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The Cox proportional-hazards model specifieswhere $h_0(t)$ is an unspecified common baseline hazard and $z_i$ is the individual's covariate vector. The hazard ratio for two fixed covariate vectors is $\exp [{\beta }^T(z_i-z_k)]$, constant in time. Assume independent subjects and independent censoring conditional on their covariates, and use the question's absence of tied events.

Let $t_j$ be an event time, $i_j$ its failing subject, and $R_j$ the risk set immediately before that time. In a short interval of length $dt$, the conditional probability that subject $i$ fails, given one failure among those at risk, isThe baseline hazard cancels. Multiplying the successive conditional event-identity factors gives the Cox partial likelihood for $\beta$:This is a partial likelihood, not the full survival likelihood for the unspecified $h_0$. Risk sets evolve with the observed history; the derivation uses their successive conditional hazards, rather than assuming the risk sets themselves are independent.

Define $w_{jk}(\beta )=e^{{\beta }^T{z}_k}/{\sum }_{\ell \in R_j}{e}^{{\beta }^T{z}_{\ell }}$ and ${\bar{z}}_j(\beta )={\sum }_{k\in R_j}{w}_{jk}(\beta )z_k$. For component $r$,Differentiating again givesThus the score and information of Cox partial likelihood are the sum of event-minus-risk-mean contrasts and the sum of risk-weighted covariance matrices, respectively. The Hessian is negative semidefinite; the likelihood is concave, with strict identifiability requiring variation in the relevant risk-set covariates.

At event time $t_j$, the vector Schoenfeld residual iswhere $\hat{\beta }$ is the fitted partial-likelihood coefficient. It compares the covariate of the person who actually failed with the covariate mean predicted for the failing person, using the fitted hazard weights over the risk set. Under the model at the true coefficient, the conditional expected failing covariate is precisely that weighted mean, so the corresponding Schoenfeld function is conditionally centered at zero. Residuals are defined at observed events, not at censoring times.

The score equation from part (a) is exactlyAt a finite interior unpenalized maximum, $U(\hat{\beta })=0$, soThis is the intended fitted-residual identity. For an arbitrary trial coefficient it is a score, not generally zero; a single event with covariates zero and one in its risk set can have Schoenfeld function $1/2$ at $\beta =0$. Penalized estimates or a boundary maximum also need their own score conditions rather than this unqualified zero-sum assertion.

Write the Cox proportional-hazards model as $h_i(t)=h_0(t)u_i$, where $u_i=\exp ({\beta }^T{z}_i)$ is the hazard multiplier and $h_0$ is the unspecified baseline hazard. At an untied event time $t_j$, let $R_j$ be the risk set immediately before the event and $i_j$ its failing individual. With independent censoring conditional on the covariates, the conditional probability that individual $i$ supplies the next event, given an event at that time and the risk set, is obtained by dividing its instantaneous event rate by the aggregate rate:More precisely this is the limit of the conditional probability for one event in a short interval; simultaneous events have smaller-order probability in the continuous model. Multiplication of these successive conditional contributions gives the Cox partial likelihoodCensored individuals remain in each risk set until their censoring time but supply no numerator. Conditioning removes the baseline hazard; this is a partial likelihood for $\beta$, rather than a complete likelihood for event and censoring times.

Ties require an observation model or an approximation. Suppose a recorded tie hides the order of $d$ continuous events, with no intervening entry or censoring. Put $D$ for the tied event set and $S={\sum }_{i\in R}{u}_i$. An exact marginal contribution for the coarsened ranks is the sum of the ordinary rank contributions over every possible order:For example, if $A,B$ fail together while $C$ remains at risk, the two possible orders giveThe risk set is depleted after the first of the two events. This expression concerns the missing event order; it is not the full likelihood of an interval in which the events occurred.

Convenient alternatives are the Breslow approximation for tied event times and the Efron approximation for tied event times:The first keeps the initial denominator at every event; the second removes an average share of the tied-event multipliers successively. In the example their denominators are $S^2$ and $S[S-(u_A+u_B)/2]$. These conventional partial-likelihood factors omit multiplicities independent of $\beta$, which do not affect its estimate; they should not be mistaken for normalized probabilities of the unordered event set.

For genuinely discrete event times there is another exact construction. If individual event odds are proportional to $u_i$, conditioning on exactly $d$ events gives the exact tied-set conditional likelihoodFor the same three individuals this is $u_A{u}_B/(u_A{u}_B+u_A{u}_C+u_B{u}_C)$. It is not generally equal to the continuous-time rank sum: the assumptions differ.

