Why 1 Minus Kaplan-Meier Overstates Risk When Patients Can Die First

อ่านฉบับภาษาไทย (Thai version)

Abstract

When a treatment lowers the cause-specific hazard of recurrence but raises the hazard of death before recurrence, its effect on the risk of recurrence cannot be read from the recurrence hazard alone. A hazard is a rate among patients still alive and event-free; the cumulative incidence is the share of everyone at the start who has a recurrence by a given time, and it depends on both hazards, because a patient who dies first can never recur. One minus a recurrence-free Kaplan-Meier curve with deaths censored treats those deaths as if recurrence were still possible. When 20 of 100 patients die in the first 6 months and 10 of the remaining 80 recur by month 12, the cumulative incidence is 10%, but 1 minus Kaplan-Meier gives 12.5%. Separate Kaplan-Meier curves for each cause cannot be added either: the real split is event-free, recurrence first and death first. This article builds each step and fixes four common misreadings.

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A regimen that slows recurrence and costs lives

An oncology team is reading the report of a new adjuvant regimen for older patients after cancer surgery. Among patients still alive and free of recurrence, recurrences come more slowly on the new regimen: its cause-specific hazard of recurrence is lower. But its toxicity is real, and more patients die before any recurrence. The team wants to tell the next patient what the regimen does to her risk of recurrence within five years.

Three readings come up at the meeting. One says the risk of recurrence must fall by the same percentage as the hazard. Another says it must rise, because the hazard of death rose. A third proposes reading the risk straight off one minus the recurrence-free Kaplan-Meier curve.

None of them holds. The risk of recurrence depends on both hazards, and seeing why starts with what can happen to one patient.

What can happen to one patient

Follow patients after cancer treatment. At the start everyone is in the same state:

$$\boxed{\text{Alive, no recurrence}}$$

From there, one of two things can happen first:

$$\text{Alive, no recurrence} \rightarrow \boxed{\text{Recurrence}}$$

or

$$\text{Alive, no recurrence} \rightarrow \boxed{\text{Death before recurrence}}$$

A patient who dies before recurrence can never have a recurrence later. That is what it means to say

$$\boxed{\text{Death is a competing risk for recurrence}}$$

because death closes the door on any future recurrence. An event that makes the event of interest impossible afterwards is a competing event; Part 1 of this series maps where competing events sit among time-to-event questions.

First, separate hazard from risk

This distinction is the heart of the matter. Two quantities sound alike and answer different questions.

The cause-specific hazard of recurrence asks: among patients who are alive and recurrence-free right now, how fast are recurrences happening? It is written $\lambda_R(t)$. It is an instantaneous rate, not the probability that a patient in the whole group has a recurrence within five years. (A hazard in general is the event rate at a given moment among patients still event-free; a cause-specific hazard counts only the events of one cause.)

The cumulative incidence of recurrence asks: of all the patients at the start, what share will have a recurrence before time $t$, in the real world where some of them die first? It is written $F_R(t)$, or $\mathrm{CIF}_{\text{recurrence}}(t)$, for the cumulative incidence function. It is an absolute probability [1].

A treatment can move the first without moving the second by the same amount. The rest of this article shows why.

Why the risk of recurrence has to take death into account

Think it through without a formula first. Take 100 patients. If nobody died, all 100 would remain able to have a recurrence. But if 40 die before any recurrence, only 60 are left who still can.

So even if the tumour's biological tendency to recur stays exactly the same, fewer patients live long enough to have a recurrence.

That is why the cumulative incidence of recurrence depends on both

$$\lambda_{\text{recurrence}} \quad\text{and}\quad \lambda_{\text{death}}$$

and not on the recurrence hazard alone [1].

