Cumulative-Dynamic ROC Curves

Why a Survival ROC Curve Needs a Time Horizon

A survival prediction is not simply a classification: at the prediction landmark, some people will experience the event later, and others may remain event-free beyond the follow-up available. To assess whether predicted risks rank people usefully, first specify a horizon \(\tau\). The resulting ROC curve compares people who experience the event by \(\tau\) with those who remain event-free past \(\tau\). Because censoring can prevent either status from being observed, the curve generally needs censoring adjustment.

Consider a validation cohort that has reached a prediction landmark, such as a clinic visit at which each person’s joint-model prediction is made. Measure \(T\) as time from that landmark to the event of interest, and \(C\) as time from the same landmark to censoring. A person’s fixed-horizon risk score, denoted \(R\), is calculated using information available at the landmark; a larger score means greater predicted event risk by \(\tau\). For example, a joint model might supply a predicted probability of event by \(\tau\) conditional on the person’s observed longitudinal history and survival to the landmark. The prediction itself is held fixed during validation.

“Cumulative” refers to the cases: they include events occurring at any time from the landmark through \(\tau\), not just events close to \(\tau\). “Dynamic” refers to the controls: a person is a control only if they remain event-free beyond the chosen horizon. These definitions are tied to the specified landmark, target population, event, and horizon. Changing \(\tau\) can change both the case and control groups and therefore the ROC curve.

What Censoring Leaves Observable

Write \(Y=\min(T,C)\) for observed follow-up time and \(\Delta=I(T\leq C)\) for the event indicator. An observed event by \(\tau\), \(\Delta=1\) and \(Y\leq\tau\), identifies a cumulative case. A person with \(Y>\tau\) is an observed dynamic control: both \(T>\tau\) and \(C>\tau\). By contrast, censoring before \(\tau\) without an event leaves the person’s case/control status at \(\tau\) unknown. Treating every such person as a control, or excluding them without adjustment, can distort the curve.

Let \(W\) denote the variables used to model censoring, measured or constructed from information available for the prediction and censoring analysis. In particular, \(W\) must include the score \(R\), or the relevant longitudinal history and predictors that determine \(R\), if censoring may depend on them. Define the censoring survival function using the convention \(G(u\mid W)=P(C\geq u\mid W)\). For controls, also define the right-limit probability \(G(\tau+\mid W)=P(C>\tau\mid W)\).

The conditional independence assumption is important for score-based comparisons, not merely for estimating an overall censoring curve. Conditional on \(W\), censoring must not carry additional information about event time. If censoring is associated with the score or longitudinal history after conditioning on the variables in the censoring model, the weights need not recover the intended case and control distributions. Positivity means the relevant \(G(T\mid W)\) and \(G(\tau+\mid W)\) are not zero; in practice, very small values also make estimates unstable.

Constructing the Weighted ROC Curve

An observed case contributes the inverse probability \(1/G(Y\mid W)\). An observed control contributes \(1/G(\tau+\mid W)\). The two weights differ because the events defining observability differ. If censoring is continuous at \(\tau\), then \(P(C>\tau\mid W)=P(C\geq\tau\mid W)\), so the distinction at the boundary has no numerical effect. If censoring can have an atom at \(\tau\), the right-limit probability is required for controls under the definition \(Y>\tau\).

With an estimated censoring survival function \(\widehat G\), the empirical weighted sensitivity and false-positive rate at threshold \(c\) are ratios of weighted counts:

Here \(I(\cdot)\) is 1 when its condition holds and 0 otherwise. Each weighted numerator counts observed cases or controls above the score threshold; each denominator represents the corresponding weighted group total. Plotting \(\widehat{\mathrm{Se}}(\tau,c)\) vertically against \(\widehat{\mathrm{FPR}}(\tau,c)\) horizontally over thresholds produces a step curve. The threshold convention \(R\geq c\) handles ties consistently. Raising \(c\) generally makes fewer people test positive; the finite-sample curve is a sequence of points, not necessarily a smooth line.

Why the Censoring Weights Work

The justification is that inverse weighting corrects for the probability that an otherwise relevant person is observed. For a case, let \(f(T,R,W)\) be any bounded contribution determined by event time, score, and \(W\). Conditional independence given \(W\) implies that, conditional on \(T,R,W\), the probability the event is observed is \(G(T\mid W)\). Thus:

The conditioning includes \(R\), or \(W\) must determine \(R\), so the argument applies to contributions such as \(I(R\geq c)\). It follows that weighting observed events recovers the case contribution that would have been available without censoring. For controls, the observation indicator is \(I(C>\tau)\), not \(I(C\geq\tau)\). The corresponding calculation is:

These identities explain the weighted ratios: their numerators and denominators estimate the target case or control totals, respectively. The ratios are finite-sample estimates, so they are not exactly unbiased in every sample; their justification is consistency under the stated assumptions and suitable regularity conditions. The censoring probabilities are typically estimated using the chosen censoring model, not assumed known.

Diagnostics, Alternatives, and Limitations

Before interpreting the curve, check whether its assumptions and estimates are credible. Inspect the fitted censoring survival probabilities at observed event times and at \(\tau+\), overall and across relevant values of \(W\). Very small probabilities create large weights, so a few people can dominate a point on the curve. Examine weight distributions and effective contribution by score range; investigate whether the censoring model has enough information and whether important censoring predictors, including longitudinal history when relevant, have been omitted. If some target-group members have essentially no chance of remaining observed, positivity fails and the data cannot reliably identify the full-cohort curve through weighting alone.

A complete-case curve uses only people whose status at \(\tau\) is observed, but it can be biased when observation depends on score or event-related predictors. Treating everyone censored before \(\tau\) as a control is also incorrect: their event status by \(\tau\) is unknown. The inverse-weighted cumulative-dynamic curve instead estimates performance in the specified target cohort under conditional independent censoring and positivity. It does not repair a misspecified censoring model or unmeasured dependence between censoring and event time after conditioning on \(W\).

The case definition also matters. A cumulative-dynamic curve includes all events by the horizon; the incident-dynamic approach uses a different case definition, focused on events around a time point. They answer different discrimination questions and should not be treated as interchangeable. The horizon-specific curve is not a summary of calibration: it describes ranking at a threshold, not whether predicted risks agree with observed event probabilities. Its estimates can be imprecise when there are few observed cases or controls, and the censoring weights add uncertainty. Choosing a different horizon changes the target comparison, rather than merely changing a tuning parameter.