Why the Evaluation Time Matters
When a covariate is incomplete, its imputation model should use observed information related to that covariate, including relevant survival information. One common summary is the Nelson–Aalen cumulative hazard evaluated at each person’s observed follow-up time. Another is a set of summaries at selected, fixed follow-up landmarks. These choices are not interchangeable: one retains a person-specific measure of accumulated hazard, while the other can represent survival history in a coarser, explicitly timed form.
Write \(T_i\) for person \(i\)’s observed follow-up time, the minimum of the event time and censoring time, and \(\Delta_i\) for event status, with \(\Delta_i=1\) for an observed event and \(\Delta_i=0\) for censoring. The cumulative hazard estimate \(\widehat H(t)\) is common to the sample; evaluating it at different \(T_i\) gives person-specific values. A censored person’s value describes accumulated estimated hazard only through their censoring time. It does not imply that the person’s unobserved event time is known.
Why Pair Cumulative Hazard With Event Status?
The rationale can be seen under a proportional-hazards model for a covariate \(X_i\):
Suppose event times are observed exactly, censoring is right-censoring and independent of the event time conditional on the modeled covariates, and the proportional-hazards model is appropriate. Ignoring factors involving only the censoring mechanism, person \(i\)’s event-time likelihood contribution is
The event indicator and the cumulative baseline hazard through observed follow-up therefore both carry information about \(X_i\). The event indicator distinguishes an observed event from censoring; the cumulative hazard reflects how much follow-up occurred in relation to the estimated event process. Replacing \(H_0(T_i)\) with an estimate such as \(\widehat H(T_i)\), and including both that summary and \(\Delta_i\) as predictors, is a practical way to convey this information to a covariate imputation model.
This is a justification for a useful summary, not a claim that the imputation model is exactly compatible with every survival model. The argument relies on the stated proportional-hazards structure, and independent censoring is needed for the survival information to be represented by the observed records in this way. With different hazard structures, informative censoring, or important nonlinearities, the two-predictor summary may be inadequate.
Worked Example: Calculating Cumulative Hazard at Observed Follow-Up
Consider five hypothetical people with follow-up in years. There is one event at year 2, one at year 4, and one at year 6; the other two observations are censored. The follow-up records are A: 2, event; B: 3, censored; C: 4, event; D: 5, censored; E: 6, event.
| Event time | Number at risk just before time | Nelson–Aalen increment |
|---|---|---|
| 2 | 5 | 1/5 = 0.2000 |
| 4 | 3 | 1/3 = 0.3333 |
| 6 | 1 | 1/1 = 1.000 |
At an event time \(t_j\), the Nelson–Aalen increment is \(d_j/Y_j\), where \(d_j\) is the number of events and \(Y_j\) is the number at risk just before that time. Thus,
For example, at year 4 the value is
The observed-follow-up values are therefore A: \(\widehat H(2)=0.2000\); B: \(\widehat H(3)=0.2000\); C: \(\widehat H(4)=0.5333\); D: \(\widehat H(5)=0.5333\); and E: \(\widehat H(6)=1.533\). In a covariate imputation model, the corresponding survival summaries are \((1,0.2000)\), \((0,0.2000)\), \((1,0.5333)\), \((0,0.5333)\), and \((1,1.533)\), in the order \((\Delta_i,\widehat H(T_i))\). A and B share a hazard value but differ in event status; C and D do also. This illustrates why the cumulative hazard should not replace the event indicator.
Fixed-Time Summaries: What They Can and Cannot Say
A fixed landmark \(t_k\) lets the analyst describe survival history relative to the same time for everyone. A safe encoding distinguishes three observed states: an event by the landmark, known event-free follow-up through the landmark, and censoring before the landmark. Define
Here \(I(\cdot)\) is 1 when its condition is true and 0 otherwise. These indicators are mutually exclusive and cover the possible observed records at that landmark. An event at \(t_k\) is classified as an event by the landmark. A person censored before \(t_k\) is not classified as event-free through \(t_k\): their event status after censoring is unknown. With several landmarks, the resulting sequence can preserve broad timing patterns, while necessarily coarsening exact event and censoring times.
A subtle but important distinction is that \(\widehat H(t_k)\), evaluated at a fixed time, is the same estimated curve value for everyone. By itself, it is not a person-varying predictor in a single imputation model. Fixed-time survival information comes from each person’s observed status relative to the landmark, or from another person-specific construction. Including the same constant \(\widehat H(t_k)\) for everyone adds no information about differences among people.
Worked Example: Encoding Two Fixed Landmarks
Use the five hypothetical records from the previous example and choose landmarks at 3 and 5 years. At 3 years, A has had an event by the landmark. B is censored at year 3, so follow-up reaches the landmark and B is classified as known event-free through it. C, D, and E are also known event-free through year 3. At 5 years, A and C have had events by the landmark; B was censored before it; D is censored at year 5; and E is known event-free through year 5.
