# Aalen-Johansen Estimator

## Contents

## 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. ^{[1]}

## Estimator

The position in state space for an entity in continuous time is a random variable taking values in the state space S (We assume a finite state space ).

The estimator is given by the expression

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 where transition events are observed (a total of K).

### Off-diagonal elements

The estimation of the transition intensities at any time where transitions are observed is simply by counting:

where is the number of transitions observed from state m to state n at time and is the number of entities in state m right before time

### Diagonal elements

The diagonal elements are given by

where is the number of transitions away from state n at time

## See Also

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

## References

- ↑ Aalen, O. O. and Johansen, S. (1978). An empirical transition matrix for nonhomogeneous Markov chains based on censored observations. Scandinavian Journal of Statistics 5, 141–150.