Kaplan–Meier and Nelson–Aalen Estimation

Suppose we follow patients after a hospital discharge and record the time until readmission. Some patients are readmitted during follow-up; others move away, withdraw, or reach the end of the study without a readmission. We cannot simply average the observed times, because the censored patients have unknown event times. We need an estimator that uses exactly the information we have: each patient contributes to the risk set until the last time we know they were event-free.

In the previous chapter we separated event times, censoring times, survival functions, and hazard functions. We now turn those ideas into two nonparametric estimators. The Kaplan-Meier estimator estimates the survival function $S(t)=P(T>t)$ directly as a product of conditional survival probabilities. The Nelson-Aalen estimator estimates the cumulative hazard $H(t)$ by adding up estimated hazard increments at observed event times.

Both estimators are built from the same object: the risk set at each event time. That shared structure lets us derive them side by side. Kaplan-Meier answers “what fraction survived past this time?”, while Nelson-Aalen answers “how much hazard has accumulated by this time?”. We will see that the two answers are closely related but not identical, especially when events are tied or the risk sets are small.

Kaplan-Meier and Nelson-Aalen make no parametric assumption about the shape of the survival distribution. We do not need to choose an exponential, Weibull, or log-normal model before looking at the data. The estimated curves follow the observed event times, so they are excellent first descriptions of a time-to-event sample. This makes them useful for exploratory analysis, for checking whether a parametric model is plausible, and for communicating results to audiences who want an empirical survival curve rather than a regression coefficient.

Both estimators handle right censoring in a principled way. A censored subject is not treated as if they survived forever, and they are not dropped from the analysis at time zero. Instead, they remain in the risk set until their censoring time and then leave. Kaplan-Meier curves are a standard way to summarize right-censored follow-up in clinical trials. The same risk-set construction can describe equipment lifetimes or time to customer churn under the stated censoring and sampling assumptions.

The estimators are also simple and fast. A production implementation sorts the observed times and computes risk-set counts and event counts, which is $O(n\log n)$ in the sample size. The output is easy to interpret: a step function for survival, a step function for cumulative hazard, pointwise confidence intervals, and medians or quantiles when the curve reaches them. The Breslow baseline cumulative-hazard estimator in a Cox model generalizes the Nelson-Aalen construction by weighting each risk-set member by its fitted relative risk; it reduces to Nelson-Aalen when all relative risks are one. Cumulative-hazard estimates also enter counting-process diagnostics, such as Cox martingale residuals, through the fitted cumulative hazard.

Kaplan-Meier and Nelson-Aalen are marginal estimators. They describe the survival experience of the entire sample, not the effect of a treatment or covariate after adjustment. If treated and untreated patients differ in prognostic factors such as age or disease severity, comparing two Kaplan-Meier curves can mix the treatment effect with those differences. Covariate adjustment requires an additional modeling, weighting, or standardization step; the Cox proportional hazards model we will meet next is one such approach.

For these unadjusted marginal curves, censoring must leave the observed risk sets representative of the event risk in the population. Independence of event and censoring times is a simple sufficient assumption. Independence only after conditioning on predictors of both failure and censoring can require stratification or weighting. Selective dropout of patients with higher subsequent event risk can leave a healthier observed risk set and bias estimated survival upward. Competing risks are another failure mode: if a patient dies from another cause, coding that death as censored and interpreting one minus Kaplan–Meier as cumulative incidence can overestimate the cumulative incidence of the event of interest.

The tails of the curves are fragile. As follow-up progresses, fewer subjects remain at risk, so tail estimates depend on less information. Absolute Greenwood standard errors and pointwise interval widths need not increase monotonically. If the largest observed time is censored, the Kaplan-Meier curve never reaches zero, and we have no direct estimate of survival beyond the last follow-up. A median may not be reached at all, even in a large study, if follow-up is short. Ties and small risk sets also make the finite-sample difference between Kaplan-Meier and the exponential transform of Nelson-Aalen more noticeable.

