Deriving the Nelson-Aalen Cumulative Hazard

Why Add Event Counts Relative to the Risk Set?

The cumulative hazard describes the accumulation of instantaneous event risk over time. In survival data, however, not every person remains under observation and event-free throughout follow-up. The number who can experience an event therefore changes over time. The Nelson–Aalen estimator accounts for this by scaling the events at each observed event time by the number of individuals still at risk then.

This scaling is central to its use with right-censored observations. Someone censored before an event time is no longer included in the risk set at that time, but does not count as an event. An individual who experiences the event is included in the risk set immediately before that event time. These conventions are the basis for the estimator derived below.

Definition: Let $t_1<t_2<\cdots<t_J$ be the distinct observed event times. Let $Y_j$ be the number of individuals who are under observation and event-free immediately before $t_j$, and let $d_j$ be the number of events observed at $t_j$. The Nelson–Aalen estimator of the cumulative hazard at time $t$ is $$ \widehat H_{\mathrm{NA}}(t)=\sum_{j:t_j\leq t}\frac{d_j}{Y_j}. $$ It is a step function: it increases by $d_j/Y_j$ at an observed event time and does not change between observed event times.

The estimate is dimensionless: it estimates cumulative hazard, not a rate per unit time. Its increments are larger when more events occur at a time, and smaller when the same number of events occurs in a larger risk set. The risk set is determined just before the event time; individuals censored earlier are excluded, while those experiencing an event at that time are included in $Y_j$.

Deriving the Estimator

Write $\lambda(t)$ for the instantaneous hazard at time $t$, conditional on having remained event-free up to that time, and write $H(t)=\int_0^t\lambda(u)\,du$ for the cumulative hazard. The derivation relies on estimating a small increment in cumulative hazard from the events among those at risk during a short interval.

For a short interval of width $\Delta t$, under a continuous-time event model, the event probability for one individual who is at risk at the start of the interval is approximately $\lambda(t)\Delta t$. More precisely, it is $\lambda(t)\Delta t+o(\Delta t)$, where $o(\Delta t)$ is a remainder that becomes negligible relative to $\Delta t$ as the interval shrinks. If $Y(t)$ individuals are at risk, the expected number of events in that interval is approximately $Y(t)\lambda(t)\Delta t$. Dividing the observed event count by the risk-set size therefore estimates the hazard accumulated over that short interval.

E\{dN(t)\mid Y(t)\}\approx Y(t)\lambda(t)\Delta t \qquad\Longrightarrow\qquad \frac{dN(t)}{Y(t)}\text{ estimates }\lambda(t)\Delta t.

Here $dN(t)$ denotes the event count in the short interval. Adding such estimated increments across follow-up approximates the integral of the hazard. At observed event times, the resulting empirical increments are $d_j/Y_j$, giving the Nelson–Aalen sum. This is an estimator, not an identity between the observed event fractions and the unknown hazard increments.

\int_0^t\lambda(u)\,du=H(t) \qquad\text{and}\qquad \widehat H_{\mathrm{NA}}(t)=\sum_{j:t_j\leq t}\frac{d_j}{Y_j}.

The approximation in the derivation is most direct when event times are continuous, so that events occur singly with probability one under the model. When several events share an observed time, the standard estimator uses their count $d_j$ in the same formula. If observation times are genuinely discrete, however, an event fraction $d_j/Y_j$ need not equal the corresponding increment in cumulative hazard: for a discrete conditional event probability $q$, that increment is $-\log(1-q)$. Thus, ties can require careful interpretation. The definition above gives the conventional Nelson–Aalen estimate; the special calculations and conventions for tied events are a separate issue.

Assumptions and Scope

The formula can be calculated from event times, censoring times, and risk sets, but its interpretation as an estimate of a population cumulative hazard requires assumptions about how those data arose. For the basic right-censored setting considered here, assume that individuals are independent, their event times follow a common event-time distribution, and censoring is independent of the event time. More generally, censoring may be independent conditional on variables included in the analysis, but the basic unadjusted estimator does not itself make that conditioning adjustment.

