Beyond time-homogeneity for continuous-time multistate Markov models

TL;DR

This paper challenges the common piecewise time-homogeneity assumption in continuous-time multistate Markov models, highlighting potential biases and proposing methods for true time-inhomogeneous likelihood computation, especially in hidden Markov model contexts.

Contribution

It introduces a framework for likelihood computation in truly time-inhomogeneous multistate Markov models, addressing biases from the piecewise assumption and enhancing Bayesian estimation methods.

Findings

Potential biases from assuming piecewise homogeneity

Proposed likelihood computation for time-inhomogeneous models

Bayesian methods for efficient parameter estimation

Abstract

Multistate Markov models are a canonical parametric approach for data modeling of observed or latent stochastic processes supported on a finite state space. Continuous-time Markov processes describe data that are observed irregularly over time, as is often the case in longitudinal medical data, for example. Assuming that a continuous-time Markov process is time-homogeneous, a closed-form likelihood function can be derived from the Kolmogorov forward equations -- a system of differential equations with a well-known matrix-exponential solution. Unfortunately, however, the forward equations do not admit an analytical solution for continuous-time, time-inhomogeneous Markov processes, and so researchers and practitioners often make the simplifying assumption that the process is piecewise time-homogeneous. In this paper, we provide intuitions and illustrations of the potential biases for parameter estimation that may ensue in the more realistic scenario that the piecewise-homogeneous assumption is violated, and we advocate for a solution for likelihood computation in a truly time-inhomogeneous fashion. Particular focus is afforded to the context of multistate Markov models that allow for state label misclassifications, which applies more broadly to hidden Markov models (HMMs), and Bayesian computations bypass the necessity for computationally demanding numerical gradient approximations for obtaining maximum likelihood estimates (MLEs). Supplemental materials are available online.