Interpreting Nelson-Aalen Cumulative Hazard Steps

Reading the Nelson-Aalen Step Curve

The Nelson-Aalen estimate is often presented as a curve rather than a table of calculations. To interpret it correctly, focus on two features: where the curve rises and how large each rise is. A rise records observed failures relative to the number still at risk; a censoring time by itself creates no rise. However, censoring can change the risk set used for later failures, and therefore affect later step sizes.

This tutorial concerns the meaning of the Nelson-Aalen cumulative hazard steps. The observations are assumed to be subject to independent censoring, so censoring does not depend on the unobserved future failure time, conditional on the information used in the analysis. The usual risk-set convention is used: an individual censored at a failure time is counted as at risk immediately before that time, and is then removed from later risk sets. If a data source uses a different convention for tied times, that convention must be stated and applied consistently.

Definition: Let $t_1<t_2<\cdots<t_k$ be the distinct observed failure times. At failure time $t_j$, let $r_j$ be the number of individuals at risk immediately before $t_j$, and let $d_j$ be the number of failures at $t_j$. The Nelson-Aalen estimated cumulative hazard through time $t$ is $$ \widehat H_{NA}(t)=\sum_{j:t_j\le t}\frac{d_j}{r_j}. $$ Each term $d_j/r_j$ is the estimated hazard increment at a failure time. Censoring times do not appear as separate terms in this sum.

The estimator is a right-continuous, nondecreasing step function. It is flat between observed failure times, including across censoring times that are not also failure times. At a failure time $t_j$, the curve rises by $d_j/r_j$. It never falls: every increment is nonnegative because $d_j\geq 0$ and $r_j>0$ at a failure time.

The numerator and denominator have distinct roles. The numerator counts failures at the time of the jump, including all failures when several individuals fail at the same recorded time. The denominator counts individuals at risk just before that time. Earlier failures and censorings are no longer in that risk set; a censoring at a later time cannot affect an earlier step.

Why Failures Create the Steps

The increment $d_j/r_j$ has a direct empirical interpretation. Just before $t_j$, there are $r_j$ observed individuals still under observation and at risk. Of these, $d_j$ fail at $t_j$. Thus $d_j/r_j$ is the observed fraction of that risk set failing at that time. The Nelson-Aalen estimate adds these local increments as time advances, which gives a cumulative quantity.

This also explains why censoring does not create an immediate jump. A censoring says that an individual's failure time is no longer observed beyond the censoring time; it does not record a failure at that time. The individual is removed from risk sets after censoring, so the denominator at a later failure may be smaller. The later jump can therefore differ from what it would have been had that individual remained under observation.

\widehat H_{NA}(t_j)-\widehat H_{NA}(t_j-)=\frac{d_j}{r_j}, \qquad \widehat H_{NA}(c)-\widehat H_{NA}(c-)=0

Here $t_j-$ means immediately before the failure time, and $c$ denotes a censoring time that is not also a failure time. The second equality describes the absence of a jump at that censoring time; it does not mean that censoring is irrelevant to subsequent calculations.

Worked Example: Locating Jumps and Plateaus

A hypothetical follow-up study begins with 12 insured devices. The observed event history is: 2 failures at time 2, 3 censorings at time 3, 2 failures at time 4, 1 censoring at time 5, and 1 failure at time 6. Find the Nelson-Aalen increments and cumulative estimate at each failure time, then describe the curve.

Method. At each failure time, calculate the risk-set size immediately before that time. Add the ratio of failures to that risk-set size to the previous cumulative estimate. Censorings reduce later risk sets but do not contribute increments.

At time 2. All 12 devices are at risk, and 2 fail. Thus $r_1=12$, $d_1=2$, and the increment is $2/12=1/6$. The cumulative estimate becomes $1/6$.

At time 4. After the 2 failures at time 2, 10 devices remain. The 3 censorings at time 3 reduce the risk set to $10-3=7$. At time 4, 2 of those 7 fail, so the increment is $2/7$. The cumulative estimate is $1/6+2/7=7/42+12/42=19/42$.

At time 6. After the 2 failures at time 4, $7-2=5$ devices remain. The censoring at time 5 reduces the risk set to 4. One fails at time 6, giving an increment $1/4$. The cumulative estimate is $19/42+1/4=38/84+21/84=59/84$.

The curve has jumps of $1/6$, $2/7$, and $1/4$ at times 2, 4, and 6, respectively. It is flat from time 2 to time 4, including at the censoring time 3, and flat from time 4 to time 6, including at time 5. The censorings affect the risk sets at the later failures, but neither censoring creates a step. As a check, the risk-set sizes $12,7,4$ agree with removing the earlier failures and censorings in sequence.

Interpreting the Height and Shape

The vertical coordinate at a given time is the sum of all estimated increments at or before that time. It is not the increment at that time alone. To find a single jump size, compare the curve's height just after the failure time with its height just before. To find the cumulative estimate at a time, add every failure increment up to and including that time.

A larger step means a larger value of $d_j/r_j$, not necessarily a larger number of failures. For example, one failure among two people at risk produces an increment of $1/2$, while three failures among 12 at risk produce an increment of $1/4$. The first step is larger even though it records fewer failures. Always read the numerator and the risk-set denominator together.

Likewise, a flat portion does not mean that the cohort has no censoring, or that nobody has left follow-up. It means that no observed failures occurred in that interval. Censoring marks may occur during a plateau, reducing the number at risk for the next failure time without changing the curve's height at the moment of censoring.

