Martingales without tears

This pedagogical paper presents a casual introduction to martingales, or fair gambling processes. Our objective is to describe the concept of a martingale and its application to common statistical tests used in the analysis of survival data, but without the mathematical rigor required for formal proofs. We use heuristic arguments to demonstrate that the logrank statistic evaluated over followup time is a fair gambling process, and introduce some mathematical notation and terminology along the way. We then employ the counting process approach to show that the logrank statistic computed over followup time can be expressed as the difference of two martingale transforms, and thus is a martingale. These ideas are first time introduced in the context of a discrete time process, and are then generalized to a continuous time process. With slight modifications, the same idea extends from the logrank to other weighted Mantel-Haenszel statistics computed over time.