Markov Renewal Proportional Hazards is All You Need

TL;DR

This paper compares semi-Markov and Markov renewal models for transition probability estimation in clinical multi-state data, showing semi-Markov models offer more nuanced insights, with the DSH estimator providing smoother probability curves.

Contribution

It introduces the application of semi-Markov and Markov renewal models to clinical data, demonstrating their advantages over traditional Markov models in capturing temporal dynamics.

Findings

Semi-Markov models better capture patient trajectory nuances.

DSH estimator produces smoother transition probability curves.

Weak convergence of the estimator is established using empirical process theory.

Abstract

Transition probability estimation plays a critical role in multi-state modeling, especially in clinical research. This paper investigates the application of semi-Markov and Markov renewal frameworks to the EBMT dataset, focusing on six clinical states encountered during hematopoietic stem cell transplantation. By comparing Aalen-Johansen (AJ) and Dabrowska-Sun-Horowitz (DSH) estimators, we demonstrate that semi-Markov models, which incorporate sojourn times, provide a more nuanced and temporally sensitive depiction of patient trajectories compared to memoryless Markov models. The DSH estimator consistently yields smoother probability curves, particularly for transitions involving prolonged states. We use empirical process theory and Burkholder-Davis-Gundy inequality to show weak convergence of the estimator. Future work includes extending the framework to accommodate advanced covariate structures and non-Markovian dynamics.