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

Azam Asanjarani, Benoît Liquet+1 more|arXiv (Cornell University)|May 29, 2020

Statistical Methods and Bayesian Inference49 references32 citations

TL;DR

The paper compares two parameterizations of semi-Markov multi-state models—sojourn times with embedded chain vs. intensity transition functions—and discusses their probabilistic links and inference advantages, with R implementations and real data examples.

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.

Motivation & Objective

• Motivate the use of semi-Markov processes in survival and reliability analysis.
• Contrast two SMP parameterizations: Approach I (sojourn times with embedded chain) vs. Approach II (intensity transition functions).
• Explain probabilistic links between approaches and implications for inference and computation.
• Showcase inference via real datasets and available R packages and a reproducible vignette.

Proposed method

• Define SMPs via two parameterizations: Approach I using sojourn time distributions and embedded chain probabilities, and Approach II using intensity transition functions.
• Derive key relationships between approaches, including how CIF relates to both parameterizations.
• Formulate likelihood functions for fully parametric models under right-censoring for both approaches.
• Demonstrate decoupled, two-state likelihood structure under Approach II enabling separate optimization per transition.
• Incorporate covariates through Cox-like proportional hazards models for both approaches and discuss interpretational differences of regression parameters.
• Provide guidance for implementation in R and discuss computational advantages of Approach II.

Experimental results

Research questions

• RQ1 How do the sojourn-time based and intensity-based parameterizations of semi-Markov multi-state models relate probabilistically?
• RQ2 What are the comparative advantages and tradeoffs in inference between Approach I and Approach II, especially under right-censored data?
• RQ3 Can the likelihood for Approach II be decomposed into independent two-state likelihoods to simplify computation?
• RQ4 How do covariates influence transition dynamics differently under the two approaches?
• RQ5 What are practical guidelines for implementing these models in R and choosing between parameterizations in real datasets?

Key findings

• Approach II yields significant computational advantages by factorizing the SMP likelihood into products of two-state model likelihoods.
• Approach II covariates affect transition hazards directly and also influence transition probabilities implicitly, unlike Approach I where covariates affect only holding-time hazards.
• There is a formal relationship linking the two approaches, enabling conversion between their parameters and their resulting CIF expressions.
• We can recover standard CTMC results as special cases of both approaches when appropriate simplifying assumptions hold.
• The paper provides reproducible R-based analyses and a detailed vignette, illustrating practical implementation on two real datasets.