Repository logo

A Procedure for Deriving Formulas to Convert Transition Rates to Probabilities for Multi-State Markov Models

Published version

Change log


Jones, E 
Epstein, D 
García-Mochón, L 


For health-economic analyses that use multi-state Markov models, it is often necessary to convert from transition rates to transition probabilities, and for probabilistic sensitivity analysis and other purposes it is useful to have explicit algebraic formulas for these conversions, to avoid having to resort to numerical methods. However, if there are four or more states then the formulas can be extremely complicated. These calculations can be made using packages such as R, but many analysts and other stakeholders still prefer to use spreadsheets for these decision models. We describe a procedure for deriving formulas that use intermediate variables so that each individual formula is reasonably simple. Once the formulas have been derived, the calculations can be performed in Excel or similar software. The procedure is illustrated by several examples and we discuss how to use a computer algebra system to assist with it. The procedure works in a wide variety of scenarios but cannot be employed when there are several backward transitions and the characteristic equation has no algebraic solution, or when the eigenvalues of the transition rate matrix are very close to each other.



Markov model, spreadsheet, transition probabilities, transition rates

Journal Title

Medical Decision Making

Conference Name

Journal ISSN


Volume Title


SAGE Publishing
Medical Research Council (G0800270)
Medical Research Council (G0700463)
Medical Research Council (MR/L003120/1)
European Research Council (268834)
British Heart Foundation (None)
British Heart Foundation (None)
Medical Research Council (G0700463/1)
Medical Research Council (G0800270/1)
inancial support for this study was provided in part by grants from the UK Medical Research Council (G0800270), British Heart Foundation (SP/09/002), UK National Institute for Health Research Cambridge Biomedical Research Centre, European Research Council (268834), and European Commission Framework Programme 7 (HEALTH-F2-2012-279233). This work was financially supported by the EPIC-CVD project. EPIC-CVD is a European Commission funded project under the Health theme of the Seventh Framework Programme, building on EPIC-Heart, which was funded by the Medical Research Council, the British Heart Foundation, and a European Research Council Advanced Investigator Award.
Is supplemented by: