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The Aalen–Johansen estimator is a product of empirical state-transition matrices over event times. In competing risks, the estimated cumulative incidence function for cause $j$ jumps by $\hat{S}(t-)d_j(t)/Y(t)$, while overall survival is updated using all event causes. Right censoring changes subsequent risk sets but does not itself create a probability jump.

Antioxidant reaction competition is competition between reactions that consume the same reactive intermediate, such as interception of a radical by an antioxidant and attack by that radical on a susceptible substrate. A large reaction rate constant alone does not establish which route dominates: the local concentrations of the reaction partners also enter the reaction rates.

For an original two-route model, let a radical react with donor $\mathrm{AH}$ or substrate $\mathrm{S}$. Hold both partner concentrations constant, with second-order rate constants $k_A,k_S$. The corresponding pseudo-first-order reaction rates per radical areThe probability of surviving without either event for time $t$ is $e^{-(a+b)t}$. Integrating the instantaneous interception probability giveswhere $\tau$ is the mean waiting time for the first event. This is a competing risks calculation for the stipulated constant-rate reaction model.

Take chosen $k_A=10^6\mathrm{L}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{s}}^{-1}$, $[\mathrm{AH}]=10\mu \mathrm{mol}{\mathrm{L}}^{-1}$, $k_S=10^2\mathrm{L}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{s}}^{-1}$ and $[\mathrm{S}]=0.10\mathrm{mol}{\mathrm{L}}^{-1}$. Both effective rates are $10{\mathrm{s}}^{-1}$, so interception has probability $1/2$ and the mean waiting time is $0.050\mathrm{s}$. Raising only the donor concentration tenfold gives $f_A=10/11$ and $\tau =1/110\mathrm{s}$.

This fraction describes the first reaction, not necessarily final protection. If substrate attack regenerates another reactive radical, radical chain propagation can continue. In a simplified chain where each event either propagates with probability $1-f_A$ or terminates by trapping with probability $f_A$, the expected number of propagation events satisfiesDepletion, additional termination and movement between phases alter this model.

An inhibited autoxidation assay measures a consequence of such chain chemistry under defined conditions. Primary work compared inhibited-oxidation measurements of lipid-soluble chain-breaking activity with independent chromatographic measurements of vitamin E in plasma and membrane preparations.^[ref] That independent comparison is stronger evidence than assigning protection from the chemical label antioxidant alone.

Competing end-of-life criteria are two or more conditions, any one of which can terminate useful operation under the stated specification. They define a remaining useful life by the earliest relevant crossing. If the prospective crossing times, measured from the present, are $T_1,\dots ,T_n$, thenThis is the event structure studied by competing risks. A condition ending one operating cycle must not inadvertently be treated as the component's aging endpoint: NASA's battery experiment separately defines a terminal-potential discharge cutoff and a30% rated-capacity-loss aging criterion.^[ref]

For an original mechanical example, suppose a component can reach either an unacceptable vibration level in12h or an unacceptable temperature in20h under a specified operating scenario. If either condition stops operation, its predicted useful life is12h. Requiring both conditions would instead define another stopping rule. An observed early crossing also does not establish when the other criterion would have been reached if the test had continued.

For uncertain crossing times, the survival function of useful life isOnly under independence does this equal the product of the individual survival functions. For an explicitly hypothetical independent pair of exponential distributions with rates $0.01{\mathrm{h}}^{-1}$ and $0.02{\mathrm{h}}^{-1}$, the product is $e^{-0.03t}$ when $t$ is measured in hours. The expected first crossing is $1/0.03\approx 33.33\mathrm{h}$, smaller than either individual mean,100h or50h.

The independence assumption is substantive. Shared temperature, load or damage can correlate crossing times, so multiplying marginal survival functions can give the wrong answer. Moreover, an intervention triggered by the first crossing can change the subsequent trajectory. The physical end-of-life criterion, dependence model and assumed future operation therefore belong with any numerical lifetime prediction.

For competing risks, a cumulative incidence function jumps by the displayed Aalen–Johansen estimator increment. A person right-censored at exactly the event time remains in the immediately preceding risk set under the usual events-before-censoring tie convention. Censoring changes the next risk set, not the current event numerator or survival jump.

A graft patency risk set contains the observed grafts still eligible to experience the specified patency endpoint immediately before a given time. It is the denominator for a time-specific failure calculation, not simply the original number of operations or the number of open grafts at the last recorded visit.

