Time-dependent ROC Analysis

• Time-dependent ROC analysis is a framework that evaluates biomarkers by tracking the evolving status of cases and controls over time.
• It employs cumulative/dynamic and incident/dynamic formulations to estimate full and partial AUC while accommodating censoring, delayed entry, and interval data.
• Advanced methods integrate flexible hazard models and machine learning techniques to optimize dynamic concordance and improve survival prediction.

Time-dependent receiver operating characteristic (ROC) analysis is a framework for evaluating the discriminative performance of a biomarker, marker, or risk score when the outcome is time to event rather than a static binary endpoint. Its central premise is that case/control status evolves with time $t$: a subject may be event-free at one horizon and become a case later. Accordingly, sensitivity, specificity, ROC curves, and area under the curve (AUC) are indexed by time, and their estimation must accommodate censoring, delayed entry, and, in some settings, time-dependent covariates or study-specific thresholds. Across the literature, two main formulations recur—the cumulative/dynamic definition and the incident/dynamic definition—and recent work has extended the framework to partial AUC, interval-censored data, covariate-specific ROC curves, left-truncated right-censored data, and meta-analytic summary ROC analysis (Hung et al., 2011, Wu et al., 2018, Sun et al., 2018, Rodríguez-Álvarez et al., 16 Jun 2025, Zhou et al., 2023, Li et al., 6 Sep 2025).

1. Conceptual basis and principal case/control formulations

In survival settings, ordinary ROC analysis is inadequate because it is designed for a binary outcome observed at the same time as the marker, whereas in time-to-event studies all subjects are event-free at baseline, event status evolves over time, and some subjects are censored before their eventual status is known (Rodríguez-Álvarez et al., 16 Jun 2025). Time-dependent ROC analysis therefore evaluates discrimination at a specified time $t$, with cases and controls defined relative to survival status by that horizon or instant.

Under the cumulative/dynamic formulation, used in several of the cited works, cases at time $t$ are subjects with $T \le t$, and controls are subjects with $T > t$ (Wu et al., 2018, Hung et al., 2011, Rodríguez-Álvarez et al., 16 Jun 2025, Li et al., 6 Sep 2025). For a threshold $m$ or $y$, one representation is

$$TP_t(m)=P(M>m\mid T\le t), \qquad FP_t(m)=P(M>m\mid T>t),$$

and the time-dependent ROC curve and AUC are

$$ROC_t(p)=TP_t\{FP_t^{-1}(p)\}, \qquad AUC_t=\int_0^1 ROC_t(p)\,dp.$$

Equivalent survivor-function formulations are also used: $$TPR_t(y)=P(Y>y\mid T\le t), \qquad FPR_t(y)=P(Y>y\mid T>t).$$ This definition evaluates how well the marker separates subjects who fail by $t$0 from those who survive beyond $t$1 (Hung et al., 2011).

Under the incident/dynamic formulation, the classification problem is posed on the survivor population at each time $t$2: a case is a subject with event at time $t$3, $t$4, and a control is a subject surviving beyond $t$5, $t$6 (Sun et al., 2018). For a scalar score $t$7,

$t$8

and

$t$9

This version is explicitly dynamic because the relevant covariate space is the survivor population at time $t$0, and in the tree-based survival framework the natural risk score is the hazard $t$1 (Sun et al., 2018).

These formulations are not interchangeable. The cumulative/dynamic definition compares failure by time $t$2 against survival beyond $t$3, whereas the incident/dynamic definition compares failure at time $t$4 against survival beyond $t$5. A plausible implication is that the choice depends on whether the scientific target is cumulative prognosis up to a horizon or instantaneous discrimination among subjects still at risk.

2. Core functionals: ROC, AUC, partial AUC, and dynamic concordance

The pointwise time-dependent AUC,

$t$6

is the standard scalar summary of discrimination at time $t$7 under cumulative/dynamic definitions (Wu et al., 2018, Rodríguez-Álvarez et al., 16 Jun 2025). In interval-censored settings, the same target is retained, but estimation proceeds through the joint distribution $t$8 of event time and marker and the marker marginal $t$9, because exact failure times are unavailable (Wu et al., 2018).