A formula to understand, not to memorise

The cumulative incidence of recurrence can be written as

$$\boxed{F_R(t) = \int_0^t S(u^-)\,\lambda_R(u)\,du}$$

In plain words: the probability of recurrence is the chance that a patient is still alive and event-free just before each moment, times the speed at which recurrence happens at that moment, added up over follow-up. Here $S(u^-)$ is event-free survival, the probability of being alive and recurrence-free just before time $u$. So there are two parts:

$$\underbrace{S(u^-)}_{\text{alive and event-free so far}} \times \underbrace{\lambda_R(u)}_{\text{recurrence hazard}}$$

And $S(u)$ is set by both $\lambda_R$ and $\lambda_D$, because a recurrence and a death each take a patient out of the event-free state. Therefore

$$\boxed{\mathrm{CIF}_R = f(\lambda_R, \lambda_D)}$$

This is the direct reason why the risk of recurrence depends on both hazards [1, 2].

Where 'one minus recurrence-free Kaplan-Meier' goes wrong

A tempting shortcut says that the risk of recurrence equals one minus the recurrence-free Kaplan-Meier curve:

$$\mathrm{CIF}_R(t) = 1 - \mathrm{KM}(t)$$

The Kaplan-Meier estimator builds an event-free curve one event time at a time: at each event it multiplies the curve by the share of the risk set (the patients still followed and event-free just before that time) who stay event-free. Part 2 of this series covers how that curve is read. Under competing risks the shortcut is not right, and the crux lies in two words: censor death.

What Kaplan-Meier does with patients who die

Suppose the interest is in recurrence only. When a patient dies without a recurrence, a naive Kaplan-Meier analysis may

$$\boxed{\text{censor at death}}$$

In statistics, censoring means roughly: we know this patient had no recurrence up to the time they left view, and after that we do not know what happened.

That is reasonable for loss to follow-up. Mr A moves to another hospital at 1 year, and after that we do not know whether he has a recurrence. He can still have one; we just do not see it.

Death is different. Mr B dies at 1 year without a recurrence. After that,

$$P(\text{recurrence after death}) = 0$$

It is not unknown. It is

$$\boxed{\text{impossible}}$$

Censoring a death as if it were ordinary censoring therefore builds a false world for the question of the actual probability of recurrence [2, 3].

Hand example: one hundred patients, step by step

Hand example. With numbers, the error shows at once. One hundred patients are followed after surgery, and nobody is lost to follow-up.

1. First 6 months

   \[ 100 - 20 = 80 \]

   Twenty patients die without a recurrence. That leaves 80 patients who are alive and can still have a recurrence.

2. Month 12

   Of the 80 who remain, 10 have a recurrence.

3. How many of the 100 had a recurrence?

   \[ \mathrm{CIF}_R = \frac{10}{100} = 10\% \]

   Ten did. The actual cumulative incidence is simply 10 out of the 100 who started.

4. What Kaplan-Meier does

   \[ S_{\mathrm{KM}} = 1 - \frac{10}{80} = 0.875 \]

   Kaplan-Meier for recurrence alone treats the 20 deaths as censored. When the 10 recurrences happen it sees a risk set of 80, and it puts the probability of remaining recurrence-free at 0.875.

5. One minus Kaplan-Meier

   \[ 1 - \mathrm{KM} = 1 - 0.875 = 0.125 = 12.5\% \]

   Flipped into a risk, the curve says 12.5%.

6. Set it beside the real count

   \[ \boxed{1 - \mathrm{KM} = 12.5\% > \mathrm{CIF} = 10\%} \]

   Yet the data show that only 10% of the patients had a recurrence.

Result: 1 minus Kaplan-Meier gives 12.5%, higher than the actual cumulative incidence of recurrence, 10%.

The contrast rests on the order of events here: all 20 deaths come in the first 6 months, before the recurrences at month 12. With another order the 1 minus Kaplan-Meier figure would change, while the cumulative incidence would still be 10 out of 100.

