From Linear Hazard Dynamics to Rayleigh Behavior: Identification of Early-Time Survival Regimes

This article establishes a mechanistic foundation for the Rayleigh distribution in event-time analysis by deriving it from a decay process governed by a linearly increasing hazard under assumptions of independence and homogeneity. More generally, when the hazard is sufficiently smooth, vanishes at the origin, and has a positive initial slope, the Rayleigh distribution emerges as the leading-order approximation at early times. A Monte Carlo sensitivity analysis comprising 114,000 realizations across sample sizes from n=50 to 10,000 quantifies the effects of departures from hazard linearity and sampling variability. Increasing observed Rλ2 is associated with decreasing Kolmogorov–Smirnov distances and greater consistency between independent estimates of the Rayleigh parameter, although its interpretation depends on sample size and does not permit establishing a universal threshold. A data-driven methodology is then proposed to operationally identify approximately linear hazard regimes and model the corresponding event times using a truncated Rayleigh distribution. Application to tuberculosis survival in guinea pigs, COVID-19 survival in healthy young men, and primate interspike intervals identified regimes up to T*=255 days, 26 days, and 18.5 ms, with Rλ2=0.95, 0.94, and 0.73, respectively. Within these regimes, the model showed strong quantile agreement (RQQ2=0.97, 0.99, and 0.98) and relative discrepancies between parameter estimates of 0.08, 0.08, and 0.19. These findings support a mechanistically grounded, data-driven framework for identifying and modeling early-time Rayleigh behavior in event-time data.