Large tie groups make naive summation over $d!$ orders or $\left(\genfrac{}{}{0ex}{}{|R|}{d}\right)$ subsets impractical. Recursion helps with the subset denominator, and Breslow or Efron avoids exhaustive enumeration. However, extensive ties can also indicate that time is measured too coarsely for an exact continuous-time ordering to be a useful target.

For example, if patients are assessed only at annual visits, use one row per patient-year still at risk and model the probability $p_{ij}$ of an event in that interval. A grouped proportional-hazards model follows directly by integrating the hazard over the interval, with covariates constant there:Fit the resulting person-period likelihood, with Bernoulli distributions for the interval event indicators and interval-specific intercepts. Many events in the same year are then ordinary observations rather than a combinatorial ordering problem. This requires a compatible interval-observation scheme and appropriate independent censoring; it avoids pretending that unobserved within-year event times are known.

Let $N(t)$ count observed events and $Y(t)$ count individuals still in the risk set just before $t$. Under a common hazard function $h(t)$ and independent censoring, the expected event count over a short interval, conditional on the observed past, isHere $H(t)={\int }_0^{t}h(u)du$ is the integrated hazard. Thus an estimating increment for $dH$ is $dN/Y$, on the range where the risk set is nonempty. Accumulating these increments gives the Nelson–Aalen estimatorThe observation times $t_j$ include both events and censorings, $r_j=Y(t_j)$, and ${\delta }_j$ is one for an event and zero for censoring. Without ties each event adds $1/r_j$; censoring changes later risk sets but adds nothing immediately. Equivalently, the local event fraction estimates the local hazard increment, and summing those estimates produces the integrated hazard. The counting-process intensity in survival analysis justifies this estimating equation; it does not imply exact finite-sample unbiasedness after the risk set can become empty.

Assume the usual cohort setup: all individuals enter at time zero, each contributes one event or censoring time, and every such observation is included. Order these distinct observation times as $t_1<\cdots <t_m$. The risk set size just before $t_k$ is then $r_k=m-k+1$. Write ${\delta }_k=1$ for an event and zero for censoring. Including the jump at an event time, the Nelson–Aalen estimator isExchange the two finite sums:Thus the sum equals the number of observed events. Each event increment appears once for each individual still at risk at that event, exactly cancelling its denominator. This is the event-count identity for Nelson–Aalen cumulative hazards. The cohort and entry convention matter: delayed entry would invalidate the simple equation $r_k=m-k+1$ and require a modified identity.

One method breaks a tied group into a sequence of distinct artificial times, by a specified or random ordering. Process tied events before tied censorings if their common recorded time means the censored individuals were still under observation when the events occurred. At each artificial event, add the reciprocal of the current risk set size and then remove that individual. This applies the untied Nelson–Aalen estimator to the resulting grid. For example, two tied failures among two individuals give successive increments $1/2$ and $1$.

On that expanded grid there is one observation time per individual. The cancellation argument in (b) is unchanged, so breaking ties preserves the identity when the sum includes every artificially separated observation. The chosen ordering can affect the estimated curve within a tied group and must not be presented as an observed event order.

A second method retains the recorded tied times. For a group with $d_j$ events and pre-time risk set size $r_j$, the grouped Nelson–Aalen estimator adds $d_j/r_j$ once. All events in the group use the same denominator; censorings in the group are removed afterwards. If two individuals fail at the same time and there are no other observations, the curve has one jump of $2/2=1$. Its sum over the one distinct time is $1$, whereas the event count is $2$. Hence the unweighted distinct-time identity does not generally survive grouping ties. On the artificially separated grid in the first method, the two post-observation values would instead be $1/2$ and $3/2$, whose sum is $2$.

There is nevertheless an exact weighted version for the grouped method. Let $n_j$ count all observations, event or censoring, at distinct time $t_j$. With everyone entering at zero, $r_k={\sum }_{j\ge k}{n}_j$. ThereforeThis grouped Nelson–Aalen event-count identity sums over individuals, with repeated values for tied times, rather than once per distinct time. Keeping that distinction resolves the apparent conflict between a valid grouped estimator and failure of the literal unweighted formula.

Define the individual background survivor function and common excess component byIntegrating the additive hazard function decomposition givesThe relative survivor function therefore compares actual survival with the survival expected from background mortality, and is common across individuals under this model. If $h_E\ge 0$, it is the survivor function of a hypothetical process having only the excess hazard. If excess mortality can be negative, the ratio may exceed one and lacks an ordinary survival-probability interpretation.

