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Exam ASTAM · Kaplan-Meier and Nelson-Aalen · Tutorial 256 of 979

Distinguishing Failure Drops from Censoring Marks

Interpret Kaplan-Meier plots by identifying which observations cause a survival step and how censoring changes the risk set for later steps.

Intermediate 10 min read

What You'll Learn

  • Define the risk set immediately before an observed time and distinguish failures from right-censoring.
  • Identify which Kaplan-Meier plot features represent failures and which represent censoring.
  • Explain how censoring changes later risk sets without directly changing the estimated survival.
  • Calculate survival steps when failures and censoring occur at different times.
  • Apply the event-before-censoring convention when failures and censoring share a recorded time.
  • Recognize common plot-reading errors and check a survival estimate against the risk-set history.

Reading the Marks on a Kaplan-Meier Plot

A Kaplan-Meier plot shows an estimated survival curve as a sequence of horizontal segments and downward steps. Many plots also mark censored observations, often with small crosses or ticks on a horizontal segment. Correct interpretation depends on keeping two things separate: failures lower the estimated survival, while censoring affects which observations remain in later risk sets.

This distinction matters when reading a plot or reconstructing an estimate from a timeline. A censoring mark does not mean that the individual failed at that time. It means that the individual’s event time was not observed beyond that point. In hypothetical insurance or health follow-up, this may happen because observation ends before an event is seen or the individual leaves the study. The event status after censoring is unknown.

Definition: At an observed time \(t_j\), the risk set consists of individuals who are still under observation and have not yet failed immediately before \(t_j\). Let \(n_j\) be the number in this risk set and \(d_j\) the number of failures at \(t_j\). A right-censored observation contributes to the risk set up to its censoring time, but is not counted as a failure.

For distinct event times \(t_j\), the Kaplan-Meier estimate of survival through time \(t\) is the product of the estimated conditional survival proportions at event times up to \(t\):

$$ \widehat{S}_{KM}(t) = \prod_{t_j\leq t}\left(1-\frac{d_j}{n_j}\right) = \prod_{t_j\leq t}\frac{n_j-d_j}{n_j}. $$

The product contains a factor for each failure time. A censoring time does not supply a factor: it does not, by itself, make the curve step down. However, a censored individual is no longer in the risk set for later times, so subsequent factors can change.

Why Failures Make Steps and Censoring Does Not

At an event time, \(d_j\) of the \(n_j\) individuals at risk are observed to fail. The estimated conditional probability of surviving past that time is therefore \((n_j-d_j)/n_j\). Multiplying these conditional proportions across event times gives the product-limit estimate.

$$ \widehat{S}_{KM}(t_j) = \widehat{S}_{KM}(t_j-) \frac{n_j-d_j}{n_j}, \qquad \widehat{S}_{KM}(t_j-) = \prod_{t_k<t_j}\frac{n_k-d_k}{n_k}. $$

A censoring time supplies no observed failure and thus no new conditional failure proportion. If the curve is at height \(s\) when an individual is censored, the estimate stays at \(s\) at that mark. The person is excluded from later risk sets, though, because they are no longer observed. These conclusions rely on the usual Kaplan-Meier assumption that censoring is non-informative for the event time, at least conditional on any variables accounted for in the analysis. Informally, censored individuals must not systematically have different unobserved survival prospects from comparable individuals who remain under observation. If that assumption fails, the curve may not estimate the intended survival distribution.

1
Locate each vertical drop.
A downward step represents one or more failures at that time. Its size depends on the number failing and the risk set immediately before the failures.
2
Locate censoring marks.
A cross or tick on a horizontal segment identifies a censored observation. The mark does not itself lower the curve.
3
Update later risk sets.
Remove censored observations from risk sets after their censoring times, along with individuals who have already failed.

The exact visual placement of a censoring mark is a plotting convention. A mark drawn at the same horizontal coordinate as a failure step can be hard to interpret from the picture alone. Use the accompanying event and censoring information, and apply the stated convention for ties. The height of the curve after a failure is not evidence that a censoring mark at that time caused the drop.

Worked Example: A Censoring Mark Between Two Failure Times

In a hypothetical follow-up of 15 devices, 3 fail at time 1, 4 are censored at time 2, and 2 fail at time 3. No other failures occur by time 3. Find the Kaplan-Meier estimate after each failure time and describe what the plot shows.

Method. Calculate a factor at each failure time, using the risk set just before that time. The time-1 risk set is \(n_1=15\), with \(d_1=3\), so the estimate drops from 1 to

$$ \widehat{S}_{KM}(1) = 1\left(\frac{15-3}{15}\right) = \frac{12}{15} = 0.8000. $$

At time 2, the 4 observations are censored. There is no failure at that time, so the curve remains at 0.8000. Those 4 observations are removed from the risk set for time 3. The time-3 risk set is therefore \(15-3-4=8\). With \(d_3=2\), the time-3 factor is

$$ \frac{8-2}{8}=\frac{6}{8}=0.75. $$

Thus the estimate after time 3 is \(0.8000(0.75)=0.6000\). The plot has a drop from 1 to 0.8000 at time 1, censoring marks on the horizontal segment at height 0.8000 at time 2, and a drop from 0.8000 to 0.6000 at time 3.

Check. The time-3 risk set has 8 people: 15 initially, minus 3 failures and 4 censorings. The two time-3 failures leave 6 in the risk set, giving the factor \(6/8\). The calculation avoids the common mistake of treating the four censorings as failures at time 2.

Worked Example: Several Censoring Marks Before a Later Drop

A hypothetical study begins with 12 units under observation. Three fail at time 1, four are censored at time 2, and two fail at time 3. Determine the curve height just before time 3, the drop at time 3, and the estimate after time 3.

At time 1, \(n_1=12\) and \(d_1=3\), so

$$ \widehat{S}_{KM}(1) = \frac{12-3}{12} = \frac{9}{12} = 0.7500. $$

There are no failures at time 2. The four censoring marks lie on the plateau at height 0.7500, and the curve remains there. The time-3 risk set is \(12-3-4=5\). Two of those five units fail, so the estimate after the time-3 step is

$$ \widehat{S}_{KM}(3) = 0.7500\left(\frac{5-2}{5}\right) = 0.7500(0.6000) = 0.4500. $$

The vertical change is \(0.7500-0.4500=0.3000\). This is a 0.3000 decrease in estimated survival, not a decrease for each censoring mark. Check. The time-3 risk set has 5 units, and 3 remain after the two failures; the conditional survival factor is \(3/5\), as used above. The common mistake avoided is subtracting censoring marks from the curve height instead of from the later risk set.

When Failures and Censoring Share a Recorded Time

A tie requires care. Under the standard Kaplan-Meier convention, individuals censored at a recorded time are included in the risk set immediately before that time’s failures are counted. The failures at that time determine the step; the censored individuals are then removed from later risk sets. This convention reflects that a right-censored observation is known to have remained event-free up to its censoring time.

Worked Example: Applying the Tie Convention

A hypothetical cohort has 10 individuals. One fails at time 1. At time 2, one individual fails and two others are censored. At time 3, two individuals fail. Use the standard convention that the time-2 censored observations are in the risk set immediately before the time-2 failure.

At time 1, \(n_1=10\), \(d_1=1\), and the estimate becomes

$$ \widehat{S}_{KM}(1)=\frac{9}{10}=0.9000. $$

Immediately before time 2, nine individuals are at risk: 10 initially, minus the one failure at time 1. Thus \(n_2=9\), including the two individuals censored at time 2, and \(d_2=1\). The time-2 failure factor is \(8/9\), so

$$ \widehat{S}_{KM}(2) = \frac{9}{10}\frac{8}{9} = 0.8000. $$

After the time-2 failure and the two censorings, the time-3 risk set has \(10-1-1-2=6\) individuals. With two failures at time 3, the factor is \((6-2)/6=4/6\). Therefore,

$$ \widehat{S}_{KM}(3) = 0.8000\left(\frac{4}{6}\right) = 0.5333. $$

The plot drops at times 1, 2, and 3; the two time-2 censoring marks do not add another drop. Check. The time-2 factor uses 9, not 7, in its denominator because the tied censored observations are included just before the time-2 failure. The time-3 risk set is 6 after all observations ending by time 2 have been removed. The calculation avoids the trap of removing tied censorings before forming the risk set for the failure at that same time.

Common Plot-Reading Traps

A Kaplan-Meier plot compresses an observation history into a curve, so a mark’s meaning should be checked against the event information, not inferred from its visual size or position alone. Keep these distinctions in view:

  • Do not count a censoring mark as a failure. A censoring mark does not create a product-limit factor or a vertical drop.
  • Do not ignore censoring when calculating later steps. A censored observation leaves subsequent risk sets, even though the curve did not change at its censoring time.
  • Do not use the risk set after the event. The denominator \(n_j\) is the number at risk immediately before the failures at \(t_j\).
  • Check the tie convention. With the standard convention described here, an observation censored at \(t_j\) is included in the risk set just before failures at \(t_j\).
  • Do not interpret a plateau as proof that no one left observation. It means there were no observed failures between the adjacent failure times; censoring can occur along that plateau.
Key takeaway: Read a Kaplan-Meier plot in two passes: first identify failure times from the downward steps, then identify censoring marks and use them to update later risk sets. Failures change the current survival estimate; censoring changes who can contribute to future estimates.

Check Your Understanding

For each problem, distinguish the curve’s failure steps from its censoring marks. Answers are given for checking.

  1. A hypothetical cohort starts with 20 individuals. Four fail at time 1, five are censored at time 2, and three fail at time 3. Find the estimate after time 3. (Answer: \(0.5818\))
  2. Fourteen units are followed. Two fail at time 2, three are censored at time 4, and two fail at time 6. What is the curve height just before time 6 and just after the time-6 drop? (Answer: \(0.8571\) just before; \(0.6667\) just after)
  3. A group of 9 has one failure at time 1. At time 2, two fail and one is censored; at time 4, one fails. Under the standard tie convention, find the estimate after time 4. (Answer: \(0.5333\))
  4. A hypothetical study begins with 16 observations. Four fail at time 2, three are censored at time 3, one is censored at time 4, and two fail at time 6. The plot is flat between times 2 and 6 except for censoring marks. Find the height of that plateau and the estimate after the time-6 drop. (Answer: plateau \(0.7500\); after time 6, \(0.5625\))
  5. In a cohort of 15, three fail at time 1, four are censored at time 2, and two fail at time 3. How many are at risk immediately before time 3, and what is the estimate after the time-3 failure step? (Answer: 8 at risk; \(0.6000\))

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