What Is the Prediction Target?
A dynamic prediction answers a question asked at a particular time for a particular patient: given what is known about this patient now, what is the probability that the event of interest has not occurred by a specified future time? The word “dynamic” matters because the information available changes as biomarker measurements accumulate. A prediction made at month 6 can use a different observed history from one made at month 12.
The target is a probability, not a hazard, a regression coefficient, or the patient’s latent biomarker value. It has three essential ingredients: the future event-free time being predicted, survival without the event up to the prediction time, and the patient information available by that time. These ingredients must be stated before interpreting a numerical prediction.
The time \(t\) is the prediction time (or landmark), and \(\Delta\) is the prediction horizon. The future calendar or study time in the probability is \(t+\Delta\), not \(\Delta\) alone. For example, a prediction made at month 8 for the next 4 months concerns event-free status through month 12.
The information set is patient-specific. Baseline covariates are values defined at or before the start of follow-up, such as treatment group or age at enrollment. The history \(\mathcal H_i(t)\) consists of the biomarker measurements and their measurement times observed up to \(t\). It does not include a measurement taken after \(t\), nor any knowledge of whether the patient experiences the event after \(t\). Using either would change the question and give the prediction information unavailable at the time it is supposed to be made.
Worked Example: Naming the Horizon Correctly
Suppose follow-up time begins at enrollment. A hypothetical patient is known to be event-free at month 6, has baseline covariates \(X_i\), and has biomarker measurements at months 1, 3, and 5. A fitted joint model gives a predicted probability of 0.72 of being event-free at month 12, conditional on this information.
Here \(t=6\), and month 12 is 6 months after the prediction time, so \(\Delta=12-6=6\). The target being reported is $$ \pi_i(6,6) = P\!\left(T_i>12\mid T_i>6,\ X_i,\ \mathcal H_i(6)\right) =0.72. $$
The corresponding conditional probability of experiencing the event between months 6 and 12 is the complement: $$ 1-\pi_i(6,6)=1-0.72=0.28. $$ Thus, under the fitted prediction, the patient’s probability of remaining event-free through month 12 is 72%, and the probability of an event by month 12, among patients with this information who are event-free at month 6, is 28%. These are predictions, not guarantees about the individual outcome.
Assumptions and Scope of the Target
The probability is meaningful only after the event and the time scale have been defined. \(T_i\) must refer to the event of interest measured from the stated time origin. The origin might be enrollment, diagnosis, or another specified starting point; changing it changes the meaning of \(t\) and \(t+\Delta\). The event definition must also be explicit. If the event is first hospitalization, for example, “event-free” means not yet hospitalized under that definition.
The target is defined for a patient who is event-free at \(t\). In observed data, it must be possible to establish that status through \(t\); a patient censored before \(t\) cannot simply be treated as known to have remained event-free until \(t\). This is a data and estimation issue, distinct from the mathematical definition of the conditional probability.
The history used in the prediction must be available by the prediction time. Measurements can occur at irregular times, and \(\mathcal H_i(t)\) includes when measurements were taken as well as the observed values. The target does not require that every patient have the same number of measurements. It does require care about what “available” means: a laboratory result collected before \(t\) but not known to the predictor until after \(t\) should not be included in an operational prediction made at \(t\).
Finally, a numerical estimate requires a statistical model and data that can support it. The probability’s definition does not by itself guarantee that the fitted model is correct or that its predictions are well calibrated. Interpretation as a useful estimate relies on reasonable model specification, adequate information in the relevant patient histories, and appropriate handling of censoring and missing observations. The conditional probability describes an association-based prediction; it does not say that changing a biomarker value would cause the event probability to change.
Worked Example: A Prediction Made at a Later Visit
Consider a hypothetical patient who is still event-free at month 10. The patient has baseline covariates \(X_i\) and biomarker values recorded at months 2, 6, and 9. A fitted model gives a probability of 0.61 of remaining event-free through month 15, conditional on that information.
The prediction is made at \(t=10\), and its horizon is \(\Delta=15-10=5\) months: $$ \pi_i(10,5) = P\!\left(T_i>15\mid T_i>10,\ X_i,\ \mathcal H_i(10)\right) =0.61. $$
The complementary prediction of an event by month 15, conditional on being event-free at month 10 and on the same patient information, is \(1-0.61=0.39\), or 39%. The month 9 biomarker measurement is eligible because it was observed by month 10. A hypothetical month 11 measurement is not eligible for this prediction, even if it is later available in the patient’s record.
The 61% prediction does not mean that the patient is expected to remain event-free for exactly 5 months, nor does it describe the probability of an event in every 5-month interval. It refers to the specific future endpoint at month 15, conditional on the stated information at month 10.
How the Target Differs from Other Summaries
Several quantities can sound like “risk” while answering different questions. The dynamic target conditions on being event-free at the prediction time and on the patient’s observed history. An unconditional survival probability instead concerns a population from its specified time origin without conditioning on this particular patient’s observed biomarker history. A hazard is an instantaneous event rate among those still event-free, not a probability of remaining event-free through a chosen future time. An interval risk can be reported as the complement of the dynamic event-free probability, provided it uses the same conditioning information and endpoint.
| Quantity | Question answered | Key distinction |
|---|---|---|
| Dynamic event-free probability | What is the probability this patient is event-free at \(t+\Delta\), given the stated information at \(t\)? | Patient-specific conditioning, including survival to \(t\) and observed history |
| Unconditional survival probability | What is the probability of surviving event-free to a time from the chosen origin in a specified population? | Does not, by itself, condition on the patient’s current biomarker history |
| Hazard | What is the instantaneous event rate at a time among those still event-free? | A rate, not a probability over a stated future interval |
| Conditional interval event risk | What is the probability of the event by \(t+\Delta\), given event-free status and information at \(t\)? | The complement of the dynamic event-free probability for the same endpoint and horizon |
Worked Example: Checking Predictions Across Horizons
For one hypothetical patient at \(t=4\) months, suppose a fitted model reports event-free probabilities of 0.86 through month 7 and 0.70 through month 10, each conditional on the same baseline covariates, biomarker history through month 4, and event-free status at month 4.
