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For absolutely continuous first-event time $T$ with event type $J$, the event-free survivor function satisfies $S(t)=\exp (-{\int }_0^t{\sum }_j{h}_j(u)du)$. The cause-specific hazard definition gives unconditional event-type density $S(t)h_j(t)$, so $F_j(t)={\int }_0^{t}S(u)h_j(u)du$. Consequently ${\sum }_j{F}_j(t)=1-S(t)$ for exhaustive disjoint causes. Independent latent cause times are unnecessary. Substituting $1-e^{-\int h_j}$ instead estimates a hypothetical net risk and generally overstates observed incidence.

If one cause-specific hazard acts at constant rate $a$ only until time $\tau$, and another acts at constant rate $b>0$ indefinitely, overall survival is $S(t)=e^{-a\min (t,\tau )-bt}$. The first cause has ultimate cumulative incidence function $a(1-e^{-(a+b)\tau })/(a+b)$; the second accounts for all remaining eventual failures. This illustrates how a transient competing hazard modifies a persistent cause's cumulative risk.

For cause-specific hazard ${\lambda }_k$, the net survival is $S_k^{\mathrm{net}}(t)=\exp [-{\int }_0^t{\lambda }_k(u)du]$. Censoring all competing events in a cause-specific Kaplan–Meier estimator targets this quantity under appropriate observation-censoring assumptions. Its complement generally exceeds the actual cumulative incidence function ${\int }_0^{t}S(u){\lambda }_k(u)du$, because all-cause survival $S(u)$ also includes the other hazards. Identifying net survival with survival after a hypothetical elimination of competing events requires extra causal assumptions.

Interpret $h_A,h_B$ as cause-specific hazards: their conditioning population consists of individuals still free of both events. WriteIn a short interval, either event removes an individual from that population. Hence the survivor function satisfies $S^{\prime }(t)=-S(t)[h_A(t)+h_B(t)]$, with $S(0)=1$. Integrating givesNo independence assumption on hypothetical latent event times is needed: this follows directly from the specified cause-specific hazards.

Among patients still alive at the same time after surgery, chemotherapy has an estimated lung-cancer death rate 3.71 times that of radiotherapy, assuming proportional cause-specific hazards. The estimated 95% confidence interval for this hazard ratio is $(1.583,8.697)$. Thus the fit suggests a higher instantaneous lung-cancer death rate on chemotherapy, with appreciable uncertainty in its size.

The null hypothesis of no treatment effect on that hazard is $H_0:{\gamma }_1=0$ against ${\gamma }_1\ne 0$. Its Wald test uses $Z=1.3113/0.4346\simeq 3.018$, approximately standard normal under the null hypothesis, with two-sided $p=0.00255$. This is evidence against equal lung-cancer death hazards in the fitted model. It is not a statement that five-year lung-cancer death probability is multiplied by 3.71 or that survival duration is divided by 3.71. The cumulative incidence function depends on both cause-specific hazards, and the model needs proportional-hazards and independent observation-censoring assumptions. In a randomised comparison the assigned treatment contrast is protected against baseline confounding, but the instantaneous comparison concerns the groups who have survived to the time in question; it does not identify a treatment effect within a common latent survivor subgroup.

Use the three-state competing risks model shown below. State 0 is alive after surgery, state 1 is dead from lung cancer, and state 2 is dead from other causes. Both death states are absorbing. Study censoring is an observation mechanism, not a biological transition to a fourth state.

For treatment $z=0$ (radiotherapy) or $z=1$ (chemotherapy), define the transition intensityThe fitted Cox proportional-hazards models areHere ${\hat{\lambda }}_{10}$ and ${\hat{\lambda }}_{20}$ represent the estimated baseline cause-specific hazards for radiotherapy. More formally, the baseline estimates can be expressed as cumulative hazard measures with increments $d_k(t)/{\sum }_{i\in R(t)}{e}^{{\hat{\gamma }}_k{z}_i}$, rather than ordinary smooth functions. All outgoing intensities from states 1 and 2 vanish, and $q_{00}=-(q_{01}+q_{02})$. The treatment hazard ratios are 3.7111 and 1.8045 respectively. This model describes the two observed causes of death, not identifiable latent failure times after death has already occurred.

