Standard Article

Aalen–Johansen Estimator

  1. Ørnulf Borgan

Published Online: 15 JUL 2005

DOI: 10.1002/0470011815.b2a11001

Encyclopedia of Biostatistics

Encyclopedia of Biostatistics

How to Cite

Borgan, Ø. 2005. Aalen–Johansen Estimator. Encyclopedia of Biostatistics. 1.

Author Information

  1. University of Oslo, Oslo, Norway

Publication History

  1. Published Online: 15 JUL 2005


The Aalen–Johansen estimator is a matrix version of the Kaplan–Meier estimator, which can be used to estimate the transition probability matrix of a Markov process with a finite number of states. The estimator is first presented for the competing risks model and the Markov illness–death model for a chronic disease. For these two simple Markov processes, the elements of the Aalen–Johansen estimator take an explicit form. Then a general finite state Markov process, modeling the life histories of individuals from a homogeneous population, is considered. It is described how the Aalen–Johansen estimator may be obtained as the product-integral of the matrix of Nelson–Aalen estimators for the cumulative transition intensities. Finally, it is briefly indicated how the product-integral formulation of the Aalen–Johansen estimator is useful for the study of its statistical properties.


  • competing risks;
  • cumulative incidence function;
  • illness–death model;
  • Kaplan–Meier estimator;
  • multistate models;
  • Nelson–Aalen estimator;
  • product-limit estimator;
  • product-integral