Estimation of Markov Chain Transition Probabilities and Rates from Fully and Partially Observed Data: Uncertainty Propagation, Evidence Synthesis, and Model Calibration

Abstract

Markov transition models are frequently used to model disease progression. The authors show how the solution to Kolmogorov's forward equations can be exploited to map between transition rates and probabilities from probability data in multistate models. They provide a uniform, Bayesian treatment of estimation and propagation of uncertainty of transition rates and probabilities when 1) observations are available on all transitions and exact time at risk in each state (fully observed data) and 2) observations are on initial state and final state after a fixed interval of time but not on the sequence of transitions (partially observed data). The authors show how underlying transition rates can be recovered from partially observed data using Markov chain Monte Carlo methods in WinBUGS, and they suggest diagnostics to investigate inconsistencies between evidence from different starting states. An illustrative example for a 3-state model is given, which shows how the methods extend to more complex Markov models using the software WBDiff to compute solutions. Finally, the authors illustrate how to statistically combine data from multiple sources, including partially observed data at several follow-up times and also how to calibrate a Markov model to be consistent with data from one specific study.

Trichotomous processes in early memory development, aging, and neurocognitive impairment: A unified theory.

“…For convenience, P is partitioned into a substate P C , in which recall succeeds, and a substate P E , in which recall fails. The number of identifiable parameters in this generic model, or in any Markov chain, can be determined with mathematical techniques that are in common use in the literature on hidden Markov models (or HMM; e.g., Bordes & Vandekerkhove, 2005;Chopin, 2007;Spezia, 2006;Welton & Ades, 2005). 2 As this model fits recall data throughout the lifespan and contains a small, fixed number of parameters, it presents a tractable solution to the identifiability problem with Equation 1-namely, convert the latter to the former by rewriting Equation 1 as the generic two-stage absorbing Markov chain, so that D i , R i , and J i become parameters of that chain.…”

State‐space size considerations for disease‐progression models

“…A number of papers in the MDM and healthcare literature examine how to handle various challenges that arise when estimating TPMs in Markov models of disease, including irregular observation times, incomplete data, and censored observations [10][11][12][13][14][15][16][17][18].…”

A Maximum Likelihood Estimator of a Markov Model for Disease Activity in Crohn’s Disease and Ulcerative Colitis for Annually Aggregated Partial Observations

“…Disaggregation and time translation is done by the same methodology. The translation from a long observation interval into a shorter perspective has been previously addressed [8,9,10,11,15]. Matrix decomposition only solves the part of our problem that is to translate into a shorter interval of time.…”

“…Translating from long to short intervals of time is fairly simple for a single risk [6]. It is more complicated for transition probabilities in models with several states that allows transition back and forth, i. e. transition probability matrices [6,7,8,9]. One such method is matrix decomposition which defines the short-term matrix as a function of the long-term matrix.…”

“…Welton and Ades [9] use a Bayesian approach in a similar situation. They present how to estimate transition rates for multi-state models from partial data, which consist of observations of the initial state and the state after a certain period of time.…”