The formulas in this chapter look different at first, one a product and the other a sum. The shared logic is easier to see if we start from the data and ask a single question at each event time: who is still eligible to have the event? The answer is the risk set. Once we can count the risk set and the events that occur inside it, both estimators follow almost immediately.

We begin with the observed data. For subject $i$,

• $T_i$ is the true event time,
• $C_i$ is the censoring time,
• $Y_i=\min(T_i,C_i)$ is the observed follow-up time,
• $\Delta_i=\mathbf{1}\{T_i\le C_i\}$ is the event indicator, equal to $1$ if we observed the event and $0$ if the observation was censored.

We assume independent draws from the target population, strictly positive event times, and right censoring. For the basic marginal estimators, independence of $T_i$ and $C_i$ is a sufficient censoring assumption, with positive probability of remaining observed throughout the time range we estimate. Random treatment assignment alone does not establish this assumption about follow-up.

At any time $t$, the risk set is the set of subjects still under observation and event-free just before $t$:

Its size is $n(t)=|\mathcal{R}(t)|$. This definition is the bookkeeping center of the whole chapter. A subject who has already had the event is no longer eligible to have it again, and a subject who has been censored before $t$ cannot contribute information about an event at $t$. Both leave the risk set. A subject who is censored exactly at $t$ is a boundary case, and we will handle it by convention below.

If the distinct observed event times are

then at event time $t_j$ we write

for the number at risk and

for the number of events at that time. The ratio $d_j/n_j$ is the empirical event fraction among those who were still eligible at $t_j$. That fraction will become the Nelson-Aalen increment, and its complement will become the Kaplan-Meier factor.

It is also where censoring enters: a censored subject remains in $n_j$ up to and including their censoring time, then leaves. Censoring does not create an event, but it changes who is available for later events.

Tied events occur when $d_j>1$. The product-limit estimator handles ties by grouping all events at $t_j$ into one conditional factor. Tied event and censoring times require a convention. We use the standard convention that events are processed before censorings at the same time. A subject censored at $t_j$ is therefore counted in $n_j$ for the event at $t_j$, and is removed only after that event time. Removing those censorings first would make the denominator smaller, increase the event fraction, and lower the estimated survival under a different convention.

This convention is worth checking in software. The tied-data check below confirms events before censorings for the installed SciPy and statsmodels implementations; consult documentation when exact tie behavior matters. The convention matters most when the risk set is small or when many subjects share the same recorded time.

Kaplan-Meier estimates survival by chaining conditional probabilities. At event time $t_j$, suppose $n_j$ subjects are at risk and $d_j$ events occur. The estimated probability of surviving past $t_j$, given that a subject was still at risk just before $t_j$, is

where:

• $d_j$: the number of events at event time $t_j$
• $n_j$: the number of subjects at risk at event time $t_j$

If no events occur at $t_j$, then $d_j=0$ and the factor is $1$; censoring times do not create factors. If all subjects at risk have an event, the factor is $0$ and the estimated survival drops to zero. In between, the factor is the fraction of the current risk set that survives the event time.

Multiplying these conditional factors across all event times up to $t$ gives the Kaplan-Meier estimator:

with the empty product defined as $1$ for times before the first event. The word product-limit refers to this product of estimated conditional survival probabilities. It is the nonparametric maximum likelihood estimator of the survival function under independent right censoring.

where:

• $\hat S_{\mathrm{KM}}(t)$: the estimated survival probability of surviving past time $t$
• $t_j$: the $j$-th distinct observed event time
• $d_j$: the number of events at event time $t_j$
• $n_j$: the number of subjects at risk at event time $t_j$
• $1-d_j/n_j$: the estimated conditional probability of surviving past $t_j$ given survival to just before $t_j$
• the product over $j:t_j\le t$: multiply the conditional factors for all event times up to and including $t$

We can read the formula as a sequence of updates. Start with $\hat S(0)=1$. At each event time, multiply the current survival probability by the fraction of the risk set that did not have an event. Censoring changes the next risk set, but it does not directly multiply the curve by anything.

