The Kaplan-Meier estimator is the standard nonparametric estimator of the survival function when data may be right-censored. It is also called the product-limit estimator because it is constructed by multiplying estimated conditional survival probabilities across event times.
Before introducing censoring, it is helpful to recall the standard estimator of a distribution function when all event times are fully observed.
Suppose we observe independent event times \[ T_1, T_2, \dots, T_n \] with no censoring. The distribution function is \[ F(t) = P(T \le t). \] The natural estimator of \(F(t)\) is the empirical distribution function (EDF): \[ \widehat{F}(t) = \frac{1}{n}\sum_{i=1}^n I(T_i \le t). \]
This estimator is simply the sample proportion of subjects whose event time is at or before time \(t\). The corresponding estimator of the survival function is \[ \widehat{S}(t) = 1 - \widehat{F}(t). \]
Thus, when there is no censoring, estimating the survival curve is straightforward: at each time point \(t\), we just count the proportion of subjects who have survived beyond \(t\).
For example, suppose the observed event times are \[ 2,\ 4,\ 5,\ 7,\ 8. \] Then \[ \widehat{F}(5)=\frac{3}{5}=0.6, \qquad \widehat{S}(5)=1-\widehat{F}(5)=0.4. \] This means that \(60\%\) of the sample has experienced the event by time 5, and \(40\%\) is still event-free beyond time 5.
The difficulty with censoring is that we can no longer directly count the proportion of subjects with \(T_i \le t\) or \(T_i > t\), because some lifetimes are only partially observed. The Kaplan-Meier estimator solves this problem by generalizing the same basic counting idea to the censored-data setting.
Let \(T\) denote the event time and let \[ S(t) = P(T > t) \] be the survival function. In practice, we usually observe \[ Y_i = \min(T_i, C_i), \qquad \Delta_i = I(T_i \le C_i), \] where \(C_i\) is a censoring time, \(Y_i\) is the observed follow-up time, and \(\Delta_i\) indicates whether the event was observed.
Suppose the distinct observed event times are \[ t_{(1)} < t_{(2)} < \cdots < t_{(k)}. \] Then we have the following decomposition of the survival function: \[ \begin{aligned} &{S}(t_{(k)})\\ = &P(T>t_{(k)})\\ = &P(T>t_{(k)}|T>t_{(k-1)})P(T>t_{(k-1)})\\ = &P(T>t_{(k)}|T>t_{(k-1)})P(T>t_{(k-1)}|T>t_{(k-2)})\cdots P(T>t_{(2)}|T>t_{(1)})P(T>t_{(1)})\\ = & \prod_{j:t_{(j)}\leq t_{(k)}}P(T>t_{(j)}|T>t_{(j-1)}) \\ = & \prod_{j:t_{(j)}\leq t_{(k)}}(1-P(T=t_{(j)}|T\geq t_{(j)}) )\\ = & \prod_{j:t_{(j)}\leq t_{(k)}}(1-P(T=t_{(j)}|T\geq t_{(j)},C\geq t_{(j)} ) )\\ \end{aligned} \]
At each event time \(t_{(j)}\), define
The Kaplan-Meier estimator of the survival function is (this is simply the plug-in principal) \[ \widehat{S}(t) = \prod_{j: t_{(j)} \le t} \left(1 - \frac{d_j}{n_j}\right) = \prod_{j: t_{(j)} \le t} \frac{n_j - d_j}{n_j}. \]
This formula has a simple interpretation. At each event time, we estimate the conditional probability of surviving past that time by \[ \widehat{P}(T > t_{(j)} \mid T \ge t_{(j)}) = \frac{n_j - d_j}{n_j}, \] and then multiply these conditional survival probabilities together. Therefore, \[ \widehat{S}(t) = \prod_{j: t_{(j)} \le t} \widehat{P}(T > t_{(j)} \mid T \ge t_{(j)}). \]
Several important features follow directly from the definition:
The most important assumption behind the Kaplan-Meier estimator is independent censoring (also called non-informative censoring). Informally, this means that subjects who are censored at a given time should have the same future survival prospects as subjects who remain under observation at that time.
In a simple formulation, the event time \(T\) and censoring time \(C\) are assumed to be independent. More realistically, in many applications we assume they are independent conditional on the variables used in the analysis.
