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An individual has neither died nor been censored by time $t$ exactly when $T>t$ and $C>t$. Under independent censoring, this has probability $F_T(t)G(t)$. Thus the probability of leaving observation by either route isThe two pieces agree at $t={\tau }_c-{\tau }_b$. The expression counts the first of death and censoring, so it does not double-count individuals who would eventually experience both.

With independent censoring, the at-risk process is $Y_i(t)=1_{T_i\ge t,C_i\ge t}$. Independence and the common censoring survivor function $G$ give

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.

Because $t^{\ast }>t_b>t_a$, the three possible complete event orders are $t_a<t^{\ast }<t_c<t_d$, $t_a<t_c<t^{\ast }<t_d$, and $t_a<t_c<t_d<t^{\ast }$. Write $K={\varphi }_a/({\varphi }_a+{\varphi }_b+{\varphi }_c+{\varphi }_d)$ and $V={\varphi }_b+{\varphi }_c+{\varphi }_d$. Their partial likelihoods, with the final singleton factor omitted, are respectivelyAdding first $L_2$ and $L_3$ gives $K{\varphi }_c/V$, and thereforeThusThis is Cox rank-likelihood deletion consistency: summing out the unobserved position of $b$ leaves the relative order information in the observed events. It is an algebraic marginalization of complete-order probabilities under time-constant hazard multipliers. It does not include the probability density function of the actual censoring time, establish the distribution of $t^{\ast }$ given all observed times, or justify informative censoring. Ignoring the censoring mechanism in the survival analysis still requires independent censoring given the modeled covariates.

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.

Let $d_j$ be the number of events at potential time $a_j$ and let $r_j$ be the size of the risk set immediately before that time, including observations censored at $a_j$. Define the discrete conditional event probabilityUnder independent censoring, and ignoring the separate censoring distribution, the survival likelihood factorizes asThis follows by multiplying a survival factor for each individual at risk who does not fail at $a_j$, and an event factor for each failure. The log factor $d_j\log q_j+(r_j-d_j)\log (1-q_j)$ is maximized at ${\hat{q}}_j=d_j/r_j$, with the corresponding boundary values when $d_j=0$ or $d_j=r_j$.

The survival function is the product of successive conditional survival probabilities. ThereforeThis is the Kaplan–Meier estimator. A product over no event times is 1. If everyone remaining experiences an event, the product becomes zero and stays zero; subsequent empty risk sets supply no additional factor.

Define the event counting process and at-risk process byUnder a common hazard function and independent censoring, the conditional event rate is $Y(u)h(u)$. Equivalently, the counting-process compensator satisfies $\mathbb{E}[dN(u)\mid {\mathcal{F}}_{u-}]=Y(u)dH(u)$. Estimate a cumulative hazard increment by the observed event increment divided by the current risk set:With no ties, $dN$ has mass $v_i$ at $x_i$, and $Y(x_i)={\sum }_{j:x_j\ge x_i}1$. Thus the Nelson–Aalen estimator isOnly event times contribute, so no empty risk set is divided into an event count. Right-censored observations contribute time to the risk sets, but no event increment.

Let $R_i=\{j:x_j\ge x_i\}$ be the risk set at an observed event time $x_i$. In the Cox proportional-hazards model, conditional on one member of this set failing then and on the preceding history, the probability that it is individual $i$ isThe baseline hazard cancels. Multiplying these conditional contributions gives the Cox partial likelihoodThe no-ties assumption avoids the need for a tied-event approximation. This construction assumes independent censoring conditional on the modelled covariates. The printed phrase calling $z$ a scalar parameter is read as a scalar observed covariate $z_i$; $\beta$ is the unknown regression parameter.

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.

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 proportional hazards assumption test assesses whether fitted covariate coefficients remain constant over time. A common implementation uses trends in scaled Schoenfeld residuals, with separate covariate tests and a joint global test. Failure to reject is compatible with proportional hazards, but does not establish the correct covariate functional form or independent censoring.

Suppose the recorded durations retain an exponential distribution of rate $\theta$, but an independent fraction $q$ are relabelled as failures and the rest as right-censored. Applying the ordinary survival likelihood gives $\hat{\theta }=d/{\sum }_i{y}_i\to q\theta$, rather than $\theta$. A censoring flag chosen independently of the recorded duration does not establish independent censoring of a latent failure time. Genuine simulation instead draws an event time and an independent censoring time and records their minimum.

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.

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.

Compute observed-minus-expected event contributions within each stratum's risk set, then add them across independent strata. Under the null, the corresponding conditional variances add. Standardizing gives an approximate normal distribution test, or its square gives a one-degree-of-freedom chi-squared distribution test. The comparison requires the usual exchangeability and independent censoring assumptions within strata.