Aalen-Johansen Estimator

Definition

The Aalen-Johansen estimator is a multi-state (matrix) version of the Kaplan–Meier estimator for the hazard of a survival process. The estimator can be used to estimate the transition probability matrix of a Markov process with a finite number of states.

Estimator

The position in state space for an entity $i$ in continuous time $t$ is a Random Variable $R^{i}(t)$ taking values in the state space S (We assume a finite state space $S={0,\dots ,D}$).

The estimator is given by the expression

$T^{mn}(s,t)=\prod _{k=1}^{K}(I^{mn}+\Delta A^{mn}(t_{k}))$

$T^{mn}(s,t)$ is the transition matrix element from time s to time t, the mn-th element of the matrix denotes the probability that the Markov process starting in state m at time s will be in state n at time t. The summation is over all times $t_{k}$ where transition events are observed (a total of K).

Off-diagonal elements

The estimation of the transition intensities $A^{mn}(t_{k})$ at any time $t_{k}$ where transitions are observed is simply by counting:

$\Delta A(t_{k})^{mn}={\frac {\Delta N^{mn}(t_{k})}{Y^{m}(t_{k})}}$

where $\Delta N^{mn}(t_{k})$ is the number of transitions observed from state m to state n at time $t_{k}$ and $Y^{m}(t_{k})$ is the number of entities in state m right before time $t_{k}$

Diagonal elements

The diagonal elements are given by

$\Delta A(t_{k})^{nn}={\frac {-\Delta N^{n}(t_{k})}{Y^{n}(t_{k})}}$

where $\Delta N^{n}(t_{k})$ is the number of transitions away from state n at time $t_{k}$

See Also

The Aalen-Johansen estimator is equivalent to the cohort method when the latter is applied to very short intervals