This assumption matters because the Kaplan-Meier estimator treats censored individuals as representative of those still at risk up to the moment of censoring. If censoring is related to prognosis, the estimator can be biased. For example, if very sick patients are more likely to drop out of follow-up, the Kaplan-Meier curve may overestimate survival.
In addition to independent censoring, it is usually assumed that:
To see how the Kaplan-Meier estimator works, consider the follow-up times of seven patients in a small clinical study. The outcome is time from treatment to relapse, measured in months.
| Patient | Time (months) | Status |
|---|---|---|
| A | 2 | Relapse |
| B | 3 | Censored |
| C | 4 | Relapse |
| D | 5 | Relapse |
| E | 5 | Relapse |
| F | 7 | Relapse |
| G | 9 | Censored |
The distinct event times are \(2\), \(4\), \(5\), and \(7\). Notice that two relapses occur at month 5. The censoring times occur at \(3\) and \(9\), but censoring times do not create jumps in the Kaplan-Meier curve.
We will use this example in the next section to show how the redistribute-to-the-right algorithm produces the Kaplan-Meier estimator by hand.
An intuitive way to understand the Kaplan-Meier estimator is through the redistribute-to-the-right algorithm.
Start by assigning each subject equal probability mass \(1/n\). If an event is observed, that subject keeps its mass at the event time. If a subject is censored, then that subject’s probability mass cannot stay at the censoring time, because the event did not occur there. Instead, the censored subject’s mass is redistributed equally over the subjects who remain under observation at later times, that is, to the subjects to the right.
This idea explains why censoring does not cause a drop in the Kaplan-Meier curve. A censored subject is not counted as a failure. Rather, its weight is passed forward to those who are still at risk.
The algorithm can be described informally as follows:
This algorithm leads to exactly the Kaplan-Meier estimator. It is especially useful pedagogically because it connects the Kaplan-Meier curve to the empirical distribution function from the no-censoring setting.
To see the connection, notice the following:
Now apply the algorithm to the dataset in Table Table 2.1. Initially, each of the 7 patients receives probability mass \(1/7\).
| Time | Observation | Redistribution step | Event mass recorded |
|---|---|---|---|
| 2 | A relapses | A keeps its mass \(1/7\) at time 2 | \(1/7\) |
| 3 | B is censored | B’s mass \(1/7\) is split equally among C, D, E, F, and G, so each receives an extra \(1/35\) | 0 |
| 4 | C relapses | C now carries mass \(1/7 + 1/35 = 6/35\) and keeps it at time 4 | \(6/35\) |
| 5 | D and E relapse | D and E each carry mass \(6/35\), so the total event mass recorded at time 5 is \(12/35\) | \(12/35\) |
| 7 | F relapses | F carries mass \(6/35\) and keeps it at time 7 | \(6/35\) |
| 9 | G is censored | G’s remaining mass \(6/35\) stays to the right of all observed event times | 0 |
Therefore the estimated distribution function places masses \[ \frac{1}{7},\ \frac{6}{35},\ \frac{12}{35},\ \frac{6}{35} \] at event times \(2\), \(4\), \(5\), and \(7\), respectively. Hence \[ \widehat{F}(2)=\frac{1}{7}, \qquad \widehat{F}(4)=\frac{1}{7}+\frac{6}{35}=\frac{11}{35}, \] \[ \widehat{F}(5)=\frac{23}{35}, \qquad \widehat{F}(7)=\frac{29}{35}. \] Since \(\widehat{S}(t)=1-\widehat{F}(t)\), the Kaplan-Meier estimates are \[ \widehat{S}(2)=\frac{6}{7}, \qquad \widehat{S}(4)=\frac{24}{35}, \qquad \widehat{S}(5)=\frac{12}{35}, \qquad \widehat{S}(7)=\frac{6}{35}. \]
These values agree exactly with the product-limit formula:
| Event time \(t_{(j)}\) | Number at risk \(n_j\) | Events \(d_j\) | Conditional survival \(\frac{n_j-d_j}{n_j}\) | Kaplan-Meier estimate |