Why 10% became 12.5%

Kaplan-Meier in effect says: the 20 patients who died are taken out of the calculation, because they count as censored. So the 10 recurrences are read as

$$\frac{10}{80} \quad\text{instead of}\quad \frac{10}{100}$$

the share of the original group who actually had a recurrence (10 in 100). But in the real world the 20 who died can never come back to have a recurrence, so they have to count in the actual probability of recurrence.

That is why naive 1 minus Kaplan-Meier tends to overstate the cumulative incidence when competing risks are present [2, 3].

Two kinds of 'censoring', side by side

A second trap: what does 'recurrence-free' count as the event?

Papers do not all define recurrence-free survival the same way, and two definitions need watching.

Definition 1: recurrence is the event and death is censored. Then $1 - \mathrm{KM}$ is not the actual cumulative incidence of recurrence, because the deaths were censored, and it tends to overstate the probability of recurrence.

Definition 2: recurrence or death, whichever comes first, is the event. Then $1 - \mathrm{KM}$ gives the probability of

$$\boxed{\text{recurrence or death}}$$

not the probability of recurrence alone, so it still does not answer the question about recurrence.

Whichever definition is meant, the shortcut fails:

$$\text{death censored:}\quad 1 - \mathrm{KM} \neq \mathrm{CIF}_{\text{recurrence}}$$$$\text{death counted with recurrence:}\quad 1 - \mathrm{KM} = P(\text{recurrence or death}) \neq P(\text{recurrence})$$

What to use for the real risk of recurrence

For a descriptive competing-risks analysis, the quantity to estimate is the

$$\boxed{\text{cumulative incidence function}}$$

and its standard estimator is the

$$\boxed{\text{Aalen-Johansen estimator}}$$

not naive 1 minus Kaplan-Meier. The Aalen-Johansen estimator knows that a patient who died before recurrence has been removed permanently from the chance of a later recurrence: like the formula above, it weights each moment's recurrences by the share of the whole group still alive and event-free. It therefore estimates the actual probability of the event when competing events are present [2].

Back to the treatment that lowers one hazard and raises the other

The new regimen does two things:

$$\downarrow \lambda_R \quad\text{(a lower cause-specific hazard of recurrence)}$$$$\uparrow \lambda_D \quad\text{(a higher cause-specific hazard of death before recurrence)}$$

What happens to the cumulative incidence of recurrence? We may not look at $\lambda_R$ alone, because

$$\boxed{\mathrm{CIF}_R(t) = \int_0^t S(u^-)\,\lambda_R(u)\,du}$$

and $S(u)$ is moved by both recurrence and death. The answer the formula gives is that the cumulative incidence of recurrence depends on both the recurrence hazard and the death hazard; the recurrence hazard alone cannot say how far it moves. Modelling each cause-specific hazard with covariates, and the Fine-Gray model for the cumulative incidence, are the subject of Part 5 of this series.

Four readings that do not hold, and their fixes

• "The risk of recurrence falls by the same percentage as the recurrence hazard."

  A 30% lower cause-specific hazard of recurrence does not mean a 30% lower cumulative incidence of recurrence. Under competing risks, a hazard and a cumulative probability have no one-to-one relation.

  Fix: Read the cause-specific hazard as a rate among patients still alive and recurrence-free, and take the risk from the cumulative incidence, which depends on both hazards.

• "Death can be treated as ordinary censoring."

  Ordinary censoring means that the outcome is still possible after censoring but unobserved; after death, a recurrence is impossible.

  Fix: Code death before recurrence as a competing event whenever the question is the actual risk of recurrence.

• "The risk of recurrence must rise because the hazard of death rose."

  It is the other way round: more deaths leave fewer patients who can have a recurrence.

  Fix: Weigh both cause-specific hazards, through the cumulative incidence, before saying which way the risk of recurrence moved.

• "The risk of recurrence equals one minus recurrence-free Kaplan-Meier."

  With deaths censored, 1 minus Kaplan-Meier acts as if the patients who died could still have a recurrence afterwards, in a hypothetical world. It is not the actual probability of recurrence, and it generally overstates the cumulative incidence.

