title: A Procedure for Deriving Formulas to Convert Transition Rates to Probabilities for Multistate Markov Models.
abstract: For health-economic analyses that use multistate 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.
authors.affiliation: University of Cambridge, Cambridge, England, UK (EJ).
authors.forename: Edmund
authors.initials: E
authors.lastname: Jones
authors.affiliation: University of Granada, Granada, Spain (DE).
authors.forename: David
authors.initials: D
authors.lastname: Epstein
authors.affiliation: Escuela Andaluza de Salud Pública, Granada, Spain (LG).|Instituto de Investigación Biosanitaria ibs. Granada (LG).
authors.forename: Leticia
authors.initials: L
authors.lastname: García-Mochón
mesh_terms: D003661:Decision Support Techniques; D018450:Disease Progression; D006801:Humans; D008390:Markov Chains; D008433:Mathematics / Q000379:methods*; D015233:Models, Statistical; D011336:Probability; D012984:Software
keywords: Markov model; spreadsheet; transition probabilities; transition rates