A major refinement is the time-dependent partial area under the ROC curve (pAUC), developed for right-censored survival data under cumulative/dynamic case/control definitions (Hung et al., 2011). For $T \le t$0, let

$T \le t$1

The target pAUC over the region $T \le t$2 is written as

$T \le t$3

with

$T \le t$4

If $T \le t$5, the time-dependent pAUC reduces to the full time-dependent AUC (Hung et al., 2011). The rescaled quantity $T \le t$6 has a probability interpretation: $T \le t$7 The same paper notes the benchmark values $T \le t$8 for a perfect biomarker and $T \le t$9 for a useless biomarker (Hung et al., 2011).

For discrete-valued prognostic scores, especially tree-based scores, ordinary time-dependent ROC curves may degenerate to finitely many points. To address this, a generalized ROC curve $T > t$0 can be defined by linear interpolation, interpreted as a randomized classification rule at ties (Sun et al., 2018). For

$T > t$1

the rule predicts $T > t$2 if $T > t$3, predicts $T > t$4 with probability

$T > t$5

if $T > t$6, and predicts $T > t$7 if $T > t$8. When $T > t$9 is continuous, $m$0 (Sun et al., 2018).

The same framework links AUC to concordance. The area under the generalized ROC curve is

$m$1

and can be written as

$m$2

An integrated functional,

$m$3

serves as a dynamic concordance index over time (Sun et al., 2018). This turns time-dependent ROC analysis from a purely evaluative device into an optimization criterion for learning algorithms.

3. Estimation under right censoring and marker-dependent censoring

For right-censored survival data with a baseline continuous marker, a nonparametric inferential framework for time-dependent pAUC under marker-dependent censoring assumes

$m$4

and estimates the joint survivor function

$m$5

using Akritas’ nearest-neighbor estimator (Hung et al., 2011). The estimator is

$m$6

where

$m$7

with

$m$8

Plugging $m$9 into $y$0 yields a closed-form estimator of time-dependent pAUC that avoids trapezoidal numerical integration (Hung et al., 2011).

This line of work also provides explicit asymptotic theory. Under regularity assumptions including

$y$1

the estimator admits a uniform asymptotic linear representation and converges weakly to a Gaussian process, with estimated variance-covariance

$y$2

Pointwise confidence intervals and simultaneous confidence bands are then constructed from this influence-function representation (Hung et al., 2011).

A distinct recent approach estimates covariate-specific cumulative/dynamic ROC curves and AUCs by combining a flexible hazard model for $y$3 with a flexible model for the biomarker distribution $y$4 (Rodríguez-Álvarez et al., 16 Jun 2025). The conditional survival is represented by

$y$5

with a flexible additive hazard specification

$y$6

where $y$7 includes smooth main effects and interactions such as $y$8, $y$9, and $$TP_t(m)=P(M>m\mid T\le t), \qquad FP_t(m)=P(M>m\mid T>t),$$0. The biomarker model is a location-scale model

$$TP_t(m)=P(M>m\mid T\le t), \qquad FP_t(m)=P(M>m\mid T>t),$$1

with

$$TP_t(m)=P(M>m\mid T\le t), \qquad FP_t(m)=P(M>m\mid T>t),$$2

Plug-in estimators of cumulative sensitivity and dynamic specificity are then obtained from finite sums over empirical residuals, with ROC computed by linear interpolation and AUC by the composite Simpson’s rule (Rodríguez-Álvarez et al., 16 Jun 2025).

This model-based approach uses the survival likelihood and piecewise exponential additive model formulation to handle censoring, assumes non-informative censoring, and relies on penalised splines with smoothing selection by REML; the reported software framework uses mgcv and pammtools, and the accompanying R package is CondTimeROC (Rodríguez-Álvarez et al., 16 Jun 2025). This suggests that conditional time-dependent ROC analysis is increasingly treated as a semiparametric regression problem rather than solely a nonparametric smoothing problem.