The endpoint must be chosen first. Primary graft patency, assisted primary graft patency and secondary graft patency use different intervention and occlusion histories. An intervention that ends one endpoint need not end another. The vascular reporting standards define these categories and discuss life-table and Kaplan–Meier estimator methods for unequal follow-up.^[ref]

For a single time-to-first-failure endpoint, let $T_i$ be graft $i$'s failure time and $C_i$ its last observable follow-up time. With a specified common time origin, defineAt an observed failure time $t_j$, the risk set size $n_j=n(t_j)$ is evaluated just before the failures. If $d_j$ qualifying failures occur then, the Kaplan–Meier estimator of patency isTies between failure and censoring times require a consistent observation convention.

In an original example, ten grafts enter follow-up. Two meet the chosen endpoint at day 10, three of the remaining grafts are last observed without failure at day 15, and one of the five still followed fails at day 20. ThenThe three censored grafts do not cause a survival-curve drop at day 15; they reduce the later denominator. The crude expression $1-3/10=0.70$ answers neither the same follow-up-adjusted question nor the question of what happened after censoring.

The censoring assumptions are substantive. If grafts are preferentially lost when failure is imminent, observed risk sets may not represent those no longer followed. Death also prevents later graft-failure observation and can be a competing risks, rather than merely an ordinary missed visit. A cumulative incidence question with death as a competing event differs from a net-patency analysis that censors deaths. Endpoint definition, time origin, observation process and treatment of death must accompany a patency estimate.

For the numerical competing risks update, the estimated probability of remaining event-free just before the tied time isThere were no intervening events, so the cumulative incidence functions did not jump between the two stated times. The Aalen–Johansen estimator uses the risk set immediately before the event time, including the subject censored at that same time under the usual events-before-censoring tie convention. Therefore the cause-$A$ increment is $0.474(2/20)$, andFor a consistency check, ${\hat{F}}_B(7.8)=0.241+0.474/20=0.2647$ and $\hat{S}(7.8)=0.474(1-3/20)=0.4029$; these sum to one. The censoring does not create an incidence jump and leaves $16$ subjects for subsequent follow-up. Dividing by $19$ would instead assume that the censored subject had already left before the event time, contrary to the stated immediately preceding risk set. This is a cumulative incidence update with tied censoring.

The last lung-cancer failure before five years is at 4.85, where the displayed net survival estimate is 0.392. The other-death net survival estimate remains 0.498 from 4.096 until 5.065. Thus the two supplied curves imply net five-year death estimates of $1-0.392=0.608$ for lung cancer and $1-0.498=0.502$ for other causes. Their sum is 1.110, already showing that they cannot be probabilities of two disjoint actual causes of death.

In competing risks, let ${\lambda }_1(t)$ and ${\lambda }_2(t)$ be the cause-specific hazards while alive, with cumulative hazards ${\Lambda }_k(t)={\int }_0^t{\lambda }_k(u)du$. Treating the other cause as censored estimates the net survival $S_k^{\mathrm{net}}(t)=e^{-{\Lambda }_k(t)}$. The actual probability of death from cause $k$ by time $t$ is the cumulative incidence functionSince $S(u)\le e^{-{\Lambda }_k(u)}$, the net probability $1-S_k^{\mathrm{net}}(t)$ generally overstates the cumulative incidence function: death from the other cause prevents a later death of cause $k$. This discrepancy exists even for independent latent cause-specific failure times. Moreover, Kaplan–Meier estimates are not generally exactly unbiased in finite samples even for their proper target. The plotted complements are not unbiased estimators of $\mathbb{P}(T\le 5,D=k)$, and are not even consistent for that target in general.

For the desired probabilities, use the Aalen–Johansen estimator ${\hat{F}}_k(t)={\sum }_{u\le t}\hat{S}(u-)d_k(u)/n(u)$, with all-cause survival $\hat{S}(u)={\prod }_{v\le u}[1-(d_1(v)+d_2(v))/n(v)]$ and independent study censoring. Removing competing deaths from subsequent risk sets is correct for estimating cause-specific hazards; the error is interpreting the separate net survival complements as cumulative incidence functions. A hypothetical elimination-of-a-cause interpretation of net survival would require additional causal assumptions.

For mutually exclusive event types, let $T$ be the first event time and $J$ its type. The cause-specific hazard for cause $j$ isIt is a rate conditional on having had no event of any cause. The cumulative incidence function, also called the cumulative risk function, is the actual probability $F_j(t)=P(T\le t,J=j)$.