The variance of the Kaplan-Meier estimator is usually estimated with Greenwood’s formula. Greenwood’s cumulative variance factor is

where:

• $\hat\sigma^2(t)$: Greenwood’s cumulative variance factor for the survival curve up to time $t$, which combines with the squared survival estimate to give the variance of $\hat S_{\mathrm{KM}}(t)$
• $d_j$: the number of events at event time $t_j$
• $n_j$: the number of subjects at risk at event time $t_j$
• $n_j-d_j$: the number at risk who do not have the event at $t_j$

Then

where:

• $\widehat{\operatorname{Var}}\bigl(\hat S_{\mathrm{KM}}(t)\bigr)$: the estimated variance of the Kaplan-Meier survival estimate at time $t$
• $\hat S_{\mathrm{KM}}(t)$: the Kaplan–Meier survival estimate at time $t$
• $\hat\sigma^2(t)$: Greenwood’s cumulative variance factor at time $t$

To see where this comes from, take logs:

For a binomial proportion $\hat p_j=d_j/n_j$, the delta method gives

where:

• $\hat p_j=d_j/n_j$: the estimated event fraction at event time $t_j$
• $p_j$: the event probability in the local binomial working model, conditional on membership in the current risk set at $t_j$
• $n_j$: the number of subjects at risk at event time $t_j$
• $d_j$: the number of events at event time $t_j$

Plugging in $\hat p_j=d_j/n_j$ yields $d_j/[n_j(n_j-d_j)]$. This binomial calculation is a sketch of the local variance contribution. The full large-sample derivation uses counting-process martingales, since earlier events change later risk sets. Summing the contributions and applying the delta method once more to $S=\exp(\log S)$ gives the factor $\hat S^2$. The denominator $n_j-d_j$ shows why Greenwood’s formula fails when $n_j=d_j$: at that point the curve hits zero and the variance formula is not finite.

For an absolutely continuous event-time distribution, at times with positive survival where the following limit exists, the hazard is the instantaneous event rate among those still at risk:

where:

• $h(t)$: the hazard function, the instantaneous event rate at time $t$ among those still at risk
• $T$: the true event time
• $\Delta t$: a small time increment
• $P(t\le T<t+\Delta t\mid T\ge t)$: the conditional probability of an event in $[t,t+\Delta t)$ given survival to time $t$

The cumulative hazard is

where:

• $H(t)$: the cumulative hazard up to time $t$
• $h(u)$: the hazard function at time $u$

This integral describes a continuous-time hazard with no initial event mass. For inherently discrete event times, cumulative hazard is instead a sum of conditional hazard jumps and need not equal $-\log S(t)$. The implementation below groups events that share a recorded time; it does not split a tied group into a sequence of separate failures.

Nelson-Aalen estimates $H$ by accumulating estimated hazard increments at the observed event times. At $t_j$, the observed event fraction among those at risk is $d_j/n_j$, so we set

where:

• $\Delta \hat H(t_j)$: the estimated hazard increment at event time $t_j$
• $d_j$: the number of events at event time $t_j$
• $n_j$: the number of subjects at risk at event time $t_j$

Adding these increments gives

with $\hat H(0)=0$, where:

• $\hat H_{\mathrm{NA}}(t)$: the Nelson–Aalen cumulative hazard estimate at time $t$
• $t_j$: the $j$-th distinct observed event time
• $d_j$: the number of events at event time $t_j$
• $n_j$: the number of subjects at risk at event time $t_j$
• the sum over $j:t_j\le t$: add the hazard increments for all event times up to and including $t$

This is a nondecreasing step function. It jumps at event times; censoring alone creates no jump. The jump size is large when many events occur among few subjects at risk, and small when one event occurs among a large risk set.

