Counting Processes and Martingales

This chapter aims to demystify the counting process and martingale approaches for readers not specializing in theoretical survival analysis. It includes more detailed heuristic derivations of key formulas than typically found in other texts, leaving formal definitions and precise technical results to Appendix B. Discussions for both homogeneous populations and regression with fixed or external covariates are presented. It is shown how the two probabilistic concepts of counting processes and martingales arise naturally in the analysis of survival data. Various formulas useful in variance estimation and asymptotic analysis are heuristically explored. Models that are previously stated in terms of the underlying distribution functions are reformulated in terms of the counting processes to illustrate the rise of intensity models. Statistical inference under intensity models and treatment effect interpretation are also briefly discussed.