Time-Dependent ROC

Survival Analysis

time-dependent-roc

auc

timeROC

ROC curves and AUC evaluated at specific follow-up times for survival predictions

Published

April 17, 2026

Introduction

Time-dependent ROC extends classical ROC analysis to right-censored survival data. At each chosen evaluation time \(t\), the analysis treats subjects observed to fail before \(t\) as cases and subjects known to survive past \(t\) as controls; subjects censored before \(t\) are reweighted via inverse-probability-of-censoring weights. The resulting time-specific AUC summarises how well a continuous risk score (such as a Cox linear predictor) discriminates failures from survivors at \(t\). It is the natural extension of the C-index to “what is the AUC at exactly one year” reporting.

Prerequisites

A working understanding of ROC curves, the C-index, Cox regression linear predictors, and inverse-probability-of-censoring weighting.

Theory

Two definitions of cases and controls compete:

Incident / dynamic (\(I/D\)): cases are subjects who fail at time \(t\) exactly; controls are those still at risk past \(t\). Useful for hazard-style discrimination.
Cumulative / dynamic (\(C/D\)): cases are subjects who fail at any time \(\le t\); controls are those still at risk past \(t\). The more common choice for clinical risk-score reporting.

Inverse-probability-of-censoring weights compensate for subjects censored before the comparison can be made; the censoring distribution is typically estimated by Kaplan-Meier on the censoring times. The result is a time-dependent AUC at each requested \(t\) with bootstrap or analytical confidence intervals.

Assumptions

Non-informative censoring (or correctly conditional on covariates if weighting = "cox" or "aalen" is used) and a continuous risk marker that is meaningful as a ranking score.

R Implementation

library(timeROC); library(survival)

data(lung)
fit <- coxph(Surv(time, status) ~ age + sex + ph.ecog, data = lung)
lp <- predict(fit, type = "lp")

tROC <- timeROC(T = lung$time, delta = lung$status, marker = lp,
                times = c(180, 365, 730), cause = 1,
                weighting = "marginal")
tROC
plot(tROC, time = 365, col = "steelblue")

Output & Results

timeROC returns time-specific AUCs with their standard errors at each requested horizon, along with empirical sensitivity-specificity pairs that can be plotted as a ROC curve. The plot shows the ROC curve at one selected time; running over multiple times reveals how discrimination evolves through follow-up.

Interpretation

A reporting sentence: “The Cox linear predictor’s time-dependent AUC was 0.72 (95 % CI 0.65 to 0.79) at one year and 0.69 (95 % CI 0.62 to 0.76) at two years; discrimination remained stable across the two-year window relevant to clinical decision-making.” Always report AUCs at clinically meaningful horizons, not just the global C-index, because discrimination can change substantially across follow-up.

Practical Tips

Report AUCs at the horizons that drive clinical decisions; a five-year AUC is irrelevant for a six-month treatment decision and vice versa.
The cumulative/dynamic definition is more common in clinical reporting; incident/dynamic is more useful when modelling hazards directly.
timeROC() supports competing risks via the cause argument; the same workflow applies with the failure of interest specified.
AUC alone does not capture calibration; pair time-dependent AUCs with calibration plots and Brier scores at the same times.
For external validation, apply the development-cohort risk score to the validation cohort and compute time-dependent AUCs there; in-sample AUCs are optimistic.
Use weighting = "cox" or weighting = "aalen" when censoring depends on covariates; the marginal weighting assumes censoring is uninformative given baseline covariates.

R Packages Used

timeROC for timeROC() with marginal, Cox, and Aalen-based weighting; risksetROC for an alternative kernel-smoothed implementation; pec for compatible prediction-error curves; survAUC for additional AUC variants including Uno’s C-statistic.