Aalen–Johansen cumulative-incidence estimator

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.