The total hazard function is $h(t)={\sum }_k{h}_k(t)$, giving $S(t)=\exp \{-{\int }_0^t{\sum }_k{h}_k(u)du\}$. Surviving every cause to time $u$ and then experiencing cause $j$ givesIn competing risks, $F_j$ is generally not $1-e^{-{\int }_0^t{h}_j}$, because competing events remove individuals before they can experience cause $j$. This formula does not require assuming independent latent failure times for the different causes.

For the survivor functions, $F_A^{\prime }(u)=-f_A(u)$ and $F_B^{\prime }(u)=-f_B(u)$. The product rule givesIntegrating from 0 to $t$, using $F_A(0)=F_B(0)=1$, yieldsThe three terms partition the possible histories: A occurs first by time $t$, B occurs first by time $t$, or neither has occurred by $t$. The first two are the cumulative incidence functions for the two competing risks, while the third is the survival probability of their minimum. The initial value assumes nonnegative event times with no atom at zero.

Recurrent postoperative event analysis studies several qualifying events occurring in the same patient after surgery, rather than retaining only a first event or the highest severity grade. It requires an event definition that distinguishes a new episode from another observation of an ongoing episode. The original Clavien-Dindo classification validation deliberately registered the most severe complication for patients with several; a recurrent-event question requires more of the event history.^[ref]

Let $N_i(t)$ count qualifying events in patient $i$ by postoperative time $t$. This is a counting process. With complete observation of $n$ independent patients through $t$, the mean cumulative postoperative event count has the simple estimatorIt describes average accumulated burden. It is different from the probability $\mathrm{Pr}(N_i(t)\ge 1)$ of any event, and neither quantity specifies the severity of the events counted.

Within-patient dependence matters for uncertainty. At a fixed time, write $K_i=N_i(t)$ and let $s_K^2$ be the usual sample variance across patients. The patient-based standard error of the mean is $s_K/\sqrt{n}$ under independent-patient sampling. An original four-patient illustration with counts $(0,0,0,4)$ gives mean $1$, sample variance $4$ and estimated standard error $1$. Treating the total count as a Poisson count would instead give $\sqrt{4}/4=0.5$. This small example demonstrates different model assumptions; it does not validate either model or a normal confidence interval for four patients.

A proper record includes patient identity, episode times, resolution rules, severity and observation periods. Resampling patients with all their events intact can preserve within-patient dependence; resampling individual events as though they came from unrelated people cannot. An event intensity model may additionally condition on previous events, evolving health and exposure time, rather than assuming a constant rate.

Incomplete follow-up and death change the question. A rate among people currently alive and observed differs from total expected burden per original operated patient. Death prevents future postoperative events, whereas loss to follow-up hides whether events occurred. These require explicit observation and competing risks assumptions instead of an automatic independent-censoring treatment. The scientific aim is to measure repeated burden without confusing events, patients and time at risk.

Secondary graft patency credits continued function of the original graft reconstruction, including qualifying restoration after an occlusion. It differs from assisted primary graft patency, which requires that patency never have been lost. The 1997 reporting standard limits what counts as salvaging the original reconstruction: a replacement that does not preserve the required original graft continuity is not simply a continuation of its secondary patency.^[ref]

Secondary patency is a durability endpoint that can credit successful salvage. It does not assert that blood flowed continuously during an interval in which the graft was occluded. An instantaneous graft patency assessment and a reconstructed follow-up history therefore need not be the same data object.

An original example follows five grafts through six months. One occludes at month 2 but is successfully reopened at month 3 with a qualifying salvage of the original reconstruction. Another occludes at month 4 and is not restored. The remaining three stay patent without procedures. At the end of this complete observation period, $4/5$ retain secondary patency while only $3/5$ have uninterrupted patency without a qualifying intervention. A newly created replacement bypass should be recorded under the stated reconstruction rules, rather than silently resetting the old graft’s clock.

A time-dependent analysis must define when secondary patency is considered irretrievably lost, how salvage episodes are adjudicated and what happens when follow-up ends before that status is resolved. Death can prevent later observation of graft failure; treating it as ordinary right censoring estimates a particular hypothetical durability quantity and requires assumptions. A real-world probability of failure before death instead calls for a competing risks analysis.

Reporting primary graft patency, assisted primary graft patency and secondary graft patency together makes the role of preventive and rescue interventions visible. The associated procedure counts, observation times and clinical outcomes remain necessary to distinguish prolonged graft function from the wider burden and consequences of maintaining it.