Uno's Concordance Under Heavy Censoring

Why Weight Comparable Pairs?

A concordance index measures how often a prediction correctly orders two people: the person who experiences the event sooner should receive the higher predicted risk. With complete event-time data, this comparison is direct. With right censoring, however, a pair can be compared only when the earlier event is observed and the other person is known to remain event-free beyond that time. Heavy censoring can leave a selected set of comparable pairs that represents the population poorly.

Uno’s concordance estimator addresses this selection by weighting each observed pair according to the probability that censoring would allow the pair to be compared. The weights can recover a specified population concordance target under explicit assumptions. They can also become very large when that probability is small, so censoring adjustment does not guarantee precise estimates.

For a prespecified upper time \(\tau\), Uno’s target is the probability of correctly ordering a randomly selected pair from the target population, conditional on the first event in the pair occurring by \(\tau\). For continuous event times, one expression for this target is

The two people in the pair are independently sampled from the target population. If risk scores tie, a common convention gives that pair one-half credit; tied event times require a prespecified tie rule and are omitted from the continuous-time derivation below. The target describes discrimination for event pairs whose earlier event occurs by \(\tau\), not calibration, causation, or accuracy at a single prediction horizon.

Assumptions and the Inverse-Censoring Weight

In the basic version, the validation subjects are independent and identically distributed, censoring is independent of the event time and risk score, and all subjects share censoring survival \(G(t)=P(C_i\ge t)\). The censoring survival function is estimated from the validation data, commonly by the Kaplan–Meier estimator with censoring treated as the event of interest. The estimator also requires positivity: \(G(t)>0\) at event times included up to \(\tau\). In practice, the relevant estimated censoring survival must not be so close to zero that the weights become unsupported by data.

The squared weight arises because both people in a pair must be uncensored through the earlier event time. Suppose person \(i\) has event time \(t\), and person \(j\)'s event time is later than \(t\). For that pair to appear as comparable, censoring must not hide person \(i\)'s event before \(t\), and person \(j\) must remain observable past \(t\). Under the common independent-censoring assumption, the probability of both conditions is \(G(t)G(t)=G(t)^2\). Therefore an observed pair at time \(t\) receives weight \(1/G(t)^2\).

The key justification is that weighting reverses this observation probability. Conditional on the two event times and scores, for a pair with \(T_i=t<T_j\),

Here, continuity of event times avoids event-time ties, and independent censoring makes the probability of observing both sides of the pair equal to \(G(t)^2\). Multiplying by the indicator \(I(r_i>r_j)\) recovers the corresponding concordant-pair contribution as well. Summing these contributions over pairs and taking their ratio yields the estimator:

If a pair has tied risk scores, replace \(I(r_i>r_j)\) by \(I(r_i>r_j)+\tfrac12 I(r_i=r_j)\) when using the half-credit convention. The formula uses ordered indices: for any pair with a uniquely earlier event, the earlier-event person occupies the \(i\) position and contributes once. The ratio is undefined if its denominator is zero, meaning the data supply no weighted comparable pairs for the chosen cutoff.

How Censoring Patterns Affect the Estimate

The correction is needed because an observed comparable pair is not simply a random pair from the target population. Earlier events are more likely to be observed when censoring survival is high; later event times can be represented by fewer pairs. Weighting gives greater contribution to the observed pairs that had a lower probability of being observed. It does not create information about unobserved outcomes: if very few people remain observable at a time, large weights can make the estimate highly variable.

The common-\(G\) assumption is not suitable when censoring depends on measured characteristics. A more general approach models censoring conditional on those characteristics. If \(X_i\) and \(X_j\) are the observed covariates used for that model, the pair's observation probability at the earlier event time \(t\) is \(G(t\mid X_i)G(t\mid X_j)\), under conditional independent censoring and independent subjects. Its inverse replaces the common \(1/G(t)^2\) weight. The conditional model must be adequate, and positivity must hold within the covariate patterns represented in the target pairs.

Uno’s Concordance and Harrell’s Concordance

Harrell’s concordance index uses observed comparable pairs and counts the fraction that are concordant, without the inverse-censoring weights. Its empirical numerator and denominator have the same pair indicators as Uno’s estimator but omit \(1/\widehat G(Y_i)^2\). It is therefore a direct summary of the ordering among pairs that happen to be observable under the validation cohort's censoring pattern.

Under censoring, that observed-pair distribution can differ from the target distribution of all eligible event pairs. Uno’s estimator instead targets the truncated population concordance \(C_\tau\), provided its assumptions hold. The two estimates can differ because they weight pairs differently, not merely because of random arithmetic variation. Harrell’s estimate is not guaranteed to be biased in one direction: the direction of any difference depends on the event-time, score, and censoring patterns. With little censoring, the weights are close to one and the estimates may be similar; with heavy censoring, differences may become more noticeable.

Neither statistic repairs an evaluation design in which predictions were fitted and assessed on the same data without appropriate protection against overfitting. Both require a meaningful validation cohort and a clearly defined risk-score direction. When a model reports predicted survival rather than risk, larger predicted survival generally means lower risk; the score must be oriented consistently before interpreting concordance.

Diagnostics, Failure Modes, and Limits

Before interpreting Uno’s estimate, inspect the estimated censoring survival over the range of event times contributing to the statistic. Report the chosen \(\tau\), the number of observed events and comparable pairs contributing, and whether the largest weights are concentrated in only a few pairs. If \(\widehat G(t)\) is near zero, \(1/\widehat G(t)^2\) can dominate both sums. A finite estimate in that situation may conceal severe instability rather than reliable correction.

The cutoff \(\tau\) should be selected in advance with the target population and available censoring support in mind. Restricting the range can avoid unsupported late times, but it changes the target \(C_\tau\); it is not a purely computational adjustment. Results also depend on the censoring model. If censoring depends on predictors but a marginal model is used, the independence assumption may fail. If the censoring model is conditional, misspecification or poor support within covariate patterns can likewise compromise the correction.

Uno’s estimator is a ratio of weighted sums, not a guarantee of low uncertainty or a substitute for examining the data. Its target is a global pairwise ordering summary up to \(\tau\); it does not by itself say how discrimination changes across prediction horizons. A common mistake is to see inverse weights and conclude that heavy censoring has been fully neutralized. The correction is justified only where censoring probabilities can be estimated and are positive; extreme weights signal that the data contain little direct information for the target pairs.