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If the final individual experiences an event at $t_1$, the Kaplan–Meier estimator acquires the factor $1-1/1=0$. ThusThe conventional fitted curve is zero thereafter. Its area gives the finite plug-in estimateThe area is a sum of rectangles: if $a_0=0<a_1<\cdots <a_m=t_1$ are the event times, it equals ${\sum }_{j=0}^{m-1}\hat{F}(a_j)(a_{j+1}-a_j)$. Censoring times within a constant segment do not alter its area.

Both terminal estimates have poor reliability because the last risk set has size one: one observed outcome determines whether the entire remaining fitted tail stays positive or collapses to zero. The zero tail is an empirical endpoint convention, not evidence that every future member of the population must fail by $t_1$. With earlier right censoring, the displayed finite area can be especially sensitive to this last event and does not remove uncertainty about the population tail. In the final-censoring case the full mean lacks a determined tail area; in the final-event case the conventional full fitted mean is finite and calculable, but should be reported with this limitation. A restricted mean survival time at a prespecified cutoff within reliable follow-up often avoids the unstable terminal extrapolation.

Let $a_1<a_2<\cdots$ be the distinct observed event times, $r_j$ the risk set size immediately before $a_j$, and $d_j$ the events there. Under independent censoring, the Kaplan–Meier estimator isIt starts at one, jumps down at events, and is unchanged by a pure censoring time; right censoring removes people from subsequent risk sets. If events and censorings coincide, the convention here processes events before censoring, so those censored at the recorded time are included just before the event. The estimated median is the first time the curve reaches or falls below $1/2$. If it never does during observation, the median is not estimable from the observed curve without tail assumptions.

With one remaining individual censored at $t_1$, no event factor is introduced. HenceThis is the terminal observed step value. With an empty risk set after $t_1$, continued horizontal plotting is a convention and supplies no information about the true later survivor function. The tail rests on one person and is imprecise; this is terminal censoring and survival-mean identifiability. If ${\varphi }_0>0$, the unrestricted mean cannot be obtained nonparametrically from this censored tail: extending the last positive step to infinity would give an infinite area, which does not establish an infinite population mean. A restricted mean survival time up to a suitably supported finite cutoff remains estimable.

A proportional hazards model assigns individual $i$ the hazard function $h_i(t)=h_0(t){\varphi }_i$, where $h_0$ is the baseline hazard for multiplier one and ${\varphi }_i>0$ is a time-constant hazard multiplier. In a Cox proportional-hazards model, ${\varphi }_i=e^{x_i^T\beta }$ for a covariate vector $x_i$ and coefficient vector $\beta$; ratios $h_i(t)/h_k(t)={\varphi }_i/{\varphi }_k$ are constant whenever the baseline is nonzero.

Assume independent individuals, independent censoring conditional on the covariates, and no tied events. Immediately before an event at $t$, condition on its observed risk set $R(t)$ and one event in a small time interval. The individual event probability is $h_0(t){\varphi }_{i}dt+o(dt)$, while the total is $h_0(t){\sum }_{k\in R(t)}{\varphi }_{k}dt+o(dt)$. Their limiting ratio is ${\varphi }_i/{\sum }_{k\in R(t)}{\varphi }_k$. Multiplication over observed events gives the Cox partial likelihoodThe unspecified baseline hazard cancels. This is the usual successive conditional event contribution defining the partial likelihood, not the full event-time likelihood conditioned simultaneously on all event times.

For the four individuals the successive event risk sets are $\{a,b,c,d\}$, $\{c,d\}$ and $\{d\}$. Individual $b$ leaves at its right censoring time and contributes no event numerator. ConsequentlyThe last event contributes one and carries no further relative-risk information.

The estimate remains $(r-d)/r$. Individuals right-censored at $t_R$ are observed throughout the preceding interval, so they contribute to each risk set until the end. They must not be removed from the denominator at $t_L$.

If events and right censoring share the recorded time $t_R$, use the usual Kaplan–Meier estimator convention: count events first, using a risk set that still includes those censored at that time, and then remove the censored observations. With no earlier censoring, the product of event factors over the interval telescopes to $(r-d)/r$.

An individual is right-censored at $t$ when follow-up establishes that the event has not occurred through $t$, but does not reveal the subsequent event time. The observed information is $T>t$ for the continuous-time convention.

Knowing only the total number censored inside an interval does not specify the risk sets at the event times. For example, start with two individuals, one event and one censoring. If censoring precedes the event, the Kaplan–Meier estimator contributes $1-1/1=0$; if the event precedes censoring, it contributes $1-1/2=1/2$. The totals $r=2,d=1,c=1$ are identical.

Therefore the timing of right censoring relative to the events is needed. Independent censoring is also required for these changing risk sets to yield a valid estimate of the underlying survivor function.

A potential event time is an allowed support point for the discrete event-time distribution used to derive the Kaplan–Meier estimator. A positive probability mass may be assigned there before fitting, even if no event was observed at that time.

This is a device for nonparametric maximum likelihood estimation from exact event times and right censoring; it does not assert that the underlying biological event time is genuinely discrete. Its estimated mass can be zero at an unobserved potential time.

