Aalen-Johansen Estimator

From Open Risk Manual


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]


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


  1. 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.

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