A grouped Greenwood-type variance estimate is

This is a binomial plug-in, Greenwood-type estimate. The Aalen counting-process estimate uses $\sum d_j/n_j^2$ instead. The two differ by the factor $1-d_j/n_j$ in each contribution. A smaller reported variance does not by itself establish better uncertainty coverage, especially with few subjects at risk.

where:

• $\widehat{\operatorname{Var}}\bigl(\hat H_{\mathrm{NA}}(t)\bigr)$: the estimated variance of the Nelson–Aalen cumulative hazard at time $t$
• $d_j$: the number of events at event time $t_j$
• $n_j$: the number of subjects at risk at event time $t_j$

A local plug-in argument treats the event count as binomial given the current risk set, with $n_j$ trials and event probability estimated by $d_j/n_j$. This motivates the contribution; it is not an exact finite-sample variance proof for the accumulated estimator. The variance of $d_j/n_j$ is approximately

Nelson-Aalen is closely related to Kaplan-Meier, but the relationship is not exact in finite samples. Since

and $\log(1-x)\approx -x$ for small $x$, we have

The approximation is useful on a time range where the event fractions $d_j/n_j$ and their accumulated second-order error are small. Large risk sets with few events per time make the individual fractions small. It is not an identity. For $0<x<1$, where $x$ represents an event fraction $d_j/n_j$,

so the product-limit estimator lies below the exponential transform of Nelson–Aalen whenever events occur:

The exact product-integral relationship is

not $\exp(-\hat H_{\mathrm{NA}})$. We should not report $\exp(-\hat H_{\mathrm{NA}})$ as if it were Kaplan–Meier. Their relative difference tends to zero in a small-increment limit where the accumulated second-order error also vanishes. Heavy ties or small risk sets can make the difference noticeable.

The finite-time gap has a simple source: for every event fraction $x$ strictly between zero and one, the product-limit factor $1-x$ is smaller than the exponential factor $e^{-x}$. We can display that inequality directly.

The formulas above imply a few structural facts that are useful when reading software output or checking a plot:

• $\hat S_{\mathrm{KM}}(t)$ is a right-continuous, nonincreasing step function with values in $[0,1]$ and $\hat S_{\mathrm{KM}}(0)=1$.
• $\hat H_{\mathrm{NA}}(t)$ is a right-continuous, nondecreasing step function with $\hat H_{\mathrm{NA}}(0)=0$.
• Both estimators change only at observed event times. A censoring time changes future risk sets but does not create a jump.
• Ties are handled by grouping: the Kaplan–Meier factor is $1-d_j/n_j$, and the Nelson–Aalen increment is $d_j/n_j$.
• The exponential transform satisfies $\exp(-\hat H_{\mathrm{NA}}(t))\ge \hat S_{\mathrm{KM}}(t)$, with exact equality only when no events have occurred at or before time $t$.
• Greenwood’s variance requires $n_j>d_j$. If $n_j=d_j$, the survival estimate reaches zero and the variance formula is not finite at that point.

The Kaplan-Meier intervals below describe uncertainty at a fixed time; they do not give simultaneous coverage for the whole curve. We use the log-log transformation because its interior limits stay within $[0,1]$, whereas a symmetric normal interval can extend outside that range. Its coverage accuracy still depends on censoring, sample support, and distance from the endpoints. Let

For $0<\hat S(t)<1$, a pointwise $(1-\alpha)$ interval is

where:

• $\hat S(t)$: the Kaplan–Meier survival estimate at time $t$
• $\hat\sigma(t)$: the square root of Greenwood’s cumulative variance factor, $\sqrt{\sum_{j:t_j\le t}d_j/[n_j(n_j-d_j)]}$
• $z_{1-\alpha/2}$: the standard normal quantile, such as $1.96$ for a 95% interval
• $\alpha$: the significance level (one minus the confidence level)

The log-log transformation is undefined at $\hat S(t)=0$ or $1$. We return missing limits at these boundaries instead of treating a zero-width display as statistical certainty. The known starting value $S(0)=1$ is separate because event times are strictly positive. These are approximate pointwise intervals. Simultaneous coverage over a specified, adequately supported time interval requires a method calibrated for the whole curve, such as a Hall–Wellner band; separate pointwise bootstrap percentiles would still be pointwise.

