Estimation of Semi-Markov Multi-state Models: A Comparison of the Sojourn Times and Transition Intensities Approaches

TL;DR

This paper compares two parameterizations of semi-Markov models—sojourn times and transition intensities—highlighting their differences, relationships, and practical implications through theoretical analysis and real data examples.

Contribution

It provides a comprehensive comparison of the sojourn time and transition intensity approaches for semi-Markov models, including their inference properties and applications.

Findings

Intensity-based approach allows efficient likelihood computation.

Sojourn time approach is more natural in certain applications.

Real data examples illustrate differences in inference and interpretation.

Abstract

Semi-Markov models are widely used for survival analysis and reliability analysis. In general, there are two competing parameterizations and each entails its own interpretation and inference properties. On the one hand, a semi-Markov process can be defined based on the distribution of sojourn times, often via hazard rates, together with transition probabilities of an embedded Markov chain. On the other hand, intensity transition functions may be used, often referred to as the hazard rates of the semi-Markov process. We summarize and contrast these two parameterizations both from a probabilistic and an inference perspective, and we highlight relationships between the two approaches. In general, the intensity transition based approach allows the likelihood to be split into likelihoods of two-state models having fewer parameters, allowing efficient computation and usage of many survival analysis tools. {Nevertheless, in certain cases the sojourn time based approach is natural and has been exploited extensively in applications.} In contrasting the two approaches and contemporary relevant R packages used for inference, we use two real datasets highlighting the probabilistic and inference properties of each approach. This analysis is accompanied by an R vignette.