The first horizon is \(\Delta_1=7-4=3\) months, and the second is \(\Delta_2=10-4=6\) months. The corresponding interval event risks are: $$ 1-0.86=0.14 \qquad\text{and}\qquad 1-0.70=0.30. $$
The six-month event-free probability is no greater than the three-month probability, as it must be for the same event and the same conditioning information: being event-free through month 10 implies being event-free through month 7. The 30% is the probability of an event by month 10, conditional on being event-free at month 4. It is not the probability of an event specifically between months 7 and 10. Computing risk only for that later subinterval would require conditioning on event-free status at month 7 as well.
Justification and Diagnostic Checks
Two basic properties follow directly from the target’s definition. First, the event-free probability and the corresponding event probability are complements when they refer to the same endpoint, horizon, and conditioning information. Given \(T_i>t\), either \(T_i>t+\Delta\) or \(t<T_i\leq t+\Delta\); these events are mutually exclusive and together cover the possibilities for an event by the future time. Therefore, $$ P(t<T_i\leq t+\Delta\mid T_i>t,X_i,\mathcal H_i(t)) = 1-\pi_i(t,\Delta). $$
Second, with the same information at \(t\), event-free probability cannot increase as the future endpoint moves later. If \(\Delta_2>\Delta_1>0\), then $$ \{T_i>t+\Delta_2\}\subseteq \{T_i>t+\Delta_1\}, $$ so \(\pi_i(t,\Delta_2)\leq\pi_i(t,\Delta_1)\). A reported increase across these horizons signals a mismatch in conditioning, endpoint definitions, or calculations, or a problem with the prediction procedure.
These checks do not establish that predictions are accurate. When predictions are assessed against data, their performance must be examined using methods appropriate to censored time-to-event outcomes. At the level of defining the target, however, several warning signs are immediate: a prediction uses biomarker values recorded after \(t\); the stated future time is confused with the horizon length; or a probability is reported without specifying the conditioning information. Each makes the target ambiguous or impossible to reproduce.
State what counts as the event and how follow-up time is measured.
Write the prediction time \(t\), future endpoint \(t+\Delta\), and positive horizon \(\Delta\).
Include baseline covariates and only biomarker history observed and available by \(t\); require event-free status through \(t\).
Report the event-free probability with its conditioning information, and distinguish it from its complementary interval event risk.
Common Pitfalls and Limitations
- Using “six-month survival” without a prediction time. A six-month horizon at month 4 refers to month 10, whereas a six-month horizon at month 10 refers to month 16. State both \(t\) and \(\Delta\).
- Leaving out survival to the prediction time. The target concerns patients who are event-free at \(t\). It is not the probability from enrollment for all patients.
- Using future biomarker information. Measurements after \(t\) cannot be used in a prediction claimed to be available at \(t\). Doing so creates information leakage.
- Calling a hazard a probability. A hazard is an instantaneous rate; it is not the chance of an event over a particular interval.
- Making a causal claim from a prediction. A patient-specific prediction describes risk conditional on observed information under a model. It does not estimate what would happen if the biomarker were experimentally changed.
The target also has practical limits. At late prediction times, relatively few patients may remain event-free and under observation, so estimates can be unstable. Some combinations of baseline covariates and biomarker history may be poorly represented in the data. In addition, a patient’s future biomarker measurements are not part of the information at \(t\); the target concerns future event status given the history so far, not a forecast conditional on knowing measurements that have not yet occurred.
Check Your Understanding
Answer each question by identifying the relevant conditioning information and time horizon. Short solutions follow.
- At month 5, a patient is event-free and has biomarker measurements through month 4. Write the target for remaining event-free through month 9.
- A prediction at month 8 is 0.67 for remaining event-free through month 14. What is the horizon, and what is the complementary conditional event risk?
- Why can a month 11 biomarker result not be used in a prediction claimed to have been made at month 10?
- For the same information at month 3, a model reports 0.80 event-free probability through month 6 and 0.84 through month 9. What should be checked?
- Does a predicted 25% conditional event risk mean that changing a patient’s biomarker would cause a 25% event probability? Explain.
Solutions. (1) The horizon is \(9-5=4\) months, so the target is \(P(T_i>9\mid T_i>5,X_i,\mathcal H_i(5))\). The history includes only measurements available by month 5. (2) The horizon is \(14-8=6\) months; the complementary conditional event risk is \(1-0.67=0.33\), or 33%, by month 14 among patients event-free at month 8 with the stated information. (3) The month 11 value occurs after the prediction time and was not available then, so using it would introduce future information. (4) With the same conditioning information and event definition, event-free probability should not increase at the later endpoint; check that the time points, horizons, conditioning sets, and reported calculations are aligned. (5) No. The prediction is conditional on observed information under a model; it is not a causal effect of changing the biomarker.