4. Extensions to interval censoring and left truncation

When the event time is interval censored, standard right-censored ROC methods cannot be applied directly because exact event times are unavailable, and naive imputation by the midpoint or right endpoint of the censoring interval is biased (Wu et al., 2018). For subject $$TP_t(m)=P(M>m\mid T\le t), \qquad FP_t(m)=P(M>m\mid T>t),$$3, the observed data are

$$TP_t(m)=P(M>m\mid T\le t), \qquad FP_t(m)=P(M>m\mid T>t),$$4

where $$TP_t(m)=P(M>m\mid T\le t), \qquad FP_t(m)=P(M>m\mid T>t),$$5, $$TP_t(m)=P(M>m\mid T\le t), \qquad FP_t(m)=P(M>m\mid T>t),$$6, and $$TP_t(m)=P(M>m\mid T\le t), \qquad FP_t(m)=P(M>m\mid T>t),$$7 encode left, interval, and right censoring, and the observation interval $$TP_t(m)=P(M>m\mid T\le t), \qquad FP_t(m)=P(M>m\mid T>t),$$8 is assumed independent of $$TP_t(m)=P(M>m\mid T\le t), \qquad FP_t(m)=P(M>m\mid T>t),$$9: $$ROC_t(p)=TP_t\{FP_t^{-1}(p)\}, \qquad AUC_t=\int_0^1 ROC_t(p)\,dp.$$0

A fully nonparametric solution estimates the joint distribution $$ROC_t(p)=TP_t\{FP_t^{-1}(p)\}, \qquad AUC_t=\int_0^1 ROC_t(p)\,dp.$$1 of $$ROC_t(p)=TP_t\{FP_t^{-1}(p)\}, \qquad AUC_t=\int_0^1 ROC_t(p)\,dp.$$2 and the marginal $$ROC_t(p)=TP_t\{FP_t^{-1}(p)\}, \qquad AUC_t=\int_0^1 ROC_t(p)\,dp.$$3 using a spline-based sieve maximum likelihood estimator (Wu et al., 2018). With B-spline basis functions,

$$ROC_t(p)=TP_t\{FP_t^{-1}(p)\}, \qquad AUC_t=\int_0^1 ROC_t(p)\,dp.$$4

subject to monotonicity and compatibility constraints. For computation, the method is reparameterized using I-splines and M-splines: $$ROC_t(p)=TP_t\{FP_t^{-1}(p)\}, \qquad AUC_t=\int_0^1 ROC_t(p)\,dp.$$5

$$ROC_t(p)=TP_t\{FP_t^{-1}(p)\}, \qquad AUC_t=\int_0^1 ROC_t(p)\,dp.$$6

with constraints

$$ROC_t(p)=TP_t\{FP_t^{-1}(p)\}, \qquad AUC_t=\int_0^1 ROC_t(p)\,dp.$$7

Optimization is performed using the generalized gradient projection algorithm (Wu et al., 2018). Plug-in estimators $$ROC_t(p)=TP_t\{FP_t^{-1}(p)\}, \qquad AUC_t=\int_0^1 ROC_t(p)\,dp.$$8 and $$ROC_t(p)=TP_t\{FP_t^{-1}(p)\}, \qquad AUC_t=\int_0^1 ROC_t(p)\,dp.$$9 follow, and under assumptions C1–C4 the paper proves

$$TPR_t(y)=P(Y>y\mid T\le t), \qquad FPR_t(y)=P(Y>y\mid T>t).$$0

Bootstrap confidence intervals using the BCa method are recommended for inference, and the associated CRAN package is intcensROC (Wu et al., 2018).

