Time-Dependent ROC Curve

The foundational semiparametric regression model for survival data. It estimates the hazard ratio—the relative instantaneous risk of an event—without requiring specification of the baseline hazard function. The model assumes that covariate effects are constant over time (proportional hazards), an assumption tested using Schoenfeld residuals. While the Cox model provides a linear predictor (risk score), the time-dependent ROC curve evaluates how well this score discriminates between subjects who will and will not experience the event by a specific horizon.

A rank-based metric measuring the discriminative power of a survival model. It evaluates the proportion of all usable patient pairs where the predicted risk ordering agrees with the actual event order. Key limitations addressed by time-dependent ROC:

• Time-invariant: Provides a single global measure, obscuring how discrimination changes over time
• Censoring handling: Relies on specific weighting schemes (e.g., Harrell's, Uno's) that may bias estimates
• No threshold analysis: Cannot identify optimal cut-points for clinical decision-making at specific time horizons

A strictly proper scoring rule that simultaneously evaluates both discrimination and calibration of survival predictions. It computes the mean squared error between predicted survival probabilities and observed event status at time t. Unlike the AUC from a time-dependent ROC curve, the Brier score penalizes models that are poorly calibrated even if they rank patients correctly. Inverse Probability Censoring Weighting (IPCW) is typically applied to account for censored observations when computing the Brier score at a specific time point.

A non-parametric statistic that estimates the survival function S(t)—the probability of surviving beyond time t—from lifetime data with right-censoring. It serves as the empirical benchmark against which model-based survival probabilities are compared. In time-dependent ROC analysis, the Kaplan-Meier estimator is often used to define the true event status at time t: a subject is considered a 'case' if their event occurs by t, and a 'control' if they remain event-free, with censoring handled via NNE (Nearest Neighbor Estimation) or IPCW methods.

The most common time-dependent ROC formulation, introduced by Heagerty and Zheng (2005). It defines:

• Cumulative Cases: Subjects who experience the event by time t
• Dynamic Controls: Subjects who remain event-free beyond time t This definition answers: 'How well does a baseline marker discriminate between patients who will fail by 5 years versus those who will not?' The resulting AUC(t) can be plotted as a function of time to reveal how a biomarker's discriminative ability changes, often peaking for short-term predictions and decaying for long-term horizons.

An alternative time-dependent ROC formulation that defines:

• Incident Cases: Subjects who experience the event exactly at time t
• Dynamic Controls: Subjects who remain event-free beyond time t This answers: 'How well does a marker discriminate between patients who will fail right now versus those who survive past this instant?' It is particularly relevant for dynamic prediction scenarios where a longitudinal biomarker is measured repeatedly, and the goal is to detect imminent risk. This definition requires estimating the instantaneous hazard rather than the cumulative failure probability.