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.

Write $r={\theta }_A+{\theta }_B$, with ${\theta }_A\ge 0$, ${\theta }_B>0$ and a finite $\tau \ge 0$. The competing risks model with transient surgical mortality has survival functionIntegrating the disease cause-specific hazard against this survival function givesThe first term accounts for disease deaths while both causes act; the second includes survivors of that period who then face only disease mortality. The probability of eventually dying from surgery isTaking the limit in $F_B$ gives $F_B(\infty )={\theta }_B(1-e^{-r\tau })/r+e^{-r\tau }$, and thereforeThe ultimate-death conclusion uses the positive continuing disease hazard. If ${\theta }_B=0$, the surviving fraction $e^{-{\theta }_A\tau }$ would instead live indefinitely in this model; the asserted conclusion is not valid in that boundary case.

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.

Let $T=\min (T_A,T_B)$ be the first event time and $J$ its type. The cause-specific hazard for A isIt is a rate among people free of both event types, not the marginal probability of eventually experiencing A. Under the given independent latent event times, their constant hazard functions give exponential survivor functions. Therefore the first-event survivor isThe combined first-event rate is $s=\alpha +\beta >0$. Subsequent formulas concerning conditioning on type A require $\alpha >0$.

Let $\gamma$ be the exponential censoring rate and assume its time is independent of the exponential death time. Regard censoring as the competing event B. The preceding conditional-density calculation givesHence the reported death-only rate estimates the combined rate:It is not, in general, an estimate of the death cause-specific hazard alone. This interpretation assumes the stated two-exponential competition without an additional administrative end of observation that truncates the conditional time distribution.

In a competing risks model, the first event has time $T$ and type $J$; observing one type prevents another type from being the first event for that individual. Its cause-specific hazard isFor absolutely continuous event times the total hazard function is ${\sum }_j{h}_j(t)$, giving the event-free survivor function $S(t)=\exp [-{\int }_0^t{\sum }_j{h}_j(u)du]$. Since an event of type $j$ near $t$ has unconditional density $S(t)h_j(t)$, its cumulative incidence function satisfiesThis cause-specific hazard to cumulative incidence formula does not require independent latent times for the competing causes. In general $F_j(t)$ is not $1-e^{-{\int }_0^t{h}_j}$, since that expression ignores removal by other causes.

Put $a={\theta }_A\ge 0$, $b={\theta }_B\ge 0$ and $q=a+b$, with $\tau \ge 0$. HereIntegrating $aS(u)$ separately before and after $\tau$ gives, for $q>0$,After $\tau$, only type $A$ can occur, so the second term is the fraction event-free at $\tau$ multiplied by their subsequent event probability. The expressions agree at $\tau$. If $a=b=0$, both incidence functions are identically zero; if $a=0<b$, this formula gives $F_A=0$.

Only times up to $\tau$ contribute to the cumulative incidence function for type $B$, because its cause-specific hazard vanishes afterwards. With $q={\theta }_A+{\theta }_B>0$,soThis increases before $\tau$ and then remains constant; it includes the protection from type $B$ events produced by earlier type $A$ events. If both intensities vanish, $F_B=0$; if ${\theta }_B=0$, it is also identically zero.

By conditional probability and continuity of event times,When $\tau >0$ and $q={\theta }_A+{\theta }_B>0$, the common factor $1-e^{-q\tau }$ cancels, givingBefore $\tau$, the constant cause-specific hazards divide the total event intensity in these proportions, regardless of how long the interval is. If $\tau =0$ or both intensities are zero, the conditioning event has probability zero and this conditional probability is undefined, not a ratio assigned by convention.

The cause-specific hazard for cause $k$ isIt is the instantaneous rate of cause $k$ among individuals still free of every competing event.