Why the Prediction Horizon Matters
An absolute-risk prediction is always tied to a time horizon. A prediction of a cause-specific event risk by two years and one by five years are different targets, even when they use the same fitted model and covariate profile. The longer horizon may be scientifically important, but it is not automatically supported by the observed data.
Suppose the available follow-up ends at 2.5 years, while the requested prediction is the probability of an event by year 4. A model may still return a number. That number does not become an empirically observed four-year risk merely because software can calculate it. The portion after year 2.5 depends on assumptions about event hazards, competing events, and how those hazards continue beyond the observed period.
The largest recorded follow-up time is a useful warning sign, not a complete assessment of support. Follow-up can be sparse near its end, and the final recorded time might belong to a censored individual rather than an observed event. The important question is whether the observations provide information about the event processes over the interval the prediction needs. A precise assessment of follow-up support considers how much information is available near the target horizon; here, the key distinction is whether the requested prediction relies on a post-observation model continuation.
Where Extrapolation Enters a Competing-Risk Prediction
Let \(T\) be the time to the first event, \(J\) its cause, and \(X=x\) a specified covariate profile. The cumulative incidence for cause \(k\) at time \(t\) is \(F_k(t\mid x)=P(T\leq t,J=k\mid X=x)\). In a continuous-time model, let \(\lambda_k(u\mid x)\) be the cause-specific hazard at time \(u\), and let \(S(u\mid x)\) be the probability of remaining free of every event through time \(u\). Then
These expressions make the role of competing events explicit: cause \(k\)'s cumulative incidence depends not only on its own hazard but also on the hazards of other causes, which affect how many individuals remain event-free and at risk. If follow-up ends at time \(\tau\) and the target is \(t>\tau\), the prediction can be split into the observed-period contribution and the post-\(\tau\) contribution:
The second term requires hazards after \(\tau\). When the data contain no relevant follow-up after that point, their values cannot be read directly from observed event experience. The model supplies them through its assumptions, such as a constant hazard, a specified parametric time trend, or a continuation of a fitted relationship. Those assumptions can be useful, but they are not equivalent to new observations.
This argument assumes a correctly specified event model for the period it describes, appropriate handling of right censoring, and a target covariate profile to which the model applies. The usual interpretation of fitted hazards also relies on the study’s assumptions about how censoring relates to future event times, conditional on the variables in the analysis. Even if those conditions are reasonable during observed follow-up, they do not establish that the hazard pattern remains unchanged after follow-up ends.
Worked Example: Classifying a Four-Year Target
In a hypothetical study, eight participants have recorded follow-up times of 0.6, 1.0, 1.4, 1.8, 2.0, 2.1, 2.4, and 2.5 years. Some experience one of two event causes; the others are right-censored. A researcher requests the probability of cause 1 by year 4.
The maximum recorded follow-up is \(\tau=2.5\) years, while the requested horizon is \(t=4.0\) years. Since \(4.0>2.5\), the target extends beyond the observed follow-up. The extra interval is \(4.0-2.5=1.5\) years, and the target is \(4.0/2.5=1.60\) times the maximum follow-up time.
That arithmetic classifies the request; it does not calculate a four-year risk. The data may inform risk up to the observed range, but the risk from year 2.5 to year 4 depends on a model for what happens after year 2.5. A responsible report would label the four-year result as model-based extrapolation and state the tail assumption. With only eight participants, the observations may also provide weak information even near the maximum time.
Worked Predictions Require an Explicit Tail Assumption
A simple illustration assumes constant cause-specific hazards. If \(\lambda_1\) and \(\lambda_2\) are constant over the entire prediction interval, then \(S(t)=\exp\{-(\lambda_1+\lambda_2)t\}\), and integration gives
This formula is valid under the stated constant-hazard model. When the target exceeds observed follow-up, extending the formula to the target also extends the constant-hazard assumption into the unobserved period.
Worked Example: A Five-Year Prediction Under Constant Hazards
Consider hypothetical fitted cause-specific hazards of \(\lambda_1=0.080\) per year and \(\lambda_2=0.040\) per year. Assume the observed follow-up reaches only two years, and request both cause-specific cumulative incidences at five years. For this calculation, assume both hazards remain constant from time zero through year 5.
The total hazard is \(0.080+0.040=0.120\) per year. Substituting into the formula for cause 1 gives
For cause 2,
Using unrounded values, the five-year probability of either event is approximately \(0.4512\), or 45.1%. The remaining event-free probability is \(\exp(-0.600)\approx0.5488\), and \(0.4512+0.5488=1.0000\), up to rounding. The arithmetic is coherent, but years 2 through 5 are beyond observed follow-up. The outputs are therefore conditional on the constant-hazard continuation, not direct five-year estimates from five years of observed data.
A model can also allow the hazards after observed follow-up to differ from those before it. Doing so does not make the tail observable; it makes the extrapolation assumption more explicit. The next example shows why that matters.
Worked Example: Two Different Hazard Continuations
Suppose a hypothetical model describes two causes with constant hazards \(\lambda_1=0.040\) and \(\lambda_2=0.020\) per year through year 2, the end of observed follow-up. Consider predictions to year 5 under two continuations. Model A keeps those hazards unchanged after year 2. Model B instead uses \(\lambda_1=0.120\) and \(\lambda_2=0.030\) per year from year 2 to year 5. Both models agree over the observed period.
At year 2, the event-free probability under either model is \(S(2)=\exp[-(0.040+0.020)\times2]=\exp(-0.120)\approx0.8869\). The cumulative incidences through year 2 are
For Model A, the three post-year-2 years have total hazard \(0.060\) per year. Its additional cause 1 incidence is \(S(2)\times(0.040/0.060)\{1-\exp(-0.060\times3)\}\approx0.09740\); its additional cause 2 incidence is approximately \(0.04870\). Thus the five-year predictions are \(F_1(5)\approx0.1728\) and \(F_2(5)\approx0.08639\).
For Model B, the post-year-2 total hazard is \(0.150\) per year. Its additional cause 1 incidence is \(S(2)\times(0.120/0.150)\{1-\exp(-0.150\times3)\}\approx0.2571\); its additional cause 2 incidence is approximately \(0.06428\). The resulting five-year predictions are \(F_1(5)\approx0.3325\) and \(F_2(5)\approx0.1020\).
Both continuations reproduce the same assumed event process through year 2, yet their cause 1 predictions at year 5 differ substantially. The difference is a consequence of the post-follow-up hazard assumptions, not evidence that one continuation is correct. These calculations are hypothetical and illustrate sensitivity; the data described through year 2 cannot by themselves select between the two future hazard patterns.
Diagnostics, Alternatives, and Limitations
Before presenting a prediction, compare the requested horizon with the range and density of observed follow-up. Look for an endpoint that is well beyond the last observed time, a sharp decline in the number still under observation near the target, or a long interval where the model is carrying its hazard pattern forward without direct follow-up. Also check whether the predicted risk changes greatly under reasonable alternative assumptions for the unobserved period. Large changes signal that the result is driven by the assumed tail.
A useful distinction is between sampling uncertainty and extrapolation uncertainty. A confidence interval can describe uncertainty in estimated model parameters under the model and analysis assumptions. It does not, by itself, capture uncertainty about whether the assumed hazard pattern remains appropriate after follow-up ends. Thus a narrow interval is not proof that a distant-horizon prediction is well supported. Where possible, present sensitivity analyses or clearly state that the interval is conditional on the chosen extrapolation model.
There are several defensible responses to a horizon beyond the data. One is to report a shorter horizon that lies within the observed range, while explaining why it is the more directly supported target. Another is to provide the requested longer-horizon model prediction, explicitly describing it as extrapolative and stating the assumptions. If a long-term prediction is essential, additional follow-up or external information about the event processes may help. A nonparametric estimate cannot supply observed event experience beyond the range of follow-up; a model can calculate a value there only by adding structure.
A common mistake is to reason, “The fitted model returns a five-year risk, and its confidence interval is narrow, so the data establish the five-year risk precisely.” The correction is that the calculation can extend beyond the observations and the interval can reflect only uncertainty represented by the fitted model. The prediction may be precise conditional on its assumptions while remaining sensitive to whether those assumptions hold in the unobserved tail.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A study’s maximum recorded follow-up is 3.1 years. Is a prediction at 3 years necessarily well supported? Explain.
- Under constant hazards \(\lambda_1=0.050\) and \(\lambda_2=0.050\) per year, calculate the four-year cumulative incidence for cause 1. If observed follow-up ends at two years, what additional caution applies?
- If a model predicts a flat cumulative-incidence curve after the last observed follow-up, does that establish that no further events would occur? Why or why not?
- Why does a narrow confidence interval not necessarily resolve uncertainty in an extrapolated prediction?
Solutions. (1) No. Although 3 years is below the maximum of 3.1, there may be few individuals under observation near year 3, so the horizon may have weak support. (2) \(F_1(4)=(0.050/0.100)\{1-\exp(-0.100\times4)\}=0.5(1-\exp(-0.4))\approx0.1648\), or 16.5%. Because the target exceeds the two-year follow-up, the calculation requires an assumption about hazards in years 2 through 4. (3) No. A flat model prediction may reflect the model’s construction or assumptions, not observed evidence about events after follow-up ends. (4) The interval may represent parameter uncertainty conditional on the fitted model, while uncertainty about whether its post-follow-up assumptions are appropriate is a separate issue.