Competing risks and cumulative incidence

Fix a finite number $K\ge 1$ of causes on a probability space. Let $T_1,\dots ,T_K$ be measurable event times with values in $[0,\mathrm{\infty })$, and assume that the minimum is attained by a unique cause almost surely. The observed first failure time and its cause are $T:={\min }_{1\le j\le K}{T}_j,D:=\min \{j:T_j=T\}\in \{1,\dots ,K\},$ so $D$ is the smallest label attaining the minimum. For each $k$, $\{D=k\}={\bigcap }_{j<k}\{T_k<T_j\}\cap {\bigcap }_{j>k}\{T_k\le T_j\},$ which is measurable; the assumed almost-sure uniqueness makes the tie-break irrelevant except on a null set. No independence among the latent times $T_j$ is assumed. When an external right-censoring time $C$ is present, the observation is $(U,\delta ,D^{\mathrm{o}\mathrm{b}\mathrm{s}})$, where $U=\min (T,C)$ and $\delta =\mathbf{1}\{T\le C\}$ as in Event-time observation, censoring, and truncation schemes, and $D^{\mathrm{o}\mathrm{b}\mathrm{s}}=D$ when $\delta =1$; the cause label is absent when $\delta =0$. Thus an event tied with censoring is recorded as an event.

The all-cause survival function is $S(t):=\mathbb{P}(T>t)$ for $t\ge 0$. For $u>0$, write $S(u-):=\mathbb{P}(T\ge u)$; this is the left limit of $S$ by continuity from above for measures (Continuity from above when one set has finite measure). Set $S(0-):=1$.

For each cause $k$, its event subdistribution measure and cause-specific cumulative-incidence function are ${\nu }_k(B):=\mathbb{P}(T\in B,D=k),F_k(t):={\nu }_k([0,t])=\mathbb{P}(T\le t,D=k)\mathrm{.}$ for Borel $B\subseteq [0,\mathrm{\infty })$ and $t\ge 0$. The cause-specific hazard measure is the measure with density $w(u):=\{\begin{array}{ll}1\mathrm{/}S(u-),& S(u-)>0,\\ 0,& S(u-)=0\end{array}$ relative to ${\nu }_k$: ${\mathrm{\Lambda }}_k(B):={\int }_{B}w(u){\nu }_k(du),d{\mathrm{\Lambda }}_k(u)=w(u){\nu }_k(du)\mathrm{.}$ The extension of $u\mapsto S(u-)$ by the constant $1$ for $u<0$ is nonincreasing on $\mathbb{R}$ and therefore Borel measurable (Every monotone real function is Borel measurable), so $w$ is a nonnegative Borel function and this measure is defined by The measure with density $f$ relative to $\mu$. It may have infinite total mass; no finite-horizon integrability is assumed in this definition.

For every cause $k\in \{1,\dots ,K\}$ and every $t\ge 0$, the cumulative incidence decomposes as $F_k(t)={\int }_{[0,t]}S(u-)d{\mathrm{\Lambda }}_k(u)\mathrm{.}$ If ${\nu }_k$ has a Lebesgue density $f_k$ and $S(u)>0$ for every $u\in (0,t]$, then $d{\mathrm{\Lambda }}_k(u)={\lambda }_k(u)du$ for Lebesgue-a.e. $u\in (0,t]$ with ${\lambda }_k(u):=f_k(u)\mathrm{/}S(u)$, and $F_k(t)={\int }_0^{t}S(u){\lambda }_k(u)du\mathrm{.}$

Input: A finite sample of $n\ge 1$ right-censored records $(U_i,{\delta }_i,D_i^{\mathrm{o}\mathrm{b}\mathrm{s}})$ for $K$ causes, using $U_i=\min (T_i,C_i)$ and ${\delta }_i=\mathbf{1}\{T_i\le C_i\}$ from Event-time observation, censoring, and truncation schemes. The observed cause label $D_i^{\mathrm{o}\mathrm{b}\mathrm{s}}\in \{1,\dots ,K\}$ is present only when ${\delta }_i=1$. Fix a horizon $t\ge 0$.

