Why Survival Outcomes Belong in the Imputation Model
Suppose a baseline covariate is partly missing, and the planned analysis relates that covariate to time until an event using a Cox model. Imputing the covariate from other baseline variables alone can weaken or distort its association with survival. The observed follow-up time and event status contain information about the missing covariate whenever that covariate is related to the event process.
A practical chained-equations strategy is to include two summaries of the survival outcome in the conditional imputation model: the event indicator and a Nelson–Aalen cumulative-hazard summary evaluated at each participant’s observed follow-up time. The summary depends on the observed event pattern and risk sets, so it carries information about how far follow-up extends and how many events have accumulated by then.
The summary \(\widehat H_{\mathrm{NA}}(T_i)\) is computed from the survival data, including the risk sets, and attached to each participant. It is not an imputed event time, nor is it generally the Cox model’s baseline cumulative hazard. In a Cox analysis with covariates, the pooled Nelson–Aalen curve is best understood here as an outcome-derived predictor that helps the imputation model represent survival information.
Assumptions and the Imputation Setup
Let \(X_i\) denote a baseline covariate with some missing values, and let \(Z_i\) denote other observed baseline predictors. A chained-equations update for \(X\) models its conditional distribution given \(Z_i\), \(\Delta_i\), and \(\widehat H_{\mathrm{NA}}(T_i)\), along with any other variables needed to make the imputation model credible. For binary \(X\), a common working model is logistic regression. For continuous \(X\), use a suitable continuous-variable imputation model; the survival predictors are added to that model as well.
This strategy needs careful interpretation. It is an approximation to the conditional distribution implied by the substantive survival model, not a guarantee of exact compatibility with every Cox model. Its use is most defensible when the Cox proportional-hazards model is a reasonable analysis model, the covariate being imputed is a baseline covariate, and the imputation model can represent its relationship with the observed outcome information.
- Right-censoring condition. Censoring should be independent of the event time conditional on relevant observed variables, or the imputation and analysis models must otherwise account for the censoring mechanism. Informative censoring can make the observed follow-up summaries misleading.
- Missingness condition. The missingness in \(X\) should be explainable by variables included in the imputation model, as required under a missing-at-random analysis. Including survival information can help, but does not establish this condition.
- Model condition. The conditional imputation model must be adequate for the type and distribution of \(X\). For a binary \(X\), logistic regression is one option; for a continuous \(X\), a model that respects its support and distribution is needed.
- Data condition. The observed data must contain enough information to estimate the imputation model, with adequate overlap across predictor patterns. A tiny teaching example can demonstrate construction but cannot support a reliable fitted model.
In a formal Cox model, the event-time contribution depends on the covariate through both the event hazard and survival up to the observed time. This explains why event status alone may be inadequate: two people can both be censored but have quite different follow-up durations. The cumulative-hazard summary adds information about the event history up to each person’s observed time.
Worked Construction of the Nelson–Aalen Summary
Worked Example: Risk Sets, Tied Events, and Participant Summaries
Consider six hypothetical participants. Their baseline covariate \(X\) is observed for some people and missing for others. Follow-up time is measured in years. The table gives only survival information and the missingness pattern.
| Participant | Observed time \(T_i\) (years) | \(\Delta_i\) | \(X_i\) |
|---|---|---|---|
| A | 2 | 1 | 0 |
| B | 3 | 0 | 1 |
| C | 4 | 1 | Missing |
| D | 4 | 1 | 0 |
| E | 5 | 0 | 1 |
| F | 6 | 1 | Missing |
At year 2, all six participants are at risk just before the event, so \(Y_2=6\) and \(d_2=1\). At year 4, A has had an event and B was censored at year 3, leaving C, D, E, and F at risk; thus \(Y_4=4\) and \(d_4=2\). At year 6, only F remains at risk, so \(Y_6=1\) and \(d_6=1\). Censorings do not create jumps in the Nelson–Aalen sum.
