Conformalized Survival Analysis

The core innovation is a censoring-adjusted nonconformity measure that accounts for the fact that we only know the true survival time for uncensored subjects. For censored instances, the nonconformity score is computed using a worst-case imputation strategy, ensuring the resulting prediction bounds remain statistically conservative and valid under the standard exchangeability assumption. This prevents the systematic underestimation of risk caused by ignoring censored data.

Instead of a two-sided prediction interval, this method outputs a lower prediction bound (LPB) for survival time. The conformal procedure guarantees that, with user-specified probability 1-α, the true survival time exceeds this bound. This is clinically and operationally more useful than a point estimate, providing a statistically rigorous safety margin for time-to-event decisions.

The coverage guarantee is marginal and distribution-free, meaning it holds regardless of the underlying survival distribution or the choice of the base survival model. It does not rely on the proportional hazards assumption or any specific parametric form. The only requirement is that the calibration and test data are exchangeable, a condition satisfied by standard random splits of i.i.d. data.

This technique functions as a wrapper around any base survival model that outputs a predicted survival time distribution or individual survival curve. Common base models include:

• Cox Proportional Hazards
• Random Survival Forests
• DeepSurv and other neural survival networks The conformal calibration step corrects any systematic overconfidence in the base model's quantile predictions.

The calibration process uses a held-out set containing both censored and uncensored instances. A censoring-adjusted conformity score is computed for each calibration point. The 1-α empirical quantile of these scores defines the threshold used to construct the lower prediction bound for new test subjects. This directly bakes the uncertainty from censoring into the final guarantee.

In medical contexts, a guaranteed lower bound on survival time provides a robust basis for treatment planning and patient stratification. A physician can state with 90% confidence that a patient's survival time will exceed the conformalized bound, enabling more informed decisions about aggressive therapies versus palliative care. This replaces heuristic risk scores with a formal statistical guarantee.

The defining challenge of survival analysis where the event of interest is not observed for some subjects during the study period. Conformalized survival analysis must account for this partial information.

• Type I Censoring: Study ends at a fixed time.
• Type II Censoring: Study ends after a fixed number of events.
• Random Censoring: Censoring time is independent of the event time.

Standard conformal prediction assumes fully observed labels, requiring specialized nonconformity measures that can handle the unknown event times for censored individuals.

A class of nonconformity scores designed to produce valid prediction bounds despite right-censoring. These scores adjust for the probability that an individual's event time is unobserved.

• Inverse Probability of Censoring Weighting (IPCW): Up-weights uncensored observations to represent censored ones.
• Conditional Survival Function: Uses the estimated survival probability at the observed time.
• Martingale Residuals: Measures the difference between observed event status and model-predicted risk.

These scores are evaluated on a calibration set to determine the empirical quantile that defines the lower prediction bound.

The primary output of conformalized survival analysis: a time value T_low such that the true event time exceeds T_low with a user-specified probability (e.g., 90%).

• Provides a statistically guaranteed minimum survival time.
• Unlike point predictions, the LPB accounts for both model uncertainty and finite-sample variability.
• Clinically actionable: "We are 90% confident the patient will survive at least 18 months."

This is a one-sided interval because the upper bound is often infinite or uninformative in survival contexts.

While marginal coverage guarantees hold on average, conformalized survival analysis can be extended to provide approximate conditional coverage across patient subgroups.

• Mondrian Conformal Prediction: Applies calibration independently within discrete strata (e.g., cancer stage).
• Conformalized Quantile Regression for Survival: Models conditional quantiles of the survival distribution directly.
• Weighted Conformal Inference: Adjusts for covariate shift between training and target populations.

Achieving exact conditional coverage is impossible without strong assumptions, but these methods mitigate fairness concerns in heterogeneous populations.

An adaptation of conformalized survival analysis for settings where a subject can experience one of several mutually exclusive events, and the occurrence of one prevents the others.

• Cause-Specific Cumulative Incidence Function (CIF): Estimates the probability of a specific event by a given time.
• Nonconformity for Competing Risks: Based on the difference between observed event type and the predicted CIF.
• Guaranteed Bounds: Produces a lower bound on the time to a specific event type, accounting for the competing risk structure.

This is critical in medical contexts where a patient may die from a cause unrelated to the disease under study.

The integration of the semiparametric Cox proportional hazards model with conformal calibration to produce distribution-free prediction bounds.

• Cox Model Output: Produces a relative risk score and a baseline hazard function.
• Calibration Step: A nonconformity score based on the Cox-Snell residuals is computed on a held-out calibration set.
• Result: Transforms the model's asymptotic confidence intervals into finite-sample valid lower prediction bounds.

This approach corrects for any misspecification of the proportional hazards assumption, providing robustness in real-world deployments.