In survival analysis, S(t) equals which expression in terms of H(t)?

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Multiple Choice

In survival analysis, S(t) equals which expression in terms of H(t)?

In survival analysis, the probability of surviving beyond time t, S(t), is connected to the hazard h(t) through the cumulative hazard H(t) = ∫0^t h(u) du. The survival probability must satisfy the relationship dS/dt = -h(t) S(t) because the chance of failing in a tiny interval is proportional to both the current survival and the instantaneous hazard. Solving this differential equation gives dS/S = -h(t) dt, and integrating from 0 to t with S(0) = 1 yields ln S(t) = -∫0^t h(u) du = -H(t). Exponentiating both sides leads to S(t) = exp(-H(t)). This form guarantees S(0) = 1 and a nonincreasing S(t) when h(t) ≥ 0, which is consistent with a survival probability. The other expressions do not satisfy these fundamental relationships, so they cannot represent the survival function.

In survival analysis, the probability of surviving beyond time t, S(t), is connected to the hazard h(t) through the cumulative hazard H(t) = ∫0^t h(u) du. The survival probability must satisfy the relationship dS/dt = -h(t) S(t) because the chance of failing in a tiny interval is proportional to both the current survival and the instantaneous hazard. Solving this differential equation gives dS/S = -h(t) dt, and integrating from 0 to t with S(0) = 1 yields ln S(t) = -∫0^t h(u) du = -H(t). Exponentiating both sides leads to S(t) = exp(-H(t)). This form guarantees S(0) = 1 and a nonincreasing S(t) when h(t) ≥ 0, which is consistent with a survival probability. The other expressions do not satisfy these fundamental relationships, so they cannot represent the survival function.

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