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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.

SUBMITTER: Jones E

PROVIDER: S-EPMC5582645 | biostudies-literature | 2017 Oct

REPOSITORIES: biostudies-literature

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A Procedure for Deriving Formulas to Convert Transition Rates to Probabilities for Multistate Markov Models.

Jones Edmund E Epstein David D García-Mochón Leticia L

Medical decision making : an international journal of the Society for Medical Decision Making 20170405 7

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 o ...[more]

PMID: 28379779

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Project description:ObjectiveAlthough Markov cohort models represent one of the most common forms of decision-analytic models used in health care decision-making, correct implementation of such models requires reliable estimation of transition probabilities. This study sought to identify consensus statements or guidelines that detail how such transition probability matrices should be estimated.MethodsA literature review was performed to identify relevant publications in the following databases: Medline, Embase, the Cochrane Library, and PubMed. Electronic searches were supplemented by manual-searches of health technology assessment (HTA) websites in Australia, Belgium, Canada, France, Germany, Ireland, Norway, Portugal, Sweden, and the UK. One reviewer assessed studies for eligibility.ResultsOf the 1,931 citations identified in the electronic searches, no studies met the inclusion criteria for full-text review, and no guidelines on transition probabilities in Markov models were identified. Manual-searching of the websites of HTA agencies identified ten guidelines on economic evaluations (Australia, Belgium, Canada, France, Germany, Ireland, Norway, Portugal, Sweden, and UK). All identified guidelines provided general guidance on how to develop economic models, but none provided guidance on the calculation of transition probabilities. One relevant publication was identified following review of the reference lists of HTA agency guidelines: the International Society for Pharmacoeconomics and Outcomes Research taskforce guidance. This provided limited guidance on the use of rates and probabilities.ConclusionsThere is limited formal guidance available on the estimation of transition probabilities for use in decision-analytic models. Given the increasing importance of cost-effectiveness analysis in the decision-making processes of HTA bodies and other medical decision-makers, there is a need for additional guidance to inform a more consistent approach to decision-analytic modeling. Further research should be done to develop more detailed guidelines on the estimation of transition probabilities.

Project description:ObjectiveTo explore the transitions of different blood pressure states based on a multistate Markov model among the Chinese elderly population.SettingA community health centre in Xiamen, China.Participants1833 elderly Chinese people.MethodsA multistate Markov model was built based on 5001 blood pressure measurements from 2015 to 2020. Research was conducted to explore the process of hypertension progression, providing information on the transition probability, HR and the mean sojourn time in three blood pressure states, namely normal state, elevated state and hypertensive state.ResultsProbabilities of moving from the normal state to the hypertensive state in the first year were 16.97% (female) and 21.73% (male); they increased dramatically to 47.31% (female) and 51.70% (male) within a 3-year follow-up period. The sojourn time in the normal state was 1.5±0.08 years. Elderly women in the normal state had a 16.97%, 33.30% and 47.31% chance of progressing to hypertension within 1, 2 and 3 years, respectively. The corresponding probabilities for elderly men were 21.73%, 38.56% and 51.70%, respectively. For elderly women starting in the elevated state, the probabilities of developing hypertension were 25.07%, 43.03% and 56.32% in the next 1, 2 and 3 years, respectively; while the corresponding changes for elderly men were 20.96%, 37.65% and 50.86%. Increasing age, body mass index (BMI) and glucose were associated with the probability of developing hypertension from the normal state or elevated state.ConclusionsPreventive actions against progression to hypertension should be conducted at an early stage. More awareness should be paid to elderly women with elevated state and elderly men with normal state. Increasing age, BMI and glucose were critical risk factors for developing hypertension. The derived transition probabilities and sojourn time can serve as a significant reference for making targeted interventions for hypertension progression among the Chinese elderly population.

Project description:BackgroundMulti-state models are complex stochastic models which focus on pathways defined by the temporal and sequential occurrence of numerous events of interest. In particular, the so-called illness-death models are especially useful for studying probabilities associated to diseases whose occurrence competes with other possible diseases, health conditions or death. They can be seen as a generalization of the competing risks models, which are widely used to estimate disease-incidences among populations with a high risk of death, such as elderly or cancer patients. The main advantage of the aforementioned illness-death models is that they allow the treatment of scenarios with non-terminal competing events that may occur sequentially, which competing risks models fail to do.MethodsWe propose an illness-death model using Cox proportional hazards models with Weibull baseline hazard functions, and applied the model to a study of recurrent hip fracture. Data came from the PREV2FO cohort and included 34491 patients aged 65 years and older who were discharged alive after a hospitalization due to an osteoporotic hip fracture between 2008-2015. We used a Bayesian approach to approximate the posterior distribution of each parameter of the model, and thus cumulative incidences and transition probabilities. We also compared these results with a competing risks specification.ResultsPosterior transition probabilities showed higher probabilities of death for men and increasing with age. Women were more likely to refracture as well as less likely to die after it. Free-event time was shown to reduce the probability of death. Estimations from the illness-death and the competing risks models were identical for those common transitions although the illness-death model provided additional information from the transition from refracture to death.ConclusionsWe illustrated how multi-state models, in particular illness-death models, may be especially useful when dealing with survival scenarios which include multiple events, with competing diseases or when death is an unavoidable event to consider. Illness-death models via transition probabilities provide additional information of transitions from non-terminal health conditions to absorbing states such as death, what implies a deeper understanding of the real-world problem involved compared to competing risks models.

Dataset Information

A Procedure for Deriving Formulas to Convert Transition Rates to Probabilities for Multistate Markov Models.

Publications

A Procedure for Deriving Formulas to Convert Transition Rates to Probabilities for Multistate Markov Models.

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