To estimate it, let $Y_i(u)$ indicate that individual $i$ is in the risk set, let $Y(u)={\sum }_i{Y}_i(u)$, and let $N(u)$ count all observed deaths. Assume appropriate independent censoring. The aggregate event intensity isConsequently the background correction must use the current risk-set averageonly on the observed range where $Y>0$. Dividing the event count by $Y$ estimates the total hazard there. Subtracting the known average background gives the offset-adjusted cumulative hazard estimatorHere $d_j$ is the event count at $t_j$ and $r_j=Y(t_j)$; under an untied scheme $d_j=1$. One possible estimate is ${\hat{F}}_{E,\exp }=\exp [-{\hat{H}}_E]$.

A product-limit version retains the full event fraction, which is especially relevant when the excess mortality is not small. Estimate the relative-survival differential equation $d{F}_E/F_E=-d{H}_E$ byAt an event time it multiplies the curve by $1-d_j/r_j$. Between events it solves ${\hat{F}}_E^{\prime }={\overline{h}}_B{\hat{F}}_E$. Starting at one therefore gives the risk-set-adjusted relative survivor estimatorThe Kaplan–Meier estimator factor accounts for observed all-cause mortality; the continuous multiplier corrects for known background mortality. The exponential cumulative-hazard version instead has event jumps $\exp (-d_j/r_j)$; it is close when event fractions are small but is not identical to this product-limit version.

Even if each background hazard is small, its accumulated contribution over long follow-up cannot simply be discarded. Nor should the excess contribution be linearized merely because the background is small. Using the known background integral inside the exponential avoids both errors. The average is over the evolving risk set: replacing it by an initial-cohort mean, or dividing by an arbitrary average of individual background survivor functions, is generally wrong when the background hazards differ. Censoring and deaths change which individuals contribute to the background correction.

Use the product-limit estimate in (b). Between consecutive event times there are no mortality jumps, soThus the estimated relative survivor curve rises between events when background mortality is positive, and is constant if it is zero. Censoring changes the risk set and can change the slope, but supplies no downward event jump. At the next event time the curve is multiplied by $1-d_k/r_k$.

Provided the curve is still positive, the ratio of the post-event estimates at the two consecutive event times isFor $d_k<r_k$, the later value is larger precisely whenFor example, suppose there are 100 individuals in the risk set after the earlier event, no intervening censoring, a 20-day gap, and one death at its end. If every background hazard is $0.001$ per day, then the integrated background contribution is $0.02$ andThe individual background hazards are small, but the interval contains less observed mortality than expected from that background. This can occur by chance in a small event sample or when the cohort is healthier than the life-table reference on other dimensions.

A nonnegative true excess hazard makes $F_E$ nonincreasing, but this unconstrained estimator need not inherit that restriction. An increase is therefore not proof of a clinical survival benefit. If the exponential cumulative-hazard version is chosen instead, its corresponding condition is ${\int }_{t_j}^{t_k}{\overline{h}}_B>d_k/r_k$; the qualitative explanation remains the same. If the risk set is exhausted by deaths, the product-limit factor is zero and no later increase is possible.

Set $\alpha =\log \{{\beta }^{(1)}/{\beta }^{(0)}\}$ and absorb the common multiplier into $h_0(t)=m(t){\beta }^{(0)}$. The model becomesIt is a Cox proportional-hazards model; the unknown baseline does not need a parametric form. The equality hypothesis is precisely $\alpha =0$. The absolute multipliers and $m$ have a scale ambiguity, but their hazard ratio $e^{\alpha }$ is identifiable when both groups supply information.

For an untied event at $a_k$, let $R_k$ be the risk set just before the event, and let $i_k$ be the subject who fails. The conditional event probability is proportional to that subject's hazard, so the common baseline cancels:Multiply these factors to form the Cox partial likelihood. One can maximize it and compare its maximum with its value at $\alpha =0$ using a partial likelihood-ratio test, or use the corresponding score test.

Explicitly, with group risk-set sizes $r_{k0},r_{k1}$ and observed event indicator $d_{k1}$ for group 1, the score at zero and its information areUnder the null the event label has exactly this Bernoulli distribution conditional on its current risk set. Hence $U^2/V$ is the usual one-degree-of-freedom log-rank test statistic, with an asymptotic chi-squared distribution when $V>0$ and enough informative events are observed. Censoring can differ between groups, provided it is independent of event time within the conditioning model. If there are recorded ties, use the appropriate tied-risk-set version: for $d_k$ events the null variance contribution is $d_k(r_k-d_k)r_{k0}{r}_{k1}/[r_k^2(r_k-1)]$.