Selecting Prediction Horizons for State Occupation

Why the Prediction Horizon Matters

A state-occupation prediction is not complete until its time point is specified. A prediction that a patient will be in a particular state “after treatment” could mean after three months, one year, or five years. Those are different questions, and the corresponding probabilities need not be similar. Selecting a horizon therefore determines both the clinical meaning of the prediction and whether the available follow-up can estimate it with useful precision.

Suppose prediction begins at landmark time \(s\), when the person is known to be in state \(j\). A horizon \(h>0\) asks about the state at the later time \(s+h\). The horizon is the elapsed time from the landmark, not the time since the original study began. For example, at a landmark of six months, a one-year horizon refers to state occupation at month 18.

The conditioning information must be stated clearly. It may consist of the current state alone or include recorded history and covariates, as specified by the prediction target and model. Whatever the conditioning set, \(\pi_k(s,h)\) describes the probability of being in state \(k\) at one particular future time. It is not automatically the probability of ever entering state \(k\) before that time.

Choose a Horizon That Answers the Question

Begin with the decision or scientific question, then translate it into a time from the landmark. A clinician deciding whether to schedule a review at six months may need a six-month prediction. A health system planning two-year service capacity may need a two-year prediction. The most convenient horizon for analysis is not necessarily the most relevant one for use.

A useful horizon should satisfy two criteria. First, it should have a clear interpretation for the intended decision. Second, the observed follow-up should provide adequate support for estimating state occupation at that time in the target population. A clinically important horizon outside the data’s support cannot be rescued by labeling it important; it requires more follow-up or a model-based extrapolation whose assumptions must be made explicit.

For ordinary Aalen-Johansen state-occupation estimation after a landmark, the analysis also relies on appropriate observation of transitions and independent censoring for the target population, along with sufficient follow-up at the requested horizon. In a model-based prediction, the assumptions of that model are also relevant. In particular, a prediction is only as credible as its treatment of the history and covariates that determine future state occupation. Earlier material on Choosing Landmark Times With Adequate Prediction Support addresses support at the landmark; horizon selection asks whether support remains adequate at the later target time.

The endpoint state probabilities for a complete set of mutually exclusive states sum to one:

This is a useful check on a reported prediction. It does not mean each probability is equally well estimated: one or more states may have sparse data or substantial censoring. Nor does the sum reveal whether a particular state was visited earlier and then left. That distinction matters for transient states.

Follow-up Support at the Chosen Horizon

Follow-up limits the horizons that can be estimated directly. If few people remain under observation near \(s+h\), the estimated state probabilities may be unstable. If nobody is observed sufficiently late, the data do not identify state occupation there without extrapolating beyond observed support. Inverse censoring adjustments or a transition model do not create information where there is no support.

Assess support in the population relevant to the prediction, not only in the full study. At a landmark, the target group may be a subset defined by current state, covariates, or history. A horizon with substantial follow-up overall can still have very little follow-up in that subset. Check numbers at risk, censoring patterns, observed follow-up times, and uncertainty at the proposed horizon. There is no universal number at risk that guarantees reliability; the consequences depend on event rates, censoring, and the state-transition pattern.

A practical diagnostic is to plot state-probability estimates over time together with uncertainty intervals and, where available, numbers at risk. Look for a sharp increase in uncertainty, unstable late changes, or horizons beyond the period of meaningful observation. A curve that appears smooth is not proof of adequate support, especially if it is based on very few people. For interval estimation, use an appropriate method and interpret the interval at the selected horizon; a narrow-looking point estimate alone does not establish precision.

How the Horizon Changes the Interpretation

Changing \(h\) changes the target quantity, even if the landmark population and model stay fixed. At \(h=0.5\), \(\pi_k(s,0.5)\) is the probability of occupying state \(k\) half a year after the landmark. At \(h=2\), \(\pi_k(s,2)\) is the probability of occupying that state two years after the landmark. One cannot describe either as the probability of the state “in general.”

For a transient state, a person may enter and later leave it. Thus, state occupation at \(s+h\) is different from the probability of having entered that state at any time in \((s,s+h]\). If the state is an absorbing competing-event state, occupation by \(s+h\) corresponds to having experienced that event by then. For non-absorbing states, the endpoint probability does not record all prior visits. The distinction between transient-state occupation and eventual absorption is important when wording a prediction.

One Horizon, Several Horizons, or a Trajectory?

A single prespecified horizon is often easiest to communicate when it corresponds to a concrete decision. Reporting several horizons can show how predictions change over time, but each is a distinct estimand and requires adequate support. A trajectory displays the state probabilities across a range of horizons; it can reveal transitions that a single endpoint hides, while also making late follow-up limitations visible.

These approaches answer related but different communication needs. A single horizon emphasizes one decision point. Several fixed horizons permit direct comparisons at selected times. A trajectory shows the pattern over time. None is universally best, and a longer list of horizons is not automatically more informative. Select a small set that addresses the question and is supported by the data, and label every estimate with its elapsed time from the landmark.

Common Pitfalls and Limitations

• Choosing the longest possible horizon by default. A longer horizon may be clinically relevant, but if follow-up is sparse, the estimate can be unstable or require extrapolation. Match the horizon to the question and assess support.
• Reporting a probability without its time point. “The probability of state \(B\) is 0.30” is incomplete. State the landmark, elapsed horizon, conditioning group, and units of time.
• Confusing endpoint occupation with an event history. For a transient state, the probability of occupying it at \(s+h\) is not the probability of ever having entered it by then.
• Treating a supported horizon as automatically precise. Follow-up at a time point is necessary but may not be sufficient. Numbers at risk, censoring, transitions, and uncertainty also matter.
• Extrapolating without saying so. Predictions beyond observed follow-up depend on model assumptions rather than direct support. Report that limitation plainly.

Horizon selection cannot eliminate limitations in the landmark population, measurement, censoring, or model specification. Nor does an appropriate horizon turn a predictive association into a causal effect. The gain is more precise alignment between the probability being estimated, the time at which it applies, and the decision it is intended to inform.