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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.

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.

Within the risk set $R_i$, the Cox proportional-hazards model makes the conditional failure probabilities proportional to $e^{\beta z_j}$. ThusThe score function sums observed minus expected covariates at event times. At a finite interior maximum of the Cox partial likelihood,The balance holds in aggregate, not separately at every event. The log partial likelihood is concave becauseWhen there is informative covariate variation in the risk sets, this gives a unique finite solution if it exists. A completely separated event pattern can instead have its maximum only as $\beta \to \pm \infty$; the question's finite $\hat{\beta }$ implicitly excludes that case.

Taking logarithms of the Cox partial likelihood givesDifferentiate each term, using the chain rule for the logarithm:This score function is the sum over failures of the observed covariate minus its current risk-weighted mean. Define that mean as ${\overline{z}}_{\beta }(x_i)$; observations with $v_i=0$ have no direct event term, although they occur in the relevant risk sets.

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.

Insert the Breslow estimator into the fitted cumulative intensities and interchange the finite sums:Every observed event contributes exactly one after summing its fitted contribution over its risk set. ThereforeThis identity is algebraic and holds for any finite coefficient used consistently in the relative hazards and the Breslow estimator; it does not require the score equation to be solved exactly. It can fail if the baseline hazard is estimated or smoothed by a different procedure.

A proportional hazards model has $h(t\mid g)=h_0(t)r(g)$, with a time-independent hazard ratio. In the Cox proportional-hazards model, $r(g)=e^{\beta g}$ and the baseline hazard $h_0$ is unspecified. At each untied event time, conditioning on the identity of the person who fails given the current risk set produces the factorMultiplying these factors gives the Cox partial likelihood. It is called partial because it uses the conditional event ordering information that identifies $\beta$ while discarding the part of the full likelihood involving the unspecified baseline hazard.

At each distinct event time $t_j$, let $d_j$ events occur among $r_j$ people in the risk set. The Kaplan–Meier estimator iscensorings remove people from later risk sets but create no factor.

Left truncation, or delayed entry, means that an individual is observed only after surviving to an entry time. A registry assembled from patients alive when a clinic opens is a practical example. Adapt Kaplan–Meier by admitting each person to the risk set only at their entry time.

Period survival analysis is useful when recent prognosis is desired but complete long-term follow-up of a recent diagnosis cohort is unavailable. For a chosen calendar year, intersect every patient's observed follow-up with that year. Express the surviving pieces on the time-since-diagnosis scale, treat the beginning of the calendar window as delayed entry and its end as right censoring, and apply Kaplan–Meier with those entry and exit times.

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 event time $t_j$, let $n_{Aj},n_{Bj}$ be the two risk set sizes, $n_j=n_{Aj}+n_{Bj}$, and let $d_{Aj},d_{Bj}$ be the event counts with $d_j=d_{Aj}+d_{Bj}$. Under the null hypothesis of equal hazards, conditioning on the risk set and total number of events gives a hypergeometric distribution, soandThe log-rank statistic and its estimated null variance areUnder the null, $U/\sqrt{V}$ is asymptotically standard normal, or $U^2/V$ is asymptotically chi-squared with one degree of freedom.

The risk set at $x_i$ contains every individual still under observation and event-free immediately before $x_i$. Since the observed times are strictly ordered increasingly, these are individuals $i,i+1,\dots ,n$, so its size is

At month 36, patients F and D are censored at treatment durations 21 and 25, after which patient A is the sole remaining member of the risk set and dies at duration 26. The corresponding Kaplan–Meier factor is $1-1/1=0$, sowithout needing the earlier factors.

At month 48, D remains at risk past duration 26 and is eventually censored at 37, so no event empties the risk set. Part ii gives

Under the null hypothesis of equal event-time distributions, every member of the combined risk set has the same instantaneous chance of being the next event. Conditional on one event at $a_j$ and on the two risk-set sizes,so

The risk set at event time $a_j$ contains individuals still under observation and event-free immediately before $a_j$. For group $k$ its size isThe use of $\ge$ keeps the individual who experiences the event at $a_j$ in the risk set just before that event.

The variance of an estimated hazard increment is large when its group has few individuals in the risk set. The log-rank weights are near zero when either $r_j^{(0)}$ or $r_j^{(1)}$ is small and are largest when both groups retain substantial information. They therefore suppress noisy late-event comparisons and weight each observed-minus-expected event by its available information. Unit weights would instead give equal influence to unstable increments from depleted risk sets.

Letand let $C$ be the common partial likelihood contribution from the first $n-3$ observations. Since $t_{n-2}>x_{n-2}$, the three possible complete-data tail orderings and their partial likelihoods areTheir sum isWhen individual $n-2$ is right-censored at $x_{n-2}$, that individual leaves the risk set before the event at $x_{n-1}$, so the directly calculated Cox partial likelihood is also $C,a/(a+b)$. Summing over the unobserved compatible event orderings therefore reproduces the censored-data partial likelihood.