Worked Example: Same Number of Failures, Different Step Sizes

A hypothetical monitoring program follows 10 sensor units. One unit fails at time 1. Two units are censored at time 2. Two units fail at time 3. Two more units are censored at time 4, and one unit fails at time 5. Calculate the estimate at the failure times and explain why the steps are not equal.

At time 1, $r_1=10$ and $d_1=1$, so the increment is $1/10=0.1$. The cumulative estimate is $0.1$.

The two censorings at time 2 do not add an increment. After the time-1 failure and those two censorings, $10-1-2=7$ units remain at risk. Thus at time 3, $r_2=7$, $d_2=2$, and the increment is $2/7$. The cumulative estimate is $1/10+2/7=7/70+20/70=27/70$.

After the two time-3 failures, $7-2=5$ units remain. The two censorings at time 4 reduce this to 3. At time 5, one of those three fails, so the increment is $1/3$. The cumulative estimate is $27/70+1/3=81/210+70/210=151/210$.

The two failures at time 3 produce a larger jump than the single failure at time 1 because $2/7>1/10$. The final jump $1/3$ is larger still, even though it records only one failure, because only three units are at risk. The curve is nondecreasing, with cumulative values $0.1$, $27/70$, and $151/210$ at the three failure times. The risk-set check is $10$, then $7$, then $3$; it accounts for all preceding failures and censorings.

Tied Failures and Censoring at an Event Time

When multiple failures occur at the same recorded time $t_j$, use one step with height $d_j/r_j$. Do not treat tied failures as a sequence of separate one-failure steps with a changing risk set within that time. The common denominator is the number at risk immediately before the tied time, under the stated convention.

If failures and censorings share an exact recorded time, apply the data's tie convention. Under the convention used here, individuals censored at $t_j$ are included in $r_j$, because they are still at risk immediately before $t_j$. The failure increment is still $d_j/r_j$; the censored individuals are then excluded from risk sets after $t_j$. They are not counted in $d_j$.

Worked Example: A Tied Failure Step After Censoring

A hypothetical group has 15 participants at the start. Three fail at time 1. Four are censored at time 2. Two fail together at time 3. Two are censored at time 4, and one fails at time 5. Calculate and interpret all cumulative steps.

At time 1, $r_1=15$, $d_1=3$, and the increment is $3/15=1/5$. The cumulative estimate is $1/5$.

After these three failures, 12 remain. Four censorings at time 2 leave $12-4=8$ at risk for time 3. The tied failures at time 3 give one increment, $2/8=1/4$, and the cumulative estimate becomes $1/5+1/4=4/20+5/20=9/20$.

After the two time-3 failures, $8-2=6$ remain. Two censorings at time 4 leave $6-2=4$ at risk at time 5. The failure at time 5 contributes $1/4$, and the cumulative estimate becomes $9/20+1/4=9/20+5/20=14/20=7/10$.

The curve rises by $1/5$, $1/4$, and $1/4$ at times 1, 3, and 5. The equal later increments arise from different event counts and risk-set sizes: two failures among eight at time 3, and one among four at time 5. The risk-set sequence $15,8,4$ confirms that earlier failures and censorings were removed before each later calculation.

Common Exam Trap: Treating Censoring as a Failure

A tempting but incorrect approach is to add an increment at every observed time, including times when individuals are censored. That changes the meaning of the estimator: a censoring is not an observed failure. A related error is to use the original cohort size as the denominator at every event time. The denominator must be the number at risk just before the particular failure time.

Common mistake: In the first example, adding a censoring increment of $3/12$ at time 3 would incorrectly make the estimate rise there. The correction is to add no increment at time 3 and use the reduced risk set of 7 when calculating the failure increment at time 4.

A further interpretation trap is to call $\widehat H_{NA}(t)$ a failure probability. It is the estimated cumulative hazard, formed by adding estimated hazard increments; its value and its steps should be described as such. This tutorial does not convert the cumulative hazard curve into a survival estimate.

Check Your Understanding

For each hypothetical cohort, identify the risk set at each failure time and calculate or interpret the Nelson-Aalen steps.

A cohort starts with 9 units. One fails at time 1, two are censored at time 2, and one fails at time 3. Give both increments and the cumulative estimate at time 3. (Answer: increments $1/9$ and $1/6$; cumulative estimate $5/18$.)
There are 8 participants. Two fail together at time 2, one is censored at time 3, and one fails at time 4. What is the increment at time 4, and does the censoring at time 3 create a step? (Answer: risk set 5, increment $1/5$; no step at time 3.)
A device cohort starts with 12 units. Three fail at time 1, three are censored at time 2, and two fail together at time 5. Find the cumulative estimate at time 5 and state the plateau interval. (Answer: time-1 increment $1/4$; time-5 increment $2/6=1/3$; cumulative estimate $7/12$; flat between times 1 and 5, including time 2.)
At one failure time, 2 of 10 at-risk individuals fail. At a later failure time, 1 of 3 at-risk individuals fails. Which step is larger, and why? (Answer: the later step is larger: $1/3>2/10=1/5$, because its failure fraction in the risk set is larger.)
At time 6, four individuals are at risk immediately before the time; one fails and one is censored at that same time. Using the stated tie convention, what is the increment, and how many remain for later risk sets? (Answer: increment $1/4$; two remain at risk afterward.)

What You'll Learn