For $\hat H(t)>0$ in a supported regime with a reliable nondegenerate variance and normal approximation, an approximate pointwise log-transformed interval for $H(t)$ is

where:

• $\hat H(t)$: the Nelson–Aalen cumulative hazard estimate at time $t$
• $\hat\sigma_H(t)=\sqrt{\sum_{j:t_j\le t} d_j(n_j-d_j)/n_j^3}$: the standard error from the chosen Greenwood-type variance estimate
• $z_{1-\alpha/2}$: the standard normal quantile for a $(1-\alpha)$ interval, such as $1.96$ for a 95% interval
• $\alpha$: the significance level (one minus the confidence level)

The displayed log interval is undefined at $\hat H(t)=0$. A no-event estimate at a positive follow-up time does not imply zero uncertainty; inference there needs a boundary-specific method. The chosen binomial variance can also vanish after every member of a tied risk set fails. A resulting zero-width interval is not justified merely because the hazard estimate is positive. The known origin $H(0)=0$ is a separate structural value.

The median survival time is the first time the estimated survival curve reaches or falls below one half:

Because the curve is a step function, the median is an observed event time, not an interpolated value between steps. If the curve never reaches or falls below $0.5$ by the last follow-up, the median is not estimable. We should report “median not reached” rather than a fabricated number, together with the last follow-up time and the number still at risk. A confidence interval for the median can be obtained by inverting pointwise survival intervals, as in the Brookmeyer–Crowley approach, or by resampling observed time–event-indicator pairs and recomputing the Kaplan-Meier median. Adequate follow-up around the median is still needed; some resamples or interval endpoints may not reach it. A point estimate alone does not show its sampling uncertainty.

Tail limitations follow directly from the risk-set definition. At late times $n_j$ is small, so each event removes a larger fraction of the remaining survival estimate. The absolute Greenwood standard error need not grow: it also depends on the survival estimate, which is falling. If the largest observed time is censored, the curve remains above zero and we have no direct estimate beyond that time. A flat empirical tail does not establish a cure fraction. Beyond the last observed follow-up, there are no further observations to support its extension. If $n_j=d_j$, the curve drops to zero and Greenwood’s denominator vanishes. Sparse late risk sets limit tail precision. Tied times require the recorded-time convention and time-resolution assumptions to be checked; informative censoring can bias the curve throughout follow-up.

The formulas are compact, so it is worth slowing down and watching the risk set change row by row. We now work through a small failure table by hand. The observed times and event indicators are:

The distinct event times are $2,3,6,8,10$. We process tied events and censorings using events first. At each event time, the first question is always the same: who is still in the risk set?

At $t=2$, all ten subjects are at risk and one event occurs:

Thus $\hat S(2)=0.9$ and $\hat H(2)=0.1$. The risk set has shrunk by one because subject 1 has now had the event.

At $t=3$, subject 1 has already had an event, so nine subjects remain at risk. Two events occur at time 3.

The updated estimates are

At $t=5$ subject 4 is censored. This does not create a step, but it removes subject 4 from later risk sets. That is the only role of censoring here: it changes who is available for future events.

At $t=6$, subject 4 is gone, but subject 6 is censored at time 6 and remains in the risk set for the event at time 6 under our convention. Six subjects are at risk and one event occurs:

Therefore

At $t=8$, subjects 5 and 6 have left after time 6, so four subjects remain at risk. One event occurs:

Thus

At $t=10$, two subjects remain at risk and one event occurs:

So

At $t=12$, the one remaining subject is censored, so the Kaplan-Meier curve remains flat at $0.21875$ after time 10. It never reaches zero because the last subject leaves the study without an observed event.

The Nelson–Aalen estimate at time 10 is $1.238889$, and

This is above the Kaplan–Meier value $0.21875$. The gap is the finite-sample effect of using $1-d_j/n_j$ rather than $e^{-d_j/n_j}$ at each event time. It is not a bug, and it is one reason we should not label $\exp(-\hat H_{\mathrm{NA}})$ as Kaplan–Meier.

