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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.

The probability of a first event of type $A$ in $(u,u+du)$ is the probability $S(u)$ of still being at risk times $h_A(u)du$. Thus its cumulative incidence function isSimilarly $F_B(t)={\int }_0^t{h}_B(u)S(u)du$. In general $F_A(t)\ne 1-e^{-H_A(t)}$: that latter expression ignores the competing removal caused by $B$ and instead describes a net survival complement.

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.