MARTINGALE INFERENCE OF MULTI-STATE MARKOV PROCESSES FOR APPLICATIONS IN SURVIVAL ANALYSIS AND EPIDEMIOLOGY

John J. Hsieh (U.S.A.)

Received May 14, 2009; Revised June 19, 2009

Abstract

This article develops martingale methods of estimation and hypothesis testing for multi-state Markov processes and applies them to make inference of multi-state survival and epidemiologic models. The approach is a generalization of the hazards processes techniques described in Hsieh [10] and Andersen et al. [1] for competing risks models to deal with intensity processes in multi-state models. The intensity processes arising from the point processes, marked processes and marked processes with covariate paths, respectively, are derived as compensators of appropriate submartingales from the corresponding deterministic intensity functions. The derived intensity processes and associated martingales are employed to obtain nonparametric estimates of various deterministic intensity, cumulative intensity and ratios thereof, as well as the survival function and its functionals. Asymptotic distributions of the estimators and test statistics are obtained, using martingale transform theorem and martingale central limit theorem. These results are illustrated with numerical examples.

Keywords and phrases: counting process, predictable process, transition rate, survival function, intensity process, covariate, censoring, competing risks, epidemic model.