Estimation of semi-Markov multi-state models: a comparison of the sojourn times and transition intensities approaches

Azam Asanjarani, Benoit Liquet*, Yoni Nazarathy

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    5 Citations (Scopus)
    18 Downloads (Pure)


    Semi-Markov models are widely used for survival analysis and reliability analysis. In general, there are two competing parameterizations and each entails its own interpretation and inference properties. On the one hand, a semi-Markov process can be defined based on the distribution of sojourn times, often via hazard rates, together with transition probabilities of an embedded Markov chain. On the other hand, intensity transition functions may be used, often referred to as the hazard rates of the semi-Markov process. We summarize and contrast these two parameterizations both from a probabilistic and an inference perspective, and we highlight relationships between the two approaches. In general, the intensity transition based approach allows the likelihood to be split into likelihoods of two-state models having fewer parameters, allowing efficient computation and usage of many survival analysis tools. Nevertheless, in certain cases the sojourn time based approach is natural and has been exploited extensively in applications. In contrasting the two approaches and contemporary relevant R packages used for inference, we use two real datasets highlighting the probabilistic and inference properties of each approach. This analysis is accompanied by an R vignette.

    Original languageEnglish
    Pages (from-to)243-262
    Number of pages20
    JournalInternational Journal of Biostatistics
    Issue number1
    Early online date6 Jan 2021
    Publication statusPublished - May 2022

    Bibliographical note

    Copyright the Author(s) 2020. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.


    • hazard rate
    • intensity transition
    • multi-state model
    • semi-Markov model
    • sojourn time


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