|---|---|---|---|---|
| 2 | 7 | 1 | \(6/7\) | \(\widehat{S}(2)=6/7=0.857\) |
| 4 | 5 | 1 | \(4/5\) | \(\widehat{S}(4)=\frac{6}{7}\cdot\frac{4}{5}=\frac{24}{35}=0.686\) |
| 5 | 4 | 2 | \(2/4\) | \(\widehat{S}(5)=\frac{24}{35}\cdot\frac{2}{4}=\frac{12}{35}=0.343\) |
| 7 | 2 | 1 | \(1/2\) | \(\widehat{S}(7)=\frac{12}{35}\cdot\frac{1}{2}=\frac{6}{35}=0.171\) |
The resulting Kaplan-Meier curve is \[ \widehat{S}(t) = \begin{cases} 1, & 0 \le t < 2,\\[6pt] \dfrac{6}{7}, & 2 \le t < 4,\\[8pt] \dfrac{24}{35}, & 4 \le t < 5,\\[8pt] \dfrac{12}{35}, & 5 \le t < 7,\\[8pt] \dfrac{6}{35}, & t \ge 7. \end{cases} \]
Notice that the curve is flat between event times and drops only when a relapse is observed. The median survival time is the first time at which the estimated survival probability falls to \(0.5\) or below. Here, \[ \widehat{S}(4)=0.686 > 0.5 \qquad \text{and} \qquad \widehat{S}(5)=0.343 < 0.5, \] so the estimated median survival is 5 months.
RThe survival package in R provides the standard tools for Kaplan-Meier estimation. The next example uses the built-in lung dataset from that package. This dataset contains survival times for patients with advanced lung cancer.
We will compare the Kaplan-Meier curves for males and females.
library(survival)
lung2 <- na.omit(survival::lung[, c("time", "status", "sex")])
lung2$status <- lung2$status == 2
lung2$sex <- factor(lung2$sex, levels = c(1, 2), labels = c("Male", "Female"))
head(lung2) time status sex
1 306 TRUE Male
2 455 TRUE Male
3 1010 FALSE Male
4 210 TRUE Male
5 883 TRUE Male
6 1022 FALSE MaleThe first few rows help us see the structure of the data:
time is the observed follow-up time in days.status is TRUE if death was observed and FALSE if the observation was censored.sex identifies the group used for comparing Kaplan-Meier curves.We can now fit the Kaplan-Meier estimator:
fit <- survfit(Surv(time, status) ~ sex, data = lung2)
fitCall: survfit(formula = Surv(time, status) ~ sex, data = lung2)
n events median 0.95LCL 0.95UCL
sex=Male 138 112 270 212 310
sex=Female 90 53 426 348 550The corresponding Kaplan-Meier plot is:
plot(
fit,
col = c("#1b9e77", "#d95f02"),
lwd = 2,
xlab = "Days",
ylab = "Estimated survival probability",
mark.time = TRUE,
conf.int = TRUE
)
legend(
"topright",
legend = c("Male", "Female"),
col = c("#1b9e77", "#d95f02"),
lwd = 2,
bty = "n"
)
lung dataset, grouped by sex.Some key points to notice when reading the Kaplan-Meier plot are:
For this dataset, the female curve lies above the male curve for most of the follow-up period. This suggests that females have better observed survival than males in these data. The estimated median survival times are approximately 270 days for males and 426 days for females.
This interpretation should be kept descriptive. A Kaplan-Meier curve summarizes survival experience in the observed groups, but by itself it does not establish a causal effect. Group differences may also reflect other prognostic variables besides sex.
Because the Kaplan-Meier curve is an estimate, it is important to quantify its uncertainty. The standard large-sample variance estimator is Greenwood’s formula: \[ \widehat{\mathrm{Var}}\{\widehat{S}(t)\} = \widehat{S}(t)^2 \sum_{j: t_{(j)} \le t} \frac{d_j}{n_j(n_j-d_j)}. \]
The corresponding estimated standard error is \[ \widehat{\mathrm{se}}\{\widehat{S}(t)\} = \sqrt{ \widehat{\mathrm{Var}}\{\widehat{S}(t)\} }. \]
Greenwood’s formula is easier to understand if we focus on what happens at each event time.
The multiplier \(\widehat{S}(t)^2\) appears because the variance is being translated back to the survival scale after combining the uncertainty from the product of conditional survival probabilities.