  Fix: Estimate the cumulative incidence with the Aalen-Johansen estimator. If the curve's event is recurrence or death, read 1 minus Kaplan-Meier as the risk of either one, not of recurrence.

Everyone is somewhere: event-free, recurred first or died first

An equation that adds up the whole group has to include the patients who have had no event at all yet. With two competing events, recurrence and death before recurrence:

$$\boxed{1 = S(t) + \mathrm{CIF}_{\text{recurrence}}(t) + \mathrm{CIF}_{\text{death}}(t)}$$

• $S(t)$: still alive and recurrence-free at time $t$;
• $\mathrm{CIF}_{\text{recurrence}}(t)$: the probability that a recurrence has happened by $t$;
• $\mathrm{CIF}_{\text{death}}(t)$: the probability that a death before recurrence has happened by $t$.

So

$$\boxed{1 - S(t) = \mathrm{CIF}_{\text{recurrence}}(t) + \mathrm{CIF}_{\text{death}}(t)}$$

This is the key that clears up most of the confusion.

Hand example: one hundred patients at five years

Hand example. Another 100 patients, counted at five years.

1. Count the three groups

   At 5 years, 20 have had a recurrence, 30 have died before any recurrence, and 50 are alive and recurrence-free.

2. Turn the counts into shares

   \[ S = 0.50, \quad \mathrm{CIF}_R = 0.20, \quad \mathrm{CIF}_D = 0.30 \]

   Each count is divided by the 100 who started.

3. Check the identity

   \[ 1 = 0.50 + 0.20 + 0.30 \]

   That is $1 = S + \mathrm{CIF}_R + \mathrm{CIF}_D$: every patient is in exactly one group.

4. Read one minus event-free survival

   \[ 1 - S = 0.50 \]

   This says that 50% had a recurrence or died, not that 50% had a recurrence.

Result: 1 minus S is 0.50, the share with a recurrence or a death; the share with a recurrence is the cumulative incidence, 0.20.

The recurrence-free curve, read against this identity

Faced with 'one minus recurrence-free Kaplan-Meier', ask at once what that curve counts as the event. There are two cases.

Case 1: recurrence is the event and death is censored. Then $1 - \mathrm{KM}_R$ is not $\mathrm{CIF}_R$, because death is handled like ordinary censoring although a patient who has died has no further chance of a recurrence. This is the reading the shortcut usually intends.

Case 2: recurrence or death is the event. The curve's event is now a composite endpoint, so

$$1 - \mathrm{KM} = P(\text{recurrence or death by } t) = \boxed{\mathrm{CIF}_R + \mathrm{CIF}_D}$$

In the five-year example that is 0.50, not the 0.20 who had a recurrence. In neither case is the shortcut the cumulative incidence of recurrence.

Is 1 = KM for death + KM for recurrence?

A natural next guess is that two Kaplan-Meier analyses, one per cause, add up to the whole:

$$1 = \mathrm{KM}_{\text{death}} + \mathrm{KM}_{\text{recurrence}}\ ?$$

No, when each of them is a Kaplan-Meier analysis of one cause that censors the other cause. The correct split is

$$\boxed{1 = S_{\text{event-free}} + \mathrm{CIF}_{\text{recurrence}} + \mathrm{CIF}_{\text{death}}}$$

It uses cumulative incidences, not separate one minus Kaplan-Meier curves.

Why separate Kaplan-Meier curves cannot be added

Kaplan-Meier for recurrence. When a patient dies, $\text{death} \rightarrow \text{censored}$. The curve is answering, roughly: what is the probability of recurrence in a hypothetical world in which competing death takes nobody away?

Kaplan-Meier for death. When a patient has a recurrence, $\text{recurrence} \rightarrow \text{censored}$. This curve answers a question about a second, different hypothetical world.

