be acted on, how long a decision-maker needs to look ahead, and whether the event has a meaningful interpretation over that interval. A horizon should have a unit and an endpoint: “risk in the next 12 months” is clearer than “short-term risk.” The time origin must also be explicit, such as diagnosis or treatment initiation, so that a landmark has an unambiguous meaning.

• Prespecify the landmarks. Choose times that correspond to intended updates or decision points. They should be fixed without using the validation outcomes to select favorable results.
• Prespecify the horizons. Use durations relevant to decisions, and state whether the endpoint is measured from the landmark or from the original time origin. In this tutorial, \(w\) is measured from \(s\).
• Check event-free availability. At each landmark, enough subjects must be event-free and have the information needed to form predictions. The eligible group will generally change across landmarks.
• Check outcome support. The validation data need meaningful follow-up over the interval through \(s+w\). A nominally interesting horizon is not useful if outcomes cannot be ascertained there.
• Limit the grid to interpretable questions. A large grid of landmarks and horizons creates many distinct targets and can leave some targets with little information.

These checks concern the definition and feasibility of the target. They do not, by themselves, resolve how to handle censored outcomes or estimate validation metrics; those require separate methods. If the event is not observed for every subject through the horizon, the distinction between the target and what is observed must be kept explicit.

Why Conditional Risk Uses Survival to the Landmark

The relationship between horizon risk and survival probabilities provides a useful check on the target. Write \(S(t\mid H(s))\) for the conditional probability of remaining event-free through time \(t\), given the history available at \(s\). For \(t\geq s\), the probability of surviving from \(s\) to \(t\), conditional on being event-free at \(s\), is \(S(t\mid H(s))/S(s\mid H(s))\), when the denominator is positive. The event by the horizon is the complement of survival through \(s+w\).

This derivation uses the conditional probability identity \(P(A\mid B)=P(A\cap B)/P(B)\), with \(A=\{T>s+w\}\) and \(B=\{T>s\}\). Since \(T>s+w\) implies \(T>s\), their intersection is simply \(\{T>s+w\}\). The ratio is defined only when the conditioning event has positive probability. The formula also makes clear why using \(1-S(s+w\mid H(s))\) alone would be wrong: that quantity includes events before the prediction time, whereas the target is conditional on reaching \(s\) event-free.

Landmarks Define Different Eligible Groups

Let \(R(s)\) denote the set of subjects who are event-free at landmark \(s\) and for whom a prediction based on information available by \(s\) can be formed. In a fully observed hypothetical data set, membership requires \(T_i>s\); in an actual validation data set, follow-up and availability of required histories also matter. As \(s\) increases, subjects who had the event earlier leave the risk set. Consequently, the target population and the distribution of histories may change across landmarks.

If the same person is event-free at two landmarks, that person can contribute two different prediction occasions. These are not duplicate versions of one outcome: the history is updated, the conditioning event differs, and the future interval moves forward. Any later analysis that combines these occasions must respect their repeated-subject structure and state how occasions are weighted. Choosing landmarks is therefore also choosing which prediction occasions the evaluation is intended to represent.

Diagnostics, Alternatives, and Limitations

Before evaluating predictions, check that each prediction occasion is aligned with its target: the subject was event-free at \(s\), the history ends no later than \(s\), and the outcome interval is exactly \((s,s+w]\). Tabulate the number of eligible subjects at every landmark and inspect how many have outcome information over the full horizon. A sharp drop in eligibility or outcome support at later landmarks signals that the corresponding target may be unstable or infeasible. Also confirm that the model can produce predictions for those histories and that the time origin and units agree between predictions and outcome records.

A fixed-horizon prediction made only at baseline is an alternative when the intended use is a single baseline decision. It avoids repeated landmark targets but cannot evaluate updated predictions made later. A single landmark with a long horizon may suit a particular planning question, while a grid of landmarks and horizons examines a broader range of uses. The grid is not automatically better: each additional combination creates another target, often with a smaller eligible group and less follow-up support.

A further alternative is to report prediction performance across event time without fixing a single horizon in advance. Such summaries can be useful, but they answer a different question and can obscure the exact clinical interval of interest. Whichever approach is used, the target should be stated in conditional terms and the evaluated prediction occasions should match it.

The main limitations are practical and interpretive. Late landmarks may leave few event-free subjects; long horizons may extend beyond reliable follow-up; and repeated landmarks can represent some subjects more often than others. A target grid chosen after inspecting validation results can also make reported performance selectively favorable. Prespecification does not guarantee adequate data, but it makes the question transparent and helps distinguish a weak prediction method from an evaluation target that the data cannot support.