Why Include a Cumulative-Hazard Summary?
When a covariate is incomplete, its imputation model should use information associated with that covariate. In survival data, the observed follow-up time and event history may contain such information. A model that ignores survival outcomes can weaken the relationship between an imputed covariate and the outcome, even when the analysis model will later relate that covariate to survival.
A useful way to represent some of this outcome information is the Nelson–Aalen estimate of cumulative hazard, evaluated at each person’s observed follow-up time. It gives a single numerical summary that can be included as a predictor in a chained-equation model for an incomplete covariate. The summary is not itself an imputed value, a probability of failure, or a replacement for the survival outcome.
This tutorial focuses on the role of that cumulative-hazard predictor. The estimate is built from the observed follow-up and event data; when event times are tied, the increment uses the number of events divided by the number at risk at that time.
Why the Summary Can Help in Imputation
Consider an incomplete covariate \(X\) that is associated with survival. If the imputation model for \(X\) uses only other covariates and omits survival information, its predictions may not reflect that association. Including \(\widehat{H}_{NA}(T_i)\) adds a summary of how much cumulative hazard a person experienced during their observed follow-up. People with different follow-up times can therefore have different values of this predictor, even if they share the same values of the other recorded covariates.
A survival-model likelihood helps explain the rationale. Under a proportional-hazards model with hazard \(\lambda(t\mid X)=\lambda_0(t)\exp(\beta X)\), an individual’s event-time contribution includes a term involving the cumulative baseline hazard \(H_0(T_i)\exp(\beta X)\). Thus survival information about \(X\) depends on cumulative hazard evaluated at follow-up time, as well as on whether an event occurred. The Nelson–Aalen summary is a data-based proxy for the cumulative-hazard part of this information.
Here \(\delta_i\) indicates whether an event was observed, and terms not involving \(X_i\) have been omitted. This expression is not generally a linear function of \(X_i\), and replacing \(H_0(T_i)\) with \(\widehat{H}_{NA}(T_i)\) does not make the imputation model exact. Instead, it motivates adding a useful survival-outcome summary to a practical conditional model. The event indicator is also outcome information; the particular role of combining it with cumulative hazard is not developed here.
The approach is most defensible when the survival summary is reasonably informative about the outcome–covariate relationship, the imputation model has an appropriate form for \(X\), and the survival process is modeled adequately for the purpose at hand. For example, proportional hazards may motivate the likelihood expression above, but the pooled Nelson–Aalen estimate is not automatically the true baseline cumulative hazard when covariate effects alter the marginal hazard.
Calculating the Predictor
The cumulative-hazard summary is constructed once from the observed survival records and assigned to each person at their observed follow-up time. The risk set \(Y_j\) consists of people still under observation and event-free immediately before event time \(t_j\). Censoring reduces later risk sets but does not add an event increment. At a tied event time, \(d_j/Y_j\) is added once for that time.
Worked Example: Evaluating the Summary at Each Follow-Up Time
Suppose six hypothetical participants have the following observed records: A has an event at time 2, B is censored at time 3, C and D have events at time 4, E is censored at time 5, and F has an event at time 6. Assume times are measured in months. The event-time risk sets and Nelson–Aalen increments are as follows.
| Event time \(t_j\) | At risk \(Y_j\) | Events \(d_j\) | Increment \(d_j/Y_j\) | Cumulative estimate |
|---|---|---|---|---|
| 2 | 6 | 1 | 1/6 = 0.1667 | 0.1667 |
| 4 | 4 | 2 | 2/4 = 0.5000 | 0.6667 |
| 6 | 1 | 1 | 1/1 = 1.000 | 1.667 |
At time 2, all six participants are at risk. Participant B is censored at time 3, so four people remain at risk just before time 4: C, D, E, and F. The tied events at time 4 contribute \(2/4=0.5000\). E is censored at time 5, leaving only F at risk just before time 6.
The evaluated summaries are therefore A: 0.1667; B: 0.1667; C and D: 0.6667; E: 0.6667; and F: 1.667. For B, the estimate at time 3 remains at its most recent event-time value because no event occurred between times 2 and 3. These values are cumulative-hazard summaries, not probabilities of experiencing the event.
Entering the Summary in Chained-Equation Models
For an incomplete covariate, include \(\widehat{H}_{NA}(T_i)\) alongside the other appropriate predictors in its conditional imputation model. The form of that model should match the type of covariate. For a binary incomplete covariate, a logistic model is one option; for a continuous covariate, a linear regression model may be appropriate if its assumptions are reasonable. The coefficients and residual or parameter uncertainty must be reflected in the imputation procedure so that imputations are stochastic rather than simply equal to fitted values.
The Nelson–Aalen predictor is calculated from observed survival records, not recalculated from the covariate value being imputed. It may be used in each chained-equation update for the incomplete covariate. Including it does not guarantee compatibility between every conditional model in the chain, nor does it correct misspecified functional forms or omitted predictors.
Worked Example: A Binary Covariate Model
Suppose \(X\) is a hypothetical binary covariate coded 1 for “present” and 0 for “absent,” and \(Z\) is a fully observed predictor. A fitted logistic imputation model is
For a participant with \(Z_i=1\) and \(\widehat{H}_{NA}(T_i)=0.6667\), the fitted log-odds are
The corresponding fitted probability is \(e^{0.1667}/\{1+e^{0.1667}\}\approx 0.5416\). In a stochastic imputation, an imputed value is drawn as 1 with this probability and 0 otherwise, subject also to the imputation procedure’s treatment of parameter uncertainty. The fitted probability is not a declaration that the participant has the covariate.
Worked Example: A Continuous Covariate Model
Suppose \(X\) is a continuous incomplete covariate measured in hypothetical units. A fitted conditional mean model is \(E(X_i\mid Z_i,\widehat{H}_{NA}(T_i))=4.00+1.50Z_i-2.00\widehat{H}_{NA}(T_i)\), with residual standard deviation \(0.800\) units. For a participant with \(Z_i=3\) and \(\widehat{H}_{NA}(T_i)=1.667\), the fitted mean is
This is the conditional fitted mean, not an imputed observation. If, for illustration, a residual draw were \(+0.300\) units, the resulting draw using fixed coefficients would be \(5.166+0.300=5.466\) units. A complete multiple-imputation procedure also accounts for uncertainty in the fitted model parameters. The linear model should be reconsidered if the covariate distribution is strongly skewed, constrained, or otherwise poorly represented by a normal-error regression.
Justification and Conditions
The motivation can be made more explicit by considering a modest covariate effect. Write \(u=\beta X_i\). For small \(u\), a first-order Taylor expansion gives \(\exp(u)\approx 1+u\). In the survival likelihood contribution, the term involving cumulative baseline hazard then satisfies
The first term does not depend on \(X_i\); the second involves \(X_i\) multiplied by cumulative hazard at the person’s follow-up time. This demonstrates why that cumulative-hazard quantity can carry information about the covariate. The approximation is local: higher-order terms matter when \(\beta X_i\) is not small, and the full survival likelihood also contains event information. Consequently, a linear predictor involving \(\widehat{H}_{NA}(T_i)\) is a practical approximation, not a derivation of an exact universally valid imputation model.
Important conditions include correctly recorded event and censoring times, risk sets that match the study’s follow-up definition, and a censoring process that is non-informative for the survival analysis conditional on relevant information. If censoring depends on unobserved risk in ways not represented in the imputation model, the observed follow-up summary may fail to represent the intended outcome information. The conditional model for the incomplete covariate must also include relevant observed predictors and use suitable functional forms.
Diagnostics, Alternatives, and Limits
Check that the calculated summary is nondecreasing over event times, is constant between event times, and has plausible values for the observed follow-up distribution. Verify risk-set counts around tied events and censoring times. In the imputation model, examine whether the cumulative-hazard term has a sensible relationship with the incomplete covariate among records where that covariate is observed. Compare observed and imputed covariate distributions across ranges of the summary, and inspect convergence and mixing across chains. Poor overlap, extreme predictions, or systematic differences between observed and imputed values can indicate that the chosen conditional model is inadequate.
The main alternative is to omit survival information from the covariate imputation model. That is simpler, but may attenuate associations needed in the substantive survival analysis. A more fully specified joint model can represent survival and covariates together, but it requires stronger modeling choices and may be more complex to fit. Adding a Nelson–Aalen summary is a pragmatic middle course: it incorporates a useful outcome feature without claiming to model every aspect of the joint distribution.
There are important limitations. A pooled Nelson–Aalen estimate is not necessarily the covariate-specific baseline cumulative hazard under a model with heterogeneous risks. A single summary can conceal departures from proportional hazards, nonlinear covariate effects, or complex dependence. It cannot repair an incorrect imputation model, unmeasured predictors of missingness, or a fundamentally unsuitable censoring assumption. Moreover, because the summary is estimated, it is a constructed predictor rather than known without error; standard implementations commonly use the calculated summary as fixed, and that simplification should be considered when uncertainty in the survival summary could matter.
- Common mistake: Treating \(\widehat{H}_{NA}(T_i)\) as the probability that person \(i\) experienced the event. Correction: It is an estimate of cumulative hazard, which may exceed 1 and is not itself a probability.
- Common mistake: Assuming that adding the summary makes any covariate imputation model valid. Correction: The conditional model still needs an appropriate outcome, predictors, functional form, and stochastic uncertainty.
- Common mistake: Adding a tied-event increment once for each event separately while keeping the original risk-set denominator. Correction: At a tied time, use the total \(d_j\) and the risk set \(Y_j\) just before that time, giving one increment \(d_j/Y_j\).
Check Your Understanding
Answer each question, then compare with the solutions below.
- At an event time, 3 of 12 people at risk experience the event. What is the Nelson–Aalen increment at that time?
- Why can a person censored between two event times have the same evaluated summary as at the earlier event time?
- A logistic imputation model gives a fitted probability of 0.70 for a missing binary covariate. Is the imputed value necessarily 1? Explain.
- Name one reason the pooled Nelson–Aalen summary may be only an approximation to the relevant cumulative-hazard information.
Solutions. (1) The increment is \(d_j/Y_j=3/12=0.2500\). (2) The estimate changes only at event times, so censoring between events changes later risk sets but adds no increment at the censoring time. (3) No. The value should be drawn stochastically, with probability 0.70 of being 1 under that fitted probability, while the full procedure should also represent model-parameter uncertainty. (4) For example, when covariate effects make individual hazards differ, a pooled estimate need not equal a covariate-specific baseline cumulative hazard; departures from proportional hazards are another concern.