These assumptions allow those remaining under observation to represent the event experience of the population still event-free. If people with a higher unobserved event risk tend to be censored earlier, the remaining risk set may become systematically lower-risk, and the unadjusted estimate can understate the population cumulative hazard. Correct recording of event and censoring times is also needed: a person must not be counted as at risk after known censoring, and an observed event must be counted at its event time.

The estimator is defined only where follow-up provides information. It can be drawn as constant after the last observed event, but that flat continuation does not show that the population hazard is zero after follow-up ends. Beyond the time when nobody remains at risk, the data provide no risk-set-based estimate of later hazard.

Worked Example: Three Event Times and Changing Risk Sets

Consider a hypothetical cohort of 12 individuals with observed events at times 1, 3, and 5 time units. The risk-set sizes and event counts are $Y_1=12,d_1=1$, $Y_2=9,d_2=2$, and $Y_3=5,d_3=1$. Suppose two individuals are censored between times 1 and 3, and two more are censored between times 3 and 5. These censorings give the stated risk-set sizes: after the event at time 1, 11 remain; two censorings leave 9 at risk at time 3. After the two events at time 3, 7 remain; two more censorings leave 5 at risk at time 5.

At time 1, the estimated increment is $d_1/Y_1=1/12=0.08333$, so $\widehat H_{\mathrm{NA}}(1)=0.08333$.

At time 3, add the increment $2/9=0.2222$: $\widehat H_{\mathrm{NA}}(3)=1/12+2/9=0.3056$.

At time 5, add $1/5=0.2000$: $\widehat H_{\mathrm{NA}}(5)=1/12+2/9+1/5=0.5056$.

The two censorings before time 3 affect the estimate by reducing the risk set from 11 to 9; they do not add hazard increments. The estimated cumulative hazard is 0.5056 at time 5, in cumulative-hazard units, not a 50.56% event probability.

Reading the Step Function and Checking the Calculation

At any time $t$, the estimate includes every observed event time at or before $t$, and no later events. If there are no events between two event times, the estimated cumulative hazard is constant over that interval. This makes the estimate straightforward to calculate from a risk table, but a plausible-looking final number does not ensure the risk sets were constructed correctly.

Order the observed event times.
List distinct event times in increasing order. At each time, identify the individuals still under observation and event-free just before it.

Record the risk-set size and event count.
For each time $t_j$, record $Y_j$ and $d_j$. Check that $Y_j$ is positive and $0\leq d_j\leq Y_j$.

Calculate and accumulate the increments.
Calculate $d_j/Y_j$, then add it to all earlier increments to obtain the estimate at $t_j$.

Check the risk-set transitions.
Confirm that events and censorings before the next event time have been removed from the next risk set, and that censorings are not counted as events.

Worked Example: Censoring Changes the Next Risk Set

In a hypothetical cohort, suppose 20 individuals are at risk at time 2 and one event occurs. Three additional individuals are censored between times 2 and 4. The next event occurs at time 4, when 16 individuals are at risk, and one event occurs. Four are then censored before time 7, where 11 individuals are at risk and one event occurs.

The event-time risk sets are therefore $20,16,$ and $11$, with one event at each time. The cumulative estimates are $\widehat H_{\mathrm{NA}}(2)=1/20=0.05000$, $\widehat H_{\mathrm{NA}}(4)=1/20+1/16=0.1125$, and $\widehat H_{\mathrm{NA}}(7)=1/20+1/16+1/11=0.2034$.

The risk-set arithmetic is consistent: after the time-2 event, 19 remain; three censorings leave 16 at time 4. After the time-4 event, 15 remain; four censorings leave 11 at time 7. The estimate increases at every event time and is unchanged at censoring times.

Relation to Survival Estimation

The Nelson–Aalen estimator targets cumulative hazard directly. A common alternative is the Kaplan–Meier estimator, which estimates the survival function by multiplying the conditional proportions remaining event-free at successive event times. With no tied events, its estimate is $\widehat S_{\mathrm{KM}}(t)=\prod_{j:t_j\leq t}(1-d_j/Y_j)$. The transformation $-\log\{\widehat S_{\mathrm{KM}}(t)\}$ is another cumulative-hazard summary, but it is not generally identical in a finite sample to the Nelson–Aalen sum.

