# Stable and practical semi-Markov modelling of intermittently-observed data

## TL;DR

This paper introduces a flexible semi-Markov modeling approach for intermittently observed data using phase-type distributions, implemented in an R package for Bayesian and maximum likelihood estimation.

## Contribution

It develops a general semi-Markov modeling framework with software implementation, overcoming previous limitations of restricted state structures and lack of software.

## Key findings

- The approach allows likelihood calculation for any state structure.
- The R package 'msmbayes' facilitates Bayesian and maximum likelihood estimation.
- Application to cognitive decline demonstrates practical utility.

## Abstract

Multi-state models are commonly used for intermittent observations of a state over time, but these are generally based on the Markov assumption, that transition rates are independent of the time spent in current and previous states. In a semi-Markov model, the rates can depend on the time spent in the current state, though available methods for this are either restricted to specific state structures or lack general software. This paper develops the approach of using a "phase-type" distribution for the sojourn time in a state, which expresses a semi-Markov model as a hidden Markov model, allowing the likelihood to be calculated easily for any state structure. While this approach involves a proliferation of latent parameters, identifiability can be improved by restricting the phase-type family to one which approximates a simpler distribution such as the Gamma or Weibull. This paper proposes a moment-matching method to obtain this approximation, making general semi-Markov models for intermittent data accessible in software for the first time. The method is implemented in a new R package, "msmbayes", which implements Bayesian or maximum likelihood estimation for multi-state models with general state structures and covariates. The software is tested using simulation-based calibration, and an application to cognitive function decline illustrates the use of the method in a typical modelling workflow.