A different extension concerns left-truncated and right-censored (LTRC) data, where subjects are observed only if

$$TPR_t(y)=P(Y>y\mid T\le t), \qquad FPR_t(y)=P(Y>y\mid T>t).$$1

In this setting, ignoring left truncation can cause serious bias because the observed sample is conditioned on surviving long enough to enter the study (Li et al., 6 Sep 2025). Under the cumulative/dynamic definition,

$$TPR_t(y)=P(Y>y\mid T\le t), \qquad FPR_t(y)=P(Y>y\mid T>t).$$2

and

$$TPR_t(y)=P(Y>y\mid T\le t), \qquad FPR_t(y)=P(Y>y\mid T>t).$$3

For independent truncation/censoring, a nonparametric regression estimator uses the left-truncation-adjusted risk set

$$TPR_t(y)=P(Y>y\mid T\le t), \qquad FPR_t(y)=P(Y>y\mid T>t).$$4

and

$$TPR_t(y)=P(Y>y\mid T\le t), \qquad FPR_t(y)=P(Y>y\mid T>t).$$5

New inverse probability weighting estimators instead weight subjects by the inverse probability of being both untruncated and sufficiently uncensored to contribute as a case or control. For example,

$$TPR_t(y)=P(Y>y\mid T\le t), \qquad FPR_t(y)=P(Y>y\mid T>t).$$6

with $$TPR_t(y)=P(Y>y\mid T\le t), \qquad FPR_t(y)=P(Y>y\mid T>t).$$7, and covariate-adjusted conditional IPW versions replace $$TPR_t(y)=P(Y>y\mid T\le t), \qquad FPR_t(y)=P(Y>y\mid T>t).$$8 and $$TPR_t(y)=P(Y>y\mid T\le t), \qquad FPR_t(y)=P(Y>y\mid T>t).$$9 by $t$00 and $t$01 (Li et al., 6 Sep 2025). The paper states that right-censored-only estimators ignoring left truncation were often severely biased, whereas conditional IPW estimators performed well when truncation dependence was explained by measured covariates and the nuisance models were correctly specified (Li et al., 6 Sep 2025).

5. Dynamic risk, time-dependent covariates, and learning algorithms

Time-dependent ROC analysis is not limited to baseline markers. In ROC-guided survival trees and ensembles, the relevant prediction target is explicitly dynamic: $t$02 and the survivor population at time $t$03 is

$t$04

A time-invariant partition $t$05 induces the tree-based hazard model

$t$06

so the same subject may move across nodes over time as $t$07 changes, while the partition remains a fixed decision rule on the current covariates (Sun et al., 2018).

A key decision-theoretic result is that among all scalar functions $t$08,

$t$09

yields the highest $t$10 (Sun et al., 2018). This establishes the hazard as the ROC-optimal time-$t$11 discriminator between incident failures at $t$12 and survivors beyond $t$13. The tree-growing algorithm is then guided by a local increment in dynamic concordance: $t$14 Pruning is based on

$t$15

which is the ROC-guided analogue of CART cost-complexity pruning (Sun et al., 2018).

Under right censoring, node-specific hazards are estimated from counting-process quantities and kernel smoothing. For a node $t$16,

$t$17

with

$t$18

The resulting framework accommodates time-dependent covariates natively, without the “pseudo-subject” device used in prior work, and yields dynamic survival prediction through the subject’s covariate history $t$19 (Sun et al., 2018).

The same paper proposes an ensemble estimator that averages martingale estimating equations rather than predicted survival curves or cumulative hazard functions. With bootstrap-grown trees $t$20, adaptive nearest-neighbor weights

$t$21

lead to

$t$22

This places time-dependent ROC analysis at the center of model construction, not merely post hoc evaluation (Sun et al., 2018).