For uncertainty, the Greenwood increments are

Their sum is approximately $0.659524$. At the final event time,

The median is the first time the curve is at or below $0.5$. At time 6 the estimate is $0.583333$, and at time 8 it is $0.4375$, so the estimated median is $8$. This is a useful check on the risk-set arithmetic: the median is read from the step function, not computed by interpolating between event times.

The times argument contains positive follow-up times in a consistent unit. It should not contain calendar dates unless they have been converted to elapsed time from a meaningful origin. The events argument is the event indicator: $1$ for an observed event and $0$ for right censoring. Missing times or missing event indicators cannot be handled by the basic estimator; they require a missing-data strategy before fitting.

The tie convention is part of the estimator, not a cosmetic detail. Our function counts every subject with $Y_i\ge t_j$ in $n_j$, including subjects censored exactly at $t_j$, and counts only observed events in $d_j$. This implements events before censorings. The level argument in km_loglog_ci controls the pointwise confidence level. The usual censoring and sampling assumptions and a sufficiently supported large-sample regime are needed for the Greenwood/delta-method approximation, particularly away from risk exhaustion and transform boundaries.

In SciPy, CensoredData.right_censored receives a Boolean censoring mask, so we pass indicators == 0; ecdf(...).sf then represents the estimated survival function. statsmodels SurvfuncRight instead receives the event indicators directly, with 1 meaning an event. Keeping these opposite indicator conventions explicit prevents a reference check from silently reversing the data.

fit_km_na returns a table at the distinct event times. The columns n_at_risk and events expose the risk-set construction, which is often more informative than the final curve alone. The columns S_km and H_na are the two estimators. The columns greenwood_sum and var_H_sum are cumulative variance components, so their square roots give standard errors after multiplying by the appropriate factors.

step_value evaluates a right-continuous step function at a scalar query time, given sorted event times and an aligned value array. The reference comparison above instead evaluates the library curves on the fitted event-time grid. km_loglog_ci returns pointwise lower and upper survival limits. For a full curve, evaluate the limits at event times and shade only the finite limits, remembering that the resulting band is pointwise rather than simultaneous. Evaluating a step function beyond observed follow-up does not establish survival there.

The SciPy survival object provides evaluate for step-function values. The statsmodels arrays surv_times, surv_prob, n_risk, and n_events expose the event-time curve and the counts used in our comparison. The from-scratch table makes the same risk-set logic visible for inspection.

The estimates are already computed. We now use plots to expose the mechanism: where the steps occur, how censorings enter, how the product-limit and exponential transforms differ, and how the shrinking risk set limits tail interpretation.

The standard-error plot describes this one simulated cohort. Its shape depends on both the risk set and the falling survival estimate; a rise followed by a decline is not a universal pattern.

The risk-set curve starts at 240 and falls as events and censorings accumulate. The standard-error curve is not monotone: its path depends on both the risk set and the falling survival estimate. Its last value is undefined and is omitted. One event removes a larger fraction of the remaining estimated survival when ten subjects are at risk than when two hundred are at risk. A flat Kaplan-Meier tail means that no further events were observed; it does not establish a cure fraction or show that the hazard has stopped. Check the number still at risk, because a flat segment can occur with either many or few subjects under observation.

Under the chapter's independent right-censoring and sampling assumptions, Kaplan-Meier provides a nonparametric descriptive survival curve. It suits summarizing time-to-event data, estimating medians or other quantiles, comparing unadjusted groups, and supplying the empirical curve behind a more structured model. Nelson–Aalen is the natural choice when the cumulative hazard is the target, when building the baseline cumulative hazard for a Cox model, or when working with counting-process representations, and the two answer complementary survival and cumulative-hazard questions. Alternatives become preferable when description is not enough: to adjust for covariates, the Cox model extends the risk-set approach through partial likelihood, while a parametric accelerated failure-time model relates covariates to survival time using a censored-data likelihood; when a competing event can preclude the event of interest, coding that competing event as censored and using one minus the ordinary Kaplan–Meier estimate can overestimate the cumulative incidence of interest, so cumulative incidence should instead be estimated with a competing-risk method such as Aalen–Johansen, or reconstructed from all relevant cause-specific hazards.

Data preparation matters more than the estimator itself. Times must be measured from a well-defined origin such as diagnosis, treatment start, or account creation, and the event indicator must be $1$ for the event of interest and $0$ for right censoring; missing times or missing event indicators cannot be silently ignored. Staggered calendar recruitment can simply produce different administrative right-censoring durations when each subject’s clock starts at enrollment. Delayed entry arises when observation begins after the chosen analysis-time origin, such as enrollment at age 60 on an age time scale. Then risk sets must include only subjects who have entered and remain observed, with an appropriate truncation assumption. This basic implementation does not handle delayed entry. The independent-censoring assumption deserves explicit discussion: in a randomized trial, administrative censoring may be roughly independent of prognosis, whereas in an observational study censoring may be informative if patients who worsen transfer care, stop responding, or are lost to follow-up. No automatic diagnostic can prove independence, but examining censoring patterns over time and across covariates, comparing censored and uncensored subjects on baseline characteristics, and performing sensitivity analyses are reasonable steps, with suitable stratification, censoring weights, or other adjustment considered when censoring and failure depend on shared covariates.

Reporting should make the risk-set mechanics visible. Include the number at risk at several follow-up times, the number of observed events, the amount and pattern of censoring, the tie convention, and the method used for confidence intervals, because those counts show how many observations support the tail. When the curve never reaches or falls below $0.5$, report the median as not reached together with the last follow-up time rather than extrapolating from a flat tail; when the curve reaches zero because $n_j=d_j$, state that Greenwood’s variance is undefined at that point instead of plotting a misleading zero-width interval. Group comparisons based on Kaplan-Meier curves are descriptive, and an ordinary unstratified log-rank test is still unadjusted: if groups differ in baseline covariates, the visual difference may be confounded, so the Cox model is the natural next step. Computationally both estimators are inexpensive. A production implementation sorts the data and processes each event time once, costing $O(n\log n)$ time and $O(n)$ memory, while the readable implementation in this chapter rescans the sample at each event time; the dominant practical cost is not computation but careful definition of the event, the time origin, the censoring mechanism, and the risk set.

Kaplan-Meier and Nelson-Aalen are the two foundational nonparametric estimators in survival analysis. Both are built from risk sets. At each event time $t_j$, $n_j$ subjects are at risk and $d_j$ events occur. Kaplan-Meier multiplies conditional survival factors $1-d_j/n_j$ to estimate $S(t)$. Nelson-Aalen adds hazard increments $d_j/n_j$ to estimate $H(t)$.

The estimators are closely related but not identical. Kaplan-Meier is the product-limit estimator, and Nelson-Aalen is the additive cumulative-hazard estimator. The exponential transform $\exp(-\hat H_{\mathrm{NA}})$ approximates Kaplan–Meier over a time range where event fractions and their accumulated second-order error are small, but it is not the same estimator. With ties or small risk sets, the difference can be material.

We used Greenwood’s formula for Kaplan-Meier and a Greenwood-type binomial plug-in variance for Nelson-Aalen, alongside the alternative Aalen variance. Intervals based on them are approximate and pointwise; transformed limits are undefined at the boundary values discussed above. The median is the first time the survival curve reaches or falls below $0.5$; if the curve stays above $0.5$ throughout follow-up, the median is not estimable. Tail estimates are limited by the number at risk, the amount of censoring, and the last follow-up time. Independent censoring and adequate follow-up support are key conditions for consistent estimation under the sampling assumptions; finite-sample bias can remain. Estimating cumulative incidence in the presence of competing events requires an appropriate competing-risk method, and delayed entry requires risk sets that account for entry times.

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