Follow-Up Support At The Prediction Horizon

Why the Prediction Horizon Needs Data Support

An absolute-risk prediction always refers to a time horizon. A prediction of a 2-year event probability asks about a different quantity from a prediction of a 5-year probability, even when both use the same model and covariates. Before interpreting a prediction at horizon \(\tau\), ask whether the study observed participants for long enough to provide information about outcomes through \(\tau\), especially for participants resembling the person or population being predicted.

This question is not answered by the largest follow-up time alone. One person observed for many years may establish that long follow-up occurred, while providing little evidence for a particular covariate group. Conversely, a cohort may have substantial support at the requested horizon even if some participants were censored much earlier. The pattern of censoring, event times, and covariates matters.

Let \(T\) be time to the first event, \(J\) its event type, and \(C\) the censoring time. The observed time is \(\tilde T=\min(T,C)\). We observe an event indicator \(\Delta=I(T\le C)\), and observe \(J\) when \(\Delta=1\). A competing event is an observed outcome, not a censoring time: it changes the possible future event history and contributes information to absolute-risk estimation. Administrative censoring occurs when observation ends because of a study rule, such as a fixed data-lock date.

For cause \(k\), the target absolute risk at \(\tau\), conditional on covariates \(X=x\), is \(F_k(\tau\mid x)=P(T\le\tau,J=k\mid X=x)\). The question here is not how to construct that prediction, but whether the study's follow-up supplies empirical support for evaluating it at the requested horizon.

Assumptions and What Counts as Support

The usual censoring-based interpretation requires an appropriate noninformative-censoring assumption: conditional on the variables used to account for censoring, censoring time must not carry additional information about the event process. In notation, a common form is \(T\perp C\mid X\), with relevant design variables such as calendar entry time included when needed. This assumption concerns the censoring mechanism; it does not mean that event times are independent of covariates.

A second condition is positivity of follow-up: in the target covariate profile, there must be a nonzero chance of remaining uncensored through the horizon. A population statement is \(P(C\ge \tau\mid X=x)>0\), or the corresponding condition after conditioning on the variables used to describe censoring. In finite samples, the number observed through \(\tau\), and the number still event-free just before \(\tau\), are practical diagnostics of how informative that support may be. Positivity alone does not guarantee precision: a handful of observations can satisfy the condition but still yield unstable estimates.

The late risk set is not the only relevant count. Participants who experience an event before \(\tau\) have provided observed information about their event type and time. Participants censored before \(\tau\), however, provide no direct observation of what happens to them after censoring. For that reason, report both the censoring pattern and late risk-set sizes rather than treating a single count as a complete measure of support.

Administrative censoring often arises from a common study end date. Participants entering earlier can have longer potential follow-up than those entering later. If enrollment dates relate to prognosis, treatment, or changing clinical practice, treating administrative censoring as independent without accounting for those factors may be inappropriate. A common calendar cutoff is a feature of the design, not automatic proof that censoring assumptions hold.

Why a Cutoff Cannot Identify Later Risk

The limitation can be seen from the information in a censored observation. If a participant is event-free through censoring time \(c\), the likelihood contribution from that record concerns survival to \(c\); it contains no observed event time or cause after \(c\). In a competing-risks setting, the record does not distinguish among possible post-\(c\) event types.

More formally, consider two candidate event-time distributions that agree on all event probabilities through \(c\), but assign different probabilities and causes to events after \(c\). For a participant administratively censored at \(c\), both distributions give the same probability of being event-free through the observed follow-up. Yet they can give different values of \(F_k(\tau\mid x)\) for \(\tau>c\). The observed record therefore cannot choose between those later-risk predictions.

This is why a prediction curve extending smoothly beyond the last well-supported follow-up does not itself establish that the data support the extension. The curve may be mathematically defined under a model, while the data do not identify its later portion without additional assumptions.

Assess Support for the Actual Prediction Target

A global follow-up summary can conceal weak support for the requested covariate profile. If the target is an individual with a particular covariate pattern, inspect whether comparable participants had follow-up through \(\tau\). If the prediction is standardized to a population, assess support across the covariate patterns represented in that population. Overall follow-up can be long while relevant strata have little or none.

Practical Diagnostics and Alternatives

Before interpreting a horizon-specific absolute-risk prediction, make a compact support assessment. State the requested horizon and the study's follow-up design; summarize the numbers censored before, at, and beyond the horizon where meaningful; and report the late risk-set size. Examine these quantities by important covariates or prediction strata. If follow-up is calendar-based, check whether entry period or calendar time is associated with the target population or care setting.

When support is inadequate, the main alternatives are to request a shorter horizon with better empirical support, wait for or collect longer follow-up, or present a longer-horizon model prediction explicitly as extrapolative and examine how sensitive it is to modeling assumptions. These choices answer different questions. Shortening the horizon changes the estimand; extending follow-up adds observations; extrapolation retains the requested horizon but relies more heavily on assumptions.

A confidence interval quantifies uncertainty under the fitted model and its assumptions; it does not create observations beyond an administrative cutoff. Even a narrow interval should not be presented as evidence of empirical support at a horizon the study did not observe. Conversely, limited support does not automatically make every model-based prediction useless: it makes the source and strength of its assumptions especially important to communicate.

Common Mistakes and Limitations

• Using the maximum follow-up time as the answer. A long follow-up for one participant does not ensure support for the target covariate profile or a useful late risk set.
• Treating competing events as censoring. A competing event is observed and contributes information about the event process. It is not an unknown outcome after loss to follow-up.
• Calling a model output data-supported because software returned it. A prediction beyond the administrative cutoff may be defined by the model but not identified by observed follow-up alone.
• Equating any positive late count with adequate support. Positivity is a condition, not a guarantee of precision, representativeness, or stable estimation.
• Assuming administrative censoring is automatically independent. Calendar entry, secular changes, or other design features can connect censoring time with prognosis; the censoring assumption must be considered in context.

Support diagnostics are not a substitute for checking model specification, calibration, or uncertainty. Nor does a large late risk set by itself prove that the prediction is transportable to a different population. The narrower conclusion is about empirical follow-up through the requested horizon, for the target to which the prediction is intended to apply.