= Piecewise-exponential mortality comparison
{title2=$P(T\leq\tau)=1-\exp(-\sum_k h_k\Delta t_k)$}
For specified exposure periods with constant death <hazard functions> $h_k$, the <integrated hazard> through the endpoint is $\sum_kh_k\Delta t_k$. 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, $\sum_jw_j(1-e^{-H_j})$, 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.
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