The pointwise population target is ${\psi }_t:=(S(t),F_1(t),\dots ,F_K(t))\in {\mathbb{R}}^{K+1}$; the algorithm returns its finite-sample estimate.

For each event time $u\in [0,t]$, define the risk count and cause-specific event counts by $Y(u):={\sum }_{i=1}^n\mathbf{1}\{U_i\ge u\},\mathrm{\Delta }{N}_k(u):={\sum }_{i=1}^n\mathbf{1}\{U_i=u,{\delta }_i=1,D_i^{\mathrm{o}\mathrm{b}\mathrm{s}}=k\},\mathrm{\Delta }N(u):={\sum }_{k=1}^K\mathrm{\Delta }{N}_k(u)\mathrm{.}$ All subjects with an event at $u$ are included in $Y(u)$, so tied observed failures are grouped before updating and $Y(u)\ge \mathrm{\Delta }N(u)>0$ at every event time. Censorings at $u$ do not contribute to $\mathrm{\Delta }N(u)$; a subject with $T_i=C_i=u$ is an event by the stated $T_i\le C_i$ convention.

Procedure:

1. List the distinct observed event times $u\le t$ in increasing order, with all causes at a common time grouped into the same counts $\mathrm{\Delta }{N}_k(u)$.
2. Initialize $\widehat{S}(0-)=1$ and ${\widehat{F}}_k(0-)=0$ for each cause.
3. At each listed $u$, set ${\widehat{F}}_k(u)={\widehat{F}}_k(u-)+\widehat{S}(u-)\frac{\mathrm{\Delta }{N}_k(u)}{Y(u)},\widehat{S}(u)=\widehat{S}(u-)(1-\frac{\mathrm{\Delta }N(u)}{Y(u)})\mathrm{.}$
4. Keep the estimates constant between event times and return $(\widehat{S}(t),{\widehat{F}}_1(t),\dots ,{\widehat{F}}_K(t))$.

Equivalently, $\widehat{S}(t)={\prod }_{0\le u\le t:\mathrm{\Delta }N(u)>0}(1-\frac{\mathrm{\Delta }N(u)}{Y(u)}),{\widehat{F}}_k(t)={\sum }_{0\le u\le t:\mathrm{\Delta }N(u)>0}\widehat{S}(u-)\frac{\mathrm{\Delta }{N}_k(u)}{Y(u)}\mathrm{.}$ The convention at $u=0$ allows a baseline event; when there is none, these formulas agree with the $(0,t]$ product-limit convention in Nelson–Aalen, Kaplan–Meier, and Greenwood estimators.

This procedure is a finite-sample definition, not a consistency or variance theorem. A common sampling setup for interpreting it as an estimator of a population law has IID subject records (Identical distribution and IID families), independent censoring $C\perp (T,D)$ (Independent random elements), and $\mathbb{P}(C\ge t)>0$, a positive censoring-support condition through $t$. These are model assumptions, not consequences of the formula. This item does not invoke a counting-process compensator, martingale bracket, or process-limit result.

Fix $n\ge 1$, finite $K$, and $t\ge 0$, and let $\mathcal{X}=([0,\mathrm{\infty })\times \{0,1,\dots ,K\}{)}^n$ carry its Borel sigma-algebra as a Euclidean subspace, where status $0$ denotes censoring and status $k$ denotes an observed cause-$k$ event. Let $\mathsf{E}$ be any statistical experiment with sample space $(\mathcal{X},\mathcal{B}(\mathcal{X}))$. Equip ${\mathbb{R}}^{K+1}$ with its Borel sigma-algebra. The output vector of Aalen–Johansen cumulative-incidence estimator at horizon $t$ is a measurable map $\mathcal{X}\to {\mathbb{R}}^{K+1}$ and hence a statistic in $\mathsf{E}$. Every event-time denominator $Y(u)$ is positive.