The indicator triples \((E_{ik},R_{ik},C_{ik})\) are:
| Person | Landmark 3: (event, event-free, early censoring) | Landmark 5: (event, event-free, early censoring) |
|---|---|---|
| A | (1, 0, 0) | (1, 0, 0) |
| B | (0, 1, 0) | (0, 0, 1) |
| C | (0, 1, 0) | (1, 0, 0) |
| D | (0, 1, 0) | (0, 1, 0) |
| E | (0, 1, 0) | (0, 1, 0) |
At year 5, the common curve value is \(\widehat H(5)=1/5+1/3=0.5333\). Giving that number alone to all five people cannot distinguish A from B, for example. The landmark indicators do distinguish their observed histories: A had an event by year 5, whereas B was censored before year 5. The price is coarsening: A’s event at year 2 and C’s event at year 4 both fall into the same “event by year 5” category.
Choosing Between the Two Encodings
The observed-follow-up pair is a compact default when follow-up time and event status are available and the survival process is reasonably summarized by a cumulative hazard. It uses the individual’s actual observed duration, rather than requiring the analyst to select a grid. Fixed landmarks can be useful when the scientific question is explicitly about particular follow-up horizons, when a smooth or proportional-hazards summary seems unsuitable, or when a coarse description is easier to include in the chosen imputation model. Neither representation is universally superior.
| Choice | Information represented | Main caution |
|---|---|---|
| \((\Delta_i,\widehat H(T_i))\) | Event status and estimated cumulative hazard through each person’s observed follow-up | Can inadequately summarize survival under a misspecified hazard structure or informative censoring |
| Indicators at fixed landmarks | Whether an event, sufficient event-free follow-up, or early censoring is observed by selected times | Depends on landmark selection and discards timing detail between landmarks |
| \(\widehat H(t_k)\) alone at a fixed time | A value of the sample-level cumulative hazard curve | Constant across people, so it cannot distinguish their survival histories |
Landmarks should be chosen before inspecting imputed values, guided by meaningful follow-up horizons and the amount of data available around those times. At late landmarks, many people may be censored earlier; their later status is unknown, not “no event.” If a large share of records falls into that early-censoring category, the landmark summary may contribute little information at that horizon. Conversely, a grid with too many landmarks can produce sparse categories and unstable imputation models.
The choice also depends on the form of the incomplete covariate and the imputation model. A continuous covariate might be imputed with a regression model; a binary or categorical covariate requires an appropriate categorical model. Including survival summaries does not repair a misspecified model for the covariate itself, nor does it establish that missingness is ignorable. The earlier material on Overlaps Missing Data addresses those broader missing-data conditions; here the focus is how to encode the observed survival outcome as predictors.
Diagnostics and Common Pitfalls
Check that follow-up times are on a common, correctly defined time scale; that the event indicator matches the event definition; and that \(\widehat H(T_i)\) is evaluated only through observed follow-up. Inspect the distribution of the cumulative-hazard values and the counts in each landmark category. Ties in \(\widehat H(T_i)\) are expected when no event occurs between two follow-up times. They are not, by themselves, evidence of an error.
A common mistake is to use only \(\widehat H(T_i)\) and omit \(\Delta_i\), reasoning that the cumulative hazard already contains the event information. It does not: in the first worked example, an event and a censoring record can have the same value. The correction is to include event status as well as the cumulative-hazard summary when using the observed-follow-up approach. Another error is to label someone censored before a landmark as event-free through it. Their status after censoring is unobserved; retain an explicit early-censoring category or use another defensible treatment rather than silently assigning a known event-free state.
Finally, do not assume that a survival summary guarantees a correct imputation model. The observed-follow-up pair is motivated by a proportional-hazards likelihood, but may be insufficient when that structure is a poor description of the data. Landmark indicators avoid imposing that same single cumulative-hazard summary, but their results depend on chosen times and lose detail. These are design choices to justify and assess, not automatic corrections.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In the hypothetical example, calculate \(\widehat H(5)\) and state which event-time increments contribute.
- A person is censored at year 4, and the selected landmark is year 6. Should this person be coded as event-free through year 6? Why?
- Why is \(\widehat H(6)\) alone not a useful person-varying predictor when evaluated at the same fixed time for every person?
- Give one advantage and one limitation of using several fixed landmarks instead of \((\Delta_i,\widehat H(T_i))\).
Solutions. (1) Events at years 2 and 4 contribute, so \(\widehat H(5)=1/5+1/3=0.5333\); the year-6 event is after year 5 and does not contribute. (2) No. Censoring at year 4 occurs before year 6, so event status between years 4 and 6 is unknown; the person belongs in an early-censoring category. (3) The estimated curve value at year 6 is the same number for every person, so it does not distinguish individual survival histories. (4) Several landmarks can represent survival status at scientifically meaningful horizons without reducing everything to one cumulative-hazard value. However, they coarsen timing and can create sparse categories, especially at late landmarks.