6. Meta-analysis, applications, limitations, and interpretation

Time-dependent ROC methods also appear in meta-analysis of prognosis studies, where study-specific cutoffs induce heterogeneity in reported sensitivities, specificities, and hazard ratios (Zhou et al., 2023). In time-dependent summary ROC (SROC) analysis, the cumulative/dynamic quantities at study-specific cutoff $t$23 are

$t$24

$t$25

A recent extension introduces a trivariate normal hierarchical model for

$t$26

then models selective publication through

$t$27

Because the expected publication proportion $t$28 is not identifiable from observed studies alone, the method is explicitly a sensitivity analysis rather than a full correction (Zhou et al., 2023).

Applications across the cited papers illustrate distinct use cases. Interval-censored ROC/AUC estimation was applied to CALGB 30801, where progression-free survival was interval censored and the markers COX-2 and pgem1 had estimated AUCs at $t$29 weeks of $t$30 and $t$31, respectively; neither was concluded to be very helpful for predicting event time (Wu et al., 2018). Time-dependent pAUC methods were applied to ACTG 175, where CD4 cell counts showed discriminatory value in broader low-FPR regions for $t$32 and $t$33, but for $t$34 the simultaneous bands suggested CD4 was essentially useless as a classifier over the considered period in both therapy groups (Hung et al., 2011). Covariate-specific ROC analysis was applied to 3488 acute coronary syndrome survivors, showing that GRACE discrimination varies with left ventricular ejection fraction and that marginal AUCs from Uno et al. (2007) were generally larger than the LVEF-specific AUCs across much of the LVEF range (Rodríguez-Álvarez et al., 16 Jun 2025). ROC-guided survival trees were illustrated on an AIDS study, where the final pruned tree used current Karnofsky score and cumulative opportunistic infections and yielded a clinically interpretable dynamic discrimination rule (Sun et al., 2018). LTRC ROC analysis was applied to the St. Jude Lifetime Cohort Study, where most AUC estimators gave similar results except that the regression-type semiparametric estimator produced smaller AUC estimates, consistent with its negative bias in simulation (Li et al., 6 Sep 2025). Time-dependent SROC sensitivity analysis was applied to Ki67 in breast cancer, where publication-bias adjustment decreased $t$35 but the substantive conclusion of statistically significant, though not especially high, prognostic discrimination remained fairly robust (Zhou et al., 2023).

Several recurrent limitations appear across the literature. Interval-censored ROC estimation establishes consistency but not asymptotic normality or analytic standard errors, so inference relies on bootstrap BCa intervals (Wu et al., 2018). Time-dependent pAUC estimation is sensitive when $t$36 is very small and when censoring is heavy, requiring adequate sample sizes and careful bandwidth selection (Hung et al., 2011). Conditional ROC estimation via penalised splines does not derive asymptotic theory in the provided text and uses bootstrap percentile intervals that were often conservative in simulation (Rodríguez-Álvarez et al., 16 Jun 2025). In LTRC settings, neither the regression-based nor the conditional IPW estimators are claimed to be doubly robust, and model misspecification in $t$37, $t$38, $t$39, or $t$40 can induce bias (Li et al., 6 Sep 2025). In meta-analysis, publication-bias adjustment depends on asymptotic approximations, estimated within-study covariance matrices, and user-specified values of $t$41, and numerical instability may occur (Zhou et al., 2023).

Taken together, these developments show that time-dependent ROC analysis has evolved from a direct survival analogue of ordinary ROC curves into a broad methodological domain. It now includes cumulative/dynamic and incident/dynamic discrimination targets, full and partial AUC summaries, estimators for right-censored, interval-censored, and left-truncated data, covariate-specific and time-dependent-covariate formulations, algorithmic learning criteria for trees and ensembles, and SROC methodology for evidence synthesis (Sun et al., 2018, Wu et al., 2018, Hung et al., 2011, Rodríguez-Álvarez et al., 16 Jun 2025, Zhou et al., 2023, Li et al., 6 Sep 2025). A plausible implication is that, in contemporary practice, the principal methodological question is no longer whether a time-dependent ROC curve can be defined, but which definition, estimand, and observation-model assumptions best match the survival prediction problem at hand.