Piecewise-exponential mortality comparison ($P(T\le \tau )=1-\exp (-{\sum }_k{h}_k\Delta t_k)$)

For specified exposure periods with constant death hazard functions $h_k$, the integrated hazard through the endpoint is ${\sum }_k{h}_k\Delta t_k$. The displayed event probability follows from the survivor function identity. For different latent exposure schedules mixed with probabilities $w_j$, average the schedule-specific probabilities, ${\sum }_j{w}_j(1-e^{-H_j})$, rather than exponentiating the average integrated hazard. When the hazards are small, $1-e^{-H_j}\simeq H_j$ gives a person-time approximation. A comparison must specify the entire custody or treatment schedule; later changes to exposure cannot be inferred from an endpoint rate alone.