For the hand-worked example above, at time \(t=7\) we found \[ \widehat{S}(7)=\frac{6}{35}. \] Greenwood’s formula gives \[ \widehat{\mathrm{Var}}\{\widehat{S}(7)\} = \left(\frac{6}{35}\right)^2 \left( \frac{1}{7\cdot 6} + \frac{1}{5\cdot 4} + \frac{2}{4\cdot 2} + \frac{1}{2\cdot 1} \right) \approx 0.024. \] Thus the estimated standard error is approximately \[ \widehat{\mathrm{se}}\{\widehat{S}(7)\} \approx \sqrt{0.024} \approx 0.156. \]
The uncertainty is still substantial near the end of follow-up because only 2 patients were still at risk just before month 7. The tied failures at month 5 also make a noticeable contribution through the term \(\frac{2}{4\cdot 2}\). This is exactly the pattern seen in practice: the right tail of a Kaplan-Meier curve is usually the noisiest part, especially when the risk set is small or multiple events occur at the same time.
In applications, Greenwood’s formula is commonly used to construct confidence intervals for the survival function. Many software packages, including survival in R, use a transformed scale such as the log-log scale to obtain intervals that remain within \([0,1]\).
The Kaplan-Meier estimator focuses on the survival function. Another important nonparametric estimator in survival analysis is the Nelson-Aalen estimator, which estimates the cumulative hazard function \[ \Lambda(t) = \int_0^t \lambda(u)\,du. \]
If the distinct event times are \(t_{(1)} < \cdots < t_{(k)}\), with \(n_j\) subjects at risk and \(d_j\) events at time \(t_{(j)}\), then the Nelson-Aalen estimator is \[ \widehat{\Lambda}_{NA}(t) = \sum_{j:t_{(j)}\le t} \frac{d_j}{n_j}. \]
This estimator is very natural: at each event time we add the observed hazard increment \(d_j/n_j\), and then accumulate these increments over time. Thus, unlike the Kaplan-Meier estimator, which is multiplicative, the Nelson-Aalen estimator is additive.
Several basic properties are worth noting:
For the hand-worked example above, the event times are \(2\), \(4\), \(5\), and \(7\), with \[ (n_j,d_j) = (7,1),\ (5,1),\ (4,2),\ (2,1). \] Therefore \[ \widehat{\Lambda}_{NA}(7) = \frac{1}{7}+\frac{1}{5}+\frac{2}{4}+\frac{1}{2} = \frac{47}{35} \approx 1.343. \]
If we want a survival estimate based on the cumulative hazard, we can combine the Nelson-Aalen estimator with the identity \[ S(t)=\exp\{-\Lambda(t)\} \] to obtain \[ \widehat{S}_{NA}(t)=\exp\big\{-\widehat{\Lambda}_{NA}(t)\big\}. \] For the same example, \[ \widehat{S}_{NA}(7) = \exp\left(-\frac{47}{35}\right) \approx 0.261. \]
This differs from the Kaplan-Meier estimate \(\widehat{S}(7)=6/35\approx 0.171\), because the sample is small and the hazard jumps are fairly large. In larger samples, or when the increments \(d_j/n_j\) are small, the two approaches are often quite close. Indeed, \[ \log \widehat{S}_{KM}(t) = \sum_{j:t_{(j)}\le t}\log\left(1-\frac{d_j}{n_j}\right) \approx -\sum_{j:t_{(j)}\le t}\frac{d_j}{n_j} = -\widehat{\Lambda}_{NA}(t), \] so the Kaplan-Meier and Nelson-Aalen estimators are closely related.
The Nelson-Aalen estimator is especially useful because cumulative hazards are often easier to manipulate mathematically than survival functions. It also appears naturally in later topics such as the log-rank test and the Cox proportional hazards model.
The Kaplan-Meier estimator is the basic nonparametric tool for estimating a survival curve from right-censored data. It is obtained by multiplying conditional survival probabilities across event times. Its validity depends crucially on the assumption of independent censoring. In practice, the estimator is easy to compute by hand for a small dataset and easy to fit in R using survfit(). Greenwood’s formula provides the standard variance estimator and helps explain why uncertainty increases when only a few subjects remain at risk. The Nelson-Aalen estimator gives a closely related nonparametric estimator of the cumulative hazard function and provides another important way to summarize censored survival data.