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The Kaplan–Meier estimator requires independent censoring: conditional on modeled covariates, censoring must carry no information about the future event time. Censoring at a common administrative data cutoff satisfies this condition when calendar entry time is independent of prognosis. Under that assumption, the month-24 analysis has valid administrative censoring despite unequal follow-up caused by staggered entry.

A period survival analysis estimates survival under the recent mortality conditions operating during a fixed calendar period. For calendar year 2022, each cancer patient's contribution begins at the later of diagnosis and 1 January 2022, which treats earlier diagnoses as left-truncated at the start of the year. Follow-up ends at death, loss to follow-up, or 31 December 2022, so surviving records are right-censored. Combining these risk-set contributions, commonly through a Kaplan–Meier estimator, constructs the 2022 period survivor function; records from several diagnosis cohorts may contribute different segments of follow-up.

In the Kaplan–Meier estimator, replace the ordinary risk set at event time $t$ byThe product-limit factor remains $1-d_j/|R(t_j)|$, but only individuals who have entered and not yet exited contribute to its denominator.

A potential event time is a support point permitted in the discrete event-time distribution used to derive the Kaplan–Meier estimator by maximum likelihood estimation. The support must include every observed exact event time. Additional points with a nonempty risk set but no observed event receive zero fitted event mass.

A grid of potential right censoring times includes every observed censoring time. Intervals between consecutive grid points have no internal censoring, so their Kaplan–Meier estimator factors telescope to a single ratio of event-free survivors to individuals observed across the interval. This gives the same estimator when endpoint risk sets are counted consistently.

The restricted mean survival time to a fixed cutoff $\tau$ is $\mathbb{E}[\min (T,\tau )]={\int }_0^{\tau }S(t)dt$ for a nonnegative event time. This follows by the tail integral formula for moments. Unlike the unrestricted mean, it uses no tail beyond $\tau$; under independent censoring it can be estimated by integrating a Kaplan–Meier estimator over adequately observed follow-up.

With a common excess hazard, let ${\overline{h}}_B={\sum }_i{Y}_i{h}_B^{(i)}/{\sum }_i{Y}_i$ be the known background hazard averaged over the current risk set. Estimate the excess cumulative hazard by subtracting $\int {\overline{h}}_B$ from the Nelson–Aalen estimator. Applying the product-limit construction gives the displayed estimator: the continuous background correction multiplies the ordinary Kaplan–Meier estimator. Its event jump is $1-d_j/r_j$, while between events it grows at rate ${\overline{h}}_B{\hat{F}}_E$. Hence the unconstrained curve can increase, even if the true excess hazard is nonnegative. Replacing its jump factor by $\exp (-d_j/r_j)$ gives the alternative exponential cumulative-hazard estimator; these versions are close for small event fractions but are not identical.

The Kaplan–Meier estimator is self-consistent under fractional allocation of censored event times according to its own conditional fitted event distribution. Adding the corresponding fractional event counts and risk set weights leaves the estimator unchanged.

If a common Kaplan–Meier estimator $\hat{S}$ is fitted and the same data are transformed to $u_i=-\log \hat{S}(x_i)$ with censoring indicators retained, event ordering and risk sets are preserved under compatible tie conventions. At a transformed event time $u_j$, the residual product-limit estimate equals $\hat{S}(x_j)=e^{-u_j}$. Consequently agreement of the pooled residual curve with the exponential target is largely built into the construction. Separate group curves or a model fitted with covariates can reveal differences concealed by the pooled fit. Residuals become infinite if the fitted survivor reaches zero.

// Target: survival-analysis.bigb

At a failure time with $d$ events among $r$ at risk, the Nelson–Aalen estimator adds $u=d/r$, whereas minus the logarithm of the Kaplan–Meier estimator adds $-\log (1-u)$. For $0\le u<1$,Thus small event fractions give close cumulative hazard estimates. Large risk sets alone do not suffice when a large fraction fail together. If all remaining individuals fail, the logarithmic estimate becomes infinite while the Nelson–Aalen jump is one.

Survival analysis studies the probability distribution of a survival time: how long from a specified starting point until a specified occurrence. The occurrence need not be death. It might be a machine's first breakdown, a customer's first renewal, or a person's recovery. The meaning of "survival" is simply that the chosen occurrence has not happened yet. Both the starting point and the outcome must be defined before interpreting a survival function.

The distinctive difficulty is incomplete observation. A machine still working when observation stops supplies a lower bound on its survival time, rather than an observed breakdown time. Right censoring lets us retain that information. The Kaplan–Meier estimator estimates a survival function using the changing risk sets, without choosing an exponential distribution or another particular survival distribution. A hazard function instead describes the current event rate among those still event-free; its cumulative hazard connects those local rates to the whole survival function.

This framework turns different follow-up lengths into usable evidence about waiting times. It supports questions such as the chance of remaining event-free for another year, or the restricted mean survival time within an observed horizon. Lu Tian's Stanford teaching materials introduce these interpretations in Survival Distributions, Hazard Functions, Cumulative Hazards, Section 1.1.

For a nonnegative survival time $T$, the survival function iswhere $F$ is the cumulative distribution function. Thus $S(t)$ is the probability that the chosen occurrence is still in the future at time $t$. We consistently use the strict inequality $T>t$: a probability mass at $t$ has already left the survival function at $t$ itself. With this convention $S$ is a right-continuous function and a nonincreasing function taking values in $[0,1]$. If $\mathbb{P}(T=0)=0$, then $S(0)=1$; if $T$ is finite with probability one, then $S(t)\to 0$ as $t\to \infty$.

For $S(a)>0$ and $b>a$, conditional probability givesFor example, $S(2)=0.8$ and $S(5)=0.6$ mean that $0.6/0.8=0.75$ of those still event-free at year two remain event-free through year five. Their intervening event probability is $0.25$, whereas the original population's probability of an occurrence in $(2,5]$ is $0.8-0.6=0.2$. Conditioning changes the denominator.

The area under the survival function up to a horizon gives the restricted mean survival time. A positive terminal plateau in a plotted Kaplan–Meier estimator does not establish that the occurrence will never happen: terminal censoring and survival-mean identifiability explains why the unobserved tail remains undetermined.

If the last observed follow-up is a censoring time and the Kaplan–Meier estimator remains positive there, the observed curve does not determine the remaining tail area. Many possible completions give different full means, including finite and infinite ones. A horizontal plotting extension is not proof that the true mean is infinite. If the final risk set is exhausted by events, its Kaplan–Meier estimator factor is zero and the conventional fitted curve has finite area. This includes a unique last individual having an event, but an event tied with terminal censoring can leave a positive fitted survivor value when events are processed before censoring. The random endpoint still does not establish a population support bound. A restricted mean survival time avoids unsupported tail completion.