For small event fractions, $-\log(1-d_j/Y_j)$ is close to $d_j/Y_j$, because $-\log(1-x)$ is approximately $x$ when $x$ is near zero. This explains why the two cumulative-hazard summaries can be close when each increment is small. The Nelson–Aalen estimator is useful when the target is cumulative hazard itself; the Kaplan–Meier estimator is the direct choice when the target is survival probability. Neither removes the need to address informative censoring or sparse follow-up.

Worked Example: Comparing Two Cumulative-Hazard Summaries

Suppose a hypothetical cohort has three event times with risk-set sizes $10,8,$ and $6$, and one event at each time. The Nelson–Aalen estimate after the third event is $\widehat H_{\mathrm{NA}}=1/10+1/8+1/6=0.3917$.

The Kaplan–Meier survival estimate at the same time is $\widehat S_{\mathrm{KM}}=(9/10)(7/8)(5/6)=0.65625$. Its negative logarithm is $-\log(0.65625)=0.4212$ to four significant figures. The two cumulative-hazard summaries differ: 0.3917 for Nelson–Aalen and 0.4212 for the negative log of Kaplan–Meier. Both are based on the same risk-set information, but they apply different calculations to it.

In this example, $1-\widehat S_{\mathrm{KM}}=0.3438$ estimates the probability of an event by the third event time under the Kaplan–Meier procedure; it is not the Nelson–Aalen estimate. Cumulative hazard and event probability are distinct quantities.

Diagnostics, Limitations, and Common Mistakes

A risk table is the most direct diagnostic for this calculation. Check that risk-set sizes do not increase over time in a single-cohort analysis, that $d_j\leq Y_j$, and that the event and censoring records imply each stated transition. A sudden large increment may be mathematically valid but deserves scrutiny: it can arise when a substantial fraction of a small risk set experiences events. Estimates late in follow-up can be especially unstable because few individuals remain at risk.

The formula itself cannot diagnose informative censoring, dependence between individuals, or errors in how follow-up times were recorded. These are design and data issues, not problems solved by adding the increments correctly. In clustered or otherwise dependent data, for example, the basic independent-individual interpretation may not apply without methods that account for that structure.

Common mistake: treating censoring as an event. Censoring removes an individual from later risk sets but does not contribute a $d_j/Y_j$ increment.
Common mistake: using the original sample size at every event time. The denominator is the number at risk just before that event time, not the initial sample size.
Common mistake: calling cumulative hazard a probability. Cumulative hazard is an accumulated hazard measure; it can exceed 1 and is not itself the probability of an event by time $t$.
Common mistake: assuming a flat estimate means no later risk. The estimate is flat after the last observed event, but follow-up ending means later event experience is unobserved, not necessarily absent.

In MICE applications, a cumulative-hazard summary may be used as information about an individual's observed survival history. The estimate still depends on a valid risk-set construction and on assumptions about censoring; it does not by itself resolve missing covariates or determine how an imputation model should be specified.

Key takeaway: At each observed event time, the Nelson–Aalen estimator adds the number of events divided by the number at risk immediately beforehand. The cumulative sum estimates cumulative hazard, subject to appropriate event-time, risk-set, and censoring assumptions.

Check Your Understanding

Answer each question, then compare your work with the solutions below.

At an event time, 24 individuals are at risk and 2 events occur. What increment does the Nelson–Aalen estimator add?
In a hypothetical study, 15 are at risk at time 2 with one event. Two people are censored before time 5, when one event occurs among 12 at risk. Calculate the estimate at time 5.
Why does a censoring before an event time affect the risk set but not add a cumulative-hazard increment?
Is a Nelson–Aalen estimate of 0.60 an estimate of a 60% event probability? Explain.
Give one reason the estimate may be unreliable late in follow-up, even if all arithmetic is correct.

Solutions. 1. The increment is $2/24=0.08333$. 2. The estimate at time 5 is $1/15+1/12=0.1500$; the stated risk set is consistent because one event leaves 14 and two censorings leave 12. 3. A censored person is no longer observed and so cannot contribute to later risk sets, but no event was observed for that person at the censoring time. 4. No. It is a cumulative-hazard estimate, not an event probability; the event probability requires a survival estimate or an appropriate transformation under its assumptions. 5. Few individuals may remain at risk, making each event a large fraction of the risk set and the estimate sensitive to individual outcomes.