The successive values are \(1/6=0.1667\) after year 2, \(1/6+2/4=0.6667\) after year 4, and \(1/6+2/4+1=1.6667\) after year 6. Therefore A and B have \(\widehat H_{\mathrm{NA}}(T_i)=0.1667\); C, D, and E have 0.6667; and F has 1.6667. These are cumulative summaries at each person’s observed follow-up time, not individual hazard estimates.
For tied events at year 4, both events contribute to \(d_4\), and the denominator is the risk set immediately before year 4. Under the usual convention, a participant censored at an event time is counted in the risk set immediately before that time. The displayed dataset has no censoring tied with an event, so that convention does not change these calculations.
Putting the Survival Summaries Into a Chained Equation
For the binary covariate \(X_i\), one working conditional model is
where \(Z_i\) is another predictor of \(X_i\), and \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). In a MICE iteration, the model is fitted using records where \(X_i\) is observed, parameters are drawn to reflect estimation uncertainty, and missing values are drawn from their predictive distributions. The event indicator and Nelson–Aalen summary are predictors in the imputation step; they are not replacements for \(X\) in the later Cox analysis.
The reason these outcome summaries are relevant can be seen from the Cox model. Suppose the hazard conditional on \(X_i\) and \(Z_i\) is \(\lambda_0(t)\exp(\beta_X X_i+\beta_Z Z_i)\), with cumulative baseline hazard \(H_0(t)\). Ignoring terms that do not depend on \(X_i\), the observed survival contribution at time \(T_i\) has the form
The event indicator identifies whether the event-hazard factor is present, while the cumulative hazard through \(T_i\) governs the survival factor. Thus \(X_i\)'s conditional distribution given survival information depends on both event status and accumulated follow-up hazard, not only on a binary event flag.
In practice, \(\widehat H_{\mathrm{NA}}(T_i)\) is a plug-in summary of that accumulated event information. It need not equal \(H_0(T_i)\), particularly when covariate effects are present, and a linear logistic predictor in \(\Delta_i\) and \(\widehat H_{\mathrm{NA}}(T_i)\) need not reproduce the exact conditional distribution implied by the Cox model. It is a useful working specification whose adequacy should be assessed, rather than a universal theorem that makes any imputation model compatible.
Worked Example: Calculating Imputation Probabilities and Making Draws
Continue with hypothetical participant C, for whom \(T=4\), \(\Delta=1\), \(X\) is missing, and \(\widehat H_{\mathrm{NA}}(T)=0.6667\). Let \(Z=0\). To demonstrate the calculation, suppose a model fitted in a suitably larger hypothetical dataset has coefficients \(\gamma_0=-1.00\), \(\gamma_Z=0.50\), \(\gamma_\Delta=0.80\), and \(\gamma_H=-0.60\). These coefficients are illustrative; they are not estimated from the six-person table.
The linear predictor for C is
Therefore the conditional probability of \(X_C=1\) is
If the random draw used for this imputation is \(u_C=0.20\), then \(u_C<p_C\), so the draw sets \(X_C=1\). If instead \(u_C=0.70\), the draw sets \(X_C=0\). The fitted probability is not itself the imputed value; it defines the distribution from which the value is drawn.
For F, the table gives \(\Delta=1\) and \(\widehat H_{\mathrm{NA}}(T)=1.6667\). If \(Z=1\), then
With \(u_F=0.60\), F receives \(X_F=0\), since \(0.60\) exceeds 0.3318. A full MICE procedure also represents uncertainty in the model parameters; repeatedly using fixed coefficients and fixed probabilities alone would understate imputation uncertainty.
Why Follow-Up Time Adds Information
Worked Example: Same Censoring Status, Different Event History
Using the same illustrative coefficients, compare B and E. Both are censored, so \(\Delta=0\), and suppose both have \(Z=0\). B has \(\widehat H_{\mathrm{NA}}(T)=0.1667\), while E has 0.6667. Their probabilities are
The difference is produced by the follow-up summary, not by event status or \(Z\). In this illustrative model, the later cumulative-hazard summary corresponds to a lower predicted probability of \(X=1\). That direction comes from the assumed coefficient \(\gamma_H=-0.60\); it is not a general survival-data rule. In real data, the direction and magnitude must be estimated, and the model must be checked.