Choose the potential right-censoring times to include every observed right censoring time up to $t$, and also the endpoints 0 and $t$. Then no individual is censored strictly inside $(c_{k-1},c_k)$. Use an explicit endpoint convention: let $r_k$ count individuals still event-free after any events at $c_k$, but before censoring at $c_k$. If $d_k$ events occur in $(c_{k-1},c_k]$, the number at the beginning of that interval, after censoring at its left endpoint, is $r_k+d_k$.

The no-censoring interval estimate from the preceding parts is therefore $r_k/(r_k+d_k)$. The possible event mass at zero contributes $r_0/(r_0+d_0)$. Multiplying yieldsFor a continuous event-time distribution, normally $d_0=0$. Within interval $k$, put $b_k=r_k+d_k$ and let its successive event counts be $e_1,\dots ,e_m$. Its Kaplan–Meier estimator factors telescope:The same calculation at zero provesThe paper's phrase “at risk at $c_k$” does not specify its side of an event at that endpoint. If it denotes the conventional pre-event risk set size $r_k^{-}$ instead, and $e_k$ events occur exactly at $c_k$, substitute $r_k=r_k^{-}-e_k$. The interval factor is then $(r_k^{-}-e_k)/(r_k^{-}+d_k-e_k)$; at zero it is $(r_0^{-}-d_0)/r_0^{-}$. This keeps the events-before-censoring convention consistent. After an empty risk set, stop the product; if the fitted survival has reached zero, retain zero rather than creating a $0/0$ factor.

In the martingale residual$v_i$ is the observed event count up to the end of the individual's follow-up: 1 for a recorded event and 0 for right censoring. The second term is the fitted counting-process compensator, obtained by integrating that individual's fitted hazard function while they remain at risk:Thus the residual is observed events minus the fitted cumulative event intensity over follow-up. The fitted intensity is not an event probability and can exceed 1. Positive values indicate more events than predicted over that observed exposure; negative values indicate fewer. A right-censored individual has a nonpositive residual. With true model parameters and independent censoring, the underlying observed-minus-compensator process is a martingale; fitted terminal residuals are not independent identically distributed errors.

$N(t)$ is the event counting process, while $Y(t)$ is the at-risk process. Their key relationship is that, conditionally on the observed past, the expected event increment is $Y(t)dH(t)$. It separates exposure to risk from event occurrence and naturally incorporates right censoring.

Before the observed endpoint, $Y+N=1$. If censoring occurs before the event, both processes are zero after the censoring time, so their sum is zero. With the inclusive endpoint conventions in the question, at an observed event time $t=T\le C$ both are one, so the sum is two at that single instant.

The expected Treatment A deaths are $1/2$ at month 1, $1/2$ at month 2, and $3(2/6)=1$ at the three tied deaths at month 5. ThusThis is not half of the five deaths because right censoring changes the treatment proportions in successive risk sets. For Treatment B, $O_B=2$ and $E_B=3$. One observed-to-expected relative-risk estimate is therefore

At the month-48 analysis, the observed time from treatment start is:For example, patient D contributes $48-12+1=37$ months. This is right censoring at a common calendar data cutoff but with staggered treatment entry.

Choose a parametric baseline hazard $h_0(t;\eta )$ and fitUnder independent right censoring, the full survival likelihood isEstimate $(\eta ,\beta )$ by maximum likelihood estimation and test $H_0:\beta =0$ with a likelihood-ratio test, Wald test, or score test. Equality of the two event-time distributions is exactly $\beta =0$ within this model.

Construct the two countries' period-specific risk sets with the same left truncation and right censoring rules, then compare their event counts by a log-rank test. This is a nonparametric comparison because it does not specify the shape of either country's hazard function.

Under independent right censoring, observation $(x_i,d_i)$ contributeswhere $d_i$ indicates an event. Multiply these contributions, take the log-likelihood, and maximize it over the parametric survival-model parameter $\eta$, numerically if the score equations have no closed form.

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.

An unobserved survival time is right-censored when it is known to exceed the last event-free follow-up time. For a potential survival time $T$ and potential observation-ending time $C$, recordWhen $\delta =1$, the occurrence was observed at $Y$; when $\delta =0$, only $T>Y=C$ is known. An intact machine at the study's closing date and an event-free subject who stops attending follow-up can both yield right censoring, but the reasons for stopping matter for independent censoring. A subject right-censored at year two has contributed two years to person-time at risk and belongs to earlier risk sets. Neither recording a failure at year two nor discarding that subject preserves this information.

Under independent censoring and separate parameters for the observation mechanism, the part of the survival likelihood involving an absolutely continuous probability distribution for $T$ is $f(Y{)}^{\delta }S(Y{)}^{1-\delta }$. Thus a right-censored observation contributes the survival function $S(Y)$, rather than the event probability density function $f(Y)$. See Lu Tian's Stanford notes, Section 2.1, equations (2.1)–(2.2). The Kaplan–Meier six-subject cohort shows exactly how this distinction changes the calculation.

The risk set immediately before an event time contains the individuals who remain observed, event-free, and eligible to experience the event. For a cohort observed from time zero, with exit times $Y_i=\min (T_i,C_i)$ under right censoring, write $R_j=\{i:Y_i\ge t_j\}$ and $n_j=|R_j|$. Individuals with an event or right censoring exactly at $t_j$ are included in this denominator; process events before removing subjects right-censored at that same recorded time. Earlier right censoring removes subjects from later risk sets. With delayed entry, use the entry times too, as in the Kaplan–Meier estimator with delayed entry.

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.