So $1 - \mathrm{KM}_R$ and $1 - \mathrm{KM}_D$ are probabilities from two different hypothetical worlds. Added together, they do not split up the real world. That is why competing risks need the cumulative incidence function [2, 4].

When 1 minus Kaplan-Meier is the right answer

None of this makes Kaplan-Meier wrong in itself. When the outcome is death from any cause, there is no competing event, because every death counts as the same event.

Kaplan-Meier then estimates $S(t) = P(T > t)$, where $T$ is the time to the event. If $S(5) = 0.80$, then 80% have not had the event at 5 years, and $1 - S(5) = 0.20$: 20% had it within 5 years. With a single kind of event,

$$\boxed{1 - \mathrm{KM} = \text{cumulative probability of the event}}$$

works well. The trouble starts only when a competing event exists, as with recurrence and death before recurrence above.

Glossary

competing event (เหตุการณ์แข่งขัน)

An event, such as death before recurrence, that makes the event of interest impossible afterwards.

censoring (การเซ็นเซอร์)

Follow-up that ends without the event while the event could still happen later, unseen.

cause-specific hazard (ฮาซาร์ดเฉพาะสาเหตุ)

$\lambda_k(t)$: the instantaneous rate of events of cause $k$ among patients still alive and event-free at time $t$; a rate, not a probability.

cumulative incidence function (อุบัติการณ์สะสม)

$\mathrm{CIF}_k(t)$: the probability, out of everyone at the start, that an event of cause $k$ happens first and by time $t$.

event-free survival (ฟังก์ชันการรอดชีพปลอดเหตุการณ์)

$S(t)$: the probability of being alive and free of any event at time $t$.

Kaplan-Meier estimator (ตัวประมาณ Kaplan-Meier)

An estimate of event-free survival that, at each event time, multiplies by the share of the risk set staying event-free, with censored patients leaving the risk set.

Aalen-Johansen estimator (ตัวประมาณ Aalen-Johansen)

The standard estimator of the cumulative incidence when competing events are present; it removes patients who had a competing event from any later chance of the event of interest.

References

1. Andersen PK, Geskus RB, de Witte T, Putter H. Competing risks in epidemiology: possibilities and pitfalls. Int J Epidemiol. 2012;41(3):861-870. doi:10.1093/ije/dyr213 https://doi.org/10.1093/ije/dyr213
2. Putter H, Fiocco M, Geskus RB. Tutorial in biostatistics: competing risks and multi-state models. Stat Med. 2007;26(11):2389-2430. doi:10.1002/sim.2712 https://doi.org/10.1002/sim.2712
3. Austin PC, Lee DS, Fine JP. Introduction to the analysis of survival data in the presence of competing risks. Circulation. 2016;133(6):601-609. doi:10.1161/CIRCULATIONAHA.115.017719 https://doi.org/10.1161/CIRCULATIONAHA.115.017719
4. Wolbers M, Koller MT, Stel VS, Schaer B, Jager KJ, Leffondré K, Heinze G. Competing risks analyses: objectives and approaches. Eur Heart J. 2014;35(42):2936-2941. doi:10.1093/eurheartj/ehu131 https://doi.org/10.1093/eurheartj/ehu131

Key takeaways

• A cause-specific hazard is a rate among patients still alive and event-free; the cumulative incidence is an absolute probability out of everyone at the start.
• The cumulative incidence of recurrence depends on both the recurrence hazard and the death hazard, so a treatment that lowers one and raises the other cannot be judged from the recurrence hazard alone.
• Censoring means the event could still happen but is not seen; a death before recurrence means it can no longer happen.
• One minus Kaplan-Meier with deaths censored is not the cumulative incidence and generally overstates it: 12.5% against 10% in the 100-patient example.
• Event-free survival plus the cumulative incidences of recurrence and of death add to one; separate Kaplan-Meier curves for each cause do not.

Related in the wiki: [[time-to-event-survival-analysis]]