Diagnostics, Alternatives, and Limitations
A useful workflow is to construct the risk sets and summaries once from the observed survival data, add \(\Delta_i\) and \(\widehat H_{\mathrm{NA}}(T_i)\) to the conditional model for the missing baseline covariate, and generate multiple completed datasets. The same observed survival summaries are used for participants with observed and missing \(X\). After imputation, fit the planned Cox model separately to each completed dataset and combine the estimates using the usual multiple-imputation rules.
Check that event counts, risk-set sizes, and cumulative summaries are internally consistent. Inspect the distribution of imputed \(X\) against event status, follow-up summaries, and other important predictors; look for implausible probabilities, sparse combinations, or lack of overlap between observed and imputed values. Also check whether the imputation model represents important nonlinearities or interactions. A single linear term for cumulative hazard may be inadequate if the relationship is strongly nonlinear. The proportional-hazards assumption is a separate concern for the Cox analysis; adding the Nelson–Aalen summary does not repair a poorly specified substantive model.
| Approach | Strength | Main concern |
|---|---|---|
| Impute from baseline predictors only | Simple and sometimes adequate when survival adds little information about \(X\) | Can weaken the covariate–survival association when outcome information predicts \(X\) |
| Include event indicator and Nelson–Aalen summary | Uses event occurrence and accumulated event history in a convenient chained-equations model | Approximate; relies on a suitable conditional model and does not ensure exact compatibility |
| Include event indicator but omit follow-up summary | Accounts for whether an event was observed | Does not distinguish participants with different accumulated follow-up histories within the same event-status group |
| Use a specialized joint or substantive-model-compatible approach | Can more directly respect the survival model and the covariate–outcome relationship | Requires additional modeling and implementation choices |
A common mistake is to treat \(\widehat H_{\mathrm{NA}}(T_i)\) as the individual’s true cumulative hazard, or as the Cox baseline cumulative hazard, and then interpret its imputation coefficient causally. It is neither. It is a sample-based summary of event experience up to observed follow-up, used here as a predictor. Another mistake is to use event status alone and assume that censoring at one year conveys the same information as censoring at five years. The correction is to preserve both the event indicator and a follow-up/event-history summary when the imputation strategy calls for them, while recognizing that these predictors do not eliminate the need for model assessment.
The method also has limits. The pooled Nelson–Aalen summary does not automatically handle a misspecified censoring model, sparse risk sets, severe nonproportional hazards, or a complicated covariate relationship. It may be unstable near the end of follow-up, where few participants remain at risk. In very small datasets, the fitted conditional model may be poorly determined; the arithmetic can be shown, but adequate imputation requires enough observed cases and a model that avoids separation and overfitting.
Check Your Understanding
Answer each question, then compare with the solutions below.
- At a distinct event time, \(Y_j=20\) and \(d_j=3\). What is the jump in the Nelson–Aalen estimate?
- Why can two right-censored participants with the same \(\Delta_i=0\) require different survival predictors in an imputation model?
- In the illustrative logistic model, calculate the probability for a participant with \(\Delta=0\), \(\widehat H_{\mathrm{NA}}(T)=0.5000\), and \(Z=1\).
- Does including \(\Delta_i\) and \(\widehat H_{\mathrm{NA}}(T_i)\) prove that the missingness mechanism is MAR? Explain.
Solutions. (1) The jump is \(d_j/Y_j=3/20=0.1500\). (2) Their observed follow-up times and accumulated event histories can differ, so their Nelson–Aalen summaries can differ even though both are censored. (3) The linear predictor is \(-1.00+0.50(1)+0.80(0)-0.60(0.5000)=-0.8000\), giving \(p=\exp(-0.8000)/\{1+\exp(-0.8000)\}=0.3100\). (4) No. These variables make observed outcome information available to the imputation model, but MAR remains an assumption about whether missingness can